Date:           Fri, 9 Jan 1998 17:16:21 +0100
From:           Francesco Brenti <brenti(at-sign)mat.utovrm.it>

Dear colleague,
  I have accepted a (tenured) position at the Associate
Professor level al the University of Rome. Therefore,
my "coordinates" have changed. The new ones are:

Dipartimento di Matematica
Universita' di Roma "Tor Vergata"
Via della Ricerca Scientifica, 1
00133, Roma, Italy

Tel. (39) (6) 7259 4671
Fax. (39) (6) 7259 4699
e-mail: brenti(at-sign)mat.utovrm.it

Sincerely yours,
Francesco Brenti

Date:           Fri, 9 Jan 1998 12:01:20 -0400
To:             bergeron(at-sign)mathstat.yorku.ca
From:           bergeron(at-sign)mathstat.yorku.ca (Nantel Bergeron)
Subject:        FPSAC98 

News Update for FPSAC98

        OnLine Registration: Visit http://www.math.yorku.ca/bergeron

        THE CALL FOR PAPERS AND POSTERS IS OVER.

        CALL FOR SOFTWARE DEMONSTRATION: DEAD LINE Jan 21.

        PARTICIPANT SUPPORT: DEAD LINE Jan 21.

-----------------------------------------------------------------------

10-th international Conference on Formal Power Series and Algebraic
Combinatorics

Sunday, June 14 - Friday, June 19, 1998

Fields Institute, Toronto

TOPICS:
       Algebraic and bijective combinatorics and their relations with other
       parts of mathematics, computer science and physics.
CONFERENCE PROGRAM:
       Invited lectures, contributed presentations, poster sessions and
       software demonstrations.
OFFICIAL LANGUAGES:
       English and French.
INVITED TALKS:
       G. Benkart (USA)
       P. Cameron (England) (not confirmed)
       P. Dehornoy (France)
       B. Derrida (France)
       P. Diaconis (USA)
       C. Godsil (Canada)
       K. Ono (USA)
       J. Y. Thibon (France)
       B. Sturmfels (USA)
CALL FOR SOFTWARE DEMONSTRATION:
       Demonstrations of software relevant to the topics of the conference are
       encouraged. People interested in giving a software
       demonstration should submit a short description of the software,
       including the hardware requirements, before January 21,
       1998, by email to

       fpsac98(at-sign)fields.utoronto.ca

PROGRAM COMMITTEE:
       I. Goulden , Chairman (U. of Waterloo), N. Bergeron (York U.),
       S. Billey (USA), F. Brenti (Italy), R. Cori (France),
       S. Dulucq (France), K. Eriksson (Sweeden), O. Foda (Australia),
       S. Fomin (USA/Russia), I. Gessel (USA), C. Greene (USA),
       A. Hamel (New Zeland), D. Kim (Korea), C. Krattenthaler (Austria),
       D. Krob (France), M. Noy (Spain), V. Reiner (USA),
       C. Reutenauer (UQAM), F. Sottile (U. Toronto), T. Visentin (U.
Winnipeg).
       M. Wachs (USA), H. Yamada (Japan), G. Ziegler (Germany).
PARTICIPANT SUPPORT:
       Limited funds are available for partial support of participants.
Requests
       should contain a letter of recommendation and include
       the estimated transportation and living expenses as well as the
amount of
       support available from other sources. All requests
       should be sent in duplicate by January 21, 1998 to the Fields
Institute.

       fpsac98(at-sign)fields.utoronto.ca
LOCATION:
       The conference will take place at the Fields Institute in Toronto,
       Canada. Registration and afternoon reception will be on
       Sunday, June 14 from 2:00 - 4:00 p.m. The first talk is scheduled on
June
       15, 1998 at 9:00 a.m.
REGISTRATION:
       Until April 15, 1998, the registration fee is 200 $CAN and 100 $CAN is
       offered for students (with verification, such as a letter
       from the advisor). After April 15, 1998 fees will be 300 $CAN and 150
       $CAN, respectively. The registration fee covers all
       lectures and presentations, program materials, refreshment breaks, a
copy
       of the abstract package, and a dinner on Thursday
       night. To register Visit http://bergeron(at-sign)mathstat.yorku.ca or contact
              FPSAC/SFCA 98
              Fields Institute
              222 College Street Toronto,
              ON M5T 3J1 Canada
       fpsac98(at-sign)fields.utoronto.ca
       web-site: www.fields.utoronto.ca
FOR MORE INFORMATION:
       Contact bergeron(at-sign)mathstat.yorku.ca . For information on Fields
Institute:
       visit www.fields.utoronto.ca
ORGANIZATION:
       N. Bergeron, Chairman (York U.), M. Delest (U. de Bordeaux), F. Sottile
       (U. Toronto), W. W hiteley (York U.).

SPONSOR:

Our sponsor are: Fields Institute of Toronto,
Centre de Recherches Mathematiques a Montreal,
University of Toronto (Connaught fund)
and York University .
We also have some technical support from
Combinatorics and Optimization at Waterloo University.

Nantel Bergeron                                   bergeron(at-sign)mathstat.yorku.ca
Associate Prof. Mathematics                          nantel(at-sign)math.harvard.edu
York University                                         nantel(at-sign)lacim.uqam.ca
http://www.math.yorku.ca/bergeron

Date:           Fri, 9 Jan 1998 12:06:37 -0400
To:             bergeron(at-sign)mathstat.yorku.ca
From:           bergeron(at-sign)mathstat.yorku.ca (Nantel Bergeron)
Subject:        SCFA98

SFCA98:

Inscription en ligne: Visitez http://www.math.yorku.ca/bergeron.

Les SOUMISSIONS DE COMMUNICATIONS ET DE POSTERS sont termines.

SOUMISSIONS DE DEMONSTRATIONS DE LOGICIELS: DATE LIMITE 21 jan.

=46INANCEMENT DE DEPLACEMENTS: DATE LIMITE 21 jan.

-----------------------------------------------------------------------

10ieme conference sur les Series Formelles et la Combinatoire Algebriques

14-19 juin 1998,

=46ields Institute, Toronto

THEMES:
       Les themes traditionnels du colloque sont la combinatoire algebrique =
et
       bijective dans leurs liens avec les mathematiques,
       l'informatique et la physique.

DEROULEMENT DE LA CONFERENCE:
       Le colloque comprend des conferences invitees de 60 minutes, des
       communications selectionnees de 30 minutes, une seance
       d'affichage de posters selectionnes et des demonstrations de logiciel=
s.

LANGUES OFFICIELLES:
       l'anglais et le francais.

CONFERENCIER INVITES:
       G. Benkart (USA)
       P. Cameron (England) (non confirme)
       P. Dehornoy (France)
       B. Derrida (France)
       P. Diaconis (USA)
       C. Godsil (Canada)
       K. Ono (USA)
       J. Y. Thibon (France)
       B. Sturmfels (USA)

DEMONSTRATIONS DE LOGICIELS:
       Les demonstrations de logiciels lies aux themes du colloque sont
vivement
       encouragees. Les personnes souhaitant faire une
       demonstration de logiciel sont invitees a envoyer une courte descript=
ion
       de leur logiciel, precisant les supports materiels
       necessaires, avant le 21Janvier 1998, par email a

       fpsac98(at-sign)fields.utoronto.ca

COMITE DE PROGRAMME:
       I. Goulden , Chairman (U. of Waterloo), N. Bergeron (York U.),
       S. Billey (USA), F. Brenti (Italy), R. Cori (France),
       S. Dulucq (France), K. Eriksson (Sweeden), O. Foda (Australia),
       S. Fomin (USA/Russia), I. Gessel (USA), C. Greene (USA),
       A. Hamel (New Zeland), D. Kim (Korea), C. Krattenthaler (Austria),
       D. Krob (France), M. Noy (Spain), V. Reiner (USA),
       C. Reutenauer (UQAM), F. Sottile (U. Toronto), T. Visentin (U.
Winnipeg).
       M. Wachs (USA), H. Yamada (Japan), G. Ziegler (Germany).

=46INANCEMENT DE DEPLACEMENTS:
       Un nombre limite de subventions sera disponible pour pouvoir particip=
er
       au colloque. Les demandes de subvention devront
       contenir l'avis d'une personnalite scientifique et inclure les
       renseignements suivants: montant des frais de transport et de sejour,
       montant des autres sources de financement. Toute demande devra parven=
ir
       en double exemplaire avant le 21Janvier 1998 a

       fpsac98(at-sign)fields.utoronto.ca.

LIEU:
       La conf=E9rence se d=E9roulera =E0 Fields Institute de Toronto,=
 Canada. Les
       inscriptions et une petite r=E9ception auront lieu le dimance
       14 juin de 14h00 =E0 16h00. La premi=E8re conf=E9rence est pr=E9vue=
 pour le 15
       juin 1998 =E0 9 heures.

INSCRIPTION:
       Avant le 15 avril 1998, les frais d'inscription sont fixes a $200 (Ca=
n)
       Un tarif reduit de $100 est prevu pour les etudiants (a
       justifier par une lettre). Pour les demandes d'inscription parvenant
       apres le 30 Avril 1998, ces frais sont portes a 50% de plus.
       Les frais l'inscription couvrent les couts pour les instructions, les
       documents de programme, les rafraichissements pendant les
       pauses, une copie des actes du colloque, et la reception du jeudi soi=
r.
       Pour s'inscrire, remplir le formulaire SUIVANT ou
       contacter Fields Institute:
              FPSAC/SFCA 98
              Fields Institute
              222 College Street Toronto,
              ON M5T 3J1 Canada
       fpsac98(at-sign)fields.utoronto.ca
       web-site: www.fields.utoronto.ca

RESEIGNEMENTS SUPLEMENTAIRES:
       Pour toute demande de renseignement supplementaire, n'hesitez pas a
       ecrire a fpsac98(at-sign)fields.utoronto.ca Pour plus
       d'information sur l'institue Fields, les directions et Toronto,
       www.fields.utoronto.ca
COMITE D"ORGANIZATION:
       N. Bergeron, Chairman (York U.), M. Delest (U. de Bordeaux), F. Sotti=
le
       (U. Toronto), W. W hiteley (York U.).

COMMANDITAIRES

Nos commanditaire sont: le Fields Institute de Toronto, le Centre de
Recherches Mathematiques a Montreal, l'Universite de Toronto
(visitez le departement de mathematiques ) et l'universite York (visitez le
departement de mathematiques ). Nous avons aussi le soutient
techniques du departement de Combinatoires et d'Optimization a l'universite
de Waterloo.

Nantel Bergeron                                   bergeron(at-sign)mathstat.yorku.ca
Associate Prof. Mathematics                          nantel(at-sign)math.harvard.edu
York University                                         nantel(at-sign)lacim.uqam.ca
http://www.math.yorku.ca/bergeron

Date:           Wed, 14 Jan 1998 22:59:44 -0500 (EST)
From:           Jim Propp <propp(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        alternating sign matrices

I'll be giving a talk on Wednesday, January 21 from 1:00 to 2:30 p.m.
in room 2-190 at M.I.T. on the history of the alternating sign matrix
formula.  It'll be directed at undergraduates, and a bit on the light
side (as you might guess from the mock-sensationalistic title of the 
lecture --- "World's Hardest Counting Problem: Solved!").

Jim Propp

Date:           Thu, 22 Jan 1998 09:03:07 -0500 (EST)
From:           Jim Propp <propp(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        MIT Combinatorics Seminar

There are still several openings for the MIT Combinatorics Seminar
in February.  If you would like to speak, please send me, by this
weekend, the _title of_ your talk, an _abstract_, and the _dates_ 
on which you could give your talk (Wednesdays and Fridays, usual 
time).  If we can't accomodate you in February, we should be able 
to schedule you in March; the spring schedule is still quite open.

Jim Propp

Date:           Wed, 28 Jan 1998 10:00:13 -0500
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        first announcement

             Come to the Twenty-eighth one day conference on

                     Combinatorics and Graph Theory

                      Saturday, February 21, 1997

                         10 a.m. to 4:30 p.m.
                                  at
                            Smith College
                         Northampton MA 01063

                              Schedule

10:00  Katalin Vesztergombi (Yale University)
        Properties of Distance-Graphs

11:10  Sheila Sundaram (Danbury, Connecticut)
        Homology of Graph Complexes and Partitions
         with Forbidden Block Sizes

12:10  Lunch

 2:00  Emily H. Moore (Grinnell College and Mt. Holyoke College)
        Extending Graph Colorings

 3:10  Dana Randall (Georgia Institute of Technology)
        TBA

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Tue, 3 Feb 1998 13:12:25 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar

Welcome back from winter break.  The combinatorics seminar will resume
this Friday with a talk by Serkan Hosten entitled "The order dimension
of the complete graph and Scarf complexes".  Abstracts and titles are
available on our web page
	http://www-math.mit.edu/~combin

We are still looking for speakers for March, April and May.  Please
let us know if you would like to give a talk and/or if you have
visitors in town who might like to speak.

FEBRUARY SCHEDULE:

      February 6: Serkan Hosten
      The order dimension of the complete graph and Scarf complexes 

      February 11: Alexander Postnikov
      Algebras of Chern forms on flag manifolds and forests 

      February 18: Harald Helfgott
      TBA. 

      February 20: Eva Feitchner
      On the cohomology algebras of complex subspace arrangements 

      February 25: Richard Stanley
      Enumerating solutions to equations in finite groups 

      February 27: Alex Burstein
      Enumeration of words with forbidden patterns 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Tue, 3 Feb 1998 21:55:02 -0500 (EST)
From:           Jim Propp <propp(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        February schedule

Here is a more complete version of the February schedule:

FEBRUARY SCHEDULE:

      February 6: Serkan Hosten
      The order dimension of the complete graph and Scarf complexes 

      February 11: Alexander Postnikov
      Algebras of Chern forms on flag manifolds and forests 

      February 18: Harald Helfgott
      Recent developments in enumeration of tilings

      February 20: Eva Feitchner
      On the cohomology algebras of complex subspace arrangements 

      February 25: Richard Stanley
      Enumerating solutions to equations in finite groups 

      February 27: Alex Burstein
      Enumeration of words with forbidden patterns 

Date:           Thu, 5 Feb 1998 11:34:57 -0500
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        second announcement

             Come to the Twenty-eighth one day conference on

                     Combinatorics and Graph Theory

                      Saturday, February 21, 1997

                         10 a.m. to 4:30 p.m.
                                  at
                            Smith College
                         Northampton MA 01063

                              Schedule

10:00  Katalin Vesztergombi (Yale University)
        Properties of Distance-Graphs

11:10  Sheila Sundaram (Danbury, Connecticut)
        Homology of Graph Complexes and Partitions
         with Forbidden Block Sizes

12:10  Lunch

 2:00  Emily H. Moore (Grinnell College and Mt. Holyoke College)
        Extending Graph Colorings

 3:10  Dana Randall (Georgia Institute of Technology)
        Two Dimensional Tilings in Two and Three Dimensions

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Fri, 6 Feb 1998 15:58:19 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Date: February 6, 1998
Speaker: Serkan Hosten
Title: The order dimension of the complete graph and Scarf complexes 
Abstract:
In this talk we will show how one determines the order dimension (of
the poset) of the complete graph on n vertices. The order dimension of
a poset is the least number of linear extensions whose intersection
realizes the poset. The order dimension of the complete graph was not
determined exactly for large number of vertices, and only rough
extimates existed. We give a precise description. This result has
implications in the combinatorics of Scarf complexes defined by Bayer,
Peeva, and Sturmfels to compute free resolutions of monomial
ideals. This is joint work with Walter Morris.

Upcoming Events:

      February 11: Alexander Postnikov
      Algebras of Chern forms on flag manifolds and forests 

      February 18: Harald Helfgott
      TBA. 

      February 20: Eva Feitchner
      On the cohomology algebras of complex subspace arrangements 

      February 25: Richard Stanley
      Enumerating solutions to equations in finite groups 

      February 27: Alex Burstein
      Enumeration of words with forbidden patterns 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

From:           IARROBIN(at-sign)neu.edu
Date:           Fri, 06 Feb 1998 23:25:56 -0500 (EST)
Subject:        of possible interest - Seminar at NU Mon Feb 9
To:             combinatorics(at-sign)math.mit.edu

 Northeastern Univ. GAS seminar, Monday Feb 9 at 1:30 PM at 509 Lake Hall:

   Vic Reiner (Minnesota):

    "Resolutions and the homology of chessboard and matching complexes"

 Abstract: [e use
           We use Lascoux's resolution for 2 x 2 minors, and some 
        generalizations of it, to calculate the homology of these
        simplicial complexes

   - AI

Date:           Mon, 9 Feb 1998 10:38:00 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        [daisymae(at-sign)math.mit.edu: APPLIED MATHEMATICS COLLOQUIUM -- Monday, February 9, 1998]

Reminder, Vic Reiner will be speaking today in the applied math
colloquium on a topic that is of interest to combinatorialists.  There
will be a dinner afterwards.  

Date:           Fri, 6 Feb 1998 14:42:07 -0500 (EST)
From:           Shirley Entzminger-Merritt <daisymae(at-sign)math.mit.edu>
To:             amc(at-sign)math.mit.edu
Subject:        APPLIED MATHEMATICS COLLOQUIUM -- Monday, February 9, 1998

A REMINDER . . .

			APPLIED MATHEMATICS COLLOQUIUM

TOPIC:		SPECTRA OF LAPLACIANS FOR SIMPLICIAL COMPLEXES

SPEAKER:	PROFESSOR VICTOR REINER
		School of Mathematics
		University of Minnesota

ABSTRACT:

This talk will discuss some recent work on the spectra of discrete Laplace
operators coming from boundary maps in a simplicial complex.  For two
families of simplicial complexes, the chessboard complexes (studied by J. 
Friedman and P. Hanlon) and matroid complexes (studied by W. Kook, D. 
Stanton and myself), these spectra are known to be integral, and
interpretations of the spectra have been given.  Why these particular
complexes should have integral spectra is still mysterious, as is the
connection to spectra of Laplacians on Riemannian manifolds.

One corollary to the interpretation of the spectra for matroid complexes
may be paraphrased as stating that one can "hear" the chromatic polynomial
of a graph. 

DATE:		MONDAY, FEBRUARY 9, 1998

TIME:		4:15 p.m.

LOCATION:	Building 2, Room 105

Refreshments will be served at 3:45 p.m. in Building 2, Room 349

Applied Math Colloquium:  http://www-math.mit.edu/amc/spring98
Math Department:  http://www-math.mit.edu

Massachusetts Institute of Technology

Department of Mathematics

Cambridge, MA  02139

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Tue, 10 Feb 1998 15:41:57 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        pretalk

Back by popular demand, there will be a pretalk tomorrow from
3:30pm-4:00pm in conjunction with the combinatorics seminar.  Alex
Postnikov will give a brief introduction to flag manifolds.  Recall
from last year, we had several pretalks aimed at graduate students and
non-experts in the particulars of the seminar that followed.  Everyone
is welcome to attend.  Questions are strongly encouraged!  Pretalks
will be held in 2-338 (seminar room).

If anyone scheduled to give a talk would like to give a pretalk,
please let me know.  

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Tue, 10 Feb 1998 14:28:40 -0800 (PST)
To:             combinatorics(at-sign)math.mit.edu
From:           Gian-Carlo Rota <gcrota(at-sign)earthlink.net>
Subject:        colloquium lectures

--============_-1324990680==_============
Content-Type: text/plain; charset="us-ascii"

Dear colleagues,
I am sending you as an attachment the draft of the Colloquium Lectures I
delivered at the annual meeting of the American Mathematical Society last
January. I will greatly appreciate any comments, corrections, suggestions
etc. before I put them on the web.

Gian-Carlo Rota

--============_-1324990680==_============
Content-Type: text/plain; name="colloquium_2=98"; charset="us-ascii"
Content-Disposition: attachment; filename="colloquium_2=98"

\documentstyle[12pt]{article}

\def\bphiz#1{\overline\phi_{\hat0}^{#1}}
\def\bPhi{\overline\Phi}
\def\phiz#1{\phi_{\hat0}^{#1}}
\def\sigalg#1{\Sigma^{\otimes{#1}}}
\def\sigalgpi#1{\Sigma^{\otimes{#1}}_\pi}
\def\finalg#1{\Sigma^{\otimes{#1}}_{\rm fin}}
\def\finalgpi#1{\Sigma^{\otimes{#1}}_{{\rm fin},\pi}}
\def\ltimes{{\rm X}}
\def\D{{\cal D}}
\def\F{{\cal F}}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\E{{\cal E}}
\def\C{{\cal C}}
\def\G{{\cal G}}
\def\R{{\bf R}}
\def\S{{\bf S}}
\def\Inpi{{I^{[n]}_\pi}}
\def\bs{\backslash}
\def\zpione{{\hat 0 \le \pi \le \hat 1}}
\def\zpilone{{\hat 0 \le \pi < \hat 1}}
\def\halmos{{Q.E.D.}} % This is temporary
\def\St{{\rm St}}

\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}

\begin{document}

\setlength{\textwidth}{6in}
\setlength{\textheight}{8in}
\parindent=0pt
\tolerance=6000
\title{The American Mathematical Society Colloquium Lectures, 1998}

\author{Gian-Carlo Rota}

\date{Baltimore, January 7,8 and 9, 1998}

\maketitle

\newpage

\begin{center} {\bf INTRODUCTION TO GEOMETRIC PROBABILITY\\

being\\

The first of three colloquium Lectures\\

delivered at the Annual Meeting of the American
Mathematical Society\\

Baltimore, January 7, 1997\\

Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\
Cambridge MA 02139-4307\\

\end{center}

\bigskip

\bigskip

I am  very happy to be here before you as the
Colloquium Lecturer for this year, and I feel deeply honored to be
given this great opportunity to share with you some of the
mathematics we love.

\bigskip

When I received from Bob Fossum the invitation to be the
Colloquium Lecturer for this year, I rushed to the library
to read in an old issue of the Notices the list of all
previous Colloquium Lecturers, going all the way  back to
James Pierpont in 1896. It is a list of distinguished
mathematicians, and I wondered how my name could ever
belong in such lofty company.  My immediate temptation was
to decline the invitation outright; but Bob Fossum assured
me that no one in the history of the Society has ever
declined the invitation to be the Colloquium Lecturer.  So I
went back to the list of previous colloquium speakers, in
search for a justification of my presence in that list. As
often happens in such situations, I soon enough found such
a justification. I computed the average age of colloquium
lecturers and discovered that this average is  somewhat
lower than my age, as a matter of fact my age exceeds by
approximately by one standard deviation the mean age of
previous speakers.  As I came to this realization, I began
to fantasize on the probable topics that my younger
predecessors might have chosen. I imagined a brilliant
young mathematician, eager to establish himself as a leader
in his field,  delivering one single  dazzling proof
beginning with the first colloquium lecture and lasting all
the way to the end of the third.  Or else, some middle aged
mathematician, anxious to have his latest theory accepted
by the mathematical world, delivering to a thrilled
audience a three-hour condensation of material that would
normally take an entire term in an advanced graduate
course. How could I, a mathematician one standard deviation
older, ever hope to match such enviable feats?

\bigskip

These fantasies came to an abrupt end when  Bob Fossum
informed me in no uncertain terms that the Council of the
Society had decided that the three colloquium lectures must
deal with three independent and unrelated topics, thereby
allowing any member of the audience  to skip one or more
lectures, without missing anything. Bob Fossum's command
deprived me of all possible role models among previous
colloquium lecturers. In a state of temporary panic, I
again scanned the list  of previous colloquium speakers,
this time looking for names of mathematicians who had not
been chosen for this honor. Sure enough, one name was
conspicuously missing: that of Hermann Weyl.

\bigskip

I hope you will forgive me if I  digress with some personal
reminiscences.

\bigskip

In the fall of 1950 I enrolled as a freshman at Princeton,
having graduated a few months before from the American High
School of Quito, Ecuador.  The principal of the American
High School of Quito was a Princeton graduate,  and he
steered me towards Princeton University.

\bigskip

In November 1950 I listened to my first mathematics
lectures. These were the three Vanuxem lectures, delivered
by Hermann Weyl and bearing the generic title "Symmetry".
They were an unforgettable experience. The lectures took
place in the old chemistry auditorium, packed with an
expectant public. As I shamelessly sat in the first row
trying to guess which of the other persons sitting in the
same row was to be the speaker, a hush fell upon the
audience:  Einstein was entering the lecture room. To my
disappointment, he sat somewhere in the middle of the
auditorium.

\bigskip

The first lecture began with an impressive and lengthy
quotation in Greek, which no one in the audience understood
except Luther Pfahler Eisenhart.  This brilliant start was
followed by a display  of slides portraying charming women
wearing the long brimmed hats fashionable at the time, and
later  by more slides showing the Alhambra and the
Pentagon. Not a word of mathematics.  The audience was left
wondering where such a sparkling display of  "Kultur" was
leading up to. Not much more mathematics was mentioned in
the second lecture, when more slides were shown of physics
experiments, for  which the lecturer provided a learned oral
commentary. Only in the last lecture did some group theory
make a modest appearance.  By that time the audience, which
had not dwindled, was enthralled with the subject, and did
not mind the fact that the speaker had said very little
about mathematics, actually he had said very little about
anything at all. What is more remarkable, the audience
seemed to be thankful to the speaker for making the
contents of the three lectures independent of one another,
thereby minimizing all memory requirements. I hazard to
guess that the success of Hermann Weyl's lectures may be in
part attributed to the speaker's astute foresight in making
his lectures  self contained,  independent and lightweight.
As I recall this distant episode, I realize that  Bob
Fossum's injunction about the independence of the present
colloquium lectures is a wise one, all the more so when the
speaker is not Hermann Weyl.

\bigskip

You may wonder why I cited my age as an aid to delivering
these colloquium lectures. What difference does one standard
deviation make? I think it makes some difference. It is a
relief, both to you and to me, to know right at the start
that the speaker does not feel the need to impress you by
stating the results of his latest research. Nor do you or I
suffer from any lack of exposure to the latest fashions in
mathematics; we'll hear enough about them in other lectures
scheduled to be delivered at this meeting.  We can therefore
afford to spend these three hours on leisurely discussion
of  some mathematics that may matter to  both
you and me.

\bigskip

We will  cover in these lectures a few items that
are not widely known, that  should be better known, and that
I vouch can be understood by anyone with a B.A. in
mathematics. I solemny promise that I will not state  any
big theorems, I will not subject you to   any ingenious
arguments, and that  I will not announce   any
revolutionary developments.

\bigskip

The title of this lecture is "Geometric Probability".
 A definition
of geometric probability might run as follows: geometric
probability is the study of invariant measures. Like all
definitions, this does not  tell us anything until  we are
shown some typical examples, and these examples are the
content of this lecture.

\bigskip

About one  hundred years ago, the properties that underlie
such notions as length, area , volume , as well as   the
probability of events were abstracted  under the banner of
the word  "measure".  Let us review the definition of
measure, since we will be using this definition in an
unusual way.

\bigskip

A measure $\mu$ is a function defined on a family of
subsets of a set $S$, which takes real values, not
necessarily positive.  The family of sets on which a
measure is defined is closed under unions and
intersections, and contains the empty set.

\bigskip

A measure is characterized by two simple axioms. Let us
take a minute to review these axioms.

\bigskip

Axiom 1.

$$ \mu ( \emptyset ) = 0,$$

where $\emptyset  $ is the empty set.

\bigskip

Axiom 2.   If $A$ and $B$ are two measurable sets, then

$$\mu(A\cup B) = \mu (A) +\mu(B) - \mu (A\cap B) .$$

\bigskip

The meaning of this  second axiom is clear from the
picture.  The axiom states that measure is additive. In
particular,  if we have two disjoint sets $A$ and $B$, then

$$\mu(A\cup B) = \mu (A) + \mu (B) .$$

\bigskip

More generally, for any finite family $F$ whose members are
sets, and for which any two members are disjoint, we  we
have :

$$\mu( \bigcup_{A\in F} A ) =\sum_{A\in F}\mu (A).$$

We most emphatically do not assume that a measure is
countably additive.

\bigskip

The best known  example of a measure is the volume
$\mu_n(A)$ of a solid $A$ in ordinary $n$-dimensional
Euclidean space.

The volume  $\mu_n (A)$  of a solid $A$ satisfies axioms 1
and 2  above, but axioms 1 and 2 do not characterize volume
among all possible measures. What additional axioms must we
add to the definition of a measure, in order to
characterize volume?

It is possible  to characterize volume among all measures
by adding to axioms 1 and 2 two additional intuitive  axioms,
namely, the following:

\bigskip

Axiom 3.

\bigskip

The volume of  a set $A$  is independent of the position of
$A$.  If a set $A$ in $n$-dimensional
Euclidean space can be rigidly moved onto a set
$B$, then $A$ and $B$ have the same volume.

\bigskip

In  other words,  volume is invariant under the group of
Euclidean motions.

Lastly, we must prescribe a normalization, as physicists
say.

This is done by taking a parallelotope $P$ with orthogonal
sides of lengths $ x_1, x_2, \dots ,  x_n $, and setting

\bigskip

Axiom 4.

$$ \mu_n (P) = x_1x_2 \dots x_n.$$

\bigskip

These axioms, together with suitable continuity conditions,
uniquely determine the volume of solids in Euclidean $n$- space.
For example,  starting from these four axioms, by a
limiting process such as one finds in an advanced calculus
textbook, one establishes the fact that that the volume of
a sphere $S_r$ of radius $r$ in
$n$-dimensional space  is given by the following formulas:

$$\mu_n(S_r) = \frac{ \pi^{n/2} r^{n/2}}{(n/2)!} $$

if the dimension $n$ is even

and

$$\mu_n (S_r) = \frac{2^n \pi^{(n-1)/2}((n-1)/2)! r^n}{n!} $$

if the dimension $n$ is odd.

\bigskip

It is still widely believed  that volume is the only
invariant measure in Euclidean $n$-space. But in point of fact, there are
other invariant
measures, defined on all reasonable subsets of Euclidean $n$-space, which
have a notable geometric significance. Our objective is to
describe all such invariant measures.

What happens if we keep the first three axioms, but tamper
with the fourth axiom, the normalization axiom ?  Will we
get something interesting, or will we get nothing new?  To
answer this question, we will appeal  to  the basic tools
of combinatorial mathematics.

\bigskip

The basic tools of combinatorial mathematics are the
elementary symmetric functions, to wit, the following
polynomials in $n$ variables:

$$e_1(x_1,x_2,...x_n) = x_1 + x_2 + ... +x_n.,$

$$e_2(x_1,x_2,...x_n) = x_1x_2+x_1x_3+...+x_{n-1}x_n,$$

$$ \dots $$

$$e_{n-1}(x_1, x_2, ..., x_n) = x_2x_3...x_n +
x_1x_3x_4...x_n+...+ x_1x_2...x_{n-1},$$

$$ e_n(x_1, x_2, ..., x_n) = x_1 x_2 ... x_n .$$

Observe an interesting coincidence.  The last of these
three symmetric functions is also the formula for the
volume of a parallelotope. Axiom 4 can be rewritten as

Axiom 4.

$$\mu_n (P)  = e_n(x_1,x_2,\dots , x_n).$$

\bigskip

Let us try an experiment, and replace the $n$-th symmetric
function by the
$n-$first symmetric function. Let us first take $n = 3$,
that is, three-dimensional space,  so that we can better
visualize what will be going on. Let  us see whether we
can  define a measure on subsets of $3$-dimensional space
by keeping three of the above axioms, but by replacing the
normalization Axiom 4 by using another symmetric function
instead of the symmetric function $e_3(x_1,x_2,x_3)$ which
gives the volume.  Let us first replace the symmetric
function $e_3$ by the symmetric function $e_2$, thereby
changing Axiom 4 to

\bigskip

 Axiom 4':

$$\mu_2(P) = x_1x_2 + x_1x_3 + x_2x_3 . $$

Does  this axiom define a measure ? Of course it does. The
right hand side is the formula for the surface area of  the
parallelotope $P$, divided by $2$.  Again we will find in
any advanced calculus textbook the explanation of the fact
that axioms 1, 2, 3, and 4' , together with some continuity
considerations, completely determine an invariant  measure
which is the surface area of solids in ordinary space.  For
example, the  following well known formula for the surface
area of a sphere $S_r$ of radius $r$ in $3$ dimensions is
obtained from these axioms:

$$\mu_2(S_r)= 4\pi r^2 .$$

Let  us take the next step.

\bigskip

Emboldened by our success with  two symmetric functions, we
now replace axiom 4 by yet another axiom, using another
symmetric function.  Let us set

\bigskip

Axiom 4".

$$ \mu_1(P) =  e_1(x_1, x_2, x_3) = x_1 + x_2 + x_3 $$

The new measure $\mu_1$ will satisfy axioms 1, 2, and 3,
and in addition it sill satisfy axiom 4". The symmetric
function of degree one plays the role that in the previous
two examples was played by the other two symmetric
functions.

\bigskip

But wait a minute: is  this  definition
consistent?

To realize that the definition of the new measure $\mu_1$ is
consistent, that is, that
$\mu_1$ as defined by axioms 1, 2, 3, and 4" really exists
and is not a dream of  reason,   look at two parallelotopes
$P_1$ and $P_2$  that have a face in common. The first
 parallelotope has sides
equal to $x_1, x_2, x_3$ , and the second parallelotope has sides equal to
$y, x_2, x_3$.   The two parallelotopes have a common face
with sides equal to $x_1, x_2$.   The measure
$\mu_1(P_1\cup P_2)$ of the parallelotope $P_1\cup P_2 $
can be computed in two ways: using the left  side of axiom
2, or using  the right side,  and the two computations had
better yield the same answer, in symbols:

$$\mu_1(P_1 \cup P_2) = \mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1
\cap P_2)$$

  Let us check this.

The  left side  is computed by observing that the
parallelotope $P_1\cup P_2$ has sides equal to $x_1$ ,$
x_2$ and $x_3 + y$. Therefore , Axiom 4" tells us that

$$\mu_1(P_1 \cup P_2) = x_1 + x_2 + x_3 + y.$

Now let us compute the right side .  We have

$$\mu_1(P_1) = x_1+x_2+x_3$$

$$\mu_1(P_2) = x_1 + x_2 + y$$

$$\mu_1(P_1 \cap P_2) = x_1 + x_2 ,$$

again by Axiom 4" applied to $P = P_1 \cap P_2$ , since
one side equals zero when the parallelotope is a flat, that
is, a rectangle. Therefore, the  right side of axiom 2
equals

$$\mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1 \cap P_2 )=
x_1+x_2+x_3 + x_1+x_2+ + y - (x_1+x_2)  = $$
$$ x_1+x_2+x_3+y ,$$

and the two sides  of our equations agree  thereby
convincing us that the definition may well be  consistent.

\bigskip

The preceding argument is convincing, even though it proves
nothing.

Actually, the   definition of $\mu_1(P) $ for a
parallelotope $P$  has a simple geometric interpretation.
When multiplied by  4, it  equals the perimeter of the
parallelotope $P$, that is, the sum of the lengths of all
the edges of the parallelotope $P$.

Just as happens for volume and area, it  can be shown by
continuity considerations that the measure $\mu_1$ can be
extended to all reasonable solids in ordinary space, for
example, to  all convex sets and to all polyhedra, convex
or non convex.

But, one may object, $\mu_1(P)$  makes sense for a
parallelotope $P$, because a parallelotope has a well
defined   perimeter.  What if $A$ is a  solid that does not
have a well defined perimeter, a sphere for example?   The
definition of the measure $\mu_1(A) $ for solids $A$ that
may  not have a well defined perimeter   flies in the face
of common sense.

\bigskip

Einstein wrote : "Common sense is the residue of those
prejudices that were instilled into  us before the age of
seventeen".  Common sense  must constantly  readjust to
reality.

\bigskip

The new  measure $\mu_1$  that we obtain in this way is
called  the mean width, a misnomer that has been kept for
historical reasons.   The mean width of a solid in space
is completely characterized by axioms 1, 2, 3, and 4".  In
particular, it is invariant , that is,  it does not depend
on position. For example, the formula for the mean width of
a sphere of radius $r$ is computed to be

$$\mu_1(S_r) = 4r. $$

Thus we see that in three dimensions each of the three
elementary symmetric functions of three variables  leads to
an invariant measure that enjoys equal rights with volume.
The first two of these measures are well known, namely,
volume and area.  The third, the mean width,  is at present
almost totally unkwnown. I know of no person who has an
intuitive feeling for the mean width, similar to the
intuitive feeling we have for volume and area.

\bigskip

Let us conjecture  a possible application of the mean
width.  A potato grower knows that a potato's volume is
important, because it determines the nutritional content of
the potato. The potato grower also knows that the surface
area of a potato is important, because it is rumored that
the vitamins in a potato are concentrated in the skin.  We
may conjecture that as soon as the potato grower will
become aware of the mean width, he or she will find a
nutritional interpretation of the mean width of a potato.
I am indebted to Steve Schanuel for this example.

\bigskip

A similar kind of  reasoning works in $n$ dimension. We
discover  $n$ different invariant measures, each of them
well defined on all polyhedra and on all finite unions of compact
convex sets. Each of the $n$ elementary symmetric functions
of $n$ variables leads to the definition of a new invariant
measure which is a different generalization of the notion
of volume. These $n$ measures  are called the intrinsic
volumes. The intrinsic volumes are first defined on an
orthogonal  polytope $P$ whose sides equal $x_1,x_2,\dots ,
x_n$ by setting

$$\mu_k(P) = e_k(x_1,x_2,\dots ,x_n),$$

where $ e_x(x_1,x_2,\dots ,x_n) $ is the $k$-th elementary
symmetric function. Here, the subscript $k$ ranges from $1$
to $n$.

One then proceeds to extend the definition of the intrinsic
volumes to more general sets, by a technique which we will
shortly see.

The intrinsic volumes are independent of each other, except
for certain inequalities they satisfy. Mathematicians are
presently working on determining these as yet unknown
inequalities among the intrinsic volumes. These
inequalities generalize the classical isoperimetric
inequality that relates volume to area.   At present, we
know very little about the intrinsic volumes;  they have
not been around  for long and very little research has been
done on them.  We do not even know the formula for the
intrinsic volumes of an $n$-simplex.

\bigskip

Now you are thinking: this is all fine and dandy, but how
is the extension of the intrinsic volumes from
parallelotopes to more general sets carried out? And
besides, isn't there any intuitive  interpretation we can
give the intrinsic volumes?

\bigskip

We will answer both these questions simultaneously. Let us
go back to three-dimensional space.  You all know that the
set of all straight lines in space - not necessarily
through the origin - forms a nice algebraic variety, called
the Grassmanian. The group of all Euclidean rigid motions
acts on the Grassmanian, and there is an invariant measure
on the Grassmanian under the action of the group of
Euclidean motions.  This invariant measure is unique except
for a constant factor. A similar statement may be made
about the set of all planes, and more generally for the set
of all linear varieties of dimension $k$ in Euclidean space
of dimension $n$. Remember that these linear varieties need
not pass through the origin.

\bigskip

In the practice of mathematics, computation with invariant
measures on Grassmanians  is  rare ; most
mathematicians would be hard put even to recall an explicit
formula for the invariant measures on Grassmanians.  Let us
take a few minutes to get a feeling for the invariant
measure on the set of all straight lines in three-space.
As is customary, we begin by giving this measure a name:
let us call it $\lambda^3_1$; the upper index $3$ stands
for three-dimensional space, and the lower index stands for
the dimension of a line, namely, one. To repeat, we use the
notation $\lambda^3_1$ to denote the invariant measure on
the set of all straight lines in three-space.

\bigskip

Consider  a rectangle $R$ placed anywhere in space, and
consider the set of all straight lines that meet the
rectangle $R$. Can we compute the measure of this set of
lines without knowing the formula for the invariant measure
on the Grassmanian of all lines in three-space? Of course
we can. A straight line meets the rectangle
$R$ either at a point or not at all; therefore, the value
of the measure of the set of all lines meeting $R$ depends
only on the area $\mu_2(R)$ of the rectangle
$R$.  If  we take another rectangle
$R'$ whose area is double the area of $R$, then the measure
of the set of all lines meeting $R'$ is double the measure
of the set of all lines meeting $R$. Proceeding along these
lines, we get to Cauchy's functional equation, and we infer
that  the measure of the set of all straight lines meeting
a rectangle $R$ equals a constant times the area
$\mu_2(R)$. Since we are at liberty to choose a
normalization of the measure, let us agree to set this
constant equal to one.

\bigskip

But instead of working with a rectangle we could have
worked with any planar figure
$C$ whatsoever, placed in an arbitrary position in space.
The measure of the set of lines meeting $C$ equals the area
$\mu_2(C)$, by the same reasoning. We stress the assumption
that $C$ must lie in a plane. To conclude: even without
knowing the formula for the invariant measure
$\lambda^3_1$, we can nevertheless compute the value of
such a measure on certain sets of lines.

\bigskip

Let us now take a more sophisticated set of straight lines. We take
a set $D$ in three-space which is the union of disjoint
sets $C_1, C_2,\dots C_n$, where each of the $C_i$ is
contained in a different plane, and we ask for the measure
of the set of all straight lines meeting $D$.  Such a
computation can be carried out, but it is a combinatorial
nightmare; so much so, that we are forced to do what
mathematicians do when confronted with combinatorial
nightmares: they change the problem ever so slightly. In this
case we take a hint from the way probabilists work.  Let
$X_D(\omega)$ equal the number of times the straight line
$\omega$ meets the set
$D$.  Instead of computing a measure, let us compute the
integral

$$ \int X_D(\omega) d\lambda^3_1(\omega) ,$$

where $\omega$  ranges over the Grassmanian, that is, over
the set of all straight lines in space. We will see that we
can compute this integral without knowing the measure
$\lambda^3_1$ on the Grassmanian.  Since

$$ D = \bigcup_{i=1}^n C_i ,$$

and since the $C_i$ are disjoint, we have

$$  \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n
\int X_{C_i}(\omega) d\lambda^3_1(\omega). $$

But we have chosen each of the sets $C_i$ to lie in a
plane, so that a straight line meets $C_i$ either once or
not at all. It follows that

$$  \int X_{C_i}(\omega) d\lambda^3_1(\omega) = \mu_2(C_i)
$$

and therefore

$$  \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n
\mu_2(C_i) . $$

What is this identity telling us? The right hand side
equals the area of the surface $D$. Nothing stops us from
passing to the limit, and making the following assertion.
Let $E$ be "any" surface in space, and let $X_E(\omega) $
be the number of times the straight line $\omega$ meets the
surface $E$. Then the integral

 $$ \int X_E(\omega) d\lambda^3_1(\omega) $$

ranging over all straight lines $\omega$, equals the
surface area of $E$.

In probabilistic language:
 the average number of times a randomly chosen straight
line meets the surface $E$ equals the surface area of
$E$.

\bigskip

Let us now retrace our steps, and repeat the same reasoning
taking the set of all planes in space, instead of the set of
all straight lines. The invariant measure on this
Grassmanian is denoted by $\lambda_2^3$, where again the
upper index stands for three dimensional space, and the
lower index for the dimension of a plane. Since a plane
meets a straight line segment either at a point or not at
all,  the same argument shows that the measure of the set
of all planes that meet a line segment $L$ equals
$\mu_1(L)$, namely, the length of the segment $L$; more
generally, if $F$ is any curve "whatsoever" in space, and
if $X_F(\omega)$ equals the number of times the plane
$\omega$ meets the curve $F$, then repeating the argument
we used for straight lines we infer that the integral

$$ \int X_F(\omega) d\lambda^3_1(\omega)  $$

equals the length of the curve $F$. The  variable of
integration $\omega$ now ranges over planes, not over
straight lines. Here again we compute  an integral without
knowing the measure.

\bigskip

We are now very close to getting an intuitive
interpretation of the mean width. Recall the parallelotope
$P$ with sides equal to $x_1,x_2,x_3$. Let us take the
curve  to be the perimeter of the parallelotope $P$.
Every plane meets the perimeter of the parallelotope at
four points.  Therefore, the measure of the set of all
planes that meet the perimeter of the parallelotope equals
four times the perimeter of the parallelotope. But a plane
meets the perimeter of a parallelotope if and only if it
meets the parallelotope. We therefore reach an important
conclusion: the measure of the set of all planes meeting a
parallelotope equals the mean width of the parallelotope,
except for a constant factor which we will again set to be
one.

\bigskip

In view of this realization, we can immediately see how to
define the mean width of any closed convex set: it equals
the measure of the set of all planes that meet the convex
set. Thus, we have shown that the mean width may be
extended to all closed convex sets in space.

\bigskip

We are now in a position to give  a probabilistic
interpretation of the mean width of a convex set.

Take two compact convex sets $A$ and $B$ in three dimensional
Euclidean space, and suppose that $A$ is contained in $B$.
Let  us begin by belaboring the obvious.  Suppose that we
take a point at random belonging to the larger set $B$.
What is the probability that the point shall belong to the
smaller set $A$?  The answer is clear: such a probability
equals the ratio of the volume of $A$ by the volume of
$B$.

\bigskip

 Instead of choosing a point at random, let us
choose a straight line at random in space.  Assuming that
such a straight line meets the larger set $A$, what is the
probability that such a straight line will also meet the
smaller set $B$?

We have already computed the answer to this question,
albeit implicitly.  Such a probability equals the surface
area of the set $A$, divided by the surface area of the set
$B$.

\bigskip

You can tell what is coming next. We now  take a random
plane in space. Assuming that the plane meets the larger
set $B$, what is the probability that it will also meet the
smaller set $A$?  The answer is the following: such a
probability equals  the mean width of $A$, divided by the
mean width of $B$.

\bigskip

In Euclidean $n$-space, we obtain by much the same
reasoning interpretations of the intrinsic volume
$\mu_k(C)$ of a compact convex set $C$ as the Grassmanian
measure of the set of all linear varieties of dimention
$n-k$ that meet the convex set $C$, and a similar
probabilistic interpretation holds.

\bigskip

What comes next? There are at least two questions still
open. First, are there any other invariant measures besides
the intrinsic volumes, and second, how can the definition
of the intrinsic volumes be extended to more general
subsets of
$n$-space than convex sets. The answers to both
these questions are closely related.

\bigskip

The answer to the first question is negative. We are
missing one measure, and to discover it, we will engage for
a minute in the kind of mathematical reasoning that
physicists find unbearably pedantic, just to show
physicists  that such reasoning does pay off.

Let us ask ourselves the question: what is the value of the
symmetric function of order zero of a set of $n$ variables
$x_1,x_2,\dots , x_n$, or $e_0(x_1,x_2,\dots , x_n)$?  I
will give you the answer, and will leave it to you to
justify this answer after the lecture is over.  The answer
is the following:  $e_0= 1$ if $n >0$, that is, if the set
of variables  $x_1,x_2,\dots , x_n$ is non empty,  and
$e_0 = 0 $ if the set of variables is empty.

\bigskip

We are led to believe that  there
may exist an invariant measure in $n$-space associated
with the symmetric function of order zero. We set

$$\mu_0(C) = 1 $$

if $C$ is any non empty  compact convex set, and of course
$\mu_0(\oslash ) = 0$. Does such a measure exist? It does indeed exist,
and the fact that it exists is, in my opinion, one of the
most remarkable discoveries ever made in mathematics.

\bigskip

We will prove that such a measure is well defined on any
set which is a finite union of compact convex sets. We do
this by employing a classical device borrowed from
functional analysis: instead of defining a measure, we
define a linear functional on all simple functions, that
is, on all real  functions $f(\omega)$ defined for $\omega
\in R^n $ which are linear  combinations of indicator
functions of compact convex sets. Let us first begin with
the case $n=1$, that is, let
$\omega$ range over points on the line. Define a linear
functional $\chi_1$ on simple functions as follows:

$$ \chi_1(f) = \sum (f(\omega) - f(\omega+)) ,$$

where the sum ranges over all real numbers $\omega$. The
meaning of the plus sign is best gleaned from an example.

Let f be the indicator function of the closed segment
$[a,b]$.  Then $f(\omega) - f(\omega+) = 0$ for all $\omega
$ except $\omega = b $, because we have $f(b) = 1$ but
$f(b+) = 0$. Thus, we see that $\chi_1(f) = 1 $ if $f$ is
the indicator function of an interval $[a,b]$.

\bigskip

Now let us go over to $n$ dimensions, proceeding by
induction. Do not worry, this won't take long. Take a straight line
$L$ and for every point
$\omega$ in
$L$ let
$H_{\omega} $ be the hyperplane through the point $\omega $
perpendicular to the line $L $. If $f$ is a simple function
defined in $n$ space, and if
$\omega$ is a point on the straight line $L$, let
$f_{\omega}$ be the restriction of
$f$ to the hyperplane
$H_{\omega}
$.   Define a linear functional $\chi_n $ as follows:

$$ \chi_n(f) = \sum \chi_{n-1}(f_{\omega}) -
\chi_{n-1}(f_{\omega +}) ,$$

where the sum ranges over all points $\omega$ on the line
$L$. There is
only a finite set of $\omega's $ for which the summand is
non zero.  When $f$ is the indicator function of a non empty
compact convex set, then an argument similar to the
preceding shows that $\chi_n(f) = 1 $.  Thus, we may define
a measure $\mu_0(G) = \chi_n(f) $, where $G$ is any finite
union of compact convex sets, and $f$ is the indicator
function of the set $G$.  We have thus proved the existence
of a
 measure $\mu_0$ which is defined on all finite unions of
compact convex sets, and which takes the value one on all
non empty compact convex sets . This measure has a long
history: it is   the Euler characteristic.

\bigskip

Now you are thinking: if this is the Euler characteristic,
then it is up to you to show that it coincides with what we
ordinarily believe to be the Euler characteristic.  Let us
conclude this lecture by deriving the formula of
Euler-Schl\"afli-Poincar\'e formula for polyhedra. As a
matter of fact, this formula can be encapsulated into a
simpler formula, one that is easy to remember.

Let $C$ be a non empty compact convex polytope of dimension
$n$, and let $int(C)$ be the interior of $C$. Then we have
the following fundamental formula for the Euler
characteristic of $int(C)$:

$$ \mu_0(int(C)) = (-1)^n .$$

Indeed, if  $f$ is the indicator function of the set
$int(C)$, we have:

$$  \mu_0(int(C))  = \sum \chi_{n-1}(f_{\omega}) -
\chi_{n-1}(f_{\omega + }) ,$$

where the sum ranges over all points $\omega$ on the line
$L$ as above. But by induction, we see that every term on the right
hand side equals zero, except when $\omega$ is the first
point on the line $L$ for which  the intersection $C \cap
H_{\omega} $ is not empty .  If $\omega_{\ell} $ is such a
first point, then we have

$$  \chi_{n-1}(f_{\omega_{\ell}}) = 0 $$

because the point $\omega_{ell} $ is on the boundary of
$C$, and

$$  \chi_{n-1}(f_{\omega +}) = (-1)^{n-1} $$

by induction hypothesis,because $ f_{\omega +}$ is the
indicator function of the set
 $int(C)
\cap H_{\omega}
$, which is the interior of a convex polyhedron one
dimension lower.

Putting all this together, we obtain

$$  \mu_0(int(C))  = \sum \chi_{n-1}(f_{\omega_{\ell}}) -
\chi_{n-1}(f_{\omega_{\ell}+}) =  - (-1)^{n-1} = (-1)^n ,$$

as desired.

We are now in a position to state the famous Euler formula
for polyhedra. What is a polyhedron? A polyhedron is a
finite union of convex polyhedra. Given a polyhedron, we
must define a system of faces. We will say that a set
$\bf{F}$ of convex polyhedra is a system of faces for an
arbitrary polyhedron $K$ when the elements of
$\bf{F}$, called faces,  are non empty compact convex sets
$F$ with disjoint interiors such that

$$ K =  \bigcup_{F \in \bf{F} }int(F). $$

Caution: the interior of a face of dimension $k$ is to
be taken relative to the linear space of dimension $k$ that
contains the face, and the interior of a point is a point.

Under these conditions we may take the Euler characteristic
of both sides, and using the fact that any two interiors of
faces are disjoint we obtain (using the fact tghat the measure of the disjoint
union of a family of sets equals the sum of the mesaures of the individual
sets):

$$ \mu_0(K) = \sum_{F \in \bf{F} } \mu_0(int(F)) = f_0 - f_1
+ f_2 - \dots +
\dots , $$

where $f_i$ equals the number of faces of dimension $i$.
This is Euler's formula.

\bigskip

We can now answer the second of the questions we had left
open: how to  extend the definition of the intrinsic volumes
from compact convex sets  to all finite unions of compact
convex sets.  If $G$ is such a finite union of compact
convex sets,  then  we set

$$ \mu_k (G) = \int \mu_0(G \cap \omega )
d\lambda_{n-k}^n(\omega), $$

where $\omega$ ranges over all linear varieties of dimension $n-k$ in
$n$-space.
The left hand side defines  is a measure, and when $G$ is a compact convex
sets it
agrees with the definition we have already given. It is
therefore the desired extension. The Euler
characteristic does all the work for us.

\bigskip

We are now in a position to state the main theorem of
geometric probability.

We will say that an invariant measure $\mu$ on Euclidean
$n$-space, defined on all finite unions of compact convex
sets, is continuous, when

$$ \lim_{C_n \rightarrow C} \mu(C_n) = \mu(C) $$

For all sequences $C_n$ of compact convex sets converging
to the compact convex set
$C$.

We have the

\bigskip

{\bf Main Theorem of Geometric Probability}

\bigskip

 The
$n+1$ intrinsic volumes  $\mu_0, \mu_1,\dots , \mu_n $ are
a basis of the space of all continuous  invariant measures
defined on all finite unions of compact convex sets.

\bigskip

The first proof of this theorem is due to Hadwiger; the
first  elementary proof was published last year by Dan
Klain of Georgia Tech.

\bigskip

In closing, let me try to answer the question you are about
to ask: what has this got to do with geometric probability,
anyway?

I will attempt a sketchy answer. Consider two compact convex sets $A$
and $B$. We imagine $B$ to be fixed in
$n$-space, and that we "drop" the rigid set $A$ at random.  What is
the probability that $A$ meets $B$? We answer this question
in three steps. First, we realize that by keeping $B$ fixed
and varying $A$ by the group of Euclidean Motions, we define an invariant
measure on
convex sets
$B$. Second, we apply Hadwiger's theorem, and infer that
such an invariant measure equals a linear combination of
the $n+1$ intrinsic volumes, with coefficients independent
of $B$. Third, we determine these coefficients by taking
suitable
$B$'s.  The end result is an identity which is known as the
kinematic formula, which has been the object of much
research in this century, still going on today.

\bigskip

Thank you for your attention.

\newpage

\begin{center} {\bf INVARIANT THEORY, OLD AND NEW\\

being\\

The second Colloqium Lecture\\

\bigskip

delivered at the Annual Meeting of the American Mathematical Society\\

Baltimore, January 8, 1997.\\

\bigskip

Gian-Carlo Rota\\ Department of Mathematics\\  MIT, room 2-351\\ 77
Massachusetts
Avenue\\
Cambridge MA 02139-4307\\

\end{center}

\bigskip

\bigskip

Invariant theory is the great Romantic story of mathematics. For one
hundred and fifty
years,
from its beginnings with Boole to the time, around the middle of this
century,  when it
branched
off into several independent disciplines, mathematicians of all countries
were brought
together
by their common faith in invariants: in England, Cayley, MacMahon,
Sylvester and
Salmon, and
later, Alfred Young,  Aitken, Littlewood and Turnbull. In Germany, Clebsch,
Gordan,
Grassmann, Sophus Lie, Study; in France, Hermite, Jordan and Laguerre; in
Italy, Capelli,
Brioschi, Trudi and Corrado Segre, in America, Glenn, Dickson, Carus (of
the Carus
Monographs), Eric Temple Bell and later Hermann Weyl. Seldom in  history
has an
international  community of  scholars  felt so  united  by a common
scientific ideal for so
long a
stretch of time. In our century,    Lie theory and algebraic geometry,
differential algebra and
algebraic combinatorics are offsprings of invariant theory. No other
mathematical theory,
with
the exception of the theory of functions of a complex variable, has had as
deep and lasting
an
influence on the development of mathematics.

\bigskip

Eventually, invariant theory was to become a victim of its own success: the
very term
"invariant
theory" is nowadays understood in such a  wide variety of senses that it
has become all but
meaningless.     It is no wonder that you are baffled by the title of this
lecture, and curious
to hear
what will be said about invariant theory in the next forty-eight minutes.

\bigskip

Like the Arabian phoenix arising from its ashes, classical  invariant
theory, once
pronounced
dead, is once again at the forefront of mathematics. The old treatises are
being dusted off
the
shelves of library basements and reread, reinterpreted and presented in a
language that
meets the
standard of rigor of our day. The program of classical invariant theory,
that had for some
time
been given up as hopeless, is again being pursued, and success may at last
be within reach.

\bigskip

We will review  two turning points in the history of invariant theory.
The first, the
"new"
one,  happened around the turn of the century, and its effects are still
being felt all over
mathematics. The second,  the "old" one, happened very early in the game,
and led to a
serious
misunderstanding that lasts to this day.

\bigskip

A pedestrian definition of invariant theory might go as follows: invariant
theory is the study
of
orbits of group actions. Such a definition is correct, but it must be
supplemented by a
programmatic statement.  Hermann Weyl, in the introduction to his book "The
Classical
Groups", was  the first in this century to  give a sweeping overview of the
program of
invariant
theory. He summarized this program in two basic assertions. The first
states that "All
geometric
facts are expressed by the vanishing of invariants" , and the second states
that "all
invariants are
invariants of tensors".

\bigskip

Let us briefly comment on  these lofty statements. What is a geometric
fact? A geometric
fact is
a fact about space  that is independent of the choice of a coordinate
system. Geometric
facts are
described by means of equations which require a choice of   coordinates. In
a vector space
$V$
of dimension $n$ one chooses a coordinate system
$ x_1, x_2,\dots ,x_n
$ . Since Descartes, we have learned to express geometric facts by
equations  in the
coordinates
$ x_1, x_2,\dots ,x_n $.  However, about  one hundred years ago,
mathematicians and
physicists  made the shocking discovery  that the usual type of equations,
that is, equations
in
the commutative ring generated by the variables
$ x_1, x_2,\dots ,x_n $, are inadequate for the description of  a lot of
geometric and physical facts.
Motivated by this discovery, they introduced  a  more general ring. This is
the ring of non
commutative polynomials in the coordinates  $ x_1, x_2,\dots ,x_n $.
Homogeneous
elements
of this ring, that is, homogeneous  non commutative polynomials in the
variables  $ x_1,
x_2,\dots ,x_n $, are called tensors. If we believe Hermann Weyl's
philosophy, then we
will be
satisfied that  equations in the tensor algebra  suffice for the
description of any geometric
fact we
will ever meet.  Furthermore, if these equations are to  express geometric
properties, then
they
must hold no matter what coordinate system is chosen; in other words,
equations that
describe
geometric facts must be invariant under changes of coordinates.    The
program of
invariant
theory, from Boole to our day, is precisely the translation of  geometric
facts into invariant
algebraic equations expressed in terms of  tensors.

This program of translation of geometry into  algebra  was to be carried
out in two steps.
The
first step consisted in decomposing tensor algebra into irreducible
components under
changes of
coordinates. The second step consisted in devising an efficient notation
for the expression
of
invariants for each irreducible component.  The first step was successfully
carried out in
this
century; the second was abandoned sometime in the twenties and only
recently has
it resurfaced.

\bigskip

The decomposition of tensor algebra into irreducible components was
discovered around
the turn
of the century   almost simultaneously by Issai Schur and Alfred Young.
The gist of this
decomposition is one of the great advances in mathematics of all times, and
it may be
worthwhile
to  present it in a form that can be made available  to undergraduates.

\bigskip

Let us consider
 functions of three variables, such as
$f(x_1, x_2, x_3)$.
Two well known classes of functions of three variables are symmetric
functions, defined to
satisfy
 the equations

$$f_s(x_1,x_2,x_3) = f_s(x_{i_1},x_{i_2},x_{i_3}) $$

for every permutation sending the indices $(1,2,3)$ to $(i_1,i_2,i_3)$ ,

and  skew symmetric functions , defined by  the equations

$$f_a(x_1,x_2,x_3) = \pm f_a(x_{i_1},x_{i_2},x_{i_3}), $$

where the sign is $+1$ or $-1$ according as the permutation sending  the
indices
$(1,2,3)$ to $(i_1,i_2,i_3)$ is even or odd.

\bigskip

It is not true that a function of three variables is the sum of  a symmetric
  function and  a skew-symmetric  function.  A third type of function is
required,  which is
called  a cyclic function,  which is defined by  the equation

$$f_c(x_1,x_2,x_3) + f_c(x_3,x_1,x_2) + f_c(x_2,x_3,x_1)  = 0.$$

Every function of three variables can be uniquely written as the sum
 of a symmetric function, a skew symmetric function, and a cyclic function,
in symbols

$$f(x_1,x_2,x_3) =  f_s(x_1,x_2,x_3) +f_a(x_1,x_2,x_3) +f_c(x_1,x_2,x_3).$$

Each of the three symmetry classes is invariant under
permutations; this fact is obvious for symmetric and skew symmetric
functions but not
quite so
obvious for cyclic functions.  These three invariant subspaces  play for
the group of
permutations
of a set of three elements  a role analogous to the  role of the
eigenvectors of a
symmetric
matrix.

\bigskip

For  functions $f(x_1,x_2,x_3,x_4) $ of four variables there are five symmetry
classes, which are defined as follows:

\bigskip

1. Symmetric functions.

\bigskip

2. Skew symmetric functions.

\bigskip

3. Cyclic symmetric functions, satisfying the three equations

$$f(x_1,x_2,x_3,x_4)+ f(x_1,x_4,x_2,x_3)+f(x_1,x_3,x_4,x_3) = 0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_4,x_2,x_1,x_3)+f(x_3,x_2,x_4,x_1) = 0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_4,x_1,x_3,x_2)+f(x_2,x_4,x_3,x_1) = 0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_3,x_1,x_2,x_4)+f(x_2,x_3,x_1,x_4)=0.$$

\bigskip

4. Functions satisfying the four equations

$$f(x_1,x_2,x_3,x_4)+f(x_2,x_1,x_3,x_4)+f(x_1,x_2,x_4,x_3)+
f(x_2,x_1,x_4,x_3) =
0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_3,x_2,x_1,x_4)+f(x_1,x_4,x_3,x_2)+
f(x_3,x_4,x_1,x_2)=
0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_1,x_3,x_2,x_4)+f(x_4,x_2,x_3,x_1)+
f(x_4,x_3,x_2,x_1)=
0,$$

$$\sum sign(\sigma ) f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma
4})= 0.$$

\bigskip

5. Functions satisfying the equations

$$f(x_1,x_2,x_3,x_4)-f(x_2,x_1,x_3,x_4)-f(x_1,x_2,x_4,x_3)+
f(x_2,x_1,x_4,x_3) =
0,$$

$$f(x_1,x_2,x_3,x_4)-f(x_3,x_2,x_1,x_4)-f(x_1,x_4,x_3,x_2)+
f(x_3,x_4,x_1,x_2)=
0,$$

$$f(x_1,x_2,x_3,x_4)-f(x_1,x_3,x_2,x_4)-f(x_4,x_2,x_3,x_1)+
f(x_4,x_3,x_2,x_1)=
0,$$

$$\sum  f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma 4})= 0.$$

\bigskip

Every function of four variables is uniquely expressible as the sum of five
functions, each
one
belonging to one of these symmetry classes.  Each symmetry class is
invariant under
permutations.

\bigskip

More generally, every function of $n$ variables $f(x_1,x_2,\dots ,x_n)$ can
be uniquely
written as
the sum of $p_n$ functions, each one belonging to a different symmetry
class. Here,
$p_n$ equals
the number of partitions of the integer $n$. Each symmetry class is defined
by equations
which are
not difficult to find.

This decomposition holds for tensors as well, after some cosmetic changes
of notation.
To this day, only
two symmetry classes of tensors have been studied in any detail.
 Symmetric tensors are ordinary commutative polynomials such as we
learned to use in analytic geometry. Skew symmetric tensors are polynomials
in the
coordinates  $
x_1, x_2,\dots ,x_n $ when the variables are assumed to  satisfy  the
equations
$x_ix_j = - x_jx_i $.
Tensors belonging to symmetry classes other than the classes of symmetric and
 skew symmetric tensors  also occur in geometry and physics. However, these
symmetry
classes
have been studied very little, and they are a long way from being understood.

\bigskip

So much for  the word   "new" in the  title of this lecture; let us next do
some  justice
to the word "old".  We will describe  the most peculiar  feature of classical
invariant theory, namely, the symbolic or umbral notation, to which Eric
Temple Bell
dedicated
his Colloquium Lectures in 1927.  We will consider the simplest group,
namely, the group
of translations of the line. The unusual  features of the symbolic method
will already be
 apparent in this  special case.

Let  $p(x)$ and $ q(x) $ be monic polynomials in the variable $x$.
 We write them in the following quaint notation:

$$ p(x) = x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} + \dots
+ {n
\choose
n-1}a_{n-1}x + a_n , $$

and

$$ q(x) = x^k + {k \choose 1}b_1x^{k-1} + {k \choose 2}b_2 x^{k-2} +
\dots + {k \choose k-1}b_{k-1}x + b_k . $$

We  assume that the polynomial $q(x)$ is of lower degree than
 the polynomial $p(x)$, that is, that $k \leq n $.

Define the translation operator $T^c$ on a polynomial $p(x)$ as follows:

$$ T^c p(x) = p(x + c) .$$

Let us write

$$ p(x+c) = x^n +{n \choose 1}p_1(c) x^{n-1} + {n \choose 2}p_2(c) x^{n-2}
+ \dots +
{n
\choose n-1}p_{n-1}(c) x + p_n(c) , $$

The $j$-th coefficient $p_j(c)$ of the polynomial $p(x+c) $ is
computed to be

$$ p_j(c) =  a_j + {j \choose 1} a_{j-1} c + {j \choose 2 }a_{j-2}c^2 +
\dots + c^n . $$

A polynomial I($a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) $ in the
 variables

$a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k $  is said to be an invariant
of the two
polynomials $p(x)$, $q(x)$ when

$$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) = I(p_1(c),p_2(c), \dots
, p_n(c),
q_1(c), q_2(c),
\dots , q_k(c) ) $$

for all complex numbers $c$. By abuse of notation, we write $I(p(x), q(x)) $
and we speak
of $I$ as
being an invariant of the polynomials $p(x)$ and $q(x)$. In this abusive
notation, a
polynomial
$I$ is said to be an  invariant of the polynomials $p(x)$ and
$q(x)$ whenever

$$ I(T^cp(x), T^cq(x) ) = I(p(x), q(x)) $$

for all constants $c$.

\bigskip

Invariant theory is concerned with the problem of finding all  invariants
of a given set of
polynomials, as well as their significance.

\bigskip

What is meant by the "significance" of an invariant? We will  appeal to
Hermann
Weyl. "Every" property of polynomials which is invariant under the group of
translations is
expressed by the vanishing of a set of invariants. In other words, "any"
set of polynomials
which
is invariant under translations is the same set as a set of polynomials
obtained by setting to
zero a
set of invariants of such polynomials.

It is impossible to understand the above  statement without examples.

Let us consider the simplest and oldest example.  The property of  a
quadratic polynomial

$$ q(x) =  x^2 + 2b_1 x + b_2 .$$

of having a double root is invariant under translations; in other words, if
the polynomial
$q(x)$
has a double root, so does the polynomial $p(x+c)$  for any constant
$c$ . Following Hermann Weyl, we look for an invariant whose vanishing
expresses this
property. Sure enough, it  is easy to check that the discriminant

$$ D( b_1, b_2) = b_1^2 - b_2 $$

is the desired  invariant.

This example, due to Boole, was the spark that led to the birth of
invariant theory.

\bigskip

One often hears the sentence "Hilbert killed invariant theory",  repeated
as an excuse to
ignore
all that went on in invariant theory after Hilbert. I don't know who made
up this infamous
sentence.  It is not true.  Hilbert loved invariant theory, and he went on
publishing striking
papers in  invariant theory  well after he proved the theorem that is
nowadays called the
Hilbert
basis theorem, the theorem that is supposed to have killed invariant
theory. Some of the
most
fascinating results in invariant theory were discovered in the first twenty
years of this
century, a
long time after Hilbert proved his basis theorem.

\bigskip

What then is the reason for the temporary demise of invariant theory in
this century? One
reason is
 the endemic  use of a notation that lacked rigor and that
amounted to little more than handwaving in print. This is the symbolic or
umbral notation.

\bigskip

Dieudonn\'e  wrote  that half the success of a piece of mathematics depends
on  a  proper
choice of notation. It would be interesting to make a list of unfortunate
notations that killed
various chapters of mathematics, as well as a list of felicitous notations
that promoted the
development of other branches of mathematics. The symbolic or umbral
notation that was
used
by invariant theorists through the nineteen twenties was a catastrophe.  A
number of
mathematicians tried to make sense of the symbolic method without success,
the three most
notable ones being Hermann Weyl,  Eric Temple Bell, and  Edward Hegeler
Carus. Eric
Temple Bell failed to properly define  umbral notation, and his book
"Algebraic Arithmetic"
remains to this day the book of seven seals.
 If Hermann Weyl and Eric Temple Bell  had lived fifty years longer,
so as to benefit of the development of what was in their time called
"modern" algebra,  they
would
undoubtedly have succeeded in properly defining umbral notation.

In our day, it does not take much work to accomplish this task.
Do not be alarmed: it  will only take a
few minutes.  Before  I start spouting out definitions, let me say what I
am not going to
say.
Umbral notation can be shown to be equivalent, or "cryptomorphic", to use a
term invented
by my
late friend Garrett Birkhoff, to another notation that has gained great
notoriety in our day: I
mean the
notation of Hopf algebras. I will not justify this Sybilline pronouncement,
not because it is
difficult
to do so, but because it would be too boring to do so.

\bigskip

Let  us go on to the definition of umbral notation.

Side by side with the polynomials $p(x)$ and $q(x)$, we consider another
polynomial
algebra
${\bf C}[ x, \alpha , \beta  ] $ in three variables $x$ , $ \alpha $ and
$\beta $,  together
with a
linear functional $E$ defined on the underying vector space
${\bf C}[x, \alpha, \beta ]$.  The  definition of the  linear functional $E$ is
the key point. It is carried out
in the following steps:

\bigskip

Step 1. Set

$$ E(x^j ) = x^j $$

for all non negative integers $j$   , in particular $E(1) = 1$.

\bigskip

Step 2. Set

$$ E(\alpha^j) = a_j ,$$

in particular, we have $ E(\alpha^j) = 0 $ if $j > n $.

\bigskip

Step 3. Set

$$ E(\beta^j) = b_j ,$$

in particular, we have  $ E(\beta^j) = 0 $ if $j > k $.

\bigskip

Step 4. This is the main step. Set

$$ E(\alpha^i \beta^j x^{\ell}) = E(\alpha^i ) E( \beta^j )x^{\ell}. $$

Following Sylvester, the variables $\alpha $ and $\beta $ are called umbrae.
In other words, the linear functional $E$ is multiplicative on distinct umbrae.

\bigskip

Step 5.

Extend by linearity.

\bigskip

This completes the definition of the linear functional $E$.

We next come to the most disquieting feature of umbral notation.  Let
$f(\alpha , \beta , x)
$ and
$g(\alpha , \beta , x) $ be two polynomials in the variables $
\alpha , \beta , x $.  We write

$$ f(\alpha , \beta , x) \cong (\alpha , \beta , x) $$

to mean

$$ E(f(\alpha , \beta , x)) = E( g(\alpha , \beta , x) ) . $$

Read $\cong $ as " equivalent to".

The "classics" went a bit  too far, they wrote

  $$ f(\alpha , \beta , x) = g(\alpha , \beta , x) $$

that is, they replaced the symbol $\cong$ by ordinary equality. This was an
excessive abuse of notation.   The "classics" were
 aware of the error, and while they avoided computational errors
 by clever artistry, they were unable to  settle on a correct notation.

\bigskip

The umbral or symbolic method consists in replacing all occurrences of the
coefficients of
the
polynomials $p(x)$ and $q(x)$ by umbrae and equivalences.  For example,

$$ p(x) \cong (x + \alpha)^n $$

and

$$ q(x) \cong (x+ \beta)^k .$$

Let us carefully check  the first equivalence.

By definition, the equivalence means the same as

$$ E(p(x)) = E( (x + \alpha)^n).$$

Since $E(x^j) = x^j $  for all non negative integers $j$, this identity can
be rewritten as

$$ p(x) = E( (x + \alpha)^n).$$

Expanding the right hand side by the binomial theorem, we obtain

$$ E( (x + \alpha)^n) = $$
$$ E(  x^n + {n \choose 1}\alpha  x^{n-1} +  {n \choose 2}\alpha^2
x^{n-2} + \dots + {n \choose n-1}\alpha^{n-1}x +\alpha^n)  $$

By linearity this equals

$$ x^n + {n \choose 1}E(\alpha) x^{n-1} + {n \choose 2}E(\alpha^2) x^{n-2}
+ \dots +
{n
\choose n-1}E(\alpha^{n-1})x + E(\alpha^n) . $$

Evaluating the linear functional $E$, we see that this in turn equals

$$ x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} +
\dots + {n \choose n-1}a_{n-1}x + a_n , $$

as desired.

The expression

$$ (x + \alpha)^n $$

is called  an umbral representation of the polynomial $p(x)$.

 In umbral notation, a     complex number $r$ is a root of the polynomial
 equation $p(x) = 0$ if and only if

$$ (\alpha + r)^n \cong 0. $$

Similarly, in umbral notation  the   polynomial $T^cp(x)= p(x+c)$ may be
represented as
follows:

$$ p(x+c ) \cong (x+\alpha + c)^n, $$

and this yields the umbral expression  for the coefficients
$p_j(c)$ of the polynomial $p(x+c)$, namely

$$ p_j(c) \cong (\alpha + c)^j . $$

Let us next see how umbral notation is related to    invariants.
 Let us assume  that the two polynomials $p(x)$ and $q(x)$ have the same degree
$n$. Then an  invariant $A$ of the polynomials $p(x), q(x)$ may be defined as
follows:

$$ A(q(x), p(x)) \cong (\beta - \alpha)^n .$$

The evaluation  of the invariant $A$ in terms of the   coefficients of
$p(x)$ and
$q(x)$ proceeds as follows:

$$ A(q(x), p(x)) = E(\beta - \alpha)^n) = $$

$$E(\beta^n -{n \choose 1} \beta^{n-1}\alpha +   \dots
+ (-1)^{n-1}{n \choose n-1}\beta
\alpha^{n-1} + (-1)^n \alpha_n) = $$

$$ E(\beta^n)  - E({n \choose 1} \beta^{n-1}\alpha) +
 \dots +  (-1)^{n-1}E({n \choose n-1}\beta
\alpha^{n-1}) + (-1)^n E(\alpha^n)= $$

$$ E(\beta^n)  - {n \choose 1} E(\beta^{n-1})E(\alpha) +   \dots +
(-1)^{n-1}{n \choose
n-1}E(\beta) E(\alpha^{n-1}) + (-1)^n E(\alpha^n) = $$

$$ b_n - {n
\choose 1} b_{n-1}a_1 + {n \choose 2}b_{n-2}a_2 - \dots + \dots +
(-1)^{n-1}{n \choose
n-1}b_1a_{n-1} + (-1)^n a_n. $$

Why is $A$ an invariant? This is best seen in umbral notation:

$$ A(T^cq(x), T^cp(x)) \cong (\beta + c - \alpha - c )^n = (\beta -
\alpha)^n. $$

The invariant $A$ is called the apolar invariant; two polynomials $p(x)$
and $q(x)$ having
the
property that $A(q(x), p(x)) = 0 $ are said to be apolar. In umbral
notation,  two
polynomials
are apolar whenever

$$ (\beta - \alpha)^n \cong 0. $$

 The concept of apolarity has a distinguished pedigree going all the way
back to
Apollonius.

\bigskip

What is the "significance" of the apolar invariant? What does it mean for two
polynomials to be apolar? This question is answered by

\bigskip

{\bf Theorem 1}. Suppose that $r$ is a root of the polynomial $q(x)$, that is,
that $q(r)= 0$.
Then the
polynomials $q(x)$ and $p(x) = (x - r)^n $ are apolar.

\bigskip

Proof. For $p(x) = (x-r)^n $ we  have $\alpha^j \cong (-r)^j $, and hence

$$ A(q(x),p(x)) \cong (\beta - (-r))^n = (\beta + r)^n \cong 0, $$

as desired.

\bigskip

Corollary. If the polynomial $q(x)$ has $n$ distinct roots $r_1,r_2,\dots
, r_n $, and if
the
polynomial $p(x)$ is apolar to $q(x)$, then there exist constants
$c_1,c_2,\dots ,c_n $ for
which

$$ p(x) = c_1 (x-r_1)^n + c_2 (x-r_2)^n + \dots + c_n (x-r_n)^n . $$

\bigskip

Proof. The  dimension of the affine subspace of all
monic polynomials $p(x)$ which are apolar to $q(x)$ equals $n$. But if  the
polynomial
$q(x)$
has simple roots, then by the above theorem the polynomials $(x-r_1)^n,
(x-r_2)^n,\dots , (x-
r_n)^n $ are
linearly  independent and apolar to $q(x)$. Hence the polynomial $p(x)$ is
a linear
combination  of these polynomials. This completes the proof.

\bigskip

Thus, we see that apolarity  gives a trivial  answer to the following
question: when can a
polynomial $p(x)$ be written as a linear combination of polynomials  of the
form $(x-
r_1)^n,
(x-r_2)^n,\dots , (x-r_n)^n $?

\bigskip

A beautiful theorem on apolarity was proved by the British mathematician
John Hilton
Grace.
We state it without proof:

\bigskip

{\bf Grace's Theorem}. If two polynomials $p(x)$ and $q(x)$ of degree $n$ are
apolar, then
every
disk in the complex plane containing every zero of $p(x)$ also contains at
least one zero of
$q(x)$.

\bigskip

Grace's Theorem is an instance of what might be called a sturdy theorem.
For almost one
hundred years it has resisted all attempts at generalization.  Almost all
known  results
about the distribution of zeros of polynomials in the complex plane are
corollaries of
Grace's
theorem.

\bigskip

We will next generalize the apolar invariant  to the case of two
polynomials $p(x)$ and
$q(x)$ of
different degrees $n$ and $k$, with $ k \leq n$. To this end, we  slightly
generalize the
definition of invariant, as follows.

\bigskip

A polynomial $ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x ) $ in the
variables

$a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k , x $  is said to be an
invariant of the
polynomials
$p(x)$, $q(x)$ when

$$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x) = $$
$$I(p_1(c),p_2(c), \dots , p_n(c), q_1(c),
q_2(c), \dots , q_k(c), x+ c ) $$

for all complex numbers $c$.

\bigskip

Sometimes these more general invariants are called covariants.

\bigskip

We   define a more general apolar invariant as follows:

$$ A(q(x), p(x)) \cong (\beta - \alpha)^k (x- \alpha )^{n-k}. $$

Again, we say that two polynomials $p(x)$ and $q(x)$ are apolar when
$A(q(x), p(x))$ is
identically zero, that is, zero for all $x$. Theorem 1 remains valid as
stated.  That is, if
$q(r) = 0$ then the polynomial $p(x) = (x-r)^n $ is apolar to $q(x)$.

Let us consider a special case.   Suppose that $q(x)$ is a quadratic polynomial
and $p(x)$ is a cubic polynomial:

$$ q(x) = x^2 + 2b_1 x + b_2 $$

and

$$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 .$$

Then we have, in umbral notation

$$ A(q(x), p(x)) \cong (\beta - \alpha)^2 (x - \alpha) =
\\ (\beta^2 - 2 \alpha \beta + \alpha^2)x - \alpha \beta^2 + 2 \alpha^2
\beta -
\alpha^2. $$

Evaluating the linear functional $E$, we obtain the following explicit
expression for the
apolar
invariant:

$$ A(q(x), p(x)) = E((\beta^2 - 2 \alpha \beta + \alpha^2)x  - \alpha
\beta^2 + 2
\alpha^2 \beta - \alpha^2) =$$
$$ E (\beta^2 ) - 2E( \alpha \beta) +E( \alpha^2)x +E ( -
\alpha \beta^2) + 2 E(\alpha^2 \beta)  -E( \alpha^2) =$$
 $$E (\beta^2 ) - 2E( \alpha)E( \beta) +E( \alpha^2)x +E ( - \alpha)E(
\beta^2) + 2
E(\alpha^2)E(
\beta)  -E( \alpha^2) = $$
$$ (b_2 - 2 a_1b_1 + a_2)x  - a_1 b_2 + 2 a_2 b_1 - a_3. $$

Thus,  a quadratic polynomial $q(x)$ and a cubic polynomial $p(x)$ are
apolar if and only
if
their coefficients satisfy the two equations

$$ b_2 - 2 a_1b_1 + a_2 = 0 $$

$$ - a_1 b_2 + 2 a_2 b_1 - a_3 = 0 . $$

Using these equations, we can prove two important theorems:

\bigskip

{\bf Theorem 2}. There is in general one quadratic polynomial which is
apolar to
a given cubic
polynomial.

\bigskip

Proof. Indeed, the above equations may be
rewritten as

$$ b_2 - 2 a_1b_1 = -  a_2  $$

$$ - a_1 b_2 + 2 a_2 b_1 = a_3. $$

The solutions $b_1, b_2$ for given   $a_1, a_2, a_3 $ is in general unique.

\bigskip

{\bf Theorem 3}.  There  is always a two-dimensional space of cubic polynomials
which are
apolar to
a given quadratic polynomial.

\bigskip

Proof. Indeed, given  $b_1, b_2$ we may solve for  $a_1, a_2, a_3 $  from
the equations

$$- 2 a_1b_1 + a_2 = - b_2 $$

$$  - a_1 b_2 + 2 a_2 b_1 = a_3 .$$

These equations always have a double infinity of solutions, as they used to
say in the old
days.

\bigskip

Theorems 2 and 3 provide a simple and  explicit  method for  solving a
cubic equation. It
goes as
follows.

Given the cubic polynomial

$$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 ,$$

first, by Theorem 2 we  find a unique quadratic polynomial $q(x)$ which is
apolar to
$p(x)$. In
general, such a  quadratic polynomial $q(x)$ has two distinct roots $r_1$
and $r_2$.  By
Theorem 1, the cubic polynomials $ (x-r_1)^3$ and $(x-r_2)^3$ are apolar to
$q(x)$.
Second, by
Theorem 3, the affine linear space of cubic polynomials apolar to $q(x)$
has dimension
two. Since
$p(x)$ is apolar to $q(x)$, we conclude  that $p(x)$ is a linear
combination of  $ (x-
r_1)^3$ and
$(x-r_2)^3$.  In symbols:

$$ p(x) = c (x-r_1)^3 + (1-c) (x-r_2)^3 $$

for some constant $c$. Observe that $c$, $r_2$ and $r_2$ are  computed by
solving
linear and quadratic equations.

In this way, the solution of the cubic equation $p(x) = 0$ is reduced to
the solution of the
equation

$$ c (x-r_1)^3 = -  (1-c) (x-r_2)^3 ,$$

and this equation is easily solved by taking a cube root.

\bigskip

This method of solving a cubic equation is the only one I can remember.

\bigskip

Let me digress with a personal anecdote. A few years ago, I was lecturing
on this material
at a symposium in combinatorics that took place at the University at
Minnesota. Persi
Diaconis
was sitting in the front row, and I could tell as I started to lecture that
he was falling asleep;
he
eventually began to doze off. But the moment  I mentioned the magic words
"solving a
cubic
equation" he woke up with a start and said: "Really! How?"

\bigskip

The  preceding two theorems are easily generalized.

\bigskip

{\bf Theorem 4}.  The dimension of the space of all (monic) polynomials of
degree
$k$ which
are
apolar to a polynomial of degree $n$ equals $2k-n$, in general.

\bigskip

{\bf Theorem 5}.  The dimension of the space of all (monic) polynomials of
degree
$n$ which
are
apolar to a polynomial of degree $k$  equals $k$.

\bigskip

Let us  try to solve an equation of degree 5 in much the same way as we
solved a cubic equation. Given
 the quintic polynomial

$$ p(x) = x^5 + 5a_1x^4 + 10 a_2 x^3 + 10 a_3 x^2 + 5 a_4 x + a_5 = 0 ,$$

Theorem 4 assures us that there is in general a unique cubic polynomial
$q(x)$ which is
apolar to
$p(x)$. In general, this cubic polynomial has three distinct roots $r_1,
r_2, r_3$. By
Theorem 1,
the polynomials $(x-r_1)^5, (x-r_2)^5, (x-r_3)^5 $ are linearly independent
and  apolar to
$q(x)$ . By Theorem 5, the dimension of the space of all polynomials apolar
to $q(x)$
equals
$3$. But the polynomial $p(x)$ is apolar to $q(x)$. Hence, $p(x)$ can be
written in the
form

$$ p(x) = c_1(x-r_1)^5 + c_2(x-r_2)^5 + c_3(x - r_r)^5 $$

for suitable constants $c_i$. Thus, we see that a generic polynomial of
degree 5 can be
written as a
linear combination of three fifth powers of linear polynomials. These are
computed by
solving
linear, quadratic and cubic equations.  This reduction to canonical form of
the quintic is as
close
as we can come to solving a quintic equation by radicals.

\bigskip

At this point, someone in the audience will raise his or her hand and say:
"Excuse me, but
the
umbral method you have introduced is not even good enough to express the
discriminant of
a
quadratic equation! "

Quite right.

The  definitions of umbrae and of the linear functional $E$ have an obvious
generalization
to
any array of polynomials, say $p_1(x), p_2(x),\dots , p_{\ell}(x) $.  One
simply
considers  the
space of polynomials

$$ {\bf C}[x, \alpha_1 ,\alpha_2, \dots , \alpha_{\ell}  ] $$

and one sets

$$E(\alpha_{t}^j )$$

to equal the $j$-th coefficient of the polynomial $p_{t}(x)$. What is
crucial, the linear
functional $E$ is again  multiplicative on distinct umbrae:

$$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell} ) =  E(\alpha_1^i
)E(\alpha_2^j) E(
\alpha_3^k )\dots x^{\ell}.$$

Now comes the catch that in the old days was turn into a notational
nightmare: the
polynomials say $p_1(x), p_2(x),\dots , p_{\ell}(x) $ need not be distinct.
In fact, the
most
important case occurs when each of the polynomials $p_1(x), p_2(x),\dots ,
p_{\ell}(x) $
is
equal to one and  the same polynomial $p(x)$. In this case, the definition
of the linear
functional
$E$ may be simplified as follows:

1. $$E(\alpha_i^j) = a_j $$  for every $i$ , and

2. $$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell}) =  a_i a_j a_k
\dots x^{\ell} $$

for all non negative integers $i,j, k, \dots \ell $.

\bigskip

Umbrae  $ \alpha_1 ,\alpha_2, \dots , \alpha_{\ell}  $ satisfying 1 and 2
are said to be
exchangeable. Thus, for exchangeable umbrae we have

$$ (x + \alpha_1)^n \cong (x+\alpha_2)^n. $$

Eric Temple Bell, who wrote $=$ in place of $\cong$, was baffled by the
fact that
two umbrae could be  be exchangeable without being equal.

\bigskip

We can now state the main theorem of invariant  theory. We will consider  a
single
polynomial.

\bigskip

{\bf Theorem 5}.  Every invariant of a polynomial $p(x)$ is obtained by
evaluating some
polynomial
in the differences $\alpha_i - \alpha_j $ and $\alpha_i - x,$ where
$\alpha_i $ and $\alpha_j
$ are
exchangeable umbrae.  Conversely, every polynomial in such differences is
equivalent to an
invariant  of the polynomial $p(x)$.

\bigskip

The proof  is extremely simple, but will be omitted.

\bigskip

Let us review some classical examples.

\bigskip

The discriminant of a quadratic polynomial $p(x) = x^2 +2a_1x+a_2 $  may be
umbrally
represented as follows:

$$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2 / 2, $$

where $\alpha_1 $ and $\alpha_2 $ are exchangeable umbrae.  Indeed:

$$ E( (\alpha_1 - \alpha_2)^2 ) =E( \alpha_1^2) - 2 E(\alpha_1\alpha_2) +
E(\alpha_2
^2) =$$
$$ E( \alpha_1^2) - 2 E(\alpha_1)E(\alpha_2) + E(\alpha_2
^2) =a_2 - 2 a_1^2 + a_2 = 2 (a_2 - a_1^2), $$

as desired.

\bigskip

Let us next consider a cubic polynomial $p(x) = x^3 + 3a_1 x^2 + 3a_2 x +
a_3 $.  The
discriminant of this polynomial, let us call it $D(p(x)) $, equals, as you
know,  the
expression

$$ D(p(x)) = \frac{4(a_2 - a_1)^2)(a_1a_3 - a_2^2) - (a_3 - a_1a_2)^2}{2} .$$

The umbral expression of the discriminant is  easier to remember.

$$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_3 - \alpha_4)^2(\alpha_1 -
\alpha_4)(\alpha_2 - \alpha_3). $$

As you know, the discriminant vanishes if and only if the cubic equation
$p(x) = 0 $ has a
double root.

\bigskip

The Hessian of a cubic polynomial can be elegantly written in umbral
notation as follows:

$$ H(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_1 - x)(\alpha_2 - x).  $$

The Hessian vanishes if and only if the
cubic polynomial is the third power of a polynomial of degree one.

\bigskip

Allow me another digression. On hearing about the vanishing of the Hessian
as the
condition
that a cubic polynomial be a perfect cube, it comes naturally to ask the
general question:
which
invariant  of  a polynomial of degree $n$ vanishes if and only if the
polynomial is the $k$-
th
power of some polynomial of degree $n/k$? Here $k$ is a divisor of $n$.
For a long time I thought the answer to this question to be beyond reach,
until one day,
while
leafing despondently through the second volume of Hilbert's collected
papers, I
accidentally
discovered  that Hilbert had completely solved it. The solution can be
elegantly expressed
in
umbral notation.  This is only one of several striking results of Hilbert's
on invariant theory
that
have been forgotten.

\bigskip

Let us consider next an invariant of the quintic. Theorem 3 tells us that a
quintic
$p(x) = x^5 + 5a_1x^4 + 10 a_2x^3 + 5 a_3x^4 + a_5 $ has a unique apolar cubic
polynomial
$q(x)$. The polynomial $q(x)$ is an invariant of $p(x)$. Does it have a simple
expression in umbral notation? Indeed it does. The expression  is the
following:

$$ q(x) \cong (\alpha_2 - \alpha_3)^2(\alpha_3 - \alpha_1)^2(\alpha_1 -
\alpha_2)^2(\alpha_1 - x)(\alpha_2 - x)(\alpha_3 - x). $$

In the classical literature this invariant is denoted by the letter $j$.

What  property  will the quintic polynomial $p(x)$ have when the invariant
$j$ vanishes?

The answer to this question is pleasing. The invariant $j$ of a quintic
polynomial is
identically
equal to zero if and only if the quintic is apolar to some non trivial
polynomial
 of degree two. But then Theorem 3 tells us  that the quintic may be
written in the form

$$ p(x) = c(x-r_1)^5 + (1-c)(x-r_2)^5, $$

where $r_1, r_2 $ are the roots of a quadratic equation. Thus, the vanishing of
the invariant $j$ is a necessary and sufficient condition that the quintic
polynomial $p(x)$
may
be written as the sum of two rather than three fifth powers of linear
polynomials. When this
is
the  case, the fifth degree equation $p(x) = 0$ can be solved by radicals.

By  similar arguments, one  can compute all invariants whose vanishing
implies that the
equation of degree five is algorithmically solvable by radicals.
Twenty-three invariants
play a
relevant role, as Cayley was first to show.

\bigskip

Hilbert's theorem on finite generation of the ring of invariants can be
recast in the language
of
umbrae, and can be given a simple combinatorial proof that dispenses with
the Hilbert basis
theorem.

\bigskip

In closing, let us touch opon another  reason for the demise of the
symbolic  method in
invariant theory.

\bigskip

In mathematics, it is extremely difficult to tell the truth. The formal
exposition of a
mathematical
theory does not tell the whole truth . The truth of a
mathematical theory is more likely to be grasped while we listen to a
casual remark made by some
expert that gives away  some  hidden motivation,  when we finally pin down the
typical examples,
or
when we  discover what the real problems are that were stored behind
the showcase
problems.

\bigskip

Philosophers and psychiatrists should explain why it is that we
mathematicians are in the
habit of
systematically erasing our footsteps.  Scientists have always viewed
askance at this strange
habit
of mathematicians, which has changed little from Pythagoras to our day.

\bigskip

The hidden purpose of the symbolic method in invariant theory was not
simply that of
finding
easy  expressions for invariants. A deeper faith was guiding this method. It
was the expectation that the expression of invariants by the symbolic
method would
eventually
guide us to single out the "relevant" or "important" invariants among an
infinite variey.  It
was the
hope  that the  significance of the ending of an invariant could be
gleaned from its
umbral expression. The vanishing of this faith was the real reason for the
demise of
classical
invariant theory, and the revival of this faith is the reason for its
present rebirth.

\bigskip

Whether or not we will succeed this second time where the classics failed
is a cliffhanger
that  will
probably be resolved in the next few years.  I would not be speaking to you
now if I did
not believe
in  success.

\bigskip

Thank you for your attention.

\newpage

\begin{center} {\bf COMBINATORIAL SNAPSHOTS\\

being\\

The third of three Colloqium Lectures\\

delivered at the Annual Meeting of the American Mathematical Society\\

Baltimore, January 9, 1998\\

Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\ Cambridge MA 02139-4307\\}

\end{center}

\bigskip

\bigskip

When I was in high school, my English teacher gave me to read   an essay by
James
Thurber, called "The secret life of Walter Mitty".  After rereading this essay
every few years,  I  decided that everyone  has a Walter Mitty complex.
One  way
to understand a person  might be to discover that person's Walter Mitty
fantasies.

Most  of our mathematical thoughts in high school or in college were Walter
Mitty
fantasies. When we learned a new piece of math, we would find ourselves
fantasizing on its possible generalizations. As soon as we understood binomial
coefficients, we fantasized about their generalization to the case when the
denominator is negative; the moment we learned about derivatives, we launched
into derivatives of fractional order.  If we were ever exposed to the Riemann
zeta function, we would  romanticize some new interpretation of this function
that would give away its secret.

\bigskip

This lecture should have been given   another title.  It should be called "The
later life of Walter Mitty". It  will  consist of a sequence of displays of
"chutzpah" by a Walter Mitty who has lost his shyness. Each snapshot will deal
with the  realization of some of youthful  fantasies that have partially
worked out.

\bigskip

FIRST SNAPSHOT: AN EXAMPLE OF PROFINITE COMBINATORICS.

\bigskip

Let  us begin with a piece of history-fiction, and fantasize how Riemann might
have discovered the Riemann zeta function.

\bigskip

Professor Riemann was aware that arithmetic density is of fundamental
importance
in number theory. If $A$ is a subset of the set of set of positive integers
$N$,
then the arithmetic density of the set $A$ is defined to be

$$ dens(A)=\lim_{n\rightarrow \infty} {1\over n} |{A\cap \{1,2, \dots , n
\}|.$$

\bigskip

whenever the limit exists. For example $dens(N) = 1$. If $A_p$ is the set of
multiples of the prime $p$, then $dens(A_p) = \frac{1}{p}$; what is more
appealing, one easily computes that $dens (A_p \cap A_q)  = \frac{1}{pq} $
for any
two primes $p$ and $q$. If density were a (countably additive) probability
measure, we would infer that  the events that a randomly chosen number is
divisible by either of two primes are independent. Unfortunately, arithmetic
density shares some but not all
properties of a probability measure. It is most emphatically not countably
additive.

\bigskip

After a period of soul searching, Professor Riemann was able to find a
remedy to
some deficiencies of arithmetic density by a brilliant leap of imagination. He
chose a real number $s > 1$, and defined the measure of a positive integer
$n$ to
equal $\frac{1}{n^s}$; in this way, the measure of the set $N$ turned out
to equal

$$ \zeta (s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$

\bigskip

Therefore, he could  define a (countably additive) probability measure
$P_s$ on
the set
$N$ of positive integers by setting

$$ P_s(A) = \frac{1}{\zeta (s)} \sum_{n \in A }\frac{1}{n^s}. $$

Riemann then proceeded to verify what he had sensed all along, namely, the
fundamental property

$$P_s(A_p \cap A_q) = P_s(A_p)P_s(A_q) = \frac{1}{pq}. $$

In other words,  events $A_p$ and $A_q$ that a randomly chosen integer $n$  is
divisible by either of two primes $p$ or $q$ are independent relative to the
probability
$P_s $.

The Riemann zeta function was good for something, after all.

\bigskip

I will now use a rhetorical device that was effectively employed by one my
undergraduate teachers, Professor Bochner.   In the classroom, Professor
Bochner
would prefix the statement of a theorem by the words: "Subject to technical
assumptions, the following is true"; without, of course, ever disclosing
what his
technical assumptions were.

\bigskip

Professor Riemann then proceeded to show that, subject to techincal assumptions
on the set $A$,

$$ \lim_{s\rightarrow 1}P_s(A) = dens (A). $$

\bigskip

Thus, even  though arithmetic density is not a probability, it is under
suitable
conditions the limit of probabilities.

\bigskip

Long after Riemann was gone, it was shown, again subject to technical
assumptions, that the probabilities $P_s$ are the only probabilities defined on
the set $N$ of natural integers for which the events of divisibility by
different
primes are independent. This fact seems to lend support to the program of
proving
results of number theory by probabilistic methods based upon the Riemann zeta
function.

\bigskip

Why didn't Professor Riemann ever publish this wonderful idea of his? The
answer
is not hard to find. True, some theorems of number theory can be proved
probabilistically by this limiting process, for example Dirichlet's theorem on
primes in arithmetic progression.  However, deeper number theoretic results
have
to this day eluded this approach, for example, no one has succeeded in proving
the prime number theorem by this method. Professor Riemann, aware of this
deficiency, threw his notes into the wastebasket and proceeded to link the
Riemann zeta function to the distribution of primes in an altogether different
way, by stating the hypothesis that bears his name and that remains unproved to
this day.

\bigskip

Why am I telling you this bit of history-fiction? Because I want to propose
another probabilistic interpretation of the Riemann zeta function that is quite
different
from the interpretation just outlined.

\bigskip

Let us consider a problem in combinatorial enumeration. Given a cyclic group of
order $r$, say $C_r $.   Every character
$\chi $ of the group $C_r$ has a kernel which is a subgroup of $C_r$. More
generally, every  sequence $\chi_1, \chi_2, \dots , \chi_s $ of characters of
$C_r$ has a joint kernel which is also a subgroup of $C_r$; the joint
kernel of a
sequence of characters is simply the intersection of the kernels of each of the
characters in the sequence  $\chi_1, \chi_2, \dots , \chi_s $ . If a sequence
$\chi_1, \chi_2, \dots , \chi_s $ of $s$ characters is chosen independently and
at random, what is the probability that the joint kernel of the sequence
equals a
given subgroup $C_n $ of $C_r $?

\bigskip

The probability of  the event that  the the kernel of a randomly chosen
character
will contain the subgroup $C_n$ equals $\frac{1}{n}$, since there are $r$
characters of the group $C_r$ and $\frac{r}{n}$ such characters  will vanish on
$C_n $. Therefore, the probability that the joint kernel of a randomly chosen
sequence  $\chi_1, \chi_2, \dots , \chi_s $  of $s $ characters shall
contain the
subgroup $C_n $ equals $(\frac{1}{n})^s $. Let us denote by $P_{C_n}$ the
probability that the joint kernel of the characters   $\chi_1, \chi_2, \dots ,
\chi_s $ shall equal  the subgroup $C_n $. Then we have the identity

$$ { 1 \over n^s} = \sum_{n|d|r} P_{C_d}.$$

Here, we use the fact that the partially ordered set of subgroups of a cyclic
group $C_r$ is isomorphic to the partially ordered set of divisors of the
integer
$r$.

We now use the M\"obius inversion formula of number theory, thereby obtaining

$$ P_{C_n} = \sum_{n|d|r} \mu (d/n){1 \over d^s}.$$

Here, $\mu(j) $ is the M\"obius function of number theory.

After the change of variable  $d = nj $  we can recast the right hand side as
follows:

$$ P_{C_n} = { 1 \over n^s}  \sum_{j}\mu (j){1 \over j^s}.$$

The variable $j$ on the right ranges over some subset of  divisors of the
integer
$r$, which we need not worry about.

\bigskip

Now if the sum on the right ranged
 over all positive integers $j$, then the right hand side would equal

$${ 1 \over n^s} \frac{1}{\zeta(s)},$$

\bigskip

 that is, it could be expressed in terms of
 the inverse of the Riemann zeta function.  If we could change our
combinatorial problem to get an unrestricted sum on the right hand side,
then we
would have a probabilistic interpretation of the Riemann zeta function.

\bigskip

This is done by replacing the finite cyclic group $C_n$ by a profinite cyclic
group.

\bigskip

Consider the  group $C_{\infty}$ of rational numbers $mod 1 $. For every
positive integer $n$, the group $C_{\infty}$ has a unique finite subgroup $C_n$
with $n$ elements.  The character group
$C_{\infty}^*$ of $C_{\infty}$ is a compact group; it has a Haar measure
which is
a probability measure $P$. The group $C_{\infty}^* $ is the desired profinite
group on which we can generalize the preceding computation.

\bigskip

The set of all characters of the group
$C_{\infty}$ (that is, the set of all elements of the group  $C_{\infty}^* $)
which vanish on a subgroup
$C_n$ of $C_{\infty}$ has Haar measure equal to
$\frac{1}{n}$.  Thus, if we choose a sequence  $\chi_1, \chi_2, \dots ,
\chi_s $
of $s $ characters of
$C_{\infty}$  independently and at random, the probability that their joint
kernel will contain the group $C_n$ equals $(\frac{1}{n})^s$ .  If we again
denote by  $ P_{C_n} $ the probability that the joint kernel of a sequence
$\chi_1, \chi_2, \dots , \chi_s $  of $s $ characters equals the group $C_n$,
then we have the identity

$${ 1 \over n^s} = \sum_{n|d} P_{C_d},$$

where the sum on the right is now infinite. Again by the M\"obius inversion
formula we obtain

$$ P_{C_n} = \sum_{n|d} \mu (d/n){ 1 \over d^s} =
 \frac{1}{n^s}\frac{1}{\zeta(s)}.$$

This is the promised   probabilistic interpretation of the  Riemann zeta
function. Some properties of the Riemann zeta function can be proved
probabilistically using this
interpretation, for example, the product formula. It remains to be seen which
other  properties of the Riemann zeta function can be proved in this way.

\bigskip

The preceding argument is as an instance of a generalization of an enumeration
problem  on a finite set to an enumeration on a profinite set.  Such a
replacement of a finite set by a profinite "set" works in other
 combinatorial problems. Will we ever have a profinite combinatorics on
profinite
sets  side by side with combinatorics on finite sets?

\newpage

SECOND SNAPSHOT: THE CYCLIC DERIVATIVE.

\bigskip

The ordinary derivative of a polynomial in one variable has been generalized by
Hausdorff to polynomials and formal power series in non commutative
variables as
follows.  Consider the associative algebra ${\bf{C}} \langle a,b,
\dots , c , x \rangle $ generated by a set of letters $\{a, b, \dots, c, x\}$.
The letter $x$ is called a variable, all other letters are called constants. A
monomial in this associative algebra is what you think it should be: it is a
word  like

$$ m = axbax^3bcxd.$$

A polynomial is a linear combination of monomials, and a formal power series is
defined as an infinite sum of monomials, with suitable restrictions on the
growth
of degrees of the summands. Formal power series in non commutative
variables form
an algebra ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $. We will
denote by $f(x)$ such a formal power series.

The Hausdorff derivative of the monomial $m$ is computed as follows:

$$ H (m) = H(axbax^3bcxd) = abax^3bcxd +3 axbax^2bcxd + axbax^3bcd. $$

This definition is extended by linearity to polynomials and to formal power
series.

\bigskip

If $m'$ is another monomial, we have the expected  rule for finding the
Hausdorff
derivative of a product:  $H(m m') = H(m)m' + m H(m'). $

\bigskip

The Hausdorff derivative suffers from a major weakness.  There seems to be no
analog of the chain rule for the differentiation of a function of a
function. For
example, the Hausdorff derivative of the polynomial $(ax)^n$, when the letters
$a$ and $x $ do not commute, is not equal to $n (ax)^{n-1}a $. It is a mess.

\bigskip

There is another notion of derivative that does satisfy a simple chain rule
under
functional composition. It is  the cyclic derivative, denoted  by the
letter $D$.

\bigskip

The cyclic derivative is defined as follows. First define  the truncation
operator $T$ as follows:

\bigskip

a. if the first letter of a monomial $m$ is not the variable $x$, set $T(m)
= 0;$

\bigskip

b. if the first letter of a monomial $m$ is the variable $x$, so that $m = x m'
$, set $T(m) = m'. $

\bigskip

c. Extend by linearity to  ${\bf C}\langle \langle a,b,\dots , c, x \rangle
\rangle $.

\bigskip

The cyclic derivative of a monomial $m$ is defined in terms of the truncation
operator follows:

\bigskip

a. Let $p$ be the polynomial obtained by adding all cyclic permutations of the
monomial $m$.

\bigskip

b. Set $D(m) = T(p).$

\bigskip

c. Extend by linearity to all formal power series.

\bigskip

 For example, the  cyclic derivative of the above monomial $m$  is computed in
the following steps:

\bigskip

Step 1. Write down all cyclic permutations on the monomial $ axbax^3bcxd$.
These
are

$$ xbax^3bcxda , bax^3bcxdax , x^3bcxdaxba ,
 x^2bcxdaxbax , xbcxdaxbax^2 ,$$
$$bcxdaxbax^3 , cxdaxbax^3b , xdaxbax^3bc , daxbax^3bcx . $$

\bigskip

Step 2. In the above list, perform one of the following operations:

\bigskip

a. if the first letter of a monomial is not $x$, remove the monomial from the
list;

\bigskip

b. if the first letter of a monomial is $x$, remove the first letter.

\bigskip

When we perform operations a. and b. on the each of the monomials in the
above list, we
obtain a shorter list, namely:

$$ bax^3bcxda ,  x^2bcxdaxba , xbcxdaxbax,  bcxdaxbax^2 , daxbax^3bc. $$

Step 3. Add the monomials thus obtained to get the cyclic derivative:

\begin{eqnarray*}D(m) & = & D(axbax^3bcxd) \\
 & = & bax^3bcxda +  x^2bcxdaxba + xbcxdaxbax+  bcxdaxbax^2 +  daxbax^3bc.
\end{eqnarray*}

Another example: the cyclic derivative of the monomial $axbxcxdx$ equals

$$D(axbxcxdx) = bxcxdxa + cxdxaxb + dxaxbxcx + axbxcxd. $$

The cyclic derivative of the monomial  $(ax)^n$ is the following:

\begin{eqnarray*} D((ax)^n) & = & D(axax \dots ax)\\ & = & axax\dots axa +
axax\dots axa + \dots axax\dots axa \\ & = & n
(ax)^{n-1} a.\end{eqnarray*}

Similarly, one computes

$$ D(x+a)^n = n(x+a)^{n-1}$$

and, for formal power series,

$$ D(e^{x+a}) = e^{x+a} $$

and

$$ D(e^{ax}) = e^{ax}a. $$

Remember,  the letters $a$ and $x $ do not commute! In  these
examples, the corresponding
Hausdorff derivative is a mess.

The  cyclic derivative enjoys all properties expected of the ordinary
derivative;
in particular, it satisfies the  chain rule for the composition of two formal
power series.

To state the rules for taking cyclic derivatives, we need one more operator,
called  the wrapping operator.

The  wrapping operator is defined as follows.
Let
$c_1,c_2,\dots ,c_n $ be any letters. If $g(x)$ is any formal power series, set

$$ \langle C c_1c_2\dots c_n | g(x) \rangle\\ = $$ c_1c_2\dots
c_n  g(x) + c_2\dots c_n  g(x)c_1 + c_3 \dots c_n  g(x)c_1c_2 + \dots + c_n
g(x)c_1c_2\dots c_{n-1}.$$

\bigskip

If $f(x)$ is any formal power series, the wrapping operator

$$ \langle C f(x) | g(x) \rangle $$

is defined by linearity.

Set

$$ \langle D(f(x)) | g(x) \rangle = T \langle C f(x) | g(x) \rangle. $$

For example:

$$ \langle D(f(x)) | 1 \rangle = D(f(x)). $$

\bigskip

The cyclic derivative of the product of two "functions"is given by the
following
identity:

$$ D(f(x)g(x)) = \langle D(f(x)) | g(x) \rangle + \langle D(g(x)) | f(x)
\rangle. $$

For example, one obtains

\bigskip

$$ D((1-ax)^{-1}(1-bx)^{-1}) & = & (1-ax)^{-1}(1-bx)^{-1}(1-ax)^{-1}a $$

$$ +(1-bx)^{-1}(1-ax)^{-1}(1-bx)^{-1}b .$$

\bigskip

No such identity holds for the Hausdorff derivative.

The cyclic derivative of the  product of any sequence  of formal power
series  is
similarly computed by the wrapping operator:

$$D(f_1(x)f_2(x)\dots f_n(x)) = $$
 $$\langle D(f_1(x)) | f_2(x) \dots f_n(x)
\rangle + \langle D(f_2(x)) | f_3(x) \dots f_n(x)f_1(x) \rangle + $$
$$ \dots\\
 +
\langle D(f_n(x)) | f_1(x)f_2(x) \dots f_{n-1}(x) \rangle. $$

\bigskip

We come now to the main property of the cyclic derivative: the chain rule.
Given
two   formal power series $f(x) $ and $g(x) $ in ${\bf C}\langle \langle
a,b,\dots , c, x \rangle \rangle $, assume that formal power series
$g(x) $ does not have a constant term.  Under these circumstances, the
composition $f(g(x))$ is well defined by replacing  $g(x) $ for every
occurrence
of the variable $x$ in the formal power series $f(x)$.

Let us write $ D_g(f(x)) $ to denote the formal power series  obtained by
substituting
$g(x)$ in place of every occurrence of $x$ in the cyclic derivative
$D(f(x))$ of
the formal power series  $f(x)$.  Then the chain rule for the cyclic derivative
goes as follows:

$$ D(f(g(x))) = \langle Dg(x) | D_g(f(x)) \rangle. $$

For example, we have

$$ D(e^{axbx}) = bx e^{axbx} a + e^{axbx}axb. $$

A more elegant example is the following:

$$ D(e^{(1-ax)^{-1}}) = (1-ax)^{-1} e^{(1-ax)^{-1}}(1-ax)^{-1} a. $$

One can prove that the cyclic derivative of a rational formal power series  in
non commutative letters is again a rational non commutative power series , and
that the cyclic derivative of an algebraic formal power series in non
commutative
letters is
again an algebraic formal power series.

Despite the  evidence that the cyclic derivative is the natural notion of
devivative for non commutative algebras, the theory as it is at present is not
satisfying. The cyclic derivative is an empirical discovery. It needs to be
ensconced in some broader algebraic theory, much like the Hausdorff derivative
has been ensconced in the theory of Hopf algebras.

\newpage

THIRD SNAPSHOT: LOGARITHMS AND THE BINOMIAL THEOREM.

\bigskip

The  Euler-MacLaurin summation formula is one of the most remarkable
formulas of
 mathematics.  For a suitable function $f(x)$ of a real or complex variable,
it is stated as follows:

$$ f(x) + f(x+1) + f(x+2) + \dots + f(x+n) =$$
$$ B_0 \int_x^{x+n+1}f(y ) dy + B_1
(f(x+n+1) - f(x)) $$
$$ + {B_2 \over 2!} D (f(x+n+1) - f(x))  + {B_3
\over 3!}  D^2 (f(x+n+1)- f(x))  + \dots .$$

The $B_n$ are the Bernoulli numbers and where $D$ is the ordinary derivative
operator.

The Euler-MacLaurin formula has proved very useful for over two hundred years.
Nonetheless, the Euler-MacLaurin formula suffers from a serious deficiency.
The
series on the   right hand side is almost never convergent, unless it
reduces to
a finite sum.

Our question is the following:  is there a vector space of functions which
contains as many of the elementary functions as possible, and a topology on
such
a vector space, relative to which the right hand side of the Euler-MacLaurin
formula is a convergent series?

The answer to this question is unexpectedly related to the answer to another
 question. What is the "right" generalization of the
binomial
coefficients ${n \choose k}$ when $k$ is allowed to be a negative integer? This
question leads in turn to a third question: how shall we know whether a
generalization of the binomial coefficients is  "right"? The answer to
this third question is easy: a generalization of the binomial coefficients is
"right" if it leads to a sensible generalization of the binomial theorem:

\bigskip

$$ (a+ x)^n = \sum_{k=0}^n {n \choose k}a^k x^{n-k} $$

\bigskip

When I was young, I used to think of the binomial theorem as trivial. I think I
have learned my lesson. A well-known philosopher, I can't remember his
name, wrote
that the whole universe can be inferred from  a grain of sand. He should have
added that a great deal  of mathematics can be derived by meditating upon the
binomial theorem.

\bigskip

Let us take the bull by the horns, and state the "right" generalization of the
binomial coefficients.  We proceed in the most pedestrian way, by first
generalizing the definition of the factorial.  Thus, let $n$ be any integer,
positive or negative. We define the Roman factorial $[n]!$ as follows:

$$ [n]! = n! $$

if $ n \geq 0  $, and

$$ [n]! = {(-1)^{n+1} \over (-n-1)!} $$

 if $ n < 0 $.

Where does this definition come from ? I could simply say that it works,
but that
would not be the whole truth. The value of   the Roman factorial $ [n]! $
for $n$
negative equals the residue of the gamma function at the integer $n$.

Using the Roman factorial, we define the Roman coefficients as follows:

$$ { n \brack k} = { [n]! \over [k]![n-k]!}. $$

When  $ n \geq k \geq 0 $, the Roman coefficients coincide
with the binomial coefficients.  For all integers $n$ and $k$, the Roman
coefficients share all elementary properties of binomial coefficients, such as
Pascal's triangle, etc.   However, there are some surprises in store, for
example, for $k$ positive we find

$$ {0 \brack -k}= {0 \brack k} = { (-1)^{k+1} \over k }. $$

\bigskip

Does this make any sense? Well, yes, because we can find a generalization
of the
binomial theorem that goes with this.  It is the following. Recall the power
series expansion of the logarithm:

$$ log(x+a) = log x + \sum_{k=1}^{\infty}{ (-1)^{k+1} \over k} {a^k \over
x^k}.$$

We can recast this power series expansion in terms of the Roman coefficients as
follows:

$$ log(x+a) = log x + \sum_{k=1}^{\infty} {0 \brack k} {a^k \over x^k}. $$

This is beginning to look like a generalization of the binomial theorem,
with the
logarithm playing the roles of zero-th power. Another  power series expansion
where the Roman coefficients make their appearance is the following:

$$ (x+a)(log(x+a) - 1) = x(log x - 1) + a log x + \sum_{k=2}^{\infty}{1
\brack   k} a^k x^{1 -k}. $$

Do we see a pattern? Well, let us try yet another power series expansion:

$$ (x +a)^2(log(x+a) - 1 - {1 \over 2}) =$$
$$ x(log x - 1 - {1 \over 2}) + {2
\brack 1}a x(log x - 1) + {2 \brack 2}a^2 log x + \sum_{k=3}^{\infty} {2
\brack k} a^k x^{2-k}. $$

Now we can leap to a generalization. For suitable functions $f(x)$, set

$$ D^{-1}f(x) $$

to be the unique indefinite integral of the function $f(x)$ which has constant
term equal to zero. Do not worry, this will make sense in a moment.

\bigskip

Define

$$ \lambda_n^{(1)} (x) = [n]! D^{-n}log x. $$

Here, $n$ is any integer, positive or negative. The functions
$\lambda_n^{(1)}(x)$
are called the harmonic logarithms of order one.  For $n$ positive we have

$$   \lambda_n^{(1)} (x) = x^n(log x - 1 - {1\over 2} - {1 \over 3} - \dots -{1
\over n}) $$

and

$$  \lambda_{-n}^{(1)} (x) = {1 \over x^n}.$$

Of course we also have

$$  \lambda_0^{(1)} (x) = log x.$$

We are now in a position to state the generalization of the binomial
theorem that
is associated with the harmonic logarithms. It goes as follows:

$$  \lambda_n^{(1)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k
\lambda_{n-k}^{(1)} (x).
$$

The  three identities above are special cases of this identity, for $n =
0,1,2$.

The generalization of the binomial theorem to harmonic logarithms gives nothing
new for negative exponents, where it reduces to the identity

$$(x+a)^{-n} = \sum_{k=0}^{\infty} { -n \choose k} a^k x^{-n-k}.$$

However, for positive exponents we obtain a genuine and baffling generalization
of the binomial theorem. It states that the functions $  \lambda_n^{(1)} (x) $,
for $n $
positive, satisfy the ordinary binomial theorem, modulo negative powers of
$x$.  In other words, we have the following identity:

$$ (x+a)^n(log(x+a) - 1 - {1\over 2} - {1 \over 3} - \dots - {1 \over n})
\cong $$

$$\sum_{k=0}^n {n \choose k}a^k x^{n-k}(log(x) - 1 - {1\over 2} - {1
\over 3} -
\dots - {1 \over {n-k}}). $$

The identity is valid modulo negative powers of $x$. Miracles of
cancellation are
occurring in this identity. I wish I knew a combinatorial or probabilistic
interpretation of this logarithmic generalization of the binomial theorem.

\bigskip

So far, we have assumed that all series converge in the topology of the complex
numbers. We will now change the topology, while retaining the convergence.

The motivation for the logarithmic topology we are about to define is the
algebra of
formal Laurent series.  This topological algebra may be defined by defining a
topology on the algebra of rational functions in the variable $x$, and then
completing this algebra relative to the topology. The topology is so chosen
as to
have $lim_{n \rightarrow \infty}x^{-n} = 0 $. Every element of the completed
algebra turns out to be a formal Laurent series, that is, a series of the form

$$ \sum_{n < d} a_n x^n. $$

We want to perform an analogous completion process on another algebra: the
algebra generated by all functions of the form $x^n(logx)^t$, where $n$ is any
integer, positive or negative, and where $t$ is a non negative integer.

In
order
to specify which elements of this algebra are to converge to zero, we need a
better behaved basis of this algebra. This basis is provided by the harmonic
logarithms of arbitrary order
$t$. They are defined as follows:

$$\lambda_n^{(t)}(x) = D^n (log x)^t $$

for every non negative integer $t$ and for every integer $n$ .  For example, we
have

$$\lambda_n^{(0)}(x) = x^n $$

for every non negative integer $n$, and

$$ \lambda_n^{(0)}(x) = 0 $$

for negative $n$.

Explicit expressions are known for the harmonic logarithms. For the harmonic
logarithms of order $2$ we have $\lambda_0^{(2)}(x) = (logx)^2 $ and for $n$
positive

$$\lambda_n^{(2)}(x) = x^n \big(  (logx)^2 - ( 2 + {2 \over 2} + \dots + {2
\over n})log x + 2 + {2 \over 2} (1+ {1 \over 2}) + \dots + {2 \over n}(1+ {1
\over 2} + \dots + {1 \over n}) \big), $$

and

$$ \lambda_{-n}^{(2)}(x) = 2x^{-n} \big( log x - 1 - {1\over 2} - \dots - {1
\over n-1} \big). $$

For  every non negative integer $t$ the harmonic logarithms of order $t$
satisfy
the same generalization of the binomial theorem that we have already seen
for the
harmonic logarithms of order $1$:

$$  \lambda_n^{(t)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k
\lambda_{n-k}^{(t)} (x).
$$

The harmonic logarithms are a basis of the algebra generated by all functions
$x^n(logx)^t$.  We define a topology on this algebra by requiring that

$$ lim_{n \rightarrow - \infty}\lambda_n^{(t)}(x) = 0 $$

 for every non negative
integer $t$.  This topology  is called the logarithmic topology.  The
completion
of this algebra relative to the logarithmic topology is the algebra of formal
power series of logarithmic type, or logarithmic algebra.

\bigskip

Every element of the logarithmic algebra is a linear combination of convergent
power series of the form

$$ f(x) = \sum_{n \leq d} b_n \lambda_n^{(t)}(x) $$

ranging over a finite set of values of $t$.

bigskip

We can now return to the Euler-MacLaurin summation formula:

\bigskip

{\bf Theorem}. For every element $f(x)$ of the logarithmic algebra  the right
hand side of the Euler-MacLaurin series converges in the logarithmic topology.

\bigskip

For example, the following infinite series is convergent in the logarithmic
topology:

$$ log x + log(x+1) + log(x+2) + \dots + log(x+n) = $$
 $$B_0 ((x+n+1)log(x+n+1) - xlogx - n - 1 ) + B_1 ( log(x+n+1) - logx ) +$$
$${B_2 \over 2!}  ({1 \over x+n+1 } - {1 \over x}) + \dots ) . $$

\bigskip

Another example is the following. As you know, the sum

$$ x^k + (x+1)^k + (x+2)^k + \dots + (x+n)^k $$

can be expressed in closed form by the Euler-Maclaurin formula. The preceding
theorem leads to analogous closed form expressions for sums of the form

$$ x^klogx + (x+1)^klog(x+1) + (x+2)^klog(x+2)+ \dots + (x+n)^klog(x+n) .$$

The harmonic logarithms have other applications, let us mention one in
closing.
\bigskip

Recall the definition of the shift operator of the calculus of finite
differences:

$$E^af(x) = f(x+a).$$

For $n$ a non negative integer, define the operator $E_1$ as follows:

$$ E_1 \lambda_n^{(0)}(x) = \lambda_n^{(1)}(x). $$

In ordinary notation, this is the same as saying

$$ E_1 x^n = x^n(log x - 1 - 1/2 - 1/3/ - \dots - 1/n). $$

One can prove the following two propositions:

\bigskip

{\bf Proposition}. The operators $E^a$ and $E_1$ commute.

\bigskip

{\bf Proposition}. The restriction of the derivative operator $D$ to the
subalgebra of the logarithmic algebra generated by the harmonic logarithms
$\lambda_n^{(t)}(x)$ for positive
$t$ (that is, excluding the non negative powers of $x$) is invertible.

\bigskip

 These two
propositions   can be used to obtain "logarithmic extensions" of special
functions. Let us conclude with the simplest example: let us compute  the
logarithmic extension of the sequence of lower factorials, namely, the
polynomials
$ (x)_n = x(x-1)(x-2)\dots (x-n+1)$. This sequence
of polynomials satisfies the difference quation

$$ \Delta (x)_n = n(x)_{n-1}.$$

This sequence can be extended to negative $n$ by setting

$$ (x)_{-n} = { 1 \over {(x+1)(x+2)...(x+n)} }, $$

and we have

$$ \Delta (x)_{-n} = -n(x)_{-n-1}.$$

For positive $n$, we may define the logarithmic extension of this sequence by
setting

$$  (x)_{-n}^{(1)} ={ 1 \over {(x+1)(x+2)...(x+n)} }. $$

For example, $(x)_{-1}^{(1)} = { 1\over x+1}$.

The elements $ (x)_{-n}^{(1)}$ belong to the submodule of the logarithmic
algebra spanned by $\lambda_n^{(1)}(x)$, as $n $ ranges over all integers.  On
this submodule, the operator
$\Delta$ is invertible, and we can therefore set

$$ (x)_n^{(1)} = \Delta^{-n-1}{1 \over x+1}$$

for all non negative integers $n$.

It  turns out that the  element    $(x)_0^{(1)}$ is given by the following
series, convergent in the
logarithmic topology:

$$ (x)_0^{(1)} = log(x+1) + {B_1 \over 1+x} - {B_2 \over 2(1+x)^2} + {B_3
\over { 3(1+x)^3 }} - \dots . $$

But this is a familiar object: it is the $\Psi$- function,  heuristically
introduced by Gauss.  Gauss motivated the
$\Psi$-function as the "right" solution of the difference equation

$$ \Delta \Psi(x+1) = {1 \over { x+1}}. $$

We have now rigorously verified  Gauss's guess. Further computations show that
the elements $ (x)_1^{(1)}$ and $ (x)_2^{(1)} $ also coincide with special
functions introduced by Gauss, namely, the digamma and trigamma functions,
which
are at last rigorously defined by infinite series convergent in the logarithmic
topology.

In a similar vein, one defines logarithmic extensions of the Bernoulli
polynomials, the Hermite polynomials, etc., and one finds that the
asymptotic expansions of these polynomials reappear naturally as members of
the
logarithmic
extensions of these functions. As a matter of fact, the logarithmic topology
allows us to replace asymptotic expansions by  series which are convergent in
the logarithmic topology.

\bigskip

In closing, two open problems may be mentioned.

First, no closed form expression is known for  the coefficients of the
expansion
of a product

$$\lambda_n^{(t)}(x) \lambda_k^{(s)}(x) $$

into a logarithmic power series.  Second, we do not know a combinatorial or
probabilistic  interpretation of the Roman coefficients ${n \brack k}$ in
general.

\bigskip

Thank you for listening.

\end{document}

--============_-1324990680==_============--

Date:           Fri, 13 Feb 1998 17:16:22 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        changes

There has been a change in the seminar schedule for next week.  

**The talk scheduled for February 20 by Eva Feitchner will be
postponed until March 11. **

February 18: Harald Helfgott
      Recent developments in enumeration of tilings 

February 25: Richard Stanley
      Enumerating solutions to equations in finite groups 
      **Pretalk at 3:30pm in 2-338***

February 27: Alex Burstein
      Enumeration of words with forbidden patterns 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 18 Feb 1998 11:22:05 -0500
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        third announcement

             Come to the Twenty-eighth one day conference on

                     Combinatorics and Graph Theory

                      Saturday, February 21, 1998

                         10 a.m. to 4:30 p.m.
                                  at
                            Smith College
                         Northampton MA 01063

                              Schedule

10:00  Katalin Vesztergombi (Yale University)
        Properties of Distance-Graphs

11:10  Sheila Sundaram (Danbury, Connecticut)
        Homology of Graph Complexes and Partitions
         with Forbidden Block Sizes

12:10  Lunch

 2:00  Emily H. Moore (Grinnell College and Mt. Holyoke College)
        Extending Graph Colorings

 3:10  Dana Randall (Georgia Institute of Technology)
        Two Dimensional Tilings in Two and Three Dimensions

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Wed, 18 Feb 1998 12:49:14 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Date: Wednesday, February 18, 1998
Speaker: Harald Helfgott
Title: Recent developments in enumeration of tilings 
Abstract: In this talk I will describe recent work with Ira Gessel on
new methods for enumerating tilings of subsets of the Aztec diamond. I
will show how these new tools allow us to solve several of Jim Propp's
open problems on enumeration of tilings, as well as explain the
connection between Eric Kuo's graphical condensation procedure and
Dodgson condensation. I will also discuss the use of our methods in
the computation of the entries of the inverse Kasteleyn matrix of the
Aztec diamond.

Upcoming Events:

      February 20: Eva Feitchner
      Postponed until March 11. 
      February 25: Richard Stanley (with Pretalk at 3:30)
      Enumerating solutions to equations in finite groups 
      February 27: Alex Burstein
      Enumeration of words with forbidden patterns 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 25 Feb 1998 13:49:29 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

                   ** Pretalk at 3:30-4:00**

Date: Wednesday, February 25, 1998
Speaker: Richard Stanley
Title: Enumerating solutions to equations in finite groups
Abstract:
We will survey such questions, going back to Frobenius and Schur, as:

In how many ways can an element w of a finite group G be written as a
square or as a commutator? If f(w) is this number, then when is f a
character of G? How many n-tuples of elements of G pairwise commute?

Special emphasis will be given to the symmetric group S_n. Techniques
involve combinatorics, character theory, and symmetric
functions. Applications to permutation enumeration will be discussed.

A link to everything mentioned in the talk (and more) may be found at
http://www-math.mit.edu/~rstan/ec/ec.html.  Note: There will be a
pretalk for non-experts from 3:30-4:00.

Upcoming events:

February 27: Alex Burstein
      Enumeration of words with forbidden patterns 

March 4: Peter Winkler
      Phase transitions on a tree 
March 6: Sara Billey
      Pattern avoidance and rational smoothness of Schubert varieties 
March 11: Eva Feichtner
      On the cohomology algebras of complex subspace arrangements 
March 13: Takayuki Hibi
      Nonregular unimodular triangulations of convex polytopes 
March 18: Alex Suciu
      Characteristic varieties of real and complex arrangements 
**Spring Break March 23-27**

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 4 Mar 1998 12:50:22 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Date: Wednesday, March 4, 1998
Speaker: Peter Winkler (Bell Labs)
Title: Phase Transitions On A Tree
Abstract:
    Although it is manifestly different from a cubic lattice in
Euclidean space, the Cayley tree (or "Bethe Lattice") offers an
opportunity to prove theorems about phase transitions, on account of
its connection with branching random walks.  Modelling hard-constraint
systems by random homomorphisms of graphs, we characterize the
constraint graphs which exhibit more than one simple invariant Gibbs
measure.

Our methods are elementary and no acquaintance with statistical
physics will be assumed.

Joint work with Graham Brightwell (London School of Economics).

Upcoming Events:

March 6: Sara Billey  **PRETALK at 3:30***
      Pattern avoidance and rational smoothness of Schubert varieties 
March 11: Eva Feichtner
      On the cohomology algebras of complex subspace arrangements 
March 13: Takayuki Hibi
      Nonregular unimodular triangulations of convex polytopes 
March 18: Alex Suciu
      Characteristic varieties of real and complex arrangements 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Thu, 5 Mar 1998 15:37:26 -0500 (EST)
From:           Jim Propp <propp(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        Winkler preprints

Copies of Peter Winkler's three articles

Graph homomorphisms and phase transitions (with Graham Brightwell)
Gibbs measures and dismantleable graphs (with Graham Brightwell)
Nonmonotonic behavior in hard-core and Widom-Rowlinson models (with
	Graham Brightwell and Olle Haggstrom)

are now available on the ledge in the public area in 2-363.

Jim Propp

Date:           Fri, 6 Mar 1998 13:36:08 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 
                ***Pretalk at 3:30 in 2-338***

Date: Friday, March 6, 1998
Speaker: Sara Billey (MIT)
Title: Pattern avoidance and rational smoothness of Schubert varieties 
Note: This talk will be a joint seminar with the algebraic geometers.

Abstract:
Let w be an element of the Weyl group S_n, and let X_w be 
the Schubert variety associated to w in the flag manifold SL_{n}(C)/B.
Lakshmibai and Sandhya showed that X_w is smooth if and only if w 
avoids the patterns 4231 and 3412.  Using two tests for rational 
smoothness due to Carrell, we show that rational smoothness of X_w
is characterized by pattern avoidance for types B and C as well.  
A key step in the proof of this result is a sequence of rules for 
factoring the Poincare polynomials for the cohomology ring of X_w 
generalizing the recent work of Gasharov.
Preprints Available:  http://www-math.mit.edu/~sara/        

Upcoming Events:

March 11: Eva Feichtner
      On the cohomology algebras of complex subspace arrangements 
March 13: Takayuki Hibi
      Nonregular unimodular triangulations of convex polytopes 
March 18: Alex Suciu
      Characteristic varieties of real and complex arrangements 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 11 Mar 1998 13:48:44 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 
              ******Pretalk at 3:30 in 2-338******

Date: Wednesday, March 11, 1998
Speaker: Eva Feichtner (MIT)
Title: On the cohomology algebras of complex subspace arrangements  
Abstract:
We show that for a complex subspace arrangement with geometric 
intersection lattice the integral cohomology algebra of the
complement is completely determined by the combinatorial data 
of the arrangement. Our approach relies on a combinatorial 
encoding of the arrangement's topology into order complexes 
of posets. We explicitly describe generating cohomology classes 
on simplicial models for the complement of the arrangement and 
derive a presentation of its cohomology algebra in the spirit 
of the combinatorial description of cohomology algebras of 
complex hyperplane arrangements by Orlik and Solomon.

This is joint work with Gunter M. Ziegler.

Upcoming Events:
      
March 13: Takayuki Hibi
      Nonregular unimodular triangulations of convex polytopes 
March 18: Alex Suciu
      Characteristic varieties of real and complex arrangements 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Fri, 13 Mar 1998 12:20:43 -0500
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        April 4th CoNE meeting

             Come to the Twenty-ninth one day conference on

                     Combinatorics and Graph Theory

                        Saturday, April 4, 1998

                         10 a.m. to 4:30 p.m.
                                 at
                            Smith College
                         Northampton MA 01063

                               Schedule

10:00  Van Vu (Yale University)
        Witness on the Upper Bound of Chromatic (Choice) Number
         of Random Graphs

11:10  Richard Stanley (MIT)
        Spanning Trees and a Conjecture of Kontsevich

12:10  Lunch

 2:00  Eckhard Steffen (Princeton University)
        Snarks

 3:10  Therese Biedl (McGill University)
        Efficient Algorithms For Petersen's Theorem

The next meeting is scheduled for May 2nd.

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Fri, 13 Mar 1998 13:01:54 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 
              ******Pretalk at 3:30 in 2-338******

Date: Friday, March 13, 1998
Speaker: Takayuki Hibi(Osaka University)
Title:  Nonregular unimodular triangulations of convex polytopes 
Abstract:
It was unknown, for a long time, if there exists a lattice
polytope which possesses a unimodular triangulation and none of whose
unimodular triangulations is regular.  The purpose of my talk is to
present a family ${\cal F}$ of $(0,1)$-polytopes such that each
polytope belonging to ${\cal F}$ has a unimodular triangulation, but
has no regular unimodular triangulation.  This is a joint work with
Hidefumi Ohsugi.

Upcoming Events:
      
March 18: Alex Suciu
      Characteristic varieties of real and complex arrangements 

April's schedule will be available soon.  The 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Sun, 15 Mar 1998 13:42:39 -0800 (PST)
To:             bjorner(at-sign)fredholm.math.kth.se, combinatorics(at-sign)math.mit.edu
From:           Gian-Carlo Rota <gcrota(at-sign)earthlink.net>
Subject:        Colloquium Lectures

--============_-1322142191==_============
Content-Type: text/plain; charset="us-ascii"

Enclosed is a dvi files of the text of the Colloquium Lectures I delivered
at the Annual Meeting of the American Mathematical Society in January of
this year. I will appreciate any comments of yours before I draft the final
text.

Gian-Carlo Rota

--============_-1322142191==_============
Content-Type: application/mac-binhex40; name="colloquium_3=14=98.dvi"
Content-Disposition: attachment; filename="colloquium_3=14=98.dvi"

(This file must be converted with BinHex 4.0)

:&Q0[E'a[FA9TG(at-sign)dJ-bma0#mj1#jNGQN!4&C*-LT849J!!!!#G9m!!!!!iJrh!J'
$NX!F1`!!!!!$k"XJ9'9B)'peG("eG#!a16Ni,M!c,M%e1M%e-6D,!!!!!3!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2rrrrqJ!R3!!)fJrE-!!+!
#,`!!MD$ppJ!!MC%j(r6c$PLV83X!$!!!!!`!!!!&Bfeb-6+j-bma0#mj1#jKGAL
(at-sign)!qUS-bma0#mj1#jKGAL6-bma0#mj1#jKGAL1MTmH!!#0NJ$pAeiaMSk-L`!!!!)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!US!+J+2(at-sign)0S2f
'e`ZJ!PXSpBfJrIT(at-sign)qif48D#qm`p%dqed!"5pF3!44ki!"(at-sign)0YFM%hZP4SCCB'3YT
"E(at-sign)9bD(at-sign)0KET00BA4SC(at-sign)eKG'PMB(at-sign)b68fq4!)UaBfPPG*(rG8jjMTmC!!#0NAZ+d80
[E'a[N3#+XA&eDA9YPJC#fNaPBh4eFQ9c,*-a16NiMTmMRr50MBf0MC)!Y``Fma"
BUe%,!!jQCJ!-!!!!"(at-sign)0YFM%bZdGTB(at-sign)iY3f&bE'q4",-[8QpdBBk1MSk1Ra`QBBf
4H8ra3Q&XG'PYEh*P,*B%Xbp+B(at-sign)k3!*ZmG(at-sign)&bHC-h,*-i,*0KEQ5615b6-6Nj1)k
1Rai!!)f5!2eIAVNbMSk-L`!!!!-!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!#qS!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if5!-j"p2-S`YC1S!!
-!!!!$!!!!!CME(at-sign)*i-6,63fpZPD!!G'9ZNh4c1SkI'i!!MC%RJ!#j-5k4"6MJ5(at-sign)k
DV(*dFQq3!&11C(9MG'P[ETB$kUKdEj0RC(at-sign)pYCA4bD(at-sign)16F(*[BQ&LD(at-sign)aTG*KjNIm
&9Lk1T"U!!)f4*i!!-Lk4"6MJ5(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)!qUSG'KPEh*jNIm
&9Lb6EfaNNf&ZC*0ZCAFZMU'0N5H!!$-ZN38ii%0[EC!!V(*LD(at-sign)jKG'pbD(at-sign)&XN32
UU(0ZBA"cD'pdFcU1RbH!!)f0MBf412r[mbNK)LbD!!`!!!!+!!!!"Q0YFhNa-03
2MSk1N86Il,P'DA*cG*B$kUKcEQ&`FfK[G$U4"6MJB(at-sign)k6CAKKEA"XCC0[CT0`FQm
-EQPdCC0MEff3!+abBQPZBA4[FQPMFbk1T"L!!)f0MBf412r[e!q1MSk440rXZ90
PBfpZC*B$kUKcEQ&`FfK[G$U4"6MJG'KPNf0jBfaTBj0NCA*TGT(r(at-sign)14KG'PfN!#
XFQ8ZMU'0MBf0N6Mrlp32MSk1N86Il,P8D'PbC*B$kUKcEQ&`FfK[G$U4"6MJE'p
RBA*TG'KYFj0KEQ56G'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)dZMSkI(J!!MC)!r9pH-ik
1M)X!!!!%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"mU!
#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N8I`!G0*6P45QU!!6d493e4*6dk(at-sign)")!!9%q
64d9268988NP$Ne"5Q%p#38**6%P8(at-sign)BkN$S!!MC)!h+$YZ(at-sign)+3!&11C(at-sign)PZCikKMC)
!L1fB9'KPPJ2UU!abFh56EfD6G'KbC(at-sign)(at-sign)6BfpXE'q3!&11FA9TG(at-sign)f66'9MG(9bCA1
1SBf4-HreC'9XDADDV(*PFQ9NPJ2UU'&dNh4SCC0"EQkBG(at-sign)&XNdePCA4TEQH6EfD
6G'KPNd&YCA*TBf&ZNdeKG'KPE(at-sign)&dD(at-sign)0KE*06Ej!!8ijMD(at-sign)9dQ(Q1SBf5!+5Re%*
KE(4TE(at-sign)pbC5b(at-sign)!qUS5Q&ZN!#XFR9KFRQ60bb6-6Nj0ikKMC)![jpQ4fPKELe$BA*
XEj%$kUK5Eh4KMU'0NJ#KlQT%CA"KFR4YC(at-sign)k3!+abG*B$kUK[CT00BA4SC(at-sign)eKG'P
MFikKMC)!hX%C68P8MU'0NJ#NITC$B(at-sign)f3!+abBR*TC'GPPJ2UU%e"Nc!b-6-j,63
c-$H1RbK&q)f4*i!!5C%%9L*KECB%9MjfQUabCA*jNfKKF(#BHC0dEj0LQP11CC0
SCA*PNf+BC(at-sign)C[FQ(at-sign)6HC!!V(*[GC0KFj0dD'(at-sign)63fpXE'qBFA9TG(at-sign)f66'9MG(9bCA+
6CQpbNh4SDA11SBf4*i!!HCUXFQ9KFLb4"*4,B(at-sign)jNPJ4bA8Q4"()kCQ9PE*0NC(at-sign)9
`E(Q6D'pZEh*PC*0dEj0LN!"6MQ(at-sign)6CfPfQ'9ZNh4SDA16Ch*PBA56Eh"`N!"6MQp
bG(9ZDA5BHC0dEj0cD'&bCBkKMC%RJ!"hDA4SPJ2UU(QDV(*[GC0cEfePNfpQNh4
SCC0YBA4SC(at-sign)eKG'PMFj0hQ'(at-sign)6E'qBGTKP,SkI&l&qMC%RJ!"AD'9ZPJ22+8Q4!mm
LFQ9MC(at-sign)PfQUabC(at-sign)56CR*[EC0#Ef+64T(r"9C[Fh0eEC0dD'(at-sign)6D(at-sign)kBGQPdBA4TEfk
6G'q6BTT6MQ(at-sign)6G'KPNd0[E'a[Q(&eDA9YNdaPBbf1SBf4*i!!G(9bCA+(at-sign)"(at-sign)I*CQp
bNh4SDA16HC!!V(*PBA)XN3A(%8Q4"(at-sign)GRFR9cD'9NNh4[Nh4SCC0XD(at-sign)*bBA*jNh4
[Nh*PB(at-sign)56D(at-sign)k6B(at-sign)k6EfaNNfPcFh9PNfpQNh4SCBkKMC%RJ!"1Eh4TBf9cPJ2,VR4
SCC0XDA0dNfpQNf&XE*0`FQ9fD(at-sign)peFj0$EfaXEj!!8ijaG(at-sign)PeEC0-C(at-sign)0dGA*PFR-
XN324i'G[D(at-sign)jRNf&XE*0dD'(at-sign)6Gj(at-sign)XFQ'6HC%$bkjLB(at-sign)16DikKMC%RJ!"dEjB$l[*
+B(at-sign)ePFj03D(at-sign)9bF*!!8ij[ET!!V(*dNfPZNc%i16BZN39&[NPdNfPcNf'6E'PcG*0
[CT0NDA0dD(at-sign)jRG(at-sign)PcD'9NNfeKG'KPE(at-sign)&dD(at-sign)0TB(at-sign)jc,*%$m!9KEQ51SBf4*i!!5C%
&LR"hPDabEfjNCA*PC*X&LYaSEj0hQ'f6HCKZB(at-sign)ePQ'0[G(at-sign)aNQ'9fNf9bQ'+3!&1
1C(at-sign)a[EQHBD(at-sign)kBFh9MNfLBE'pQG*0jQ'0[EA"KET0jNIm&9Lk4#KPl6AQ1SBf4*i!
!D(at-sign)eYC(at-sign)4TBA4PPJ4q"A4PEA"dBA4TEfk6GjUXFQ&cNh4[Nf4PBfaTEQ(at-sign)6G'KPNfP
ZQ(CTG'&dD(at-sign)pZNfpeG(*TCfLBG$Z4"-HdBR9dNd*[BT0'NIm&9Qpc,BkKMC%RJ!"
cG(at-sign)f(at-sign)"%q1BA0cGA*PC*0YCC0dD'&dNfj[NfpZCC0TET0dD'(at-sign)6D'PcG'pbHC0[CT0
dD'(at-sign)68fq3!&11BfPPG*UXFRQ6D'&cNf9fQ'9bNf4PBfaTEQ9NMU'0N5H!!(4SCCB
%(at-sign)B&TET!!V(*fDA4KG'P[ET0dEj0LQP11CC0dD'(at-sign)63fpXE'qBFA9TG(at-sign)f66'9MG(9
bCA)ZN3D&De0[NdQ4"&PPGj(at-sign)XFQ9ZNh5E"&Q"BQ&MNfZBG'qBG'KPQ'aTFh5BEfD
1SBf4*i!!F(*PGQP[GA1(at-sign)""8*BfpXE'qD8ijaG(at-sign)PeEC0cF*KPB(at-sign)ZDV(*PFR-XN33
IS(at-sign)PZNh0PBA*MQ'L6CQpbNf'6DR9cG'N-Bf&dD(at-sign)pZNfpQNffBHC0`FQ9cC(at-sign)jMCC0
TESkKMC%RJ!"dD'&dPJ5!DQaTFh3ZN3Ek*d&cNfpQG'9ZNfKKF(#D8ijPER16D(at-sign)k
6Fh9MN!#XFQL6FfPdG(at-sign)&dD(at-sign)pZFbb4"+AE5C%%J%4cEjK[ET0PEQpeCfL6CQpeEQ5
6Fh9MN!#XFQL6BBkKMC%RJ!"UGA0dD3aMBA4TEfiZN38(at-sign)"8Q4!i(pBfpYF(9dC(at-sign)5
(at-sign)!i)BG'KPNf'9V(*fNf9bB(at-sign)GPPJ1#''&RCC0[CT0MEfaXEj!!8ijaG(at-sign)PeEC0XC(at-sign)0
dGA*PFR16B(at-sign)jNNf4TFf0[N!#XFRBYMU'0N5H!!'9bC(at-sign)5(at-sign)"+%-G'KKG*0dD'PcNf'
9V(*fNf9bB(at-sign)GPPJ5K$'PcNh0[E(at-sign)9hD'&dNfa[PDabGj0PFTB%S3adD'&ZNff3!+a
bHC0KCf8XN361T(at-sign)&cNf'6E(at-sign)&dG'9bNfpQNfCKBh51SBf4*i!!ECUXFRQ(at-sign)"0*JB(at-sign)G
PNf9iBf9PC(16BTKjNf&`F(*[Q(KTE(at-sign)&dC(at-sign)ajNfpZCC0cG'&ZC'&bC*0NCACTBA4
TEfk6G'KPNfePB(at-sign)k6B(at-sign)GPNfpQMU'0N5H!!("bCACTEh9cPJ2d0h0`N!"6MQ9KDj!
!V(*PFR-ZN399M8&cNdQE!r3dBf&YCC0dEj0dD'PcNh*PB(at-sign)aTHQ&dD(at-sign)pZ,*%$pTY
*Q'+3!&11C(at-sign)GKET0dEj0QB(at-sign)k3!+abG'&cDATPNfpZNh4SCBkKMC%RJ!"`FQpLB(at-sign)*
XCCB$5-jdEh"TBh16G'KKG*0YQUabHC0jQ'peEQGPFT0`FQ9NC(at-sign)0PFh0[FR16E(at-sign)P
RD*KdNfKKQ(DBCC0MQ'K[Ff9Z,T%&!Ze*N30)T(at-sign)PYB(at-sign)GTEQ9NMU'0N5H!!''(at-sign)!k-
[BR*TE'aTB(at-sign)kDV(*dNhQBEh9ZCj0YBA4SC(at-sign)eKG'PMD(at-sign)&Z,*%$XATPB(at-sign)GPFT0dEj0
PFh4KBQaTFfL6D'PYFf9XCT0KFj0KNfaPB(at-sign)4PFT0TET0SDA11SBf4*i!!$'9XC#b
4!j9`C'9XDAD3!+abCA*TEQH(at-sign)!i!LEfjPNh0TEQGXCC0NBATkE'PZCj0`FQqD8ij
[CT0LQ'9RD(at-sign)jZD(at-sign)jRNhGTG'L6G'KPN`abFh56BfpXE'qBFA9TG(at-sign)f1SBf4*i!!E'9
MG(9bCCB$e0&KEQ56E'&cG'PZCj0KE'b6G'KPNhH9V(*KNhQ(at-sign)!p64G'q6G'KPNf9
ZC*0[CT0dD'(at-sign)6G'KTFQ3ZN38aQ%pbNf9XFf8XN32C-(0[E(at-sign)(at-sign)6E(at-sign)PNC'aPMU'0N5H
!!'&RC(at-sign)5(at-sign)!bZFE(at-sign)&dD'9YBA4TBfPKELb4!e(5B(at-sign)jiD(at-sign)peFj0dEj0SBC(at-sign)XFRD6CCB
$+jaSDA16E'&dCA0dNh4SC(at-sign)pbHC0KBf0PF(4PC*0LN!#XFRQ6G'KPNfeKG'JYMU'
0N5H!!'9YBA4TBf&XQ`-e'hH9V(*[FQaN,*%$(at-sign)(at-sign)TNC(at-sign)aTGT0PFQPZCjKdEjKKQ(4
SFQPXE'9NQ'&eC'PPEQ0PQ''BG'KbC(at-sign)8YD'peFTKMEfjNC(at-sign)jcBA4TEfkBEfD1SBf
4*i!!E(at-sign)&dCA*TB(at-sign)b(at-sign)"'X6G'KKG*0hQUabEh9XC*0ZEh*YB(at-sign)aXHC0dB(at-sign)ZBCC0KET0
PETKdDA*PNh4PFQf6D(at-sign)k6B(at-sign)k6B(at-sign)4fNIpBj'&ZBf9NNfGbB(at-sign)4eBA4PMU'0N5H!!'0
[GA*cC5k4#-JC5'qDV(*hPJ8DCQ0[G(at-sign)aNNdNXNf'6E(at-sign)&dD'9YBA4TBfPKET0[EQ(at-sign)
6Fh4KEQ4KFQ56C'9fD(at-sign)&dD(at-sign)pZNfpXC'9b,*%&CP9PGTKPFSkKMC%RJ!"SEh#3!&1
1CCB$kUKdEj0YBA4MQUabD*0cG(at-sign)1BD*0PETKfD(at-sign)&LE'(at-sign)6CQ9KG(-rMTmAXAk0N5H
!!&4SCA0PPJ6GbfCKET!!V(*dBA0TCA16Bf&YCC0dEj0KET0KBR*eF(56C(at-sign)jNNhG
SC(at-sign)k63QpLNdD4r`9(at-sign)Eh0cG(at-sign)f6D(at-sign)jQEh*YC(at-sign)56E(at-sign)(at-sign)6D(at-sign)k1SBf4*i!!EQq(at-sign)"A9rG(at-sign)j
MCA*dB(at-sign)PZNh4PFQecNh4SBA56G'KPNd0[G(at-sign)jMD(at-sign)b6EfD6G'KPNe0[N!"6MQ0TCA5
3!+abHC0SB(at-sign)56C'9MD(at-sign)4PC*0dD'&dNh4SCBk1Rai!!)f5!2eIAM51MSb,!!!!"3!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"##J!U!SpBfJrBE
A#k!#(at-sign)bMeMD$pbGF,MC%RJ!#jG'KbC(at-sign)(at-sign)(at-sign)"0GcBfpXE'qD8ijaG(at-sign)PeEC0XC(at-sign)0dGA*
PFj0YN!#XFR9cG*0NC(at-sign)&XNhGTG'L6G'KbC(at-sign)(at-sign)6D(at-sign)jNCA#BC(at-sign)jNC(at-sign)k3!+abG*0KEQ5
6G(at-sign)jbC(at-sign)aKG'9NMU31J!#0N5H!!(4[F'PMFbb4"(at-sign)kNG'KPFQ9LPDabHCX&)3YKE'a
[NhGTEQHBB(at-sign)k6HCKYC(at-sign)f6BT!!8ijPFTK[CTKdD'(at-sign)BBA9ND(at-sign)9ZBf(at-sign)BG'qBFfYTF*K
[EQ(at-sign)BEh+BE(at-sign)pbCBkKMC%RJ!"XC(at-sign)0dGA*PFbb4!r&8GfPdD'peG*B$lrpYDA0cD(at-sign)j
RNf&ZQUabHA4SD(at-sign)jR,T%&514#Ef+64T(r"9C[Fh0eE5GcNf0[E(at-sign)eKEQ56C'9`FQP
fQ'9NNfePNfpQMU'0N5H!!'&XE*B%I+e`QP11Eh0cD(at-sign)*XCC0bEfaPNfe[Q'4PE(1
6B(at-sign)e[EQH6F(*PGQP[GA16BfpXE'qBFA9TG(at-sign)f6E'9MG(9bCA*c,T%'lZp*ET0KNh0
dBA4PNfpQMU'0N5H!!(4PEA#D8ij[FQ&bHCB%VDG`B(at-sign)jTBbb4"0jR5C%%VA9KCf&
TET0cBf&ZEQ9NNh4SCC0XDA0dNfpQNh"bCACTEh9cNf0[E'a[Q(&eDA9YNh0`Q'9
KDj!!V(*PFR-XMU'0N5H!!(4SDA1(at-sign)"ACcG'PYCC0XEjT6MQpVD(at-sign)jRNfC[FT0ZB(at-sign)e
PFj0[CT0YBA4SC(at-sign)eKG'PMD(at-sign)&ZFj0hD'q6D'&NNfj[G*0LQ'9PET0MN!#XFQK[Ff9
ZMU'0N5H!!'C[FTB%Q0CdD'PcNfK[EQpb,T%(3fY6GA*PNf9ZEh9RD#b4"-4LEfj
PNfjKE(at-sign)(at-sign)6Gj!!V(*KFj0MEfjcF'PMG(at-sign)peFfajNfeTFh0TEQFkN3D92A4SBA56EfD
1SBf4*i!!5'9bE(at-sign)&ZET%$kUKANIm&9Q9jE#k1T"Q+2Bf4*i!!5CB$kUKSEh#3!&1
1CC0jQUabEh(at-sign)6GfPXE*0QEh*RDADBCC0YCC0TCT0*Nf4TCh*PFh16GfPdD*0cEfe
PNh#3!&11CA*cEfjKE*0bC(at-sign)eTEQPcBf9ZBf9c,SkKMC%RJ!"*ETB$U%TdD'(at-sign)6CQ&
XE*0[CT-a168`NdQ4!kJjC(at-sign)jbEfaXC(at-sign)56BA16BC0QFQ9cD'eKET0KG*03FQPZBf9
dEfiXN31eN!"SBC!!V(*fD(at-sign)jRNfGbB(at-sign)4eBA4PC*0KMU31J!#0N5H!!'CPGjB%0p4
YEfkDV(*dD(16BT!!8ijPCQpbCC0QFQpYNh4SCC0"E(at-sign)9bD(at-sign)0KET0)D(at-sign)GSNe0MQ'K
[N!"6MQpXNfpQNe&eDA4[,*%%5ap&Bh9KC'pb,T%')'48D'(at-sign)1SBf4*i!!F(*TEQ0
TF'&XPJ5E+fpQNh4SCC0"E(at-sign)9bD(at-sign)0KET0)D(at-sign)GSNe0MQUabD'q3!&11Efb6EfD68A9
TG'q6GjKKFj0KNe"bD(at-sign)jMCA4[ET0RFQ&NG(at-sign)&dC5b1SBf4*i!!B(at-sign)jNPJ2UU'KPNh0
dC(at-sign)9bC(at-sign)56E(at-sign)(at-sign)6G'q9V(*hNf&bC(1(at-sign)!qUS8(*TEQ0PG'pZNe9ZDAD9V(*PFR0TG*0
jNIm&9Lk1U4Q+2Bf4*i!!5(at-sign)kE!ZJl6Qq9V(*fNf9YNf+3!&11CA+B-6Ne-*K*N3,
RqQaTFh4PEQ9NQ(4[Q'f6HCJ-FR0dQ'eKG'KPE(at-sign)&dD(at-sign)0cQ'aPBh4eFQ9c,T%%iVa
8D'9cCCKhNf9bCCKdD'(at-sign)1SBf4*i!!G'KbC(at-sign)(at-sign)(at-sign)!dm#9T(r"9CKETUXFR9iC(at-sign)f6E'9
MG(9bCA-XN30Z)f4PE'PfQ'9bC(at-sign)56BTKjNdKPFQeKEQk69j(r"9CPH(at-sign)b6B(at-sign)jNNf+
3!&11C(at-sign)&bD(at-sign)jRNh4SCC0RC(at-sign)jPFQPMMU'0N5H!!(4TG'aPPJ9B3#*6H(at-sign)eYCA4bH5)
ZQ`Q"U&4SCA0PNfaPBh4eFQ9cNhH3!+abCA*PNf&ZNh9ZCQpbCf9dG'&LE'(at-sign)6CAK
`N!"6MQ9bD(at-sign)9ZBf8ZQ&4SCBkKMC%RJ!"XC(at-sign)0dGA*PFjB#Z4adEj!!8ij[Dj0`E'&
MCC0TET0dD'(at-sign)6EfaNNf1DV(*SC(at-sign)eTFh4bHC0KG(at-sign)4TG'pbDA9Y,*%#pMK`B(at-sign)1BDjK
PC*0hDA4SNf&ZNf9iF*!!8ijPBh4KETKdMU'0N5H!!("eBQaTBbk4"1q[3A1(at-sign)!`m
(at-sign)5C%$$YjcD'&YC(at-sign)aPFh0XHC0cBA56D(at-sign)k6G'KPN`abFh56FQqDV(*hNh4bH(at-sign)PZCj0
dEj0RG(at-sign)9cFj0hD'PMQ'L6EfD6G'KPNfpdD'9bMU'0N5H!!(#3!&11CA*cEfjcPJ3
TXh0TG(4TEQH6D(at-sign)k6G'KPNh0KE(at-sign)(at-sign)6FQqDV(*hNhHBBA16G'q6BTT6MQ(at-sign)6G'KPNh0
`Q'9KDjUXFQ9b,*%%1A9KNfLBGA0SNfCPE'b6GA#3!&11Efk6G'KPMU'0N5H!!'&
eC'PPEQ0P1T%'jIC&D(at-sign)jcG'9TETB%`60hQUabBA16C(at-sign)kBG'9bD(at-sign)jRNh4SCC0XC(at-sign)0
dGA*PNh*[N!"6MQpY,T%([)&8NIm&9Qq6ECKjNf4TFf&`F*!!8ij[D(at-sign)kBG'ePETK
d,)kKMC%RJ!"SCCB$kUKcBA56FfpYCAGSCA*PNfPZNh4SCC0YD(at-sign)4NE'(at-sign)6EfD6G'K
PNf&eC'PdEh*TG(at-sign)dZMUD0N5H!!&4SCCB%2UN-FR0dNfaPBh4eFQ(at-sign)6BT!!8ijPCf&
ZNhGTG'L6B(at-sign)k6D(at-sign)e`FQ9cFfPfQUabCC0KEQ56E'9ZCh4SQ(Q6FA9[G'&dD(at-sign)pZNfP
ZNdGbC(at-sign)9V,)kKMC%RJ!"hD'PMN!#XFQL(at-sign)")&%EQq6EfjPNfPZNh4SCC0KG(at-sign)4TC(at-sign)j
MCC0eEQ4PFR0dEj96MQq6C*B%J84PH'0PF(566(9dD'9bNe"QB(at-sign)KXCA+64(at-sign)PcC(at-sign)j
SBA*d,SkKMC%RJ!"8D'PcPJ5!9f*bD(at-sign)aXD(at-sign)&ZQUabG*0cG'&bG*0hQ'&cNfC[E'a
[Q(HBC(at-sign)56BTKjNf'6C'PcF'aKQ(Q6EfD6FfaTC'9cNh#3!&11Eh*dFQ'BH(at-sign)PZCj0
MQ'KKFQeTEQH1SBf4*i!!Gj(at-sign)XFQpYC(at-sign)kE!lbXGj0PBA*TEQHBG'KPQ'a[EQHBBR*
TE(at-sign)ePC*KSBA4cQ'CKFfKTEfjKBQaPQ'&dQ(4SCCKdD(at-sign)eP,*%$aGpKEQ5BE'&dCA+
BBT0jMU'0N5H!!'e[FQ(at-sign)(at-sign)"$&2FfaTC'9cNh0SEjUXFRGTEQH6G'KPNd&XD'&YQ'*
bBC0KEQ56G'KPNe#BC(at-sign)kBG'&REfiZN3B-e%j[G*0KNhHBEh*NNfpQNfeKG'JYMU'
0N5H!!'9YBA4TBh-ZN39GS&4SCCB$pZKKG(at-sign)4TC(at-sign)jMCC0hQUabBA16E'9QG*0hQ'p
ZC'9bD(at-sign)jRNhGSCA*PNh0eBjKSNf'6Fh"KFQYXD(at-sign)jRNf4TFh"XBCKjNfpQMU'0N5H
!!#*,G(at-sign)adGA)LPJ6RSAHDV(*KFj0XC(at-sign)&ND(at-sign)jRNh9`Nh4[,T%),mY1Eh56ECKeBjK
SNfe[FQ(at-sign)6E(at-sign)&dD'9YBA4TBh16GjKKFj0YC(at-sign)kBG'P[EQ9NMU'0N5H!!'PZPJ3a3R4
SCC0cC(at-sign)0[EQ56E'9MG(9bC5b4"%,TGfKPET0YEh*PNh0XD(at-sign)4PFj0hQUabCA*PNh0
SEjKhET0[CT0`D*KjFfPMFj0PH(#3!&11CA*TE(at-sign)9ZQ(4c,)kKMC%RJ!"QEh+(at-sign)!ql
IGfKTBjUXFQL6G'KPNfaPBh4eFQ9bNh"bEjKfD(at-sign)4PC*0KNfaPBA*ZC(at-sign)56Eh*KE*0
MEfeYC(at-sign)kBG'&bHC(r"9BZN39&K%pZE(Q6D(at-sign)k6G'KPNfaKFh51SBf4*i!!E'9MG(9
bCCB$U'9ND(at-sign)56FfpYCC0RFQpeF*0dD'9[FRQ6E(at-sign)&VN!#XFQ(at-sign)6BC0YEjT6MQ4PFh5
6BA"`Q'9KFQ&ZBf8ZN38LbN*jNh4SBA56G'PYCC0dD'(at-sign)1SBf4*i!!BA9ND(at-sign)9ZBf8
XQ`2K$RGSD(at-sign)13!+abD*B$hUGSB(at-sign)56EQpdNf4hD(at-sign)jNE'9N,*KhQUabBA16C(at-sign)kBG'K
bB(at-sign)aXC(at-sign)56GfPdD*0dD'(at-sign)6Fh9LN3#R('TPBh3XN32K$Q&ZC*0ND(at-sign)51SBf4*i!!EQp
dPJ414'eTEQ56G'KPNfCKBh56G'KKG*0dD'(at-sign)6Fh#3!&11C(at-sign)&VQUabCA+6D'&NNh0
KD(at-sign)56GTKPFRQ6E'PdG'aPNf&LN!"6MQpeG*0YBA4SC(at-sign)eKG'PMFbb1SBf4*i!!B(at-sign)0
dG(at-sign)&XE(Q(at-sign)"A0iD'(at-sign)6D'&NNh0KD(at-sign)56GTUXFQ9bHC0XDA4dE'(at-sign)6B(at-sign)+3!&11Eh9dNf&
ZQ(PdD'PZCj0KG*0KE'`ZN3R66eGSBA56DA16E(at-sign)pbCC0bC5f1SBf4*i!!E(at-sign)&bDj(
r(at-sign)14KBQaP,*%$ZSTdD'(at-sign)(at-sign)!kk$BA9ND(at-sign)9ZBf(at-sign)6Ff9PE(at-sign)9NNh4[Nf+D8ijPNh4SB(at-sign)j
VCR9XNh4[Nh4SCC0cF*KPB(at-sign)Z3!+abCA+6CQpbNfeKDfPZCj0dD'(at-sign)1SBf4*i!!Bfp
ZPDabG'9ZNh4cPJ32BfpQNh4SCC0dD(*PCC0XC(at-sign)0dGA*PFj0TEQ4PF*!!8ijPEQ4
PETUXFR56EfD6EfjPNf&ZEh4SCA)XN33BNR4SCA*PBTKjNfeTEQPYDASYMU'0N5H
!!'PZCjB$N!!hB(at-sign)aXNfePE(at-sign)pbHC0bCA&eDA*PE(at-sign)9ZN!#XFR4c,T%&'VT*N313!#"
SBATKFQ56G'q6Ch9PFh16G'KKG*0dD'(at-sign)6Fh9MBf9cFj0[CT0)CA*YB(at-sign)jZMSkI(J!
!MC)!r9pH0Bk1M)X!!!!'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!3+D!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,PANIm&9Q9jE#GcPJ1
!3(at-sign)aPBh4eFQ9cNfeKN!#XFRQ6BTT6MQ(at-sign)6D(at-sign)k6F'&bG*0KG(4bD(at-sign)*eG'9NNh4[Nh4
SCC0cF*KPB(at-sign)ZDV(*PFLGcNf&cG(9dCC0QEh*PFfPRD*KdNfPZMU31J!#0N5H!!'e
KDfPZCjB%+2*SDA16E'9MG(9bCA16Ff9XCT0MEfk9V(*dB(at-sign)PZC(at-sign)3XN33iK(at-sign)PZC'9
`N!"6MQ9ZC'9ZNh5(at-sign)"#MbB(at-sign)jNNfaTCfL9V(*dNhH6C(at-sign)PRD*0d,T%&mlj"Fj%%+2*
*N33SiR*PBf&XE)kKMC%RJ!"dD'PcPJ3VJf4TFh4KET!!V(*dNf9`DA0[QP11C'8
XN33lZNQ4"#YbFQ9KE'PkCC0dD'&dNd*[BT0'NIm&9QpcFh9Y*h16D(at-sign)jUG(at-sign)jMG'P
[ET0KBTK[GA56G'KPNfPZC'8YMU'0N5H!!(#D8ijPEQ4PEQ0PPJ6J!'pQNh4SCC0
`FQ9cC(at-sign)k3!+abG*0MEfaXEjKaG(at-sign)PeEC0XC(at-sign)0dGA*PFj0TFj0KNhGTFf(at-sign)6EfjP,*%
&(99KE'b6G'KPNfe[FQ(at-sign)6Ffq1SBf4*i!!GfKPETB$kUKdD'(at-sign)6Fh#3!&11C(at-sign)&VN!#
XFQ9bNfPcNfj[G*0)CA*YB(at-sign)jZNeH4r`9(at-sign)CAPX,SkI'214MC%RJ!"CNIm&9QpePJ0
ULQeKQUabHC0hQ'pZC'9bNhGSQ(Q65C%$DQPMDA4PC*0YQ(Q6B(at-sign)GPNf&cNf&ZNf&
TC*0dEj0NC(at-sign)aTGTKPFQPZCj0dD'9cCC0MEfaXEj!!8ijaG(at-sign)PeEBkKMC%RJ!"XC(at-sign)0
dGA*PFbkE"04X9fKKG*B#[8aND3YPFQ9ZBf(at-sign)6C'q3!&11CA16EfjPNh0dB(at-sign)jNBA*
NNf4PGQPKG'P[ET0YB(at-sign)Z3!+abC6qB5C%#[3"dD'PZDj0TG*0YB(at-sign)Z3!+abCA11SBf
4*i!!FfpYCCB%C[*ND3YPFQ9ZBf8ZN3DY[8PdNfPcNf'6FQ9XD(at-sign)9Q,*X%KJ4LN!"
6MQpdD*0dEj0jN!#XFQpeNf&ZC*0dEj0YC5bBG'q6Dfj[QUabGj0bD(at-sign)GSQ(56BA5
6G'KPMU'0N5H!!(0dBA*dPJ4Q2(4SBA56G'KPNh0`QP11C(at-sign)&VN!#XFQ9bNf4[Q'9
cNfj[G*0QC(at-sign)9XNh4SCC0ZC(at-sign)9NNh4[NfPYF(*PFh16HCUXFQpeNf+BHC0cG'&dD(at-sign)j
RNh4SCBkKMC%RJ!"bCA0eE(4cPJ-qifpQNfKTFj0XBA4PFh56FQ9cC(at-sign)&bBjUXFQJ
ZN36rRNj[FT0NEj0jQ'peNfpbNdQ4!ckhFh8,CA+6CR*[EC0KETKjNfaKBjKVNfp
QNf9iF*!!8ij[Fh9bCBkKMC%RJ!"dEjB$FDTdD'(at-sign)6E'&dCA0dNfCKFfKTEfjcNfP
ZNfeKG'KPE(at-sign)&dD(at-sign)0c1j%$QIphN!#XFQ8RE'b6D'9KFT0PEQpeCfL6B(at-sign)+3!&11Eh9
dNh4SC(at-sign)f6D(at-sign)k6Eh4SCA+1SBf4*i!!E'9MG(9bCA1(at-sign)"'e!Ff1DV(*SC(at-sign)4eE'9NNh4
[Nf+3!&11CC0NC(at-sign)aTGTKPFQ9NNf&dNh4SDA16E(at-sign)9PG'PZCbk4"X#T9j(r"9CPNf0
KET0dD'9bC(at-sign)C[FQ(at-sign)6B3Y[FQ51SBf4*i!!G'q(at-sign)!md8Fh#3!&11C(at-sign)jNNh4SCA0PNh4
SFQ9PNfK[GA*cNfpZNfaPDA0eFQ9XHC0NDA0MGA0cD(at-sign)pZNfpQNh0[E(at-sign)(at-sign)6E(at-sign)&dD'9
YBA4TBh16G'KKG)kKMC%RJ!"YBCUXFRQ(at-sign)!qUSE(at-sign)&dG'9bNh4[Nf+3!&11Eh4SNhQ
BEh(at-sign)6B(at-sign)jNNfeP,SkT'214MC%RJ!"ANIm&9Q(at-sign)(at-sign)"#&-GfPXE*0MEj(at-sign)XFRD6CA+(at-sign)"#&
-D(at-sign)k6G'KPFf(at-sign)6E'9MG(9bCA16BC0QCAH6DA4PEA16G'KKG*0KFQ(at-sign)6EQpdNhGTC'9
XHC0VEQq3!+abGfiXN33ZpA4SBA51SBf4*i!!FfK[G(at-sign)aNPJ0VJ'+D8ijPNf+BCA4
dCA+6Dfj[QUabGfiXN31%lQ&ZC*0dD'&dNdQ4!fYIGTK[G(at-sign)1BD*0MB(at-sign)k6BTT6MQ(at-sign)
6G(at-sign)jNCA*cG'qBEjKNNf+DV(*jNf&ZQ(QBEfjPNhGTG'L1SBf4*i!!BCB&8Y4#,N%
ZNfPZNfeKG'KPE(at-sign)&dD(at-sign)0c,T%*F(at-sign)9*Q`95H(0[E'9YEQajNh"bEfeTFf(at-sign)6G'KKG*0
*Q(GTE'b6EQpdNh0dBA4PNf&ZN!#XFRQ6BQPRMU'0N5H!!(4SC(at-sign)pbC(at-sign)ec,*%$eL0
*N323qhGTE'b(at-sign)!p%#EQpdNh0eBT%!TaaUC(at-sign)0dNhQDV(*[GC0dEj0KETKjNfPZCf9
ZD(at-sign)peFj0KFQGeE(at-sign)9ZQ(4c,*%$eL0KEQ56G'KKG*0*N323qhGTE'b1SBf4*i!!EQp
dPJ2UU'&ZEQpeEQ0PNf&ZQUabHC0bCADBEfaeG'P[EQ&bHC0NCADBC(at-sign)a[F'ePETK
dFbk1TSf4*i!!9'KPPJ2mK(4TG'aPNfpQNh4SDA16E'9MG(9bCC0TFj-L4f9[E(at-sign)9
dFQPMNe"bEf*KBQPXDA53!+abH5)ZN39ZFd'4!rarC'8-EQPdD(at-sign)pZNfpQNfGPEfe
PG#f1SBf4*i!!FQPMPJ6kmh"bEf*KBQPXDA5DV(*jNfeTCfLBG*0bG(at-sign)k6BA16CQp
XE'qBGh-kN3GCG(at-sign)GPEfePG(*TBj0`FQpLB(at-sign)*TE'PdQ(Q6DA16G'KPNh0dG(at-sign)4jNfp
QMU'0N5H!!'PZPDabGT(r(at-sign)14KFQPKET0dQ`43j'ePBA0eFQ9c,T%'Dj9-D(at-sign)Z6CCK
KE'bBC'8-EQPdD(at-sign)pZFbb4"'TcG'KTFjKNEj!!8ijPFjKZEh5BG'9XE*KeFjKKET0
jG'KTEQHBG(at-sign)k6G'PXMU'0N5H!!(HDV(*PPJ2qHQ&bCC0cD'qBGfk6FfpYCC0dQ(P
`D(at-sign)0KE*0PH'&YF'aPFbb4"!0[B(at-sign)jNNh4SCA0PNf9iB(at-sign)e`E'9cNf&bCC0dD'(at-sign)6Bfp
ZQ(4PETKdNfpQMU'0N5H!!(4SDA14!qUSE'9MG(9bC5k1TSf4*i!!3(at-sign)+3!&11Eh9
dPJ5'"QpZCC0SQUabG(at-sign)jNFQ9NNhQBC(at-sign)&bFj0KCfmXN35XhA4SCC0`FQp`N!"6MQ9
bG'PPFj0dD'&dNh9ZC'9bE'PPNh0eBjKSNfj[G'P[ER16BA11SBf4*i!!E'9ZCh4
S,*B&PN0KFQ9K,*0fQUabEfaeE(at-sign)8XNf&cPJ9![AHBC(at-sign)aXNf&cNh4SCC0`FQpLB(at-sign)*
TE'PdQ(Q6EfD6CADBC(at-sign)kBG(16GjKPFQ(at-sign)6B(at-sign)*cG(*KBh4PC)kKMC%RJ!"eEQ4PFTB
&,30dD'(at-sign)6BQ&ZEQ9bNfpQNh4SCC0hN!#XFQpbC*-LE(at-sign)9KFh9bC5)ZN3MrmNaPG*0
eFj0bCACTCAH6G'KPNf4P$'jTG'P[ET0[CSkKMC%RJ!"YC(at-sign)&cGA*P,*B$kUKcD(at-sign)j
MCC0hQUabCC0hD(at-sign)aXNf+3!&11CC0eFfPZCj0dD'PcNf4P$'jTG'P[ET0TET0KET0
eETKeFh9KE*0hQ''BHC(r"9BZMUD0N5H!!%'4")M%E(at-sign)9KFh9bCCB%L1cc,,IKCk-
!$!!!!!`!!!!'BfeYD6%beaD6Z(at-sign)PcNf'6CR9ZBh4TEfk6C'8-EQ9NNfpZNf'6CQ&
YD(at-sign)ajNfpQNh0eBR0PG(16EfD6BC0cCA56ee14!,,AZ5b4","pGfKTBj!!V(*SMU'
0N5H!!(4KDjUXFQ9cPJ9"YA*PB(at-sign)b6GT(r(at-sign)14KE(9PFbb4"CGiEQpdNfjPBf9cFf&
bD(at-sign)ajNh#3!&11Eh0TG'PfQ'8ZN3Nq"P4SCC0QB(at-sign)eTE(Q6EfD6Ff9dFj0[ET0hD'P
MQ'L6BBkKMC%RJ!"YC(at-sign)&cGA*PPJ1)FfPcNf4P$'jPC*0TFj0ME'pcC(at-sign)56G(at-sign)jNCA+
6G(at-sign)jTEfjcNf&ZC*0TETUXFR4PFR0PBh4TEfjc,*%$R"GKEQ56BfpZQ(4KD(at-sign)jcNh4
SCBkKMC%RJ!"PEA"dN!#XFRQ4!qUSFf9d,SkQMC%RJ!""N34Y"(at-sign)ePBA0eFQ(at-sign)(at-sign)"'d
RDA16BjUXFQKKFQ&MG'9bDATPC*0LQ(Q6G*KhQ'q6FfPYF'aPNf&iD(at-sign)pYFbk4"X"
F6'9dNh9cNh4KDjKPNf'6E(at-sign)PZQ(9dCC0dEikKMC%RJ!"bCACTCAH(at-sign)!qUSG'KPFf(at-sign)
6BAKTEfec,SkQMC%RJ!""H'P[EC%$kUJa,Sk1Rai!!)f5!2eIAMD1MSb,!!!!"`!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(F(at-sign)J!U!SpBfJrBE
A#k!#(at-sign)bMeMD$pbGF,T!k!!)f5!0,XJ0F(at-sign)Z5M81lNTPJ098Mf6-0FlMTm9!!#0N5H
!!,PhD'9bCCB$kUM81j1jDA16G'KPNf9YF(53!+abHC0cCA3ZMTmDJ!#0N5H!!%&
iD(at-sign)pYPJ2UU$)ZN38ii%PQNpG"NlPKEQ56ed+4")(at-sign)ZZ(at-sign)&bCC0dPDabGj0[PJ2UU'e
PBA0eFQ&LE'(at-sign)6Ff9dFbb6G'KPESkKSBf5!)NLj0F(at-sign)Z5MA3CB#UUM8(at-sign)j2A3TX!Q`D
j+CB$99)pNpF(at-sign)Z5MA3ENTPJ+UU#Z6eaDj+0G#Q,NTNp3!NpF(at-sign)Z5MA3C28A*2A3TL
j+GFkMUNK!!#0N5H!!,P8D'(at-sign)(at-sign)"%jFE(at-sign)9KEQPZCj0[CT0dD'PcNh0PBfpZC*0KH'P
[EC0TFj0ME'9KFLk4"Q2p9'KPNf&iD(at-sign)pYNh0dBA4PFj0dD'&dNfePBA0eFQ(at-sign)1SBf
4*i!!DA1(at-sign)!qUSB(at-sign)4NDA4TGTUXFQ8ZN38ii%PZNh"KFR4TBh9XBA)XNfPQNhHBCC0
SBCKfQ'(at-sign)6G*KhQ'q6C'PcDQpTETKdNh0PG(16ed'6Z(at-sign)&ZC*2A3T%!Q`Dj,*0dD'9
ZMU'KMC)!U'N1eaDj+0G"PJ+UU04ENpG#Q`#E"VNTPJ098Mf6eaDj+0G"Z5Q(at-sign)!UU
S+j2A&VNSed+BZ5RA1SkQMC%RJ!#j6(at-sign)pbCCB#l(at-sign)pRC(at-sign)jPFQ&XE(Q4r`9(at-sign),*%$)"4
QEh+6B(at-sign)kDV(*jN`aZDA4PNfCKE(at-sign)PXHC2A4T%%Mc(at-sign)jGfK[Ff(at-sign)6E(at-sign)9YQ'+3!&11CA*
cNf&bCC0cCA4c,*%$)"4KEQ56CQpbNhGSD(at-sign)1BD)kKMC%RJ!"KET(at-sign)XFRQE!qUSG*0
hNfqBE(at-sign)9YNf+3!&11CA*cQ'&bCCKNDA0UEfPZNh3XQ(H6CCKSBC0fNf(at-sign)B1SkKSBf
5!,0E$pF(at-sign)Z5L0MBf4!lfiRrArr2-!qV&e%J!+!!!!#J!!!!CME(at-sign)9i-6#V(at-sign)ik1R``
L*Bhc,GF"&c)!#!!!!!J!!!!&BfeYD6MB3I-[[N[)#`!)!!!!#!!!!!9MEA0j10S
bf%D1MT%8Pq(A3ENTPJ098Mf0MBf4"(at-sign)KIRrArr+YBMSkI$#)PMC2B3GSbf%D1MT%
Al62A&VNSed'j+GFkMTmGH-'0N5H!!,PANIm&9Q(at-sign)(at-sign)!qUSE(at-sign)pcG*0PEA"SBA4TBf&
XE(Q6C'q6EQpdNf&cFh9YCC0dD'&dNf'6E(at-sign)9KFh9bCC0TFj0MEh9ZQUabG'&LE(Q
6B(at-sign)4NDA4TGTKP,SkI'S!!MC%RJ!"8D'(at-sign)(at-sign)"AYDBT!!8ijPFh56Dfj[QUabGfk6CAK
KEA"XCC0[CT0KNfePBA0eFQ(at-sign)6DA16G'KPNhDBEfaeE(at-sign)(at-sign)6eaD0R`(-c0KZMT%&U&#
j+0G"Z5Q6EfD6BC0cEfaTC*2A3BkKMC%RJ!#jD(at-sign)k(at-sign)"9FGEh*ND(at-sign)jKFRQ6efkj,(at-sign)4
TE(at-sign)9ZFfP[EQ&XNd9eBfaTC'9KET0cF'&MC5k4#Aiq9'KPNhD3!+abEfaeE(at-sign)(at-sign)6eaD
0R`(-c0KZMT%&U&#j+0G"Z5Q6EfD6BC0cEfaTC)kKMC%RJ!$A3CB%i(+jFf&dDA-
-CA16BAKTEfecNc'6B(at-sign)jNNc+6B(at-sign)+3!&11Ej(at-sign)XFRD6C5b4"4hPBR9dPJ6JFQ&iD(at-sign)p
YFj-aNf&ZC*-bNf4[Nfj[G*0MN!#XFQKKFQ&MG'9bDATPMU'0N5H!!(DDV(*[E(9
YCCB%*eeKE(at-sign)pZCj0KE'b6F*!!8ij[Fh0TBQaPNfePBA0eFQ9c,T%&l`"AD'&dNf&
NC'PdD(at-sign)pZB(at-sign)b6BAKTEfecNffBGA0dNhHBCC0KC'51SBf4*i!!G'q(at-sign)!i3ZG'KPNf4
P$'jTG'P[ET0[CT0KNfePBA0eFQ8XN31BV(at-sign)PZNfpbC'9bNh4[Nf1DV(*SBA*KBh4
PFQPkCC0fQ'pXG(at-sign)eP2j%&&VG*G*0TFj0`N!"6MQpcFfPLE'(at-sign)1SBf4*i!!G'q(at-sign)!jp
rBjUXFQKKFQ&MG'9bDATPNhDBEfaeE(at-sign)(at-sign)6B(at-sign)e[EQH6B(at-sign)aXNfePBA0eFQ9cNf+BHC0
KC'4TEQH6G'q6BAKTEfecNc'6B(at-sign)jNNc+6G*KhQ'q1SBf4*i!!B(at-sign)4NDA4TEfjKE*X
$kUKTET(at-sign)XFR4eDA4TGT0PQ'&iD(at-sign)pYFbbBEQ&YC(at-sign)ajNIm&9LbBG'KPQ'C[E'a[NhG
TEQFkMU3DJ!#0N5H!!%&iD(at-sign)pYN32UU$-ZMU'0N5H!!&4SCCB&#6&fQUabEfaeE(at-sign)(at-sign)
6EfD6BC0cCA56ed'6Z(at-sign)PcNfPZC'9`N!"6MQ9ZC'9ZQ(56EfD6G'KPNh#3!&11Eh0
TG'P[ET0[CT2A3ENZN3L8HdPQNf'6Ff9dNpG"NlPTESkN$S!!MC%RJ!$AEVNYC'P
YC(at-sign)jcD(at-sign)pZB(at-sign)b(at-sign)"-A54A9ME'PNC(at-sign)&ZNh0`B(at-sign)0PNf0KET0LN!"6MQ(at-sign)6FQPRD(at-sign)4XHC0
YEj(at-sign)XFRD6C(at-sign)5E"-A5Efk6G'qBBCKcCA5Bed+4!*X'Z5b4"2bFG'KPETMA3BkKMC%
RJ!#jB(at-sign)jNQ`2UU0G#N35&VVPSBC(at-sign)XFRD6CCKdD'(at-sign)BFf&YCCKfNfpXG(at-sign)eP,SkT'S!
!MC%RJ!"*ETB$kUK[G'KPFT0hQUabEh*NFbb6GTK[E(9YCC0TFj0TETKfNIpBj'&
bD(at-sign)&ZQ(56G(at-sign)jNCA+6G'KPNfGbEh9`NfpQNd9eBfaTC'9KET0YEh4TEfjc,SkKMC%
RJ!"-BA0dE(Q4r`9(at-sign),*%$[%4hPDabCCX$X+aYNh9cG*K`FQ9cBh*TBT!!8ijPQ''
BEQpbE(at-sign)&XDATKG'P[ELb4!la%BA1BF'L6HA0TBfPcG(1BFf'6HC(r"9BZN38PM&4
SDA1BDA1BC'pZCCKLNhQ1SBf4*i!!G'&VD(at-sign)jRPJ2Zkf'6F'&bB(at-sign)aXC(at-sign)a[G'p`N!"
6MQ(at-sign)6ee#4"C!!XEPhDA4SNfpbG'K[CfpZB(at-sign)b6FfPNCA16EfD6E'9ZCh4SFj2AH)f
I!Fc-mbTmHeN(!!J!!!!)!!!!"'0YFMM9-Bk(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpF
lMCJkQ$UB1Sk4%FUF1jKiMCm"c-cBESk4"DK3Z5b4!qrlB(at-sign)jNMU'0N5H!!(0PG(4
TEQH1TSf4*i!!3AKTEff4!qUS0#k1MTmH!!#0NJ$pAeihMSk-L`!!!!J!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#RfS!+J+2(at-sign)0S2f'e`ZJ!PX
SpBfJrFRA#k31J!#0NJ#fm*6A&SfI!Fc-f'k1Q`(at-sign)S8,NSee#4!D('Z5Q(at-sign)!e952C2
AH)fI!Fc-e6'1N36!"0GiMCm"c-c9-Sk0N3E!!YFkPJ(rrMU61Sk4&SUJH)fI!Fc
-f'k1Q0FkMTmH5'(at-sign)0N5H!!,P8D'9cCCB%9p4KH'P[EA-XQ`4c(h4[Cf9dD'9bNhG
TG'L6Fh9TG'&LE'(at-sign)6BfpZPDabG'PZNh9TG*0jPJ4Ae'0[EQ4TG'P[ER-XQ(9ZDA&
eC(at-sign)ajNf4PG'9b,BkKMC%RJ!"YD(at-sign)jPPJ2#(at-sign)R4SCC0fN!#XFQpXG(at-sign)ePNfpQNh0[E'P
NFj0TET0&G(at-sign)0XD(at-sign)4PB(at-sign)k6efkj,C0cF'&MC5k4"5Ya4T(r"9C[FT0PH'&YF'aP,*%
$bQPcG'&bG'PZCj0QFQpYMU'0N5H!!(4SCA0PPJ6fXfC[GA+6BAKTEfec,*%&1EC
LQUabHC0KNfaTE(at-sign)PdD(at-sign)jRNh"bEj!!8ijMCA0cNh0eBjKSNf&cNfpZCC--EQ4cNfP
ZNf&ZNf&NGT(r(at-sign)14KEQ0PC)kKMC%RJ!"MB(at-sign)aMG(at-sign)aeFjX%dlGdCAKdBT96MQq6EfX
XN380qfpZCCKPFh4KBQaTFfKPFjKdD'(at-sign)BCQ&MG*KdD'&dQ(4SCCKfN!#XFQpXG(at-sign)e
PQ'pQQ''BBQ&XE*MA8ifI!Fc-f(+1N3PQ'EP[CSkKMC%RJ!"bB(at-sign)4TGA1(at-sign)!qUSeh+
4"$ifZ(at-sign)PZNpGZZ5eND(at-sign)ePER0TEfjKE*0cF'&MCC0TFj0RDADDV(*PET0LQ(Q6G'K
PNfC[E'a[Q(GTEQH6CQpbECKeE'&c1SkKU40j#Bf5!-"-RGF(at-sign)MCm"c-cBESk4"DK
3Z5MA8ifI!Fc-f(+1N355BVNTN3098Mf0MBfIpq(&MC%%L)AA'Bf3!'ijRrZP-YK
Z2G8bMT%1PT(AFSf3!&11RrZP-YKZMSk4")L&R`94EBN!!'CP!##)4Tm,#!Z0N3#
ZfENSefipZ6)T)Bk1MSk1RaRQG)f4*i!!D(at-sign)D(at-sign)!qUSG'KPNf4TE(at-sign)9ZFfP[ET2AET1
jDA16CAD3!+abC(at-sign)k1SBf4*i!!B(at-sign)jNMU'QMC)!NeqGeaD0R`(-c0KZMTB&U&#j+0G
6MCm"c-cBFSk4"**LZ5Q4!e952Bf0MCrhiF(at-sign)0N35)K6+0RrZP-YKZMT2A'Bf3!'i
jRrZP-Y8Sf'lD!08a+GJpe6+1N5!2KENS+0GZPJ+UU03!NlNa+GFpZ6)T)GGbMC!
!8ikIqk8bf'k1MT%%L)(at-sign)I"9&YL3!!CQ8!HQ*'R`X)#if41!fCefkj)Bk1MSk1RaE
QG)f4*i!!D(at-sign)D(at-sign)!qUSG'KPNf4TE(at-sign)9ZFfP[ET2AET1jDA16Ej!!8ijNC#k1U4MbJif
4*i!!5A5(at-sign)"++MDA16Fh4TE'b6GfPNC(at-sign)ajNf+3!&11C(at-sign)aTCADDV(*PC*0dD'&dNhD
BEfaeE(at-sign)(at-sign)6DA16G'KPNfpZE(Q6D(at-sign)kBGT(r(at-sign)14KFQPKETKdNfePBA0eFQ(at-sign)6D(at-sign)k64A8
YMU'0N5H!!'0XD(at-sign)4PB(at-sign)k(at-sign)"(at-sign)jIefkj,A0`B(at-sign)0P,T%*a!4#GA56D(at-sign)k6F*!!8ij[D(at-sign)k
DV(*dNfpQNfCKBh56G'KPFQ(at-sign)6BA*PNfpdD'9bNfPZQ(D4reMNBA*TB(at-sign)kBG*0YC(at-sign)&
cGA*PFbb1SBf4*i!!C'8-EQ9NPJ(at-sign)4EQpZNf&XE*0bC(at-sign)&cEfjKBQaPNh0eBR0PG(1
6EfD64A9ME'PNC(at-sign)&ZNpGZZ5ecF'&MC5b4"IXIGfKTBjUXFQL6D''BGTKPNf'6EQm
YMU'0N5H!!(4KBQaPPJ4fTfGPEfePG(*TBj0cD(at-sign)GZD3aMB(at-sign)jMC5k4"YcG6h9bNfp
LN3#R('TPBh4TGTUXFQ(at-sign)6DA16G'q6C'9cBh*TBT!!8ijPNf&XE*0cG(at-sign)1BD*0TETK
fNIpBj'&bD(at-sign)&ZQ(51SBf4*i!!E(at-sign)9KFh9bCA-ZMUD0N5H!!&GSBA5(at-sign)!lm,D'&`F*!
!8ijPER16D(at-sign)D6GjUXFQ(at-sign)6DjKPCA#6G'KPN`abFh56G'KbC(at-sign)(at-sign)6BAKTEfec,*%$am4
LGA56G'&YF*!!8ijPFT0hDA4SNh4SCC0QEh9bG'L1SBf4*i!!BAKTEfdXN3261h4
SCCB$c(at-sign)"ZEh*YB(at-sign)aTHQ&dD(at-sign)pZNf&iD(at-sign)pYNcq4"5mG9fPXE*0hQUabCC0RCA56Ffp
YCA4SD(at-sign)jRNfPZQ(4PFQ9cG'PZCbb4!p-lEh+6GfPXE)kKMC%RJ!"hQUabCCB%`1Y
RCA56EQpdD'PZCj0ZCAFrN3HlU954r`9(at-sign)Ej0KER0hQ'9bNh4SDA16FA9PFh4TEfi
XN36fI(HBCC0hD(at-sign)aXNf&`F*!!8ijPB(at-sign)b6G'q6G'KPNf*KFfPMMU'0N5H!!(4[N!"
6MQpXFjB$kUK[CT0MEff3!+abBQPZBA4[FQPKE*0YBA4SC(at-sign)eKG'PMFbk1TSf4*i!
!9'KPPJ1cm'*KFfPMNh4[N!"6MQpXFj0[CT0MEffDV(*LD(at-sign)jKG'pbD(at-sign)&XNfeKG'K
PE(at-sign)&dD(at-sign)0cNf&bCC0dD'(at-sign)6C(at-sign)aPE(at-sign)9ZQ(4KFRQ6FhPYE(at-sign)9dFQPMMU'0N5H!!'CeEQ0
dD(at-sign)pZFbb(at-sign)!qUSG'q6GfPd,*0dD'(at-sign)6CQpXE'q3!+abGfPZCj0`N!"6MQpXH(at-sign)j[E(at-sign)P
KE(16D(at-sign)k6efk6ZAD4reMNBA*TB(at-sign)*XCA-kMU'KMC)!La8%ef(at-sign)0R`(-c08aMTB%`!5
j+0GiMCm"c-c9-Bk6ecZE!IrqH)fI!Fc-e6+1NpFlQ$Sk1MZBH)fI!Fc-f'k1N3(at-sign)
S8,NTPJ098Mf6ehL0R`(-c08aMTB(DUbj+jX#UUMAH)fI!Fc-e6+1NlNVQ0Fk1MU
BZ5ZBehL0R`(-c0KZMT%&U&$A1MZ1U429iU'0NA(at-sign)`C(at-sign)(at-sign)0R`(-c08bMTB%`!5j+0G
iMCm"c-c9-Bk6ecZE!IrqH)fI!Fc-e6+1NpFlQ$Sk1MZBH)fI!Fc-f'k1N3(at-sign)S8,N
TPJ098Mf6ehL0R`(-c08aMTB%`!6AH)fI!Fc-e6+1Q`GUV,NVN3+UU0GiMCm"c-c
9-Bk6ehL0R`(-c08cMTLj+jB#UUMA1MSkNlNVNpGiMCm"c-cBEYS!e6'1N4#%c0G
iMCm"c-cBESk4"DK3ecZ1TU'0MC)!ijU`1TB"rrikNcU1MUDI&mKPMC%kC)aPMCm
"c-cBEYS!e6'1N4#%c,NSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZ
0Q03"Q!'B!Bk4%IrkecZBH)fI!Fc-f'k1N3(at-sign)S8,NTPJ098Mf6ehL0R`(-c08bMTB
%`!6AH)fI!Fc-e611MC%'`!,8!CJ"Q!'1N4DrrYGiMCm"c-cBESk4#&,iZ5Z4!UU
SehL0R`(-c08aMT2AH)fI!Fc-e611NpGiMCm"c-c90)k0N3E!!Y3"Q!'B!Bk4&Vr
qehL0R`(-c0KZMT%)8[Lj+jB#UUMA1MSkNlNVNpGiMCm"c-c9-Bk4"-!%ehL0R`(
-c08bMSf4"X!#e!'B!CJ"MT%(at-sign)[rlAH)fI!Fc-f'lD!08aMT%3K-cA1ik1Rai!!)f
5!2eIAVNiMSk-L`!!!!N!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!$-5S!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#k31J!#0NJ#D`T(ACBfI!Fc-f'k
1N3(at-sign)S8,NSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZ0Q03"Q!'B!Bk
4%IrkecZBH)fI!Fc-f'k1N3(at-sign)S8,NTPJ098Mf6ehL0R`(-c08aMT%%`!6AH)fI!Fc
-e6+1MC%'`!,8!CJ"Q!'1N4DrrYGiMCm"c-cBESk4"DK3ecU1Ra6SpSf4*i!!Z8p
LFf9bGTUXFQ(at-sign)(at-sign)!VH(B(at-sign)k6D(at-sign)kBG'9bCA0dD(at-sign)jRNf0[D(at-sign)jMD(at-sign)4PEQ0P,T%%dS"8D'(at-sign)
6E'&cG*0[CT0dD'9cCC2AET1jFhPYE(at-sign)9dFQPMNfCeEQ0dD(at-sign)pZFj0TFikKMC%RJ!"
KE(0[PJ0ePA4SCC0QEh*YQUabG(at-sign)aKNfC[FT0dD'(at-sign)6GTK[E(9YCC0[CT0KNh"KFQ&
XE'9XEh4[F*T6MQ8ZN384fN&iD(at-sign)pYNc56Bf&ZNf+BCC0bCAGbDA4dC(at-sign)k1SBf4*i!
!BA11SBf4*i!!3AKTEff4!qUS0#k1SD'0NJ#PdCVA&SfI!Fc-f'k1Q`(at-sign)S8,NSee#
4!D('Z5Q(at-sign)!e952C2ACBfI!Fc-f'k1Q,NSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm
"c-c9-Sk6ecZ0Q03"Q!'B!Bk4%IrkecZBH)fI!Fc-f'k1N3(at-sign)S8,NTecU1Rb$+2Sf
4*i!!Z8aPG*B%V69eFj0dFRQ6B(at-sign)k6CAK`N!"6MQ9bD(at-sign)ePETUXFR3XN36Gf'&ZC*0
bCA"XB(at-sign)0PNh4SCC2AEVNYG'L6FhPYE(at-sign)9dFQPMNfCeEQ0dD(at-sign)pZNf+BHC0dD'(at-sign)1SBf
4*i!!efk930A8!*1j-5ecG*B#[#jcH(at-sign)eYCA4bD(at-sign)16CR9ZBh4TEfiZN368$8aPG*0
eFj--FR0dNh4KDj!!V(*PNpGZPJ098VNpNc-XPJ,iVA4SBA54!V`ZDA-XNh4SFQ9
P,(at-sign)4TE(at-sign)9ZFfP[EQ&XMU'0N5H!!(0`B(at-sign)0P,*%%4J&cEjB%-ladD'&dNhH3!+abCC0
MB(at-sign)k6BT!!8ijPG(4PFT0fDA0eB(at-sign)aTHQ(at-sign)6GfKKG*0TFj0REfPZCj0[ELk4"K3F6'9
dNh9cNh0PCC0hD'9dD'9bMU'0N5H!!(HDV(*PPJ2fL(at-sign)0KET0NC3aZCC0KNfePBA0
eFQ(at-sign)6Efk6Fh9LFf9dFj0[CT-c,(at-sign)4TE(at-sign)9ZFfP[EQ&XNh0`B(at-sign)0PNf+BHC0VQ'9PF'P
ZCj0dD(*PCBkKMC%RJ!"[CTB%GTadD'(at-sign)6B(at-sign)+3!&11Ej(at-sign)XFRD6CCB%GTaKH'P[EA-
XN35CQ(at-sign)*eG*0LQUabHC0bCA"XB(at-sign)0TEQH6G'KPNfj[FQeKE'PkBA4TEfk63AKTEff
60*0LQ(Q6GA0TEQH1SBf4*i!!B(at-sign)j[G'KPFTB%-A9cH(at-sign)eYCA4bD(at-sign)16CR9ZBh4TEfk
6D(at-sign)jcG'9KC*0[CT0dD'(at-sign)6FhPYE(at-sign)9dFQPMNfCeEQ0dD(at-sign)pZNpGPMCm"c-c9-ik(at-sign)"-!
%Z5MAH)fI!Fc-e6'1NpFlQ`(rrRL0R`(-c08bMT2A1jKiMCm"c-c9-ik6Z5Q1SBf
4*i!!GfKTBj(at-sign)XFQLE"-QfCfPfNf9cQ(4SCCKfNfpXG(at-sign)eP,T%(eJP-CA5BGA1B$(*
cG*KbCA"XB(at-sign)0PQ(4SCCKcH(at-sign)eYCA4bD(at-sign)1BCR9ZBh4TEfkBef(at-sign)0R`(-c08cMT%*LEU
jBT0jMU'0N5H!!(4SCCB$kUKcH(at-sign)eYCA4bD(at-sign)16CR9ZBh4TEfk6ef(at-sign)0R`(-c08bMT%
%`!5j,*0dD'9bC(at-sign)+DV(*jNf1BD'&ZCfPZCj0"H'P[EC-dNh4[MTmDB8L0N5H!!%&
iD(at-sign)pYN32UU$3R1SkKSBf5!+%K9GF(at-sign)MCm"c-c9-SkE"-!%Z5MA8*%"SFDj+CB$99)
pNpGiMCm"c-c9-BkBehL0R`(-c08bMTB(DUbj+j%#UUMAH)fI!Fc-e6'1Q0GiMCm
"c-c9-ik6Z5Z4!UUSehL0R`(-c08bMTMAH)fI!Fc-e611Q0FkMTm8k2D0N5H!!,P
%EjT6MQ9cPJ2$+A4SDA16BAKTEff6C'8-EQ(at-sign)6BC0YC(at-sign)&cGA*P2j%&+lC2CT0MEh9
bFf(at-sign)6DA56C'qBCA-ZN38VYP4SCC0bD(at-sign)GSN!#XFR56D'&ZC*0cD(at-sign)4PNfPcMU'0N5H
!!(4SCCB$q'pQEh*YQUabG(at-sign)aKNfC[FT0dD'(at-sign)6Fh9bCQ&MCC0KFQ9KNfpQNh4SCC0
`BA*KE'aPE'pdEh#3!&11CC2A8*%"SFDj,*%$qq&NDACTC'9NNf+BHC-b,T%&BM4
"Cf&TESkKMC%RJ!"hQUabCCB%Uj*hD(at-sign)aXN`aZC*0TET0KETKjNf&NGT(r(at-sign)14KEQ0
PC*0MB(at-sign)aMG(at-sign)aeFj0dCAKdBT96MQq6EfZ(at-sign)"+Z5G'KPNf9iF'aKEQ&dD(at-sign)pZNfpQNh4
SCC0QB(at-sign)0dMU'0N5H!!(4SBA5(at-sign)"2VcBAKTEfecNc%XPJ8r"6)XNc-XNf&ZC*X%q[-
d*bb6G'pRCA4SCA+BGfPdD*KcEfePQ'0[ET(at-sign)XFR4TET0eDA56HCKMEfjcD(at-sign)4PFQ&
dD(at-sign)pZFbb1SBf4*i!!BfpYF'aPG'9XHCB$1ceNCA4PFQeTEQ(at-sign)6B(at-sign)k6D(at-sign)k9V(*fNIp
Bj'&bD(at-sign)&ZNh5(at-sign)!cXpE(at-sign)9KFh9bCC0hD'PMN!#XFQL6DA16G'KPNh0eFQCKBf(at-sign)6BA*
PBC0[CT0cEfaTC(11SBf4*i!!D(at-sign)k(at-sign)"3&fEh*ND(at-sign)jKFRQ6Fh"KBf8ZN3Kp58D4r`9
(at-sign)Eh+6CAKKEA"XC5b4"8FTG'KPNfC[E'a[QUabGfPZCj0hQ'9XE*0VEQqBGfk6CQp
bECKeE''6CQpbNh4SCBkKMC%RJ!"cGA*QB(at-sign)0PPJ4R%(at-sign)&bC(at-sign)'6EfD6BC0LB(at-sign)aXNpG
6MCm"c-cBFSk4#2PcZ(at-sign)pQNh*KC'PeFj2AFT%%ZTqjD(at-sign)k6-j0ND(at-sign)ePER0TEfjcNfP
cNfpLG'&TEQ9NNfCbEff6G'KPFf(at-sign)1SBf4*i!!BAKTEfec1SkKSBf5!-6UK0F(at-sign)MCm
"c-c9-Sk4"-!%Z5MA8ifI!Fc-f(+1N355BVNTPJ098Mf600FCN!"Z1A+0N!"6MTr
l#jR9-Sk4"415ecU1Ra6SpSf4*i!!Z8aPG*B$kUKeFj0dB(at-sign)Z3!+abCC0dD'(at-sign)6EQ9
iG*0cG'9`,SkT'Q&)MC%RJ!"&EC(at-sign)XFQ+3!&11EfaNC(at-sign)jPC*X%UB*LNhQBEh9bQ(0
eBf0PFh1BGfPdD*KdNhH6EjKcH(at-sign)eYCA4bD(at-sign)1BCR9ZBh4TEfjc,*%%f6KhNf(at-sign)BEQq
6GjKbCA"XB(at-sign)0PMU'0N5H!!'&iD(at-sign)pYPJ10X$56BTUXFRQ6HCKPG*0KEQpdD'9bNf&
iD(at-sign)pY,*%$S%KeFfPZCj0KEQpdD'9bNh0jE(at-sign)ePG(*TBj0QG(at-sign)jMG'P[ELk4"4RM6'9
dNh9cNh0PG)kQMC%RJ!""H'P[EC%$kUJd)Lk1SD'0NJ#,lDEA&SfI!Fc-e6'1Q`6
!",NSee#4!D('Z5Q(at-sign)!e952C2ACBfI!Fc-e6'1Q,NSehL0R`(-c08aMTMA1jB"rrj
iMCm"c-c9-SkBecZ6H)fI!Fc-e611Q,NTPJ098Mf6ehL0R`(-c08aMTB(DUbj+jX
#UUMAH)fI!Fc-e6+1NlNVQ0GiMCm"c-c9-ik1MTmH!!#0NJ$pAekj1Bk1M)X!!!!
+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!pBk!#S#MeMD$
pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,P8D'(at-sign)(at-sign)"#R4EQ9hNfePBA0eFQ(at-sign)6eaD0R`(-c08
aMT%)kG(at-sign)jGfPXE*0cBA4TFfCjNd&iD(at-sign)pYFj-a,*B%1CXb,*0KEQ5E"#R4-bb6B(at-sign)j
NQ'PZQ'&NC'PdD(at-sign)pZQ'PdQ(GTE'b1T!k!!)f4*i!!Ff&dDA0QHCB$,DpKH'P[EC-
d)Lk4"2RL9'KPNh0jE(at-sign)ePG(*TBj0QG(at-sign)jMG'P[ET0[CT0NC(at-sign)GbC(at-sign)(at-sign)6EfjPNh"XBC!
!V(*jFj0dD'(at-sign)6FQpXCC0dD'&dNfPZMU'0N5H!!(4SCCB$B6G`FQ9fD(at-sign)peFj0dPDa
bGj0[PJ0K0f9iB(at-sign)e`E'9cNhHDV(*KFj0`E''BHCKPC*0LQ(Q6G'KPNfpdD'9bNh5
BGjK[Nh0jE(at-sign)ePG(*TBj0QG(at-sign)jMG'P[ER-ZMU3DJ!#0N5H!!%*eG*B$kUKhQUabB(at-sign)P
dNf'6E(at-sign)PZQ(9dC6U4"6MJDA16G'KTFj0NC3aZDA4TEfk6BfpZFfPcG'9ZQ(3rMU'
0N5H!!&54r`9(at-sign)EjB%hUGbC(at-sign)&XDATPNh4SBA56G'KPNf4P$'jTG'P[ET0[CT0dD'(at-sign)
6EQ9hNfePBA0eFQ(at-sign)6eaD0R`(-c08aMT%*RUZjDA16BfpZFfPcG'9ZN!#XFR3XN38
ETh4SBA56DA-XMU31J!#0N5H!!(4SBA5(at-sign)!`a+eaD0R`(-c08aMT%(c%kjBA16C'8
-EQ9NNf+3!+abHC0KH'P[EA16-5b(at-sign)!cM$-Lb6-bb6B(at-sign)jNPJ--5M3LNh*PB(at-sign)aXHC0
PH'PcG(16B(at-sign)jNNfPcNfj[G*0KNf4bC(at-sign)&YNfpQMU'0N5H!!(*PBA0[ELb4!a6BE'q
D8ij[DjB#hf4KG*0dPDabGj0[PJ,IC("KFQ&XE'9XEh4[F*KPFj2A8)fI!Fc-e6'
1Q`HID,PKEQ56ee#0R`(-c08bMTLjG'KKG*0SBC(at-sign)XFRD6CCB#hf4KNfCKBf(at-sign)6D(at-sign)k
6BfpYE(at-sign)pZ,T%%hmP8D'(at-sign)1SBf4*i!!$(*cG*B$Z8G`BA*KE'aPE'pdEh#3!&11CC0
SBA16FfPNCA16CA&eB(at-sign)b6G'q6ehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk
6ecZBH)fI!Fc-e611N3Kj5lNXN32$+'&ZC*B$Z8GdD'(at-sign)6Ff9MEfjNNh"KFQ&XE'9
XEh4[F*!!8ijPMU'0N5H!!'KKFjB$LRjcD(at-sign)4PFj0PFA9KE*0dEj2AH)fI!Fc-e6'
1PJ6!"0FlQ`(rrRL0R`(-c08bMT2A1jKjN!"Z1ENZN38BdP4SCCX$LRjdPDabGj0
[Q("KFQ&XE'9XEh4[F*!!8ijPFjKSBC0fNf(at-sign)BBCKMEfeYEfkBCQ&MCCKhDA4SMU'
0N5H!!(0TC'9cPJ4'T'9aG(at-sign)&XNh4[NpGiMCm"c-c9-Bk(at-sign)"-!%ecZ4!IrqH)fI!Fc
-e6+1NlNZN3C-e94SCCB%4U4YC(at-sign)&cGA*PNpF(at-sign)MCm"c-c9-Bk(at-sign)"-!%Z5MA8)fI!Fc
-e6'1Q`HT6G4EN3,T5GG3MCm"c-c9-Sk6Z5Q(at-sign)"%DNEfD6G'KPNh"KFQ&XE'9XEh4
[F*!!8ijPNpG3MCm"c-c9-BkBe&Z4!ZP*ee#0R`(-c08bMSkKMC%RJ!#jBf&ZPJ5
I+Q+3!&11CC0MEfe`GA4PC*0TET0dPDabGj0[Q`5I+RH6BC0jFcU4"U(PGA0TEQH
BG'KPQ'aPCR5BFfPNCCK[CTKKH'P[ECJb,*%%c%Y[FTKeFfPZCjKdD'(at-sign)1SBf4*i!
!FQPRD*UXFR5(at-sign)"*NTFfPNC5b4"-6*B(at-sign)jNNh4SCC0dQ(HBEj0MEfe`GA4KG'P[ER1
6D'&NNf+3!&11CA4dCA+6H(at-sign)PPE'56G'KPNh0KE(at-sign)(at-sign)6B(at-sign)jcGjKPFLb4"-6*D(at-sign)k1SBf
4*i!!FhPYN!#XFQ+3!&11Efac1SkKSBf4GLcVeaD0R`(-c08aMTB%`!5j+0G3MCm
"c-c9-Bk4"fUXe&ZE!UUSee#0R`(-c08bMT1j+CB$99)pNpF(at-sign)MCm"c-c9-Bk(at-sign)"-!
%Z5MA8)fI!Fc-e6'1NlNTQ#ZBeaD0R`(-c08aMT1j+0G3MCm"c-c9-Sk6Z5QBe!#
BeaD0R`(-c08aMT1j+0G3MCm"c-c9-Bk4"fUXe&bBee#0R`(-c08bMT1j+GFkMUN
9!!#0N5H!!,P-CA5(at-sign)!qUSGA16Bj(at-sign)XFQKPBj0VN32UU(4SDA-ZMTmDJ!#0N5H!!&4
SCCB%i[YXC(at-sign)CdNh0TC'(at-sign)6DA16BfpYF(9dC(at-sign)56BT!!V(*jNfpLFf9bGQPZCj0dD'&
dNh4SCC0`BA*KE'aPE'pdEh#3!&11CC2A8)fI!Fc-e6'1N3J6[Y4EN306ZYG3MCm
"c-c9-Sk4#D,rZ(at-sign)KKFikKMC%RJ!"cD(at-sign)4PFjB$kUKPFA9KE*0dEj2AH)fI!Fc-e6'
1N36!",NXehL0R`(-c08bMT%)UUbjB(at-sign)jNNpGiMCm"c-c9-ik4"fUXZ5Z4!UUSehQ
3!'ijZ5k4"6MJ9'KPFQ9QEh*P,*0"H'P[EC-d)T0dC(at-sign)aXFj0eFj0dD'&dMU'KMC)
!QC&(eaD0R`(-c08aMTB%`!5j+0G3MCm"c-c9-BkE"fUXe&Z4!UUSee#0R`(-c08
bMT1j+CB$99)pNpGiMCm"c-c9-BkBZ5Z(at-sign)!UUSehL0R`(-c08bMTLj+j2AH)fI!Fc
-e611Q,NVNpGjN!"Z16U1TSf4*i!!Z8j[QUabGjB$kUKXCA56GA16BfpYF(9dCC0
dD'(at-sign)6FQPRD*KdNh0TC'8ZN38ii&H4r`9(at-sign)CC0SBCKfQ'(at-sign)1SD'0NJ#b9rhA&SfI!Fc
-e6'1PJ6!",NSee#0R`(-c08aMT1j+CB$99)pNpGiMCm"c-c9-Bk(at-sign)"fUXZ5ZE!UU
SehL0R`(-c08bMT1j+jMAH)fI!Fc-e611MUDKMC)!Y2SjeaD0R`(-c08aMTB%`!5
j+0G3MCm"c-c9-Sk6Z5Q(at-sign)!e952C2AH)fI!Fc-e6'1PJGUV,NVQ`+UU0GiMCm"c-c
9-Sk6Z5ZBehQ1TU'0NJ#`e6N(at-sign)MCm"c-c9-Bk(at-sign)"-!%Z5MA8)fI!Fc-e6'1Q`GUV04
FN3+UU0G3MCm"c-c9-Sk6Z5Q(at-sign)!e952C2AH)fI!Fc-e6'1Q,NVN3+UU0GiMCm"c-c
9-Sk4"-!%ecZ1TSf4*i!!Z(at-sign)&RB(at-sign)PZPJ3*Xf+3!+abHC0"H'P[EC-d)T0KF("XD(at-sign)9
NNh4[NpG3N38VllNpN31++GG3MCm"c-c9-Bk4"hr2e&b4!Vr,ee#0R`(-c08bMT%
%`!5j,*%%%ACcD(at-sign)jMCC0[EQ(at-sign)6FfPNCC0PFA9KE(16HQ9bEj0hD'9ZMU'0N5H!!(4
SCCB%PJY`BA*KE'aPE'pdEh#3!&11CC0TFj0KN`eKG#bE"-$MG'KKG*0TFbbBBC0
bC(at-sign)0dB(at-sign)jRE'8ZN3Fl#&4SCA*PCQpbC5bBG'KPNh*TCfL3!+abG*0cD(at-sign)4PNfpQMU'
0N5H!!%&iD(at-sign)pYPJ2UU$+6CA&eB(at-sign)acMU'N'S!!MC%XTE(A&SfI!Fc-e6'1PJ6!",N
See#0R`(-c08aMT1j+CB#UUJVNpF(at-sign)MCm"c-c9-Bk(at-sign)"-!%Z5MA8)fI!Fc-e6+1NlN
TPJ+UU03!NpF(at-sign)MCm"c-c9-BkE"-!%Z5MA8)fI!Fc-e6'1N3GUV04FNpG3MCm"c-c
9-SkBZ5Q(at-sign)!e952C2AH)fI!Fc-e6'1PJGUV,NVQ`+UU0GiMCm"c-c9-Sk6Z5ZBehL
0R`(-c08cMT1j+jMAH)fI!Fc-e6'1NlNVQ0GiMCm"c-c9-Sk6Z5ZBehQ4!aMKe!#
BZ5MAH)fI!Fc-e6'1NlNVQ0GiMCm"c-c9-Sk4"-!%Z5Q4!e952BkKMC)![[#'ehL
0R`(-c08aMTB(DUbj+jX#UUMAH)fI!Fc-e6+1NlNVQ0GiMCm"c-c9-ik6Z5ZBehQ
3!'ij1ik1Rai!!)f5!2T[B,Na-)k1M)X!!!!,!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!"(qD!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,P
KEQ5(at-sign)"(at-sign)PBG'KPNh59V(*hNfq(at-sign)"(at-sign)PBFfPNCA16EfD6Eh9bNf9aG(at-sign)&dD(at-sign)pZFj0KCh*
PC5b4"FN%G'KPFQ9LQUabHC0MEfkBGQPZBfPZCj0eFj0dD'&dNh4SCBkT$S!!MC%
RJ!"NC3aZDA4TEfk(at-sign)!qUSE(at-sign)'DV(*jNhHBC(at-sign)aXNf+3!&11CC0MEfjcDA0dC(at-sign)kBG#k
1T"QeHSf4*i!!9'KPPJ2UU("bC(at-sign)0PC'PZCj0KFQGeE(at-sign)9ZQUabG*0TFj0MEfkBGQP
ZBfPZCbb6CADBC(at-sign)k6G'K[G(at-sign)GSNfPdNh"bEjKfQ'9cNfj[G'KTEQFZMU'0N5H!!%&
MG(9KE'ajNIm&9Lb4!f)IG'KPPJ-rr'4P$'jTG'P[ET0[CT2A&SfI!Fc-e6'1N36
!",NSee#4!D('Z5Q6CQpbNf'6F'&bB(at-sign)aXC(at-sign)a[G'p`N!"6MQ(at-sign)6ee#4"1(#Z(at-sign)KKFj0
KNh0TEA"XCC0RC(at-sign)pYCA4bD(at-sign)11TSf4*i!!D(at-sign)k9V(*dCA*`FQ9dBA4TEfiZN3E!ePG
SC(at-sign)kE"'e2EC0eE(4TF'aTC(at-sign)5BBT0jQ$3XN350q(at-sign)PdQ'9aG(at-sign)&XFjKdD'(at-sign)BF*!!8ij
PFQPYCA4PFTK[CTKdD'(at-sign)BF'&bB(at-sign)`YMUD0N5H!!'aPE'pdEh#3!&11CCB$)F[A8*%
"SFDj,*X$5IGdD'&dNfPc,*KdD'(at-sign)6Fh9YNfpQNh4SCC0XC(at-sign)jRG'KcNfpQNf&XE*0
dD'(at-sign)6C(at-sign)4RCA16EfD6G'KPNh"KFQ&XE'9XEh4[F*!!8ijPMUD0N5H!!0G3N3'KaVN
ZMUD0N5H!!%TeFh5(at-sign)"4ERBA16D'&`F*T6MQ9ZFj0QEh+6GT!!V(*[E(9YCC0KEQ5
6BA*PB5b4"(at-sign)(fDA56Bf&ZNf+BCC0cD'qDV(*hET0LQ(Q6BfpZQ(4TETKeDA5BHC0
MEfiYMUD0N5H!!(0TC'9bBA4TEfjcPJ4bBR4SBA56G'KPNfePBA0eFQ(at-sign)6eaD0R`(
-c08aMT%*-QDjBf&ZNf+3!&11CC0PH(4PEQ4PC*0dEj0KE'b6FQ9KFfpZB(at-sign)*XCC0
cEfaTC(16D(at-sign)k1TSf4*i!!Eh*ND(at-sign)jKFRQ(at-sign)""U*Fh"KBf8XQ`3QJ(at-sign)C[FT0PH'&YF'a
P,*KdEj0KE'b6BfpZPDabGT0PH*B%'SPcCA4cNf&ZC*0dEj0KE'b6F*!!8ij[E(P
SC(at-sign)4bB5bBBfpZPDabGT0PH)kQMC%RJ!"[FT%$kUKZEfjMEfk9V(*fNf9i,SkKMC%
RJ!"#GA3XQ`96!fpZCCB&#[&YBC!!V(*jNfpLN3#R('TPBh3XQ0F(at-sign)MCm"c-c9-Bk
4"-!%Z5MA8*X"SFDj+C0YB(at-sign)Z3!+abCA16Ff9ZFf(at-sign)6CQpbNf'6F'&bB(at-sign)aXC(at-sign)a[G'p
`N!"6MQ(at-sign)6ee#BZ5b4"9-$BT!!8ijPBf&eFf(at-sign)6BBkQMC%RJ!"`BA*KE'aPE'pdEh#
D8ijPPJ0E(at-sign)(at-sign)KKFj0KNhH3!+abC(at-sign)aXNf4P$'jPC*0`Q'9bD(at-sign)ePG'9b,T%&#4YAD'&
dNfPQNpG"NlPTFj0KNh0[E'PNNh4SBA56C'qBCA16EQpdMUD0N5H!!'KKPDabGT0
PPJ50(''6Gj!!V(*PE'b6C'8-EQ9NNh#3!&11CA*TE(at-sign)9dCA)XN35eZ(at-sign)'6Fh"SCA*
PNfC[FT0PH'&YF'aP2j%()$e8D'(at-sign)6C'8-EQPdD(at-sign)pZNfpQNh4SCBkQMC%RJ!"YC(at-sign)&
cGA*PPJ33XYF(at-sign)MCm"c-c9-Bk4"-!%Z5MA3ENTNfC[FT0cEfaTC(16ed'6ZA4SBA5
6E(at-sign)'DV(*jNfj[G*0SBCKfQ'(at-sign)6BC0hQ'9XE*0NC3aZC(at-sign)56F*!!8ijPFQPYCA4PFT-
0D(at-sign)9cMUD0N5H!!'PZPJ2UU(4SCC0QB(at-sign)0PNfpQNf0[E(at-sign)e[ET0cC(at-sign)jcC5k1SBf4*i!
!4(at-sign)PZFh4PD(at-sign)k(at-sign)!rLVGh*[G'8kN398jL*$EfeYEfk6Ff9ZFf(at-sign)6DA16G'KPNh*PFfP
NG(at-sign)(at-sign)6EfD6G'K[Ff(at-sign)6F(*PDR9ND(at-sign)0PFj0dD'&dNhH3!+abCA*PMUD0N5H!!'PZFh4
TE'aPC*B$&'eTETUXFR4[Nh9cNf+3!&11C(at-sign)C[FQ(at-sign)6G'KPNf&RCC0[CT0cCADBC(at-sign)k
BG'9PEL)ZN36aGd0[E(at-sign)e[ET0cC(at-sign)jcCC0YQ(9cG*0MEfjcG'&ZQ(4XHBkQMC%RJ!"
bC(at-sign)&NDR9cG*B$kUKdEj0bC(at-sign)&XDA53!+abHC(r"9BZMU'0N5H!!&4SCCB%!49ZCAH
6E(at-sign)9KFh9bCC2A&SfI!Fc-e6'1N3M"'EPdD'&dNhHDV(*PNfpLG'&TET0TET0dD'P
cNhHBBCKjNfPcNf0KE'aPC*0dD'(at-sign)6E(at-sign)9KET0hD(at-sign)4dD#b4"!D`BBkQMC%RJ!"YDA0
ZEfePFTB#a9TdD'&dNfKKFj0LN!"6MQ9PET0VN!#XFQ9`G*0QEh+6D'PcG'pbD(at-sign)0
KE*0bC(at-sign)&cEfjc,T%%eaY8D'(at-sign)6E(at-sign)9KET0hD(at-sign)4dD*0[CT0KNh0[E'PNMUD0N5H!!'P
ZPJ1AHA0`B(at-sign)0PNfPcNf0[EA"XCA4PE(Q6BjUXFQKKFQ&MG'9bDATPC*0LQ(Q6BAK
TEfecNc%XPJ1S($)XNc-XNf&ZC*B$PhNd)Lk4"4dQ5(at-sign)k6F'&bG'PMG(at-sign)aKFLb1TSf
4*i!!DA5(at-sign)"8&ZDA16D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh3XPJ(at-sign)A(h4SBA5E"8&ZDA-XNfP
dQ'4[P911CA1BEQpdQ'4PF*0PEQ5BEfkBF*0[FfPdD(at-sign)pZ,T%*26&'NIm&9QpbQ'9
iB(at-sign)e`E'8XN3(at-sign)A(h4SCBkQMC%RJ!"QEh*YN!#XFR9XBCB$kUKQEh+6G'KPNfePB(at-sign)k
6GfPNG'L6EfD6BC0cF'KPFQ(at-sign)6EfD6FQ&NDA9cNpGbN33q0VPTFj0MEfe`GA4PC*0
dEj0LN!"6MQ(at-sign)1TUD0NJ$,+PIA&SfI!Fc-e6'1N36!",NSee10R`(-c0KbMT%%NQ+
j+CB$99)pNc6AFT!!V()kMTm8D"Z0N5H!!,P8D*(at-sign)XFR9cQ`-Z!AH6CCKcC(at-sign)(at-sign)BG'K
KG*KTETKdD(*PCCKND(at-sign)ePER0TEfjcQ'9KBj0SQ'pQQ(4SCCKdD(*PCCKPE'9YC(at-sign)k
6G'&bHCKcH(at-sign)eYCA4bD(at-sign)11TSf4*i!!CR9ZBh4TEfjcPJ4&''pQNh4SFQ9PNhDEreM
NBA*TB(at-sign)*XCA16E'9KC(16G'q6B(at-sign)k6D(at-sign)k9V(*fQ'&bD(at-sign)&ZNh5(at-sign)"%8BE(at-sign)9KFh9bCC0
dD'&dNf9ZDQq3!+abHA16CA&eB(at-sign)b1TSf4*i!!FQPRD*UXFR4cPJ1eDhGTG'L6GTK
[E(9YC5k4"5FK9'KPN`abFh56G*KhQ'q6EfD6G'KPFf(at-sign)6E(at-sign)9KFh9bCA16BA*PNhH
BC(at-sign)aXNfYZEjKhELb4!m!4EQ&YC(at-sign)ajNIm&9Lb1TSf4*i!!GT!!V(*[E(9YCCB%dZT
KEQ56BA*PB5k4"r'R9'KPNh4SDA*N,*X&$2YdD'(at-sign)6E(at-sign)9KET0hD(at-sign)4dD#bBDA16BA5
6F(*PFf9ZN!#XFR56B(at-sign)aYEh0dNh4[G'&XE(Q1TSf4*i!!G(at-sign)jVEQq9V(*hELk4#-I
r5C%&'JpVEQq6GjB&'Pe[CT0ZEj0`N!"6MQ9bFfpZNhGSEj0SBA16B(at-sign)k6D(at-sign)k9V(*
dG(at-sign)PdDAD6CCB&'PeQC(at-sign)9XD(at-sign)jRNfC[FT0dD'(at-sign)6E(at-sign)9KESkQMC%RJ!"hD(at-sign)4dD#b(at-sign)!qU
SFfPYD(at-sign)aKFT0dEj0dD'(at-sign)6D(at-sign)k9V(*dG(at-sign)PdDAD6CCB$kUKQC(at-sign)9XD(at-sign)jRNhHDV(*PNfK
KQ(DBCC0QEh+6GTK[E(9YCC0KEQ56BA*PB5k1SBf4*i!!6'9dPJ1*LA9cNf0[EQT
PBh4eFQ(at-sign)6BC0`QP11Eh0cD(at-sign)*XCC0KF("XD(at-sign)0KG'P[ET0[CT0dD'(at-sign)6E(at-sign)9KET0hD(at-sign)4
dD#k4"4L!3C%$LA"`Q'pdBA4[NfGbEj(at-sign)XFRH6CA+1TSf4*i!!Dfj[QUabGh1(at-sign)!p+
iG'KKG*0KNh#3!&11Eh4KG'mRFj0fQ'pXG(at-sign)ePNfPcNfPYF*96MQpbG'&ZQ(3XN32
AJQ+6C(at-sign)0KGA0PPJ25Z'PdNf4PG'9bE(at-sign)PZCA16G'KPNfkBGA4bD5f1TSf4*i!!G'P
[EQ&XQ`2H+f0[ET(at-sign)XFR4PET0dQ'pQQ(4SCCK`P911Eh4KG'mZN38dYP4SCCK`Nfp
dBA4[Q'GbEj(at-sign)XFRH6CA+BB(at-sign)acEjKVEQq6Gh1BG'KKG*KdD'(at-sign)BFh9bCQ&MCBk1Rai
!!)f5!2T[B$%aMSk-L`!!!!`!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!&0`S!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if4*i!!Z(at-sign)&bC(at-sign)'(at-sign)"&65EfD
6BC0`QP11Eh4KG'q6DA16D(at-sign)e`Q'pbG'&ZN!#XFR3XN34[A(at-sign)+BC(at-sign)0KGA0PNfPdNfP
cNh*eE(at-sign)pbC(at-sign)56G'KKG*0dD'(at-sign)6GQPdB(at-sign)eTER16D(at-sign)k6BBkN$S!!MC%RJ!"`N!"6MQp
dBA4[PJ2`if&bCC0MEfjMC(at-sign)kDV(*dFQ&dC(at-sign)56D(at-sign)k6G'KPNh0VD(at-sign)iZN39,NPH4r`9
(at-sign)CC0YBCKjNf0[EQTPBh4eFQ(at-sign)6G'KKG*0KFj0cEj!!8ij[ET0KFj0dD'(at-sign)1SBf4*i!
!F*!!8ij[G'&dEjX%hc0RFQq9V(*hNf9bQ(GTE'bBBT!!8ijPBfpYCCKKNhH6BA*
PQ'pQQ(4SCCKYC(at-sign)&ZQ(GTC(4S,*%&(&CSCCK[FTKcD'(at-sign)BGfPXE*J-EQ5BBBkKMC%
RJ!"ZPDabGA4bDA4TEfjKE*X%Ci0TET0dCA*`FQ9dBA4TEfkBEfDBG'KPQ'ePB(at-sign)k
BGfPNG'LBEfDBBCK`N!"6MQpdBA4[,T%'Vh**N34RBf&YQ'PZC'9LG'9NQ(4[MU'
0N5H!!&0dCAD9V(*PQ`2UU&0MNfKKET0eC(at-sign)bBCQpbQ(4SDA1BCAKKEA"XC5k1RaU
!!)f4*i!!3C%%6SacD(at-sign)eTE'&bPJ41TQYTEQ56EfD6FQ9KFfpZD(at-sign)jRNhHDV(*[FQY
cNfPZNpGZNlPND(at-sign)ePER0TEfjc,T%'C0TANIm&9Q(at-sign)6C'PcBfqBGTKPFT2AET1jC'N
,CA*PETKdMU'0N5H!!'PZPDabGT(r(at-sign)14KFQPKET0dQ`8-P(at-sign)ePBA0eFQ9c,*%&94"
PB(at-sign)16D*K[CTKdD'9YQ(H6C(at-sign)aXQ'4P$'jPC*K[ETKKE'bBF*!!8ij[E(PSC(at-sign)4bBCK
KEQ5BEfkBB(at-sign)aXMU'0N5H!!!aZDA4PPJ5IDA9ZD(at-sign)pZFj0[CT0MEfe`B(at-sign)0dNf0[ET(at-sign)
XFRD6CALE"*pTFf9dFbk4"eFN4(at-sign)&MNfLBEfDBG'KPQ0GZQ,PPE'9YC(at-sign)k6G'&bHCK
cH(at-sign)eYCA4bD(at-sign)11SBf4*i!!CR9ZBh4TEfjcPJ6LTfpQNpGZNlPfQrpBj'&bD(at-sign)&LE'9
cNfaPB(at-sign)4cNh4[Nh4SCC0NC3aZDA4TEfk6EfD6BC0ZCAH6D(at-sign)k9V(*fQ'&bD(at-sign)&ZNh5
4"1+RE(at-sign)9KFh9bCBkKMC%RJ!"hD'PMQUabD*B$9'&TFj0KNf4T#f9bC(at-sign)kBG*0RC(at-sign)j
PFQ&XDATKG'P[ET0[CT0dD'(at-sign)6EQpdD(at-sign)pZNfpQNhDBEfaeE(at-sign)8ZN38'b&4SCA0PNpG
ZNlPYC(at-sign)&cGA*PFikKMC%RJ!"KFQ(at-sign)(at-sign)!f9#Bf&XE'9NNh4SCC0TETUXFR4bD(at-sign)jcD(at-sign)1
6GTK[E(9YCA-ZN38-D94SCC0TETKdFQPZFfPMNhDBEfaeE(at-sign)9cNf&bCC--FR0dNf4
P$'jPC*0[ET0KESkKMC%RJ!"[FR4SEfG[EQ&XQ`2UU(#98ij[E(PdEh#6CCMA8*%
&M'kjGfK[Ff(at-sign)BFfPNCA1BCA&eB(at-sign)bBehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c
9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ(L0R`(-c0KZMT%*N[LjBT!!V(*jN32UU(0PG(4
TEQH1SD'0NJ#QF3IA&SfI!Fc-f'Z1Q`8MNVNSee#4!D('Z5Q(at-sign)!e952C2ACBfI!Fc
-f'Z1Q,NSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%
4bT`lQ(L0R`(-c0KZMT%&U&#j+GFlMTm9!!#0N5H!!,PhD'9bCC%$$P,ACBfI!Fc
-f'Z1N38MNVNSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZ0Q$UB1TJ
kMT%4bT`lQ(L0R`(-c0KZMT%&U&#j+CB$$P*TFj0dD'(at-sign)6efZ3!'FGZ5edD*0PE'9
YC(at-sign)k3!+abG'&bHC0cH(at-sign)eYCA4bD(at-sign)16CR9ZBh4TEfiZN36[ENKPFQ8XN3-kBh4SCBk
KMC%RJ!"cG(at-sign)*cBh*TF(5(at-sign)!qUSefZ4"&(&ZA*KEQGPFj0QFQpYNc'6G'q6efkj,Sk
T'S!!MC%RJ!"2EQ(at-sign)(at-sign)"!LNG'KPET0`FQq3!&11Bf9PC(16G'q6CAKdC(at-sign)jNNh4SCC0
NC3aZDA4TEfk6EfD6G'KPNfPZQUabG(*TER0TBj0fQ'pXG(at-sign)ePFj0dEj0YEh*PMU'
0N5H!!'GPEQ9bB(at-sign)b(at-sign)!qUSFf9dFbb6BTUXFRQ6BC0dC(at-sign)1BD'jTFA9PNhGSD(at-sign)1BD*0
hQ'(at-sign)6GfPXE*0cD'pbG'ajNh0PC5k1TSf4*i!!9'KPPJ5C2(at-sign)PZQUabG(*TER0TBj0
fQ'pXG(at-sign)ePFj0KFQ(at-sign)6D(at-sign)jNCA#3!&11C(at-sign)jNC(at-sign)kBG*0[CT0PB(at-sign)1BD*0[G'KPFLb4"-6
LCAKMCA"dNfC[FT0MCA*dB(at-sign)PZNfPZ,BkKMC%RJ!"PFA9KE'PdD(at-sign)9cPJ4[UR4SCAQ
6Ff&dDA0QHC(r"9BZN3E(jdeKG'KPE(at-sign)&dD(at-sign)0TB(at-sign)jcNf&bCC0`FQ9cC(at-sign)kDV(*dE(Q
6GjK[FQYTEQH6Efk6C'9dCA*YD(at-sign)iYMU'0N5H!!'PZCjB%e9PdD'9cCC0KFj0jQUa
bCA56G(at-sign)jVEQqBGfk6D(at-sign)jPFA9KE'PdD(at-sign)9cNf&YEfjRNh4SCC0TETKdFQPZFfPMNhD
BEfaeE(at-sign)9c,T%(q208D'9cCBkKMC%RJ!"TEQ9aG(at-sign)&XDA4TCA1(at-sign)"$,*Cf9ZCA*KE'P
kCC0dD'(at-sign)6BfaKFh0TBf&XNfPcEh#3!&11CA*TE(at-sign)9dFQPMNfPZCA&eB(at-sign)aTG*UXFRQ
6G'KKG*0bC(at-sign)aKG'9cNhDBEf`YMU'0N5H!!(9YCCB%h1jdEj0KFQ9K,T%)$l&"QUa
bG*0`FQ9cC(at-sign)kBG#b4"4PrGjKPNfYZEjKhNhDBCA*jNfaTG(4XCC0KBT!!8ij[GA5
6G'KPNfPZQ(4bD(at-sign)jcD(at-sign)16GTK[E(9YCA-lMU'0N5H!!(4SCAQE"+K0D''9V(*fNf(at-sign)
BEQpdQ'+3!&11C(at-sign)9ZQ'&bEh9ZC*KQEh+BE'pZCjKKEQ5BGT0PFRQBE'PdG'aPQ(*
PFf9KFQ16D*KSBA1BBT!!8ijPC(at-sign)kBC'pZCBkKMC%RJ!"[ETB%8*CdD'9Y,T%'DUY
ANIm&9Q(at-sign)6C'q6EQpdNf9fQUabC(at-sign)k6Dfj[Q(H6G'KPNfC[FQfBG(at-sign)aKNfC[FT0dD'(at-sign)
6D(at-sign)kBG(*TER0TBj0fQ'pXG(at-sign)ePFj0[CT0KESkKMC%RJ!$AEVNYFfPYF'aPH#k1TSf
4*i!!6Qq9V(*hQ`1`l(Q6Eh(at-sign)BBA*PQ(4SD(at-sign)jVD(at-sign)jR1T%&(!*dD'PcQ'PcQ'&XE*J
-EQ(at-sign)BB(at-sign)jNQ'4KEQ4jNIm&9Lb4!laiBR9dQ'K[NhHBDA1BG'KPQ'9iG'9ZFfP[ETK
[CSkKMC%RJ!"dD'(at-sign)(at-sign)",*RD(at-sign)kDV(*dFQPZFfPMNhDBEfaeE(at-sign)9cNfCbEff6F'&bB(at-sign)a
XC(at-sign)a[G'p`N!"6MQ9cNh4[Nfe[FQ(at-sign)6Cf9ZCA*KE*0cCA4cNf0KFR*TC(at-sign)56Eh9d2ik
KMC%RJ!""EQ5(at-sign)!ikpBT!!8ijPFfPNCA-XN31K(fPcELGdNh4SCA*PNf&ZQUabHC0
TETKdG(at-sign)PdDADBCC0TETKdCA*`FQ9dBA4TEfk6GjKPNf0KET0RDADBCC0dD'(at-sign)6D(at-sign)k
BG(*TER0TBikKMC%RJ!"fN!#XFQpXG(at-sign)ePFcq1TSf4*i!!9j(r"9CPPJ0"4RGTE'b
6B(at-sign)jcGjUXFQ9bNf+3!&11Eh4SNh4SCA0PNh&eCA0dD(at-sign)pZFj0cD(at-sign)fBG(at-sign)adB(at-sign)jPEh9
cE(Q4r`9(at-sign),T%&!'T-CA56GA16Cfq6BQ&MQ'Z6G'q6G'KbC(at-sign)8YMU'0N5H!!'4TE(at-sign)9
ZFfP[EQ&XPJ6(jh0`B(at-sign)0P,T%(d*eCNIm&9QpeNf&XE*0VEQqDV(*hNh4SBA56G'K
PNh0PG*0[CT0KE'b6Fh4bB(at-sign)PRD*KdNfaTEQ9cNfPZNh0`B(at-sign)0PMU'0N5H!!#f(at-sign)"-F
#EQpdNfjPBf9cFf&bD(at-sign)ajNh4SFQpeCfL6G'KPNfpbD(at-sign)GTET-YNfC[FQecNf'6EQP
MCC0KE'GPBR*KD(at-sign)16GT(r(at-sign)14KFQPPG*!!V(*jNIm&9Lb4"2iCBf&XE'9NMU'0N5H
!!(4SCCB&5"4(FQ&cFfeKEQjTB(at-sign)iZN3P4*94SCC0RFQpeF*0[CT0KE'b64A9ME'P
NC(at-sign)&ZNh*TCfPNNfe[G'P[ER16B(at-sign)0dFj0[ET0dD'(at-sign)1SBf4*i!!4h*KFh0YB(at-sign)jZD(at-sign)&
Z,*%$(hYKEQ5(at-sign)!Zb`G'KPFQ(at-sign)6DA16B(at-sign)k6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)!Zb`E(at-sign)9
KFh9bCC0[ET0dD'(at-sign)64h*KFh0YB(at-sign)jZD(at-sign)&ZNh9ZC'9bMSkI(J!!MC)!qQpJ-6+1MSb
,!!!!$3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!AkQJ!U!
SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#jG'KPPJ(at-sign)*m'&MG'P[ET0[CT0dD'(at-sign)6Ch*
[GA#6EfD64A9ME'PNC(at-sign)&ZNfe[G'P[ER-ZN3S(at-sign)Z94SDA16D(at-sign)k9V(*fNIpBj'&bD(at-sign)&
ZNh5(at-sign)"BR`E(at-sign)9KFh9bCC0TFikN$S!!MC%RJ!"eEQPaG(at-sign)(at-sign)(at-sign)!e)cCAKMCA"dNfC[FT0
KNf0[ER0dB(at-sign)kDV(*dNfCKBh4[FLk4"3B13C%$8JacD(at-sign)eTE'&bNh0dBA4PE(at-sign)9ZQ(5
6E(at-sign)'BHC0LQP11CC0YB(at-sign)4PNf&LQ'peG)kKMC%RJ!"dD'(at-sign)(at-sign)"%hFFf9dNfpQNf&XE*0
`E'&ZCA-XN34QU(at-sign)&ZC*0YEh*PNfGPEQ9bB(at-sign)aXHC0QEh+6G'KPNh0PG*0[CT0KE'b
6E'PZC(at-sign)&bNhD4reMNBA*TCA4TCA16EfD1SBf4*i!!C'PYC(at-sign)jcD(at-sign)pZPJ1I1pGVN33
'(at-sign),PTET0&G(at-sign)0XD(at-sign)4PB(at-sign)k6Fh"KBf(at-sign)6EfD6C'PYC(at-sign)jcD(at-sign)pZNpGZZ5k4"4qm8Q9YC(at-sign)f
3!+abBT!!8ijPFT0dD'&dNh4SCA0PNfaTEQ9KFSkKMC%RJ!"fNIpBj'&bD(at-sign)9dD(at-sign)9
cPJ2UU'jPC(at-sign)56EQpdNh"KFh16G'KbEh9RD*0dD'(at-sign)6Eh*TCfPZ,SkI'BSpMC%RJ!"
*ETB&c%KdD'(at-sign)6F(*KBh4TBf(at-sign)6EfD6E(at-sign)&dD'9YBA4TBh-XN3C%X'0[EA"eG'&dD(at-sign)p
ZNhGTG'L6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)"Fa)E(at-sign)9KFh9bCA16Efk1SBf4*i!!4h*
KFh0YB(at-sign)jTB(at-sign)jcPJ0CE'PcNh*KFQ8lN31*e(at-sign)e[Fh56E(at-sign)&dD'9YBA4TBfPKER16GjU
XFQpeE'56BT!!8ijPNfKKFQ56F(9dNf9fQ'9ZNh4[Nh*PBf&XE)kKMC%RJ!"KETB
#dN"PH("XD(at-sign)0TG*0QEh*YQUabG(at-sign)aKNfC[FT0dD'(at-sign)6D(at-sign)kBGT(r(at-sign)14KFQPKETKdNfe
PBA0eFQ9cNfpZNdGbBA0cE(at-sign)&ZD(at-sign)&ZFbk4"0YS6'9dNh9cNh4KDjKPNf'1SBf4*i!
!CQ9hPJ-'h(at-sign)eTETUXFR9dCA16G'q6Cf9dNf'6CQ9PE'PZCj0QEh+6G'KPNfPZQ(D
4reMNBA*TB(at-sign)kBG*0YC(at-sign)&cGA*PNfpZNh4SCC0cCA56EfD6B(at-sign)aXNh0dFQ&TCfLBG)k
KMC%RJ!"XD(at-sign)jPFjB#Z&KTET0dD(*PC5ecF'&MC5k4"0,&3A16DA16Bh9cG'pYBA*
jNIm&9Lb4![(at-sign)EGjUXFQ(at-sign)6BT!!8ijPCfPZNf+BHC0RDACTEQH6G'KTFj0YC(at-sign)&cGA*
PNf'6EQ&YC6U1SBf4*i!!E'9dPJ4iN!"eFj0MB(at-sign)aXNfPdNpF9MCrlT6+0e611R`G
53BdaMSk4"-!%Z6Z4",q%G'KPNh9`F*!!8ijPFT0TEQ4PH*-cNh0dB(at-sign)jNFj0QEh+
6G'KbC(at-sign)8YC'PYC(at-sign)jcD(at-sign)pZB(at-sign)b6Fh"KBf8XN35F#Q&ZC)kKMC%RJ!"dD'(at-sign)E!`f2E'q
9V(*hNf9bQ'PZC'9iQ(0dB(at-sign)jNFjKQEh+BG'KPQ'4TE(at-sign)9ZFfP[ETK[CTKKQ'aTEQ8
XPJ-jb'jKE(at-sign)9XHC[r"9BXNfpZC5k4"1mY9*K[N3-0Mh*PF*!!8ijPBA3XNhH3!+a
bCBkKMC%RJ!"eFf(at-sign)(at-sign)!j4ZG'KPNfj[G'&dD(at-sign)pZNpF9MCrlT6+0e611R`G53BdaMSk
4#&4bZA4[Nf4PEQpdCC0dD'(at-sign)6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)!j4ZE(at-sign)9KFh9bCC0
[ET0dD'(at-sign)6Ff9dNfpQNf&XE*0cG(*KD(at-sign)GSN!#XFR51SBf4*i!!E'PZCA1(at-sign)!qUSD(at-sign)k
6G'KbC(at-sign)8YFh"KBf8ZMTmCLMf0N5H!!%0[ER0TC'9bPJ1rPQ'6FQ9MG'&ZCfaPNpG
5N32Bi,P`E'&MC(at-sign)56B(at-sign)k3!+abHAGSCA*PNfPZNh0`B(at-sign)0P,*%$b$0KEQ56BfpZFfP
NCA+6G'KPNh0PG*0[CT0KE'b1SBf4*i!!Fh4bB(at-sign)PRD*UXFR5(at-sign)"-X5E'PZCA16G'K
KG*0YC(at-sign)9dNh4SCC0bC(at-sign)0dB(at-sign)jRE'(at-sign)6ee+3!"P+Z5k4"pSH3f&ZNhHBCC0MEfe`GA4
PNh4SCC0YC(at-sign)&cGA*PNfpQMU'0N5H!!(4SDA1(at-sign)!aFpFf9dNfpQNfaTEQ9cNhGTG'K
[GA56Dfj[QUabGfPZCj0dD'(at-sign)6CQpbECKeE''6CQpbNh4SCC0TETKfNIpBj'&bD(at-sign)&
ZQ(56E(at-sign)9KFh9bCC0[ET0dD'(at-sign)1SBf4*i!!4h*KFh0YB(at-sign)jZD(at-sign)&ZPJ3QNQpQNf&XE*0
XD(at-sign)jPFj0TET0dD(*PC5ecF'&MC6qE"HbG6fD6BfpeFR0PNhH3!+abCC0MB(at-sign)iZQ%'
4"#D#Fh4bB(at-sign)PRD*!!V(*dNfaTEQ(at-sign)1SBf4*i!!E(at-sign)9PG(1(at-sign)""qhG'KPNh*PBh4KEQG
XCC2A8T%%13'jC(at-sign)PdD'9bNf&dNf'6F*!!8ij[D(at-sign)k3!+abG*0[FT0ZEh56BA56B(at-sign)a
X1j%%1MpdD'9bC(at-sign)C[FQ8XN33Xqh4SCC0fNIpBj'&XG(at-sign)(at-sign)6EfD1SBf4*i!!G'KPPJ1
81QePBA0eFQ(at-sign)6EfD6G'KPNh0PG*0[CT0KE'b6E'PZCA16E(at-sign)9PG'PZCj2A8T%$VB5
jC'9`N!"6MQ9ZC(16EfjXHC0[ET0dD'(at-sign)6BA*PBC2A&SfI!Fc-e6+1N36!",NSee+
3!"P+Z5Q1SBf4*i!!EfD(at-sign)!p63G'KPNh*PBh4KEQGXCC2A8T!!'8Uj,T%&-CK*CT0
hQUabCC0dB(at-sign)ZBCC0KEQpdD'9bNh*PBh4KEQGXCC2A8Sf3!"P+RrZP-YS`MT%'[&1
jGfK[Ff(at-sign)6BA*PBC0TFj0NEh9LE'(at-sign)6G'KPMU'0N5H!!'&bC(at-sign)'(at-sign)"'&aEfD6ee+D'8U
j,*%%Ib0dD'9ZNh4SCC0YC(at-sign)&cGA*PNfpQNh4SCC0cCA56EfD6B(at-sign)aXNfaTEQ9cNfe
PCA4TEQH6ee+0Q*rlT6,D-)k4"dMdZ(at-sign)PcNf4[G(at-sign)*XCC0dD'(at-sign)1SBf4*i!!E(at-sign)9KFh9
bCCB%Gi0[CT0dD'(at-sign)6Ff9dNfpQNf&XE*0XD(at-sign)jPFj0YC(at-sign)9dD(at-sign)jRNpG5N!!C5VNZN3E
IF&"bEj!!8ijMC(at-sign)9ND(at-sign)jRNf&XEfjRNh4SCA0PNfaTEQ9c,*%%QVPhN!#XFQ(at-sign)1SBf
4*i!!Cf9dPJ0G#h4[Nd0KG(at-sign)19V(*SNhNRFjB$A3YQG(at-sign)jMG'P[EQ&XNf9aG(at-sign)&dD(at-sign)p
Z,*%$H9jKEQ56Gj!!V(*PNfPZCQ9bNh4SBA56G'KPNfePBA0eFQ(at-sign)6EfD6G'KPNh0
PG)kKMC%RJ!"[CTB%E)"KE'b6Fh4bB(at-sign)PRD*UXFR56E'PZCA16E(at-sign)9PG'PZCj0KNh*
PBh4KEQGXCC2A8T%%KFUjCA&eB(at-sign)acNf'6BfpZFh4KETKdNh4TE(at-sign)9cNh4SCC0KFQ9
KMU'0N5H!!0F(at-sign)MCm"c-c9-Sk4"-!%Z5MA8T!!'8Uj+5k4"9,m8fPZBf(at-sign)(at-sign)!r0FGjU
XFQ(at-sign)6BA*PNf&dNfaTBT!!8ijPFR5BHC0dEj0MQ'K[N!"6MQpcCC0KNfj[FQeKE'P
kBA4TEfk6EfD6G'KPNfePBA0eFQ8XN32eL(at-sign)aPG)kKMC%RJ!"eFjB$kUKKCh*PCC0
dEj0cCA56G'KTFj0MEfjcG'&ZN!#XFR56CA&eB(at-sign)b6G'q6EfjP,SkI'BSpMC%RJ!"
#GA5(at-sign)"D,-D(at-sign)jcG'9KC*0[CT0hQUabEh*VD(at-sign)jRNhGTG'L6BC0bC(at-sign)0dB(at-sign)jRE'(at-sign)6GjK
PNf0[G(at-sign)aNNfKKQ(DBCC0hQ'pbDjKPC*0hDA4SNf&ZQ(Q1SBf4*i!!F'aKEQ&bPJ6
6D!aRGA*PNpG$N3(at-sign)`"VPhD'&dFfqD8ijPGT!!V(*PFLb4"3fBF'aKBf9NNfPZNf&
ZNf&bBQPdFQ&bHC0`Q'pcDA4TEfk6D(at-sign)k6Fh"KBf8ZN3Ic)94SCBkKMC%RJ!"YC(at-sign)&
cGA*PPJ6J9QpQNh4SCC0cCA56EfD6E'PZCA16E(at-sign)9PG'PZCj2A3j%&[25jCA&eB(at-sign)a
cNh4SCC0KFQ9KNpF(at-sign)MCm"c-c9-Sk4"-!%Z5MA3j%!h*kj+5b4"4h"BT!!V(*jNh4
SCC0cB(at-sign)ePMU'0N5H!!(*PBA0[EQPZCbkE##UV9j(r"9CPPJ6Pl(0dFQ9cFj0dD'(at-sign)
6BA0cG(at-sign)e`G'P[ET0dD'&dNpG$N3A#LVPYN!#XFR9cG*0XD(at-sign)(at-sign)6D(at-sign)k6BC0`E'&ZC5k
B9*(r"9C[Nf0[ELf1SBf4*i!!BfaeC'8kN3Dikf9fQUabC(at-sign)k(at-sign)"+UZGfPdD'peG*0
VEQqBGfPZCj0dD'(at-sign)6CQpbECKeE''6CQpbNh4SCC0TETKfNIpBj'&bD(at-sign)&ZQ(56E(at-sign)9
KFh9bCC2A&BfIqk8bMG8cMTm(8N'0-Bk1N36!",NXN36DVhHBCBkKMC%RJ!"MB(at-sign)k
(at-sign)!d"IEQ9fQUabCA*dD'9XCA0cNf0[EA"eG'(at-sign)6G'KPNhD4reMNB(at-sign)aeCC0[CT0cG(at-sign)1
BD*0KNfePBA0eFQ(at-sign)6Efk6Bf9bG'&TET0cCA4cNfpQNfaTEQ9c,SkI'BSpMC%RJ!"
-CA5(at-sign)"#&LGA16EQqDV(*hNh4KDjKPNf'6E(at-sign)pbCC0cEh"SDA0dD(at-sign)0KG'9NNh0PG*0
[CT0cG(*KD(at-sign)GSQ(56E'PZCA-ZN3AG$eH4r`9(at-sign)CC0dB(at-sign)ZBCC0KNh0PG*2A4)kKMC%
RJ!#jD(at-sign)k(at-sign)"0ieG'KbC(at-sign)8YFh"KBf(at-sign)6G'KKG*0TFj0dD'(at-sign)6G(at-sign)jTEfk6EfD6C'PcDQp
TET!!V(*dNh0PG(16ed10R`(-c08aMTB%`!6A1jX"rrj$MCm"c-c9-Sk6ecZ0Q03
"Q!'B!Bk4%IrkecZB3ifI!Fc-f'k1N3(at-sign)S8,NXN38E'(GSCA*PN36H0(at-sign)9KBj!!V(*
SMU'0N5H!!'pQPJ8)CA4SCC2A3ifI!Fc-f'Q1N3KY2lPTFj0MEfkDV(*dB(at-sign)PZC(at-sign)5
6D(at-sign)k6BC0ND3YPFQ9ZQ(56F'aKEQ8XN392e'&ZC*0hQ'(at-sign)6BA0VNfC[FT0dD'(at-sign)6E(at-sign)9
KFh9bCC0[CSkKMC%RJ!"dD'(at-sign)(at-sign)")0IFf9dNfpQNf&XE*0cG(*KD(at-sign)GSQUabG*0XD(at-sign)j
PFj0YC(at-sign)9dD(at-sign)jRNpG%N!"6MVNZN3F$"90eBjKSNf'6BfpYF(9dBA4TEfk6Bf&ZNf+
3!&11CC0MBA*bD(at-sign)9NMSkI(J!!MC)!qQpJ-611MSb,!!!!$J!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!E%HJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF
,MC%RJ!#jEh9d,*%%H!YLGA5(at-sign)"&[%DA56DA16BC0MEffDV(*LD(at-sign)jKG'pbD(at-sign)&XNfj
TCfLBG'eKFQ8lN3588h0[NffBG(at-sign)1BD*0cEbb4"(J,G'KKG*0hQ'(at-sign)6BA*PNfC[FQ0
PC*0dEikN$S!!MC%RJ!"NEjB$(`jhD'&dNfeKG'KPE(at-sign)&dD(at-sign)0TB(at-sign)jcNf4[NhGSC(at-sign)k
6BfpZCR*[ETUXFR4PC*0hDA4SNf0[ECKLD(at-sign)jKG'pbD(at-sign)&XNfjTCfLBG'eKFQ9c1Sk
KMC%RJ!"dD'9jPJ00p'1DV(*SB(at-sign)jRCC0dD'(at-sign)6F(*[BQaPEC0PGTKPFT0cEj0cE'P
RD*KdE(Q4r`9(at-sign),T%&"+4*ET0dD'PcNf0KFf(at-sign)6GjKPNh4KDjKPNf'6D'PZQ(56CR*
[EC0dD'(at-sign)1SBf4*i!!Gj(at-sign)XFQ'6HCB%cJT`FQpLB(at-sign)*TE'PcG(16GjUXFQpbDbk4"q-
&6'9dNpGBMCm"c-cB4)k4"l(at-sign)NZ5MA)C!!EMQj+C0PFA9KE*0dD'(at-sign)6ETKeECKLN!"
6MQ9bNfpQNh4TE(at-sign)9cNh4SCC0cG(*KD(at-sign)GSQ(51SBf4*i!!E'PZCCB$[r[A)C%%,M5
jE(at-sign)9PG(16G'KPNh0PG*2A4*!!8ikj,T%&+UC*ER0dC(at-sign)&NNfpQNf0[EA"eG'PZCj0
KNfePBA0eFQ8XN32)K'aPG*0eFj0MEfe`GA4PNh4SCBkKMC%RJ!"TET!!V(*dC(at-sign)G
bB(at-sign)b1SD'0MC)!`'(at-sign)4Rr(MKkYDMT)!c'(at-sign)3!0GBMCm"c-cB4)k4"l(at-sign)NZ5MA)C9Z1EN
Tef39MCrl#jQ0e611R`IVfSdaMSk4"-!%Z5MA)C1j+GFlMTm(at-sign)m'Q0N5H!!,PhD'9
bCCB%LGhA)C%%q"DjFQ&ZCf9cNfq9V(*fNf9bPJ5*hA4SCC0(FQ&cFfeKEQjTB(at-sign)i
XQ`5aUR4SBA56DA-XQ'q9V(*fNf9bPJ5*hA4SCC0cCA56EfD6B(at-sign)aXNh0dFQ&TCfL
3!+abG)kKMC%RJ!"XD(at-sign)jPFjB#pj*TET0cF'&MC5k4"1IC9j(r"9CPNhGTE'b6Ff9
PNh4SBA56GjUXFQ(at-sign)6Bf&ZNf0[EA"eG'(at-sign)6G'KTFj0TETKdC(at-sign)GbB(at-sign)b6GfPdD'peG*0
VEQqBGfPZCikKMC%RJ!"dD'(at-sign)(at-sign)!qUSE(at-sign)9KFh9bCC2A&BfIqk8bMG8cMTm(8N'0-Bk
1N3LUV,P[ET0dD'(at-sign)64h*KFh0YB(at-sign)jZD(at-sign)&Z,T%&11"6D(at-sign)jMCBkKU4*abSf5!-iPJpG
%N31Si,NpMCrbrrf0N3HKeGKZMTm0!!10MC%%Tm1IpIrmUeZ1MTm,p*D0N3098YK
Te6daMSk4%aDQed10R`(-c0KTMT%$C0VA1ikI'mL&MC%RJ!#jB(at-sign)jNPJ2UU(0TEQ0
PNh4SCC2A3ifI!Fc-f'Q1N3G2JVPKFQ(at-sign)6C'PcDQpTETUXFR3XNhHBCC0SBCKfQ'(at-sign)
1SDD0MC)!KXS(at-sign)Rr(MKkYDMT)!NXS9eeL0R`(-c0K%MT%(YD5j+0FKP(at-sign)ijZ5RAC"(at-sign)
0RrX,QBh9-ikI"q[DM6'1MTX%`!5j+0FKNlNTPJ098Mf0Rr,rrBf4"rS2f'k1R`d
!!if0NjrerrbV(at-sign))k1R`[dPSf4!kf-f'R926'1MSf4%mFCRr(MKkYDMT%IaaMA(at-sign))f
I!Fc-f%10R`%i1[-Z%-fqcJ!'!!!!"J!!!!9ME(at-sign)eT0YPTMSk4#EY8Z5MA)C9Z1EN
Tef39MCrl#jQ0e611R`IVfSdaMSkBZ5MA)C1j+GFkMTmEb)(at-sign)0N5H!!,P#GA5(at-sign)"'R
AGjUXFQ(at-sign)6D''BGTKPNf1BD'pcC(at-sign)k6C(at-sign)&MQ'L6EfD6G'KPNh0PG(16ed10R`(-c0K
TMT%(cV'jG'q6E'PPNfPZNf'6F'aKEQ8XN35*Sh0[Nh4SBA56BC0cG(*KD(at-sign)GSQ(5
1SBf4*i!!E'PZCCB$kUKYC(at-sign)9dFj2A3ifI!Fc-f'Q1N3G2JVPPDA4SCA+6EfjMCC0
[FT0ZEh56BA56B(at-sign)aX,T%&11"*G*0QEfaXEj!!V(*hFj0dD'&dMU'KMBf5!+Lm6Tr
aiiHV(at-sign)Sk5!,5m6GGBMCm"c-cB3ifI!6Jkf(at-sign)Q1MT%*Ze5j+0FKP(at-sign)ijZ5RAC"(at-sign)0RrX
,QBh9-ikI"q[DM6'1MTX%`!5j+0FKNlNTPJ098Mf6eaD0R`(-c08bMTLj+0G$MCm
"c-cBDBk4!f6DZ5Q1RaE`DBf4*i!!B(at-sign)jNN32UU(4SCA*PCQpbCBkKTSf0NJ#Ij(b
ImH1(UeU1NJ#Vj([A(at-sign))fI!Fc-f%51N3HeT,NSeb'9EMQj+GGN&BfIq`ZCMG8cMTm
(kpU0-Bk1Q`6!",NSeb'6Z5Q(at-sign)!e952BfIm[rpMC%(qJrBESkI$3!$MBf6RrArr+Y
BMSkI#r5(at-sign)MC%$VBcBDG8p-Bk1N42('GF(at-sign)MCm"c-c9-SkBZ5MA3ifI!Fc-f'Q1N30
NfVNTecU1Ra[)KBf4*i!!Z9GSBA5(at-sign)"C'-DA16G'KTFj0TC'9ZPDabG'PdNhQ(at-sign)"C'
-G'9XE'PZCj0eFcq4#Lf09'KPNh*TCfL3!+abG*0SB(at-sign)jNNh0TC'(at-sign)6CA&eB(at-sign)acNh4
SCC0KFQ9KNfpQMU'0N5H!!(4SCCB%4YacGA*QB(at-sign)0PNpG%N!"6MVNZN3C0I%j[G'K
TEQH6Fh4[F(16GA16CR*[EC0`BA0cD(at-sign)jRNh4[Nh4SCC0XD(at-sign)eTG#b4"&hTB(at-sign)jNNfe
KDfPZCj0dD'(at-sign)1SBf4*i!!CQpXE'qDV(*hD(at-sign)jRPJ3iH'&cFf9bG'P[ELk4"L*46'9
dNpG&N36XMlPLN!"6MQ(at-sign)6)Q&ZQ(NLNh0eFQCKBf(at-sign)6D(at-sign)k6Fh"KBf8XN34,l(at-sign)&ZC*0
XCA56eeL0R`(-c0K&MT%(,BLj+0FKN!"Z1ENTNf+3!&11CC0dD'(at-sign)1SBf4*i!!ET(at-sign)
XFR9YNf+3!&11CA+(at-sign)!qUSEfD6G'PYCA16G'KPNh0dFQ&TCfLDV(*dNfaTEQ(at-sign)6eb'
4"&MKZ(at-sign)ePCA4cNh4SCC0cGA*QB(at-sign)0PNpG&N3#d&lNZN38ii&4SC(at-sign)k6G'KPNfPZQ(4
PCh*KE)kKSBf0NJ$#5f(at-sign)ImH1(UeU1NJ$15f6A(at-sign))fI!Fc-f%(at-sign)1N3FYL,NSeb'9EMQ
j+GGN&BfIq`ZCMG8cMTm(kpU0-Bk1N36!",NSeb'6Z5Q1RaE`DBf4*i!!FQ&ZCfP
ZCjX$kUK[PDabGT0PFTKKE'bBFh4bB(at-sign)PRD*0dQ'aTEQ9cQ0FKN!"Z1ENXQ'9aG(at-sign)&
XFjKdD'(at-sign)BFh9bCQ&MCCKKFQ9KQ'pQQ0G&N3#d&lNZMU'0N5H!!%PZPJ5bNR"bEf*
KBQPXDA0dD(at-sign)16E'&ZCh9KCf8kN3E)Y(4SCC0KPDabGT0PFQ&RCCX%XT*ZNh9YNf+
3!&11CA+BEfDBG'PYCA1BBCKbB(at-sign)jNEfeXHCKMNfK[Ff9ZMU'0N5H!!(0dFQ&TCfL
3!+abG*B$kUKXD(at-sign)jPNfePCA4cNh4SCC0cGA*QB(at-sign)0PNpG&N35H[lPPFA9KE(16G'K
PNh0eFQCKBf(at-sign)6BA*PBC0[CT2A4C%!Y"Hj,SkI'5M1MC%RJ!"-CA5(at-sign)"%4)GA16EQq
3!+abGj0bCA4bB(at-sign)0PNfpeFT0cG'9`Fbb4"&U`B(at-sign)jNNh*PF*!!8ijPBA56G'KPNh0
KE(at-sign)(at-sign)6FQ9KFfpZD(at-sign)jRNh4KDfPZCj0dD'(at-sign)6Ff9dMU'0N5H!!'pQPJ4q3Q&XE*0`E'&
ZCA16D(at-sign)k6Fh"KBf8XN35M+'PZFh4PB(at-sign)56EfD6G'KPNh0PG*0[CT0KE'b6Fh4bB(at-sign)P
RD*UXFR56E'PZCA-ZN3EcVP4SCC0TETKfNIpBj'&bD(at-sign)&ZQ(51MTmH!!#0NJ$kEf!
a0)k1M)X!!!!2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"
l%k!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,PYC(at-sign)&cGA*PPJ-)d(at-sign)pZNh4SDA1
64h*KFh0YB(at-sign)jZD(at-sign)&ZNfPcNf4PEQpdC(at-sign)56BT!!V(*jNpF9MCrlT6+0e611R`G53Bd
bMSk4"-!%Z5b4!cAmGfKPFQ(at-sign)6B(at-sign)GKD(at-sign)k6G'KPNh9`F*!!8ijPFT0TEQ4PH)kN$S!
!MC%RJ!"cG'&ZC(1(at-sign)"0m0CQpbNh4SFQ9PNf4TE(at-sign)9ZFfP[EQ&XNh0`B(at-sign)0P,*%&(#G
KEQ56G'KPNfa[PDabGj0PFTB%h`eTEQ4PH*0QEh+6G'KPNf4TE(at-sign)9ZFfP[ESkKMC%
RJ!"[CTB%5BaKNh"XB(at-sign)jP,T%'9BY6D(at-sign)jMCC0KNh"XB(at-sign)jPNfePCA4cNf'6Fh4bB(at-sign)P
RD*UXFR56E'PZCC0cC(at-sign)GYC(at-sign)kBG*0PDA4SCA+6BA56BC0`N!"6MQpTETKdNfpbMU'
0N5H!!'j[G*B$Jf"KG*0KE'`XN31B#(4SCC0cB(at-sign)ePNf&bCh9YC(at-sign)kDV(*dNh0SEjK
hFj0dD'&dNh4SCC0YC(at-sign)&cGA*PNfpQNh4SCC0cCA56EfD6B(at-sign)aXNh"XB(at-sign)jPFikKMC%
RJ!"dD'&dPJ2aJ'ePCA56BC0XD(at-sign)jPNh0PCfePETUXFR56edb6Z(at-sign)9aG(at-sign)&XFj2A&Sf
I!Fc-e6'1N36!",NSedbj+5b(at-sign)!r-fEQ&YC(at-sign)ajNIm&9Lb6G'KPPJ2aJ'aPEQGdD*0
[CT0dD'(at-sign)6Ff9RE(at-sign)9ZQ(51SBf4*i!!edbj1j%&NFGYEh*PPJ8%[(at-sign)GPEQ9bB(at-sign)aXHC(
r"9BXN39,3QPQNpG'N3DQJlPTFj0KETUXFRQ6Bh9bGTKPNb*hD'&dFfq3!&11CAD
BCA)LNfPZNh0`B(at-sign)0P,*%&5d*KEQ56D(at-sign)D6eeL0R`(-c0K'MT%($dqj+0FKN!"Z1EN
TMU'0N5H!!'9aG(at-sign)&XFjB%Ga"dD'(at-sign)6ET(at-sign)XFR9YNf+D8ijPFTB%Ga"[CT0dD(at-sign)ePFj0
dD'(at-sign)6F'aKEQ(at-sign)6eb'4"19*Z(at-sign)ePCA4cNh4SCC0MGA*fN!#XFQ(at-sign)6edD4!D('Z5b4"*S
UG'KPET0bCA#BC(at-sign)&dD(at-sign)jRMU'0N5H!!(4SCCB$kUKKFQGeE(at-sign)9ZQUabG*0hQ'(at-sign)6GA0
PC*0QEh+6Fh4bB(at-sign)PRD*KdNfaTEQ9cNhHBCC0TEQCPFT0dD'&dNh4SCC0TETKdC(at-sign)G
bB(at-sign)b1SD'0MC)!`PU"Rr(MKkYDMT)!cPU!eeL0R`(-c0K'MT%($dqj+0FKP(at-sign)ijZ5R
AC"(at-sign)0RrX,QBh9-ikI"q[DM6+1MT%%`!5j+0FKNlNTMTmAmFk0N5H!!'9aG(at-sign)&XFjB
%4Q"dD'(at-sign)6E'9ZCh4SNfpQNh4SCC0MGA*fQUabCC2A4T%"SFDj,T%'6!K8D'(at-sign)6GT(
r(at-sign)14KFQPKBQaPNfpQNfPZQ(4PCh*KG'P[ET2A)C%%Y*QjEQqBGj0bB(at-sign)jRCA11SBf
4*i!!Ej(at-sign)XFRD6CA+(at-sign)"GL*F'aKEQ9c,*%'9!&ZEh56Ej(at-sign)XFRD6CA+E"GL*Fh4bB(at-sign)P
RD*0dQ'aTEQ9c,T%,!S0)CA*PQ'&RB(at-sign)PZQ(H6CCKMEfe`GA4PQ'&ZQ'PZNh4PCh*
KE)kKMC%RJ!"hDA4SEh9dPJ2UU'YZEj!!V(*hD(at-sign)jRNh4SCC0YC(at-sign)&cGA*P,SkI'S!
!MC%RJ!"ANIm&9Q(at-sign)(at-sign)"Ab-BA*PNfj[QUabGj0fQ'9bHC0ME'pcCC0dEj0RCA4dD(at-sign)j
RNf&ZNfPZQ(4eDA4TGTKPNfPZQ(4PFR"bCA4KG'P[ET0[CT0dD'(at-sign)6E(at-sign)9KESkKMC%
RJ!"hD(at-sign)4dD#k4"8lZ8Q9MB(at-sign)aXPJ2b!R4SCC0`BA*KE'aPE'pdEh#3!&11CC2A8*%
&NmLjGfPdD*0cD(at-sign)4PFj0PFA9KE*0dEj2AH)fI!Fc-e6'1PJ6!"0FlQ`(rrRL0R`(
-c08bMT2A1jKiMCm"c-c9-ik6Z5k4"8lZ9*(r"9C[N32b!QePBA0eFQ(at-sign)1SBf4*i!
!G'KPPJ2J%("XB(at-sign)jPFj0YC(at-sign)9dD(at-sign)jRNh4SCC0`BA*KE'aPE'pdEh#3!&11CC2A8*%
"SFDj,*%$iLjhN!#XFQ(at-sign)6$(*cG*0MEfjcD(at-sign)4PFT0KNfCKE(at-sign)PXHC0[CT2c1CZlL%!
!$!!!!!`!!!!'BfedD6%bj(#(at-sign)rfCQBA+6B(at-sign)b4!*QCE'9XMU'0N5H!!,P`E'&ZCA-
XQ`031(at-sign)&XE*B$+CecD'&bD(at-sign)jRNh4SCC0cB(at-sign)ePN`aiC(at-sign)56G(at-sign)jTG*0ZEh*YB(at-sign)b6eh(at-sign)
j,T%%q)G*ET0[G'KPFT0hN!#XFQpbC(-XQ'0[ER0TC'9bNh4SCBkKMC%RJ!"cCA5
(at-sign)"*NHEfD6B(at-sign)aXNh"XB(at-sign)jPFj0`BA*KE'aPE*0dEj0dD'(at-sign)6F'aKEQ(at-sign)6eh(at-sign)0RrZP-YS
rMT%(((Lj,T%(4%*ADA4SEh9dNfa[Fh16EfD6Cf9ZCA*KE'PdN!#XFRQ4r`9(at-sign),*%
%a,Y`E'&MCBkKMC%RJ!"dD'(at-sign)(at-sign)!c%-F'&bB(at-sign)aXC(at-sign)a[G'p`N!"6MQ(at-sign)6D(at-sign)k6Fh"KBf(at-sign)
6Ffq6G'KKG*0[EQ(at-sign)6EfD6G'KPNhD3!+abCA*dD(at-sign)0PFj0[CT2A8*%%dY+jDA16BA5
6G'KPNfpbD(at-sign)GTELb4!eBVB(at-sign)jNMU'0N5H!!(0eBjUXFQL(at-sign)!h)cG'KKG*0dD'(at-sign)6GTK
PBh4[FT2AGC1jE'PPNfPZNh4SCC0[N!"6MQ0dB(at-sign)kBG*0[CT0cF'&MCC0[F(#D8ij
[FfPdCC0dEj0dD'(at-sign)6F'&bB(at-sign)aXC(at-sign)a[G'p`Q'(at-sign)1SBf4*i!!ee#4!D('Z5k4"5VA+&H
Er`9(at-sign)CCB$`)aMB(at-sign)k6C'q6G'KTFj0RC(at-sign)jPFQPMB(at-sign)aXHCJZ+C%&+YG%C(at-sign)j[G'(at-sign)6G'K
PNf9NCf9cNfpQNpG3N39L8VPdD'&dNfePCA56G'KPNfpbD(at-sign)GTESkKMC%RJ!"LN!#
XFRQ4!qe,ehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZBH)fI!Fc-e61
1NlNZN39!bNGTGTUXFQ9ZPJ2Y5h4SCC--H'9NNh9ZDA56GTKPBh4[FT2AGENXN32
Yp'&ZC*0TG(16CQ&YD(at-sign)ajNfpQNfj[FQeKE*0`E'&ZCA-XMU'0N5H!!'aPG*B$kUK
eFj0dB(at-sign)ZDV(*PNh4SCC0MGA*fQ'(at-sign)6edD4"BaZZA4[Nf+3!&11CC0KNh"KG'L6B(at-sign)a
[EQH6G'KPNf9NCf9cNbKXD(at-sign)jPNh0PCfePETKdFbQ1SD'0N9k'"eX`ecZE!IrqH)f
I!Fc-e6'1PJ6!",PGecZ4"993Z9[AH)fI!Fc-e6'1NpFlQ,NSehL0R`(-c08aMTX
(DUbj+j%#UUMAH)fI!Fc-e6+1NlNTAGFlMC%*2rLjB(at-sign)jNMT%MEKjE+0GiMCm"c-c
9-BkBZ5Z4!UUSehL0R`(-c08bMT1j+GFlN3(rrVNSehL0R`(-c08aMTLj+jB#UUM
AH)fI!Fc-e6+1Q,NVNpGiMCm"c-c9-ik4"-!%Z5PGecZ1Ra8!!)f4*i!!Z(at-sign)PZPJ1
)mR4SBA56Eh*NCA)ZN38B6N'4!iMDF'aKEQ(at-sign)6F'&bB(at-sign)aXC(at-sign)b6G'q6eh(at-sign)0RrZP-YS
rMT%+T(at-sign)UjE(at-sign)9PG(16G'KPNh"KFQ&XE'9XEh4[F*!!8ijPNpG3N38UZ,PTCT0KEQ5
6EfjXHC0TCSkKMC%RJ!"TG*B$!dPYC(at-sign)9dFj0dD'(at-sign)6Bh9bGT!!V(*PNpG'N35P$lP
[ET0dD'(at-sign)6F'&bB(at-sign)aXC(at-sign)a[G'p`QP11CC0KG*0PH'&MG'ajNfpZCC0`Q'pTET!!V(*
d,T%%km"8D'9bC(at-sign)C[FQ8XN3-aMh4SCBkKMC%RJ!"YC(at-sign)&cGA*PPJ2AY'pQNh4SCC0
cCA56EfD6B(at-sign)aXNh"XB(at-sign)jPFj0`BA*KE'aPE*0dEj2AGBfIqk8bfMq1N3Vd,,PdD'&
dNfePCA56G'KPNh"KFQ&XE'9XEh4[F*!!8ijPNpG3MU'0N5H!!,PTFjB$%pj`FQp
`N!"6MQpbG'P[EQ&XNh4[Nh4SCC0XC(at-sign)jRG'L6EfD6G'KPNf0eFRDDV(*PNpG'N3'
KaVNZN36a4d&fQ'9bB(at-sign)GTEQH6EjKfQ'9bNf&XE*0eEQPdNhDBC(at-sign)0dEh*cNpGeMU'
0N5H!!,NSB(at-sign)jNQ`-8a(at-sign)KPEQ0P,*%$2ia[PDabGT0PFTKKE'bBCQ&YD(at-sign)aTCA1BEfD
BF'&bB(at-sign)aXC(at-sign)bBF'aKEQ9c+5b4!cq-Gj0PQ'0[EQ0XG(at-sign)4PQ(4SBA5BG'KPQ'ePBA0
eFQ(at-sign)1SBf4*i!!EfD(at-sign)!m+eG'KPNh0PG*0[CT0KE'b6F'aKEQ9cNfePCA4TEQH6BC0
`BA*KE'aPE'pdEh#3!&11CC0PFA9KE(16G'KPNfePB(at-sign)k6GfPNG'L6EfD6G'KPMU'
0N5H!!("KFQ&XE'9XEh4[F*!!8ijP,*%$Z44PH'0PF(5(at-sign)!kb[CQpbNf'6BfpZFh4
KETUXFR56CQ&MG'pbNhGSD(at-sign)1BD*0hQ'(at-sign)6GfPXE*0KCf&TET0cCA56G'q6BT!!8ij
PNfpZC5k1RaU!!)f4*i!!5(at-sign)k(at-sign)"&p`GQPPGj0[CT0dD'PcNh*PB(at-sign)aTHQ&dD(at-sign)pZ,*%
%I+*hQUabCC0MB(at-sign)k6D(at-sign)eYC(at-sign)4TBA4PE(Q6Ff9PNfK[Q(H6G'q6C'8-EQ(at-sign)6G'KPNfe
PB(at-sign)k1SBf4*i!!GfPNG'L(at-sign)!mCMEfD6B(at-sign)kDV(*jNf0XEh0PC*0MEfkBGTKPH*0cCA3
kN38Q[QPdNf9aG(at-sign)&XFj0dD'(at-sign)6E(at-sign)9KFh9bCC0[CT0dD'(at-sign)6Ff9dNfpQNf&XE*0`E'&
ZCA11SBf4*i!!G'KKG*B%KEpYC(at-sign)9dNh4SCC0MEfk9V(*fNf9iQ`5&[h0PG#k4"`S
N9'L6GA-XN35XK(H6CCKSBC0fNf(at-sign)BFfK[NhGZQ(4SBA5BG'KPQ'ePB(at-sign)kBGfPNG'L
BE(at-sign)'6HBkKMC%RJ!"LN!"6MQ(at-sign)(at-sign)!qUSCAKdC(at-sign)jNC(at-sign)56G'q6B(at-sign)aXNf0XEh0PC*0MEfk
9V(*fNf9iPJ2UU(0PG(16D(at-sign)k6Fh"KBf8ZMSkI(J!!MC)!qQpJ-6(at-sign)1MSb,!!!!%!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!KLLJ!U!SpBfJrBE
A#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j9j(r"9CPPJ4KVf&bCC0ZEjUXFRH6D(at-sign)k6BC0`N!"
6MQpcDA4TEfk6G'q6CfPfQ'(at-sign)6BC0`FQpLB(at-sign)*TE'PcG'PMNfPZQ(4PFR"bCA4KG'P
[ET0[CT0dD'(at-sign)6E(at-sign)9KESkN$S!!MC%RJ!"hD(at-sign)4dD*B$kUK[CT0KNf0[ET(at-sign)XFRD6CAL
4!qUSFf9d,SkT'S!!MC%RJ!"8NIm&9Q&VPDabCCX#mZadNhH6EjKMEfe`B(at-sign)0dQ'0
[ET0fNf9iQ(0PG(1Bed'BZ(at-sign)&ZC*MA3T%$MI+jD(at-sign)kBG'KbC(at-sign)(at-sign)BC'PYC(at-sign)jcD(at-sign)pZB(at-sign)b
B4A9ME'PNC(at-sign)&ZQ(0`B(at-sign)0P,)kKMC%RJ!"KEQ5(at-sign)!V+fFh9`F*T6MQpcCC0dD'&dNpG
"NlPTFj0MEfk3!+abG'&TEQ9NNfPZNpG#N3#E"VNZN363j8aPG*0eFj0LQ'9RD(at-sign)k
6BT!!V(*jNf+BC(at-sign)aKBTK[FQPZCj0dD'(at-sign)6Ef+3!+abGQP[GA-ZMU'0N5H!!&0eF(#
3!&11Eh0PPJ0$ER4SBA56GjUXFQ(at-sign)6G'&VQ'(at-sign)6BC0`N!"6MQpTETKdNf&dNh*KEQ4
[EC0LN!"6MQ9XEfjRD(at-sign)jRNh4[Nh4SCC0XBA*RCA+6Ff9dNpG#N3#E"VNZN38")PG
SBA51SBf4*i!!DA1(at-sign)"AE2G'KPNh"bEf*KBQPXDA5DV(*jNh4SBA56G'KPNh#3!&1
1EfPZQ(56FfKKE'b6BT!!8ijPE'pZCj0dEj0dD'(at-sign)6FfeKE'aPFT0cCA56ed'j2j%
*h9C8D'(at-sign)1SBf4*i!!B(at-sign)jcGjUXFQ9bPJ0i''PcNf0XC(at-sign)&b1T%%rjKcG(at-sign)1BD*0KNh"
bEf*KBQPXDA5BHC0PFA9KE(16G'KPNh*KG'P[NfpQNh4SCC0fQ'pXG(at-sign)ePNfpQNpG
"NlPLQ(Q6G'KPMU'0N5H!!(D3!+abEfaeE(at-sign)(at-sign)(at-sign)!qUSEfD6ed+4!*X'Z5k1TSf4*i!
!5(at-sign)jcG'9KC*B$#L4[CT0MN!#XFQK[QP11Eh0TEQH6BC0`Q'pTETUXFR56BA56FQ&
ZC'pY,*%$0`YXCA56GA16BjKSEj!!8ij[Ff(at-sign)6BC0cG(*KD(at-sign)GSQ(56E'PZCC0KG*0
bB(at-sign)jNEff1SBf4*i!!D(at-sign)k(at-sign)!q8'Fh"KBf8ZN38frd&cFh9YD(at-sign)jRNh4SBA56Fh9MQUa
bD*0KNh0dFQ&TCfLBG*0XD(at-sign)jPNfePCA4cNh4SCC0XBA*RCA+6Ff9dNpG#N3#E"VN
XN32Q*RGSBA56DA11SBf4*i!!G'KPPJ2UU("bEf*KBQPXDA5DV(*jNh4SBA56Fh9
MQ'L6BC0cG(*KD(at-sign)GSQ(56E'PZCC0hD(at-sign)aXNf&XFfq6E(at-sign)9PG*0dD'(at-sign)6FfeKE'aPFT0
cCA56ed'j2ikKMC%RJ!"ANIm&9Q(at-sign)E!U0hD''9V(*fNf(at-sign)BB(at-sign)abC(at-sign)&NHCKMEfe`GA4
PC*KdD'(at-sign)BB(at-sign)jcGj0PFTKdEjKdD'PcQ(&eCA0dD(at-sign)pZ,*%#j1GKE'+3!&11C(at-sign)PdQ'P
YF'aTBfPdE(Q4r`9(at-sign),T%%bp"6G(at-sign)16D)kKMC%RJ!"KPJ0D%("bEf*KBQPXDA5DV(*
jNf9aG(at-sign)&XFj0dD'(at-sign)6Fh9bCQ&MCC0KFQ9KNfpQNh4SCC0cCA56ed'j,*%$G[YNDAC
TC'9NNf+BHC0dD'(at-sign)6Fh9bCQ&MCC0KFQ9KMU'0N5H!!'pQPJ2UU(4SCC0cCA56ed+
4!*X'Z5k1TSf4*i!!(at-sign)C[r"9C[GCB%Xq"MB(at-sign)k6G'9XE*0hD'&dNfPcNf0[E(at-sign)PZCj0
ZCAKd,T%(P)KAQ'(at-sign)6EQqDV(*hNh4KDjKPNf'6FQ&ZC'pYNh"XB(at-sign)jPNfPZNh0`B(at-sign)0
P,SkKMC%RJ!""Fh0eE(at-sign)PZCjB$FdYdD'&dNh4SCC0`E'&ZCC0YC(at-sign)9dFj0dD'(at-sign)6E'&
bCf9bNh0PG*2A3T%!Q`Dj,*%$LbYhD'&dNfPcNh4SCC0`FQpLB(at-sign)*TE'PdN!#XFRQ
6G'KKG)kKMC%RJ!"TG*B&5bGhD(at-sign)aXNf&XFfq6E(at-sign)9PG*0dD'(at-sign)6FfeKE'aPFT0cCA5
6ed'j2j%*(at-sign)Pe8D'(at-sign)6B(at-sign)jcGjUXFQ9bNfPcNh4SCC0QEfaXEjKhD(at-sign)jR1T%(qGjcG(at-sign)1
BD*0KMU'0N5H!!("bEf*KBQPXDA5DV(*jPJ2UU'9aG(at-sign)&XFj0dD'(at-sign)6E(at-sign)9KET0hD(at-sign)4
dD*0[CT2A3ENXNf4TGQPNC(at-sign)56BTKjNh4SCC0YC(at-sign)&ZNhGTC(4SNfpQNpG#N3#E"VN
ZMUD0N5H!!%PZPJ11"d9eBfaTC'9KET2AEVNYFh"KBf8XN31JMRHDV(*PNfpLG'&
TET0LQ(Q6ECKeBjKSNh4SCC0cB(at-sign)ePNh*PBA0[EQPZCj0TETKdCA*`FQ9dBA4TEfj
cMU'0N5H!!'pQPJ,jBh4SCC0TETUXFR4bD(at-sign)jcD(at-sign)16GTK[E(9YCC2A&SfI!Fc-f'Z
1N38MNVNSed14!0bHZ5Q6EfD6BC0MEfe`B(at-sign)0dNf0[ETKfQ'9iNh0PG*2A3j%$eJ'
jBA16G'KPNdGbBA0cE(at-sign)&ZEQPKESkKMC%RJ!"YC(at-sign)&cGA*PPJ6'(fpQNh4SCC0cCA5
6EfD6B(at-sign)aXNfaTEQ9KFT0fNIpBj'&bD(at-sign)9dD(at-sign)9cNfpQNf4TE(at-sign)9ZFfP[ET2AETB$3"6
8!*2ADj%&,6bjG'KKG*B%aKpYC(at-sign)9dNh4SCBkKMC%RJ!"MEfk9V(*fNf9iPJ2UU(0
PG*2A3j%!h*kj,*0KEQ56BC0cD(at-sign)eTE'&bNh"bEf*KBQPXDA0dD(at-sign)16D(at-sign)k3!+abG'9
bF(*PG'&dD(at-sign)pZNfK[E'4c,SkQMC%RJ!"AD'&dPJ6b$'0[E(at-sign)9cNfjPH(3rQ`K2$&4
SCA*PNf&bCC0KG*0XC(at-sign)&cG*0dPDabGj0[PJ6b$(&eCA0dD(at-sign)pZFj0cG'PXE*0[F*!
!8ijPELkB4QPbFh3XN38cj(at-sign)&bCBkKMC%RJ!"dD'9bCCB$-fCKETUXFRQ6Eh4SCA+
6D(at-sign)kBGT(r(at-sign)14KFQPKETKdNfePBA0eFQ9cNf+3!&11CA0TC'9cNh4SCC0TETKdFQP
ZFfPMNhDBEfaeE(at-sign)9c,*%$(at-sign)!eKEQ56Ff9MEfjN,)kKMC%RJ!"SEjUXFRH(at-sign)"!-4Bf&
ZNh4SCC0NC3aZDA4TEfk6EfD6G'KPNfPZQ(4bD(at-sign)jcD(at-sign)16GTK[E(9YCA16BT!!8ij
PNf9iG'9ZC'9NNh4[Nfe[FQ(at-sign)6Cf9ZCA*KE)kKMC%RJ!"cG(at-sign)*cCA4cPJ04EfpQNpG
ZZ5ecF'&MCC0dD'&ZNf0[ET(at-sign)XFRD6CAL(at-sign)!e&[Ff9dFbk4"3A09'KPNf&ZFhH3!+a
bCA*cNh4[Nf+3!&11Eh4SNh4SCA0PNh&eCA0dD(at-sign)pZFj0KFQ(at-sign)1SBf4*i!!Bfa[Ff9
XHC%$kUKbC(at-sign)aKG'9N,SkQMC%RJ!"8D'(at-sign)(at-sign)"+E%B(at-sign)jcGjUXFQ9bNh4[Nh4SCC--FR0
dNh&eCA0dD(at-sign)pZNfPcNfjPCf&dDADBC5k4"fde9j(r"9CPNf&bCC0YDA0cD(at-sign)jRNfp
ZCC0YC(at-sign)&cGA*P,)kKMC%RJ!"KEQ5(at-sign)"$8SG'q6C'PcBfq9V(*fNf9bQ`3e+'Pd,*%
%4mPhNf(at-sign)BGfPXE*KPEQGKCf(at-sign)BCQpbQ''BE(at-sign)PZNh9dCCKTETKdD'(at-sign)BDfPZC*K[CTK
YBA4SC(at-sign)eKG'PMB(at-sign)b1SBf4*i!!FQ9KFfpZD(at-sign)jRPJ6%D(4SBA56F'LDV(*jFfPMDA0
dFj--EQ56G(at-sign)kBBTT6MQ9KFQ&LE(Q6F*KPC'&ZQUabG'PM,*%%qYGUGA0dNh4[Nh0
SEjKhNh"SQ(PcD(at-sign)0TFh4cMU'0N5H!!(4SBA5(at-sign)!qUSFh9MQUabD*0bC(at-sign)&cEfjTEQH
6C'q3!&11CA16F''BHC0[#bk1TSf4*i!!6'9dPJ,Q$(9cNf&cDj0[GA*cC(at-sign)afN!#
XFQ9cNh4SCC0aG(at-sign)9cG'P[EMU4",D5GfKKG*0TFj0dD'(at-sign)6GT(r(at-sign)14KE(9PNfpQNh4
SCC0cH(at-sign)eYCA4bD(at-sign)16CR9ZBh4TEfk1SBf4*i!!EfD(at-sign)",50Eh*NCA+6HQ9bEj0[CT0
KNh0PG*0[CT2AET1jGT(r(at-sign)14KFQPKBQaPFj2AH)fI!Fc-e6'1PJ6!"0FlQ`(rrRL
0R`(-c08bMT2A1ifB1TJkQ$U1N4(+R$ZBH)fI!Fc-f'k1N3(at-sign)S8,NXN36R"h0KN!#
XFRQ4",50ef(at-sign)0R`(-c08`MT1j+0GiMCm"c-c9-Bk6ecZBH)fI!Fc-e6+1NpFlMCJ
kQ$UB1Sk4%FUF1jKiMCm"c-cBESk4"DK3Z5NrN3H(at-sign)N!"*MU'0N5H!!(GTE'b(at-sign)!r#
hCfPfQUabCC0jQ'peNh4SCC0KER0hQ'9b,*%$mMYKEQ56GfPXE*0XC(at-sign)'BGTKPNfP
dNh4[NhQBEh(at-sign)6G'q6DR9cG'PQHC0dD'PcNf&ZFhHBCA+6B(at-sign)CdCA+1MTmH!!#0NJ$
kEf!a0Sk1M)X!!!!4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!#8Fk!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,PdD'(at-sign)(at-sign)!c0(at-sign)E'9MG(9bCC0
TFj0[PDabGT0PFLk4"2[&9'KPQ`-c9Q&ZFhH6CA+BDA1BG'KPQ'C[E'a[NhGTEQF
kN36G0pGPMCm"c-c9-)k4#"9(at-sign)Z6f(at-sign)!e95-CKTCTMAET-qNlN`,*B$(at-sign)!"dD'&dQ'P
c,*0TCTKdD'(at-sign)1T!k!!)f4*i!!Ff9dPJ2SG(at-sign)pQNhD4reMNBA*TB(at-sign)*XCA16ehL0R`(
-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ(L0R`(-c0K
ZMT%*N!$&Z(at-sign)PcPJ2SG(at-sign)j[ET0PEA"dN!#XFRQ4r`9(at-sign),*%$k19KEQ56ef(at-sign)0R`(-c08
`MT%)&9Dj2C%$99)`NfPQNh4SCC0cCA56EfD6GT(r(at-sign)14KFQPKBQaPFikKMC%RJ!"
TFj%$kUKPEA"dN!#XFRQ4r`9(at-sign),SkI'S!!MC%RJ!"ANIm&9Q(at-sign)(at-sign)"&e!BA*PNfaPC*0
dEj0LN!"6MQ9XD(at-sign)9fQUabCC0dD'&dNh4SCA*PNfeKQ(Q6CAKTFh56B(at-sign)k6D(at-sign)kBGT(
r(at-sign)14KFQPKETKdNfePBA0eFQ(at-sign)6D(at-sign)k6efkj,A0`B(at-sign)0PMU'0N5H!!'&cFfq3!&11BfP
KG'9NPJ2UU(GTG'L6G'KPNh0jE(at-sign)ePG(*TBj0QG(at-sign)jMG'P[ET0[CT0[FQ4PFT0kCA*
[,T%&11"ANIm&9Q(at-sign)6Ff9dMU'KMC)!d)[[eaD0R`(-c08`MT%%`!5j+0G$N3$FRVN
TPJ098Mf6-BkI&3!!MC%RJ!"TCTB%M-EA3j%&D(at-sign)5jDA16B(at-sign)kDV(*jNfj[ET0PEA"
dQ(Q6BfpYF'&MG*0MEfkBGTKPH*0cCA3XN35e6Q&ZC*0[CT0MEh9bFf(at-sign)6eaD0R`(
-c08`MT%%`!5j+03lZ5Q(at-sign)"'P%2C-`,T%((cT%Ej!!8ijPFikKMC%RJ!"cG(at-sign)13!+a
bD*B#m+KKNfePBA0eFQ(at-sign)6CAKTFh3rN36PLdPdNf4[N!"6MQ9cNfPZC'9PC*0PH'P
cG#bE!b+SB(at-sign)jNNh4SCC0QB(at-sign)0dNh4SBA56DA56CAKTFh4cNfPc,*KTET0YN!#XFRQ
1SBf4*i!!Eh"TEQP[ELb(at-sign)!qUSEfjPNfpQNh4SCC0YEh0dNh*PE(at-sign)&bDj(r(at-sign)14KBQa
PNf4TFf0[PDabGT0PFQPPFjX$kUKPGT0PFTKYB(at-sign)4PQ'PZQ'eKG'KPE(at-sign)&dD(at-sign)0c,Sk
I'S!!MC%RJ!"ANIm&9Q(at-sign)(at-sign)!i`mGfPXE*0`FQq9V(*fNf(at-sign)(at-sign)!i`mG'KKG*0cG(at-sign)1DV(*
SNf'6E(at-sign)9KFh9bCC0TFj0hQ'9XE*0NC3aZC(at-sign)56Efk6B(at-sign)kBHC0cCA56GfKTBjKSNfP
cNf'6$'jTG'(at-sign)1SBf4*i!!G(at-sign)jTEfk(at-sign)"-cfEfD6BfpYF'&MG*0MEfk9V(*fNf9iPJ6
-pR0PG(-ZN3IIbeH4r`9(at-sign)CC0NEj0dD'PcNf+DV(*jNf9YF'a[Q(PTEQH6BC0ME'&
cFfPMB(at-sign)b6C'9fD(at-sign)0PMU'0N5H!!'+3!&11Eh*bEj(at-sign)XFRH6C(at-sign)5(at-sign)"#EFCR*[EC0QG(at-sign)j
MG'P[EQ&XNf&ZB(at-sign)ajFfPc1T%&X8PTER0dC(at-sign)&NNfpQNf4P$'jTEQH6BC0YC(at-sign)&cGA*
P,*%%0HPhN!#XFQ(at-sign)6C'8-EQ(at-sign)1SBf4*i!!BCB$cL4XD(at-sign)jPBA+6CR9ZBh4TEfjKE*0
[ET0KE'b6FfPYF'aPNfCeEQ0dD(at-sign)pZFbbE!p2BG'KKG*0TFbbBEfk6B(at-sign)aXNh*PB(at-sign)b
6CR9ZBh4TEfjcNpGQN3&(rlNSeb'3!'ijZ5Q1SBf4*i!!C'8-EQ9NPJ5URQC[FT2
A)C%&#NR8-T%%R"$A8Sf3!"P+RrZP-YKZMT%+E$LjGfKTBjUXFQL6BA*PNfaTEQ9
KFT0MEffBBQPZBA4TEfjcNfpQNfPZC'PMBA4[FT0QG(at-sign)jMG'P[ER16EfD1SBf4*i!
!BfpYF'&MG*X&+M4MEfk9V(*fNf9iQ(0PG(-ZN3MhK8aPG*KeFjJ-FR0dQ'+3!&1
1C(at-sign)GTETKhDA4SQ(4SCCKMBA0PQ0GZPJ9e1lNpNc%XPJ9k&h4SBA5BDA-XNfaPG)k
KMC%RJ!$A)C%&kZ1jFQ&ZCf(at-sign)E"AbUEj(at-sign)XFRD6CA+BF*!!8ij[D(at-sign)k6G(1BEfkBG'K
PQ'aTEQ8ZN3RZj84P$'jPQ''BE'PZC(at-sign)&bQ'CeEQ0dD(at-sign)pZB(at-sign)bBeaq0R`(-c08aMT%
+2+kjEfkBFfPYF'aPMU'0N5H!!'CeEQ0dD(at-sign)pZFjB$kUKKFj0QEfaXEj!!V(*hFcU
1SD'0NJ#M*U(A(ifI!Fc-e6'1N36!",NSefDE!8IrZ5Q(at-sign)!e952Bf0MC1IpIrmUeL
1MSk4%FFEZ5MACTLj+0FKN!"Z1ENTPJ+UU03!NpGQQ,NSeb'3!'ijZ5XT+GFlMTm
9!!#0N5H!!,PhD'9bCCB$TXTdD'(at-sign)6Fh9YNh*KEQGPFj0[PDabGT0PFTB$TXTKE'b
6FQ9KE*0ZPDabG(at-sign)f6BT!!8ijPFR1(at-sign)!kE+eb'3!'ijZ5k4"5*"9'KPNfePB(at-sign)jTEQH
6EfD6G'KPNh"XGA16FfPRESkKMC%RJ!"TFjB$kUKLN!"6MQ9cG*0RE'9KEQ9NNfC
bEff6B(at-sign)k6CAKKEA"XC5k1SBf4*i!!6'9dPJ6N*pGQN3BX*VPLN!"6MQ(at-sign)6G'KPNfP
ZC'PMBA4[FT0QG(at-sign)jMG'P[ET0[CT0dD'(at-sign)6Bfa[Ff9NNh0PCfePET!!V(*dNe[AB6Z
4!IrqBVPG,T%)*9j8D'9ZNpGQN3&(rlNSeb'3!'ijZ5Q4!e5(e!#1SBf4*i!!efD
4!8IrZ5MA)C!!EMQj+bQ(at-sign)!fY[2C-`PJ2hTQC[FT0KE'b6eb'4"'AIZ(at-sign)9iBf9`G*2
A)C%$fDLj2C%$DfrABVNXN32kj(at-sign)+3!&11C(at-sign)0KGA0PNhHDV(*PNfKKQ(DBCC2ACTX
"4rqj+0GLZ5Q(at-sign)!fY[2C-aPJ2hTQ*eG*2ACTLj+0GLZ5XTPJ0VEcf6-#k1SBf4*i!
!9'L9V(*eFbb4!iT'Gj0PPJ0b,A0PCC0dD'&dNpFIMCm"c-c9-Bk4"-!%Z5MACT%
"4rqj+CB$99)pNc'(at-sign)!h)YD(at-sign)D6efD4",SXZ(at-sign)PcNh4SCC0TEQ4TBf&dEh+6CR9ZBh4
TEfk6EfD6B(at-sign)k6D(at-sign)k3!+abG'9bGT(r(at-sign)14KE*0Eef%lN3(rrQ+jA5k1RaU!!)f4*i!
!6QqDV(*hPJ2$TfaPG*0eFj0REj0[Q(DBCA+6G'q6efk6Z(at-sign)4TE(at-sign)9ZFfP[ER-XN32
,G("bEj!!8ijMC(at-sign)9ND(at-sign)jRNf+BHC0TEQ4eBh4TEfiZN38Vi%4[Nfj[G*0hQ'pbFRQ
4r`9(at-sign),)kKMC%RJ!"dD'PcPJ5#eAHDV(*[ELGdNh4KDjKPNfa[EQFZN3F"CP54r`9
(at-sign)B(at-sign)ZBCC0KNh0dFQ&TCfLBG*0XD(at-sign)jPNpG-NlPKEQ56CQpbNf9fQ'9bHC0`N!"6MQp
TETKdNpFKN36a$VPTET2A6*1jE'9dMU'0N5H!!0G)MCm"c-cB)Bk4#J!kZ(at-sign)+D8ij
PPJ2YZ(4SCC0SN!#XFRP`Q'9bF'aKEQ(at-sign)6G'KbEh9RD*0dD'(at-sign)6F*K[D(at-sign)k3!+abG*2
A)C%%(at-sign)r'jF*KPFR#BC(at-sign)jND(at-sign)0eE'&bNh4[Nh4SCC0XD(at-sign)jPNpG-Z5k4"8)25(at-sign)D6efD
1SBf4*i!!Z(at-sign)PcPJ5XCf'6FfPYF'aPNfCeEQ0dD(at-sign)pZNf4P$'jPC*0TET2AET1jFh"
KBf8XN36Fef&ZC*0TCT2A)C%&'U#jDA16BC0`N!"6MQpTETUXFR56Efk6G'KPNh0
dFQ&TCfLBG)kKMC%RJ!"XD(at-sign)jPPJ4ZZpG-Z5b4")qrE'9dNpGQMCm"c-cB)Bk4#S%
pZ(at-sign)+D8ijPNh4SCC0bCA0dFQPMG'P[ET0[CT2ACT%&YVUjG'q6G'KPNfL3!+abHA#
BCA*`E'&ZCC2A5)fI!Fc-f#'1N3B5JVNZN3E&'%4P$'jPNf'6E'PZC(at-sign)&bMU'0N5H
!!'CeEQ0dD(at-sign)pZB(at-sign)b(at-sign)!qUSeaq0R`(-c0KZMT%*N[LjBA16CQpXE'q3!+abGh-kMU'
KMC)!NTY2eaq0R`(-c0KZMT%&U&#j+0GQN3&(rlNTPJ098Mf0MBf6RrArr+YBMSk
1N42('GFIMCm"c-cBEYS!e6'1Qa#%c,NSefD0R`(-c0JKMT%'%S+j+CB#UUM8!*2
A(ifI!Fc-f'lD!08aMTLj+0GQMCm"c-cB)C!!5A[9+ik4$+lkZ5RA1ikI&3!!MC%
RJ!#jGfKPFQ(at-sign)(at-sign)![f[G'KPNh0eEC0bB(at-sign)jRCA16Ej(at-sign)XFRD6CA+(at-sign)![f[B(at-sign)aXNh#3!&1
1EfPZN!#XFR4cNpFKN30Vk,P[ET0dD'(at-sign)6E'PZCC2A6,NZN36TiP4SCA*PNfPcNfp
ZE(Q6BC--EQPdCC0cCA51SBf4*i!!EfD(at-sign)!a'6eb'0N!"Z1CrlT6,D-)k4!cabeh1
6Z(at-sign)C[FT0hD'PMN!#XFQL6G'KPNh0eE(at-sign)eKEQ56DA16EQpZHQ9bEbk4"2#%9fKPET2
ACT%%(at-sign)C+jDA16G'KPNfPZC'PMBA4[FT0QG(at-sign)jMG'P[ET0[CSkKMC%RJ!"KPJ1J(at-sign)(at-sign)j
[ET0PEA"dQUabHC0MEfe`B(at-sign)0dNf0[ETKfQ'9iNh0PG#b4!kmfG'KPET0KET0KFQG
eE(at-sign)9ZQ(56FfPYD(at-sign)aKFT0dEj0dD'(at-sign)6F(*PBf9ND(at-sign)jRMSkI(J!!MC)!qQpJ-6H1MSb
,!!!!%J!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!S5QJ!U!
SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#jFfK[QUabGh1(at-sign)!c52G'KKG*2A(ifI!Fc
-f'k1N3(at-sign)S8,NSefD4!8IrZ5Q(at-sign)!e952C-a,T%%r#e8D*KeFbb4!eMlGjKPPJ-dMfe
KQ(Q6C'8-EQ(at-sign)6BC0YC(at-sign)&cGA*PNpF(at-sign)MCm"c-c9-)k4"-!%Z5MA4lNTPJ098Mf6eaq
0R`(-c0KZMT%&U&#j+0GQN3&(rlNT,*%$(at-sign)2YhD'9bCBkN$S!!MC%RJ!$A4jB$ZR(at-sign)
jDA16B(at-sign)kDV(*jN`aZDA4PNh9ZD(at-sign)pZNfpQNf0[EA"KBh56BfpZQ(DBCAL6Ff9dFbb
4!m3CB(at-sign)jNNpGQN38#G,PTFj0dD'(at-sign)6D(at-sign)jND(at-sign)0KG'pbNfCeEQ0dD(at-sign)pZMU'0N5H!!'p
QPJ5QmR4SCC0cCA56edHj,T%(EEeANIm&9Q(at-sign)6D''9V(*fNf(at-sign)E"+EbG'L6GA1BF(*
[NhD6C(at-sign)5BG'KPQ'9iDA0dC(at-sign)jMCCK[CTKKQ'ePBA0eFQ(at-sign)BeaD0R`(-c08`MT%*C[D
jGfKTBj0SQ'PcMU'0N5H!!'4P$'jPC*B$1aT[ET0KE'b6$'jTG'(at-sign)6G(at-sign)jTEfjcNfp
QNf0[EA"KBh56BfpZPDabGT0PH*B$1aTcCA4c,*%$AMCKEQ56GfKTBjUXFQL6G'&
VQ'9cNh4SCC0fNIpBj'&XG(at-sign)(at-sign)1SBf4*i!!EfjPPJ1H*(at-sign)pZNf&XE*0ZEfk6C(at-sign)e`G*U
XFRQ6BfpYF'&MG*0MEfkBGTKPH*0cCA4c,T%&(ep8D'PcNfePBA0eFQ(at-sign)6D'&cNf'
6E'pZCj0SDA0dEh*j1SkKMC%RJ!"TG*B$kUKTFj0dD'(at-sign)64A9XCA+6Bj!!V(*SBA*
KBh4PFQPcG'PM,SkT'S!!MC%RJ!"1Ej(at-sign)XFRHE!lq4HC0[GCKKFQ(at-sign)BG'KTEQYTEQF
kN38M9'PQQ(4SDA1BDA1BG'KPQ%9eE'9bQ'16D'&bB(at-sign)0dCA*TFh4TBbb4!mJ[G'K
PETKTG*KTFjKeF*KdEjKjNfpeMU'0N5H!!(4[PJ8BdA0SEjUXFRH6G'KKG*0TG*0
MEfPZBfPNCA16GfPdD*0hD'&dNhHBCC0[FQ4TEQ&bD(at-sign)ajNf+3!&11C(at-sign)aTCADBCC0
dEj0LN!"6MQ(at-sign)6G'KPNd9eE'9bMU'0N5H!!'1DV(*SBA*KBh4PFQPcG'PM,T%&*#a
-CA5(at-sign)!kb-GA16BfpZBfaeC'(at-sign)6G'KTFj0XC(at-sign)0dGA*PNf+BHC0NCA*TGQPZCj0dD'(at-sign)
6CQpbECKeE''6EfD64A9XCA)YMU'0N5H!!&0MPDabD'arNISJ"'%0D5e3NfpTEQ0
KFT-6NITcNQ(at-sign)(at-sign)"9*#CQpbNh#D8ij[E(PSC(at-sign)4bB5k4#(at-sign)qZ3A16BC0YBA4dCA+6EfD
6CQ&MG#b4"D`SG'KTFj0QEh*YN!#XFR9XBC0MB(at-sign)k6BTKPMU'0N5H!!'9ZBf&`Fh9
XBA4PC*B$kUKTETUXFR4[Nf'6FfPYF'aPFT0QEh*YQ(9XB5b6EfjPNh4SBA56DA1
6C(at-sign)&cHC0dEj0bC(at-sign)ePECKLN!"6MQ9b,SkQMC%RJ!"-CA5(at-sign)"8H-ed14"L3UZ(at-sign)+3!&1
1CC0KNfj[ET0PEA"dQUabHC0MEfe`B(at-sign)0dNf0[ETKfQ'9iNh#98ij[E(PdEh#6CCB
&4ia[CT0ND(at-sign)ePER0TEfk6efkj,*%&RX9KEQ56E'9dMU'0N5H!!0GTER5j+0G$Q`$
FRVNTPJ-YIf+3!&11CC0dD'(at-sign)6D(at-sign)k3!+abG'9bD(at-sign)pbNfpQNpG$Q,NZN36jdP4SC(at-sign)k
6GjUXFQ(at-sign)6D''BGTKPNh4SCC0QEfaXEjKhD(at-sign)jRNfCeEQ4KE(at-sign)9ZQ(4KE*0QEh*YQ(9
XBBkKMC%RJ!"QEh+(at-sign)!qUSG'KPNd9eE'9bNf13!+abD'&bB(at-sign)0dCA*TFh4TBj0[CT2
AD(at-sign)jdZ5MA3j%!h*kj+6U1SD'0NJ#fTG[A&SfI!Fc-e6#1N36!",NSefPZG,NSed1
4!0bHZ5NTPJ098Mf6+03!Z6%TMCrl#jRBESk4"DK3ecU1U48!!)f4*i!!Z8PZC'9
PC#b(at-sign)!qUSD(at-sign)D6efD4"6+RZ(at-sign)PcNh4SCC0TEQ4TBf&dEh+6CR9ZBh4TEfk6EfD6G'K
PNh0PG*2AD(at-sign)jdZ5MA3j%!h*kj+5b6GjUXFQ(at-sign)6D''BGTKP1SkKSBf5!)AS%GF(at-sign)MCm
"c-c9-)k4"-!%Z5MAD(at-sign)jdZ5MA3j%!h*kj+5Q(at-sign)!e952Bf0MC1IpIrmUeL1MSk4%mF
Ceaq0R`(-c0KZfJ$9-BkE%)6-Z5MACSfI!Fc-f#'1N3B5JVNTPJ+UU03!NpFIMCm
"c-cBEYS!e6'1Q,NSefD0R`(-c0JKN!"*Hp8VMT%-V[Uj+GFlMUD0N5H!!,PhD'9
bCCB&K(YdD'(at-sign)6Fh9YNh*KEQGPFj0[PDabGT0PFTB&K(YKE'b6F*T6MQpTET!!V(*
dFj2A)C%&mV5jEfk6G'KPNfaTEQ(at-sign)6edb6Z(at-sign)&cNf&LQ'q9V(*fNf8ZN3S'(at-sign)8*eG*%
&K(YLNhQ1SBf4*i!!D(at-sign)jNG(at-sign)0dD(at-sign)pZ,*%$h&ehQUabCCB$f-YcC(at-sign)(at-sign)6G'KKG*0PGTK
PFRQ6G'9bEC0[ET0dD'(at-sign)6FQPRD*KdNfKKEQ56FfPNCC0PFA9KE(16HQ9bEbb4!pa
GCAKMCA"dMU'0N5H!!(GSC(at-sign)k(at-sign)"$K#eb'4"+ClZ(at-sign)PcNh4SCC--FR0dNh#3!&11EfP
ZQUabG*0[ET0dD'(at-sign)6E'PZCC2A6*1jCQpbNhGSD(at-sign)1BD*0dD'(at-sign)6D(at-sign)kBG'9bFf9MG'P
[ET2A3j%$["c8A*%#hhlA5)fI!Fc-f#'1N3T+a,PTFikKMC%RJ!"ZEh5(at-sign)!qUSC(at-sign)e
`G*UXFRQ4r`9(at-sign),T%&11"*CT2A)BfI!Fc-f'#1N3IMMVPTFj0cG(at-sign)1BD*0KN`abFh5
6F*!!8ij[D(at-sign)kBG#b6G'KPET0hQ'(at-sign)6D''BGTKPMU'KMC)!aijleaq0R`(-c0KZfJ$
9-Bk4%)6-Z5MACSfI!Fc-f#'0R`&T2pPJMSk4#(at-sign)9jZ5Q(at-sign)!e952C-`MUD0N5H!!'+
D8ijPBf&eFf(at-sign)(at-sign)!qUSG'KPNh#BEfPZN!#XFR56eb'0R`(-c0KJMT%(iikjDA16Efk
6G'KPNf+BEh9ZC'&bHC0[CT2A3j%!h*kj,*0KEQ51SD'0NJ#b`(at-sign)RA(ifI!Fc-f'l
D!08aMT%3K-bj+0GQMCm"c-cB)BfI!(at-sign)Nrf(at-sign)#1N31FFY8VMT%3!I'j+CB$99)pNbM
8!,Na+BfIq`ZCf'lD!08aMSkQMC%RJ!#jBTUXFRQ(at-sign)!i&%G'KPNfPZC(9MG'P[ET0
SQ(P`P911Eh4SCA0TFbb4!jCBBT0PBf&eFf(at-sign)(at-sign)!i&%efD0R`(-c0JKMCm"D6rCB)k
4!jabe5Z1N41$0EPTFj0dD'(at-sign)6D(at-sign)jND(at-sign)0KG'pbNfCeEQ0dD(at-sign)pZNfpQNh4SCC0cCA5
1SBf4*i!!efPZG,NSed14!0bHZ5Q(at-sign)!`qRe&b6edL0R`(-c0JKMCm"D6rCB)k4!ja
be5Z1N4!"mENXN35N&(GSD(at-sign)1DV(*SPJ4qrfPcNh4SCC0TETKdCA*TEh+6EfD6BC0
MEfkBGTKPH*0`N!"6MQpXH(at-sign)KPC(*[ET0[EQ(at-sign)6C'PYC(at-sign)jcD(at-sign)pZMU'0N5H!!'a[PDa
bGj0PFLk1SBf4*i!!8(9dG'PZCjB$kUKKE'b6G'KTFj0dEfGPG'KPFLb6Gj!!V(*
PNfpLG'&TESkKSBf45UcDeaD0R`(-c08`MT%%`!5j+0GTER5j+0G$N3$FRVNT+CB
$99)pMBf0NjrerrbV(at-sign))k1MT%6aaRA(ifI!Fc-f'lD!08aMTX3K-bj+0GQMCm"c-c
B)BfI!(at-sign)Nrf(at-sign)#1MT%*CAQj+CB#UUM8!*2A(ifI!Fc-f'lD!08aMTLj+0GQMCm"c-c
B)BfI!(at-sign)Nrf(at-sign)#1N31FFY8VMT%3!I'j+CB$99)pNp3!Z5M8!,Na+BfIq`ZCf'lD!08
aMT%6fKkj2C-Se!#j-5Q0RrX,QGKZMT%&U&$A1ikQMC%RJ!#jBA14!qUSC'9cDA*
PC#k1MTmH!!#0NJ$kEf!a1)k1M)X!!!!6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!#ZUU!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,PANIm
&9Q(at-sign)(at-sign)"(E[BA*PNfj[QUabGj0TET0KNh#3!&11Eh0TG'P[ET0dEj0cG'&dCC0dD'(at-sign)
6CQ&YEh9cNd9eE'9bNfC[FQfBG(at-sign)aKNfC[FT0`N!"6MQpXH(at-sign)KPC(*K,SkN$S!!MC%
RJ!"AD'&dPJ3GP'PcNf'6F*96MQpXH(at-sign)KPC(*[EMq4"G'N3C%%(BG`NfpXH(at-sign)KPC(*
[ETB%(C4TFj0KN`aZDA4PNh9ZD(at-sign)pZNfpQNf0[ET(at-sign)XFRD6CAL4""f8F*!!8ij[E(P
SC(at-sign)4bB5k1SBf4*i!!4fPfQUabC(at-sign)k(at-sign)"8QQBC0`N!"6MQpXH(at-sign)KPC(*[ELb4"D&QGjK
PNffBGA0dNf4P$'jPNf'6FhPcG'9YNfpQNfCKBf9cNbK[CT0KE'b6C'PYC(at-sign)jcD(at-sign)p
ZFbb1SBf4*i!!FQ&ZCfPZCjB%%S"QFQpYNf4TE(at-sign)9ZFfP[ET-`NbKKNh#3!&11EfP
ZQUabG#Q6G'q6C'PYC(at-sign)jcD(at-sign)pZNpGZZ5NZN3(at-sign)`CeH4r`9(at-sign)CC0hD(at-sign)aXNh0KQ(Q6G'K
KG*0KNh0PG)kKMC%RJ!$64TB%efbjEfD6BfpZPDabGT0PH*B%efa`QP11EfajD'9
NFQ'6DA16BC0cHA0dC(at-sign)f6EfD6CQ&MCA16CQpbNf&ZNf&bBQPdFQ&bHC0`Q'pXH(at-sign)K
PC(*[ET2A5ikKMC%RJ!#jGfKPETB$qrpdD'(at-sign)6C(at-sign)aPE(at-sign)9ZN!#XFR4cNfpQNp0'Z5b
E"!"8Bf&XE'9NNfCKBf9c,*KKFQ(at-sign)6EQpZNf9YF(5DV(*jNf0[EA"KBh56BfpZQ(D
BCAL6Ff9dFj2A4SkKMC%RJ!#jGfPdD*B$kUKNDA0UEfPZQUabG*0TETKdCA*TEh*
cNh0eBjKSNh4SBA51SD'0NJ$!KZ6A5j%%-I#j2Bf0MC%'pdfIpIrmUeZ1MTm-*pD
0N3098YK'N3%Z1YSbmc%bad$*!!J!!!!)!!!!"(at-sign)0YBRJih%D1MT%AYERAD(at-sign)jdZ5M
A4T%"SFDj+GFkMUNGIR+0N5H!!,P$BA9dD(at-sign)pZ1T%&m#KdD'(at-sign)(at-sign)"%C-D(at-sign)kDV(*dCA*
TEh+6EfD6BC0QB(at-sign)0PNfpQNf4TE(at-sign)9ZFfP[ET2ADj%%V(at-sign)QjDA16G'q6BT!!8ijPNh4
KDjKPET0bC(at-sign)aKG'PfQ'(at-sign)6G'q6G'KPMU'0N5H!!'aTEQ9KFTB$ECjcF'&MCC0[CT0
ND(at-sign)ePER0TEfk6efZ4!p5lZA4SBA56BfpZQUabG'&TER16G'KPNfCKBf8XN31'S'&
ZC*0dD'(at-sign)6D(at-sign)kBG'9bD(at-sign)pbNfpQNf'6F*!!8ij[D(at-sign)kBG)kKMC%RJ!"TFjB$kUKKNh#
3!&11EfPZN!#XFR3ZMTmDJ!#0N5H!!&9ZC'9bPJ58rh4SCA0PNf0[EQ4TG'P[ER1
6GjUXFQ(at-sign)6E(at-sign)'BHC0dB(at-sign)ZBCC0dD'(at-sign)64A9XCA+6BjKSBA*KBh4PFQPcG'PMNfpQNf+
3!&11Eh4SNh0TC'9c,)kKMC%RJ!"KEQ5(at-sign)!aNeGA0TEQH6G'KPNfCKBh56G'KKG*0
KETUXFRQ6G*KhQ'q6D(at-sign)kBG'9bD(at-sign)pbFj0[CT0QB(at-sign)0PFj0KFQ(at-sign)6C'PcDQpTETKdNhH
BCC0[BR4KD(at-sign)k6+(9cD(at-sign)jRMU'0N5H!!(4SCCB$j*!!CQ&MG*0dD'&dNh4SCC0YC(at-sign)&
cGA*PNfpQNh4SCC0NDA0UEfPZN!#XFR56G(at-sign)jTEfk6EfD6BC0QB(at-sign)eTE(Q6EfD6Ff9
dFj0PFA9KE(16G'KPMU'0N5H!!(0eECB$kUK[CT0dD'(at-sign)6E(at-sign)9KFh9bCA16EfD6G'K
PNfPZC'PfD(at-sign)4eB(at-sign)b6Ff9dFbNkMU'KMC&Rl&rA&SfI!Fc-e6#1Q`6!",NSedZ4!0b
HZ5Q(at-sign)!e952Bf0MC%&6++IpIrmUeL1MTm-*pD0NpK'N3%Z1YSbh%D1MT%AYERA&Sf
I!Fc-e6#1Q,NSefPZG,NSedD4!D('Z5NTNcf6efD0R`(-c08`MTB(DUc8!*X#UUM
ACSfI!Fc-e6'1NlNVQ0GQMCm"c-c9-Sk6e!#0Q0FkPJ(rrMU61Sk4%ar`Z5Z0Q0F
kPJ(rrMU61Sk4%R9'1ikQMC%RJ!#jGfKPFQ(at-sign)(at-sign)!qUSefD0R`(-c0KTMT%(6i+jCA&
eB(at-sign)acNh4SCC0ZPDabG(at-sign)f6BT!!8ijPFTB$kUK[CT0QB(at-sign)0PFj0[CT0ND(at-sign)ePER0TEfk
6efQj,T%&11"8D'PcNfPcNd9eE'9b*h16CQpbEC!!V(*eE'%ZMTmDJ!#0N5H!!&H
4r`9(at-sign)CCB%bL&MB(at-sign)k6EQqDV(*hNf&ZFhHBCA+6G'KPNh0PBfpZC*0[CT0dD'(at-sign)6FA9
PFh4TEfjcNhHBCC0SB(at-sign)56E'9QG*0[F*!!8ijPEMU4"[I6D'qBGj0dEikKMC%RJ!"
PH(4PEQ5(at-sign)!d4*G'KPNf4P$'jTG'P[ET0[CT0dD'(at-sign)6D(at-sign)kDV(*dFQPZFfPMNhDBEfa
eE(at-sign)9cNfCbEff6BfpYF'&MG*0MEfkBGTKPH*0cCA4cNh4[Nf&XE)kKMC%RJ!!-EQP
dCCB%,!eeEQP[ER16EfD6BfpYF'&MG*0MEfk9V(*fNf9iPJ3X$A0PG(-ZN3Ap$NP
QNpG(NlPTFj0cG(at-sign)13!+abD*0KN`aZDA4PNh9ZD(at-sign)pZNfpQNf0[EA"KBh51SBf4*i!
!BfpZPDabGT0PH*B$kUKcCA4c,*0dD'9ZNhH3!+abCC0cCA51SD'0NJ#DS,(A&Sf
I!Fc-f'Z1N38MNVNSedHj+CB$99)pMC1ImH1(UeU1N3p98GF(at-sign)MCm"c-c9-)k4"-!
%Z5MA4jB#UUM8A*2A)C9Z1ENTef39MCrl#jQ0f'k1R`IVfSeZfJ$BDik1N4$S(at-sign)VN
Seb'6Z5RA1ikI&r(1MC%RJ!#jGfKPFQ(at-sign)(at-sign)"%R8eb'4",J0ZA*KEQGPFj0[PDabGT0
PFTB%5G4KE'b6E'PZC(at-sign)&bNhD4reMNBA*TCA4TCA16EfD6C'PYC(at-sign)jcD(at-sign)pZNpGZPJ,
VG03!NpGVN35`mEPTET%%5G6AEVNYFh"KBf8ZN3C(at-sign)Be4SCBkKMC%RJ!"XC(at-sign)CdPJ9
"ZQKKEQ56FfPNCC0NC3aZCA16BC0YC(at-sign)&cGA*P,*%&PhjKEQ56GfKPET2A4j1jDA1
6BC0MEfe`B(at-sign)0dNf0[ET(at-sign)XFRD6CAL(at-sign)"8'kFf9dNfPdMU'0N5H!!'&RFQ9PFjB%1L9
hDA4SNh4SCC0NC3aZDA4TEfk6GjUXFQ(at-sign)6D''BGTKPNf&XFQ9KC(Q6CfPfQ'9Z,T%
'*eK*G*0TFj0dD'9bC(at-sign)C[FQ(at-sign)6G'KPNf4PFfPbC(at-sign)51SBf4*i!!CAKdC(at-sign)jcD(at-sign)pZ,T%
&11"8D'(at-sign)(at-sign)!qUS4A9XCA+6BjUXFQKKFQ&MG'9bDA0dD(at-sign)16C'q3!&11CA16B(at-sign)aXNh4
SCC0hQ'pbDj0QEh+6GA-ZMTmDJ!#0N5H!!&H4r`9(at-sign)CCB$EA*KFQ(at-sign)6EQqDV(*hNfP
ZNf'6F*!!8ij[FfPdD(at-sign)pZNh4[Nh0dBA4PNh4SCC0YB(at-sign)PZNh4SC(at-sign)pbC(at-sign)f6EfD6Cf9
[E(at-sign)9dFQPMNh"bEf*KBQPXDA5BHC(r"9BZMU'0N5H!!&H4r`9(at-sign)CCB$F`YhD(at-sign)aXNh0
KQUabHC0dD'&dNf&ZNfPZQ(D4reMNBA*TB(at-sign)kBG*0YC(at-sign)&cGA*PNpF(at-sign)NlP[ET0&G(at-sign)0
XD(at-sign)4PB(at-sign)k6efkj,A0`B(at-sign)0P,*%$L[GNC3aZC(at-sign)56Efk6B(at-sign)aXMU'0N5H!!!aZDA4PPJ2
UU(9ZD(at-sign)pZFj0[CT0MEfe`B(at-sign)0dNf0[ET(at-sign)XFRD6CAL(at-sign)!qUSFf9dFbb6DA16BfpZPDa
bG'PZNh9[GA-XN32UU(GSC(at-sign)k1SD'0MBf5!,f3!$pXD(at-sign)f1R`FL)Bf5!,L5`0K$MCm
"!!$CESk4"4ihfL(B3ik1NJ$8hhIA&VNSed10R`(-c0KZMT%&U&#j+CB$99)pNpF
(at-sign)Z5MA3j%!h*kj+Bk1Rai!!)f5!2T[B$%jMSk-L`!!!"3!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,T1S!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if
4*i!!Z(at-sign)C[FTB#k54KE'b6Ff9aG(at-sign)9ZBf9cNpG$MCm"c-cBESk4#*&dZ(at-sign)pQNf0[EA"
KBh56BfpZPDabGT0PH*B#k54cCA4cNf0[ET(at-sign)XFRD6CA*RD(at-sign)jRPJ,T*(4[Nh4SCC0
MEfe`B(at-sign)0dNf0[ET(at-sign)XFRD6CAL1T!k!!)f4*i!!Ff9dN32UU0G$N3$FRVNZMU'0N5H
!!&H4r`9(at-sign)CCX$kUKSBC(at-sign)XFRD6CCKdD'(at-sign)1T"U!!)f4*i!!ddeKD(at-sign)k(at-sign)")!!9'KPEh*
PEC0[CT0(C(at-sign)pYCA4bD(at-sign)168(*[BQ&LD(at-sign)aTG*!!S!"jMU'0N5H!!,P8D'(at-sign)E"DAeefk
(at-sign)!pKkZ5Z6-CKTET(at-sign)XFR4bD(at-sign)jcD(at-sign)1BGT0[E(9YCA1BeaD0R`(-c08`MTB%`!6A1jX
"rri(at-sign)MCm"c-c9-Bk6ecZ0Q$UB1TJkMT%4bT`lQ"D0R`(-c0KZMT%,6N(at-sign)jBA*PPJ(at-sign)
Pp(at-sign)'6BQ&cDA16EfD6G'KPNh0`B(at-sign)0PNfpQNf&XE)kN$S!!MC%RJ!"MEfk9V(*dD(at-sign)k
6G(at-sign)peFjX$KdGTET0fNIpBj'&bD(at-sign)&ZNh5BE(at-sign)9KFh9bCA1BC'8-EQ9NQ'pZQ'&XE*J
-EQPdCCKeEQP[ER1BEfDBBfpYF'&MG*KMEfk6GT0PH)kKMC%RJ!"cCA4c,SkT'S!
!MC%RJ!"8D'(at-sign)(at-sign)!fH$$(*cG*0`FQqD8ij[CT0[CT0dD'PcNh4SC(at-sign)pbC(at-sign)f6DA16C(9
PNh4[NdKKC(GTCf9b1j%$NcTdD'(at-sign)6$(*cG*0PE'9YC(at-sign)k3!+abG'&bHC0`FQqBEfD
1SBf4*i!!GjUXFQ&cPJ2UU("eBQaTFfKPC*0XBA0dNhQBC(at-sign)&bNf+BHC0%B(at-sign)k65fa
KD(at-sign)k6EfD64f9[FQGTBC08NIm&9Q9MQ'JZMUD0N5H!!%PZPJ1[`'0XEh0TEQFXN31
lL'aPG*0YCC0dFRQ6G'q6B(at-sign)jcGjUXFQ9bNh4SCC0aG(at-sign)9cG'P[ET0jQ'peNf&bCC0
KBT!!8ij[GA56G'q6BA0V1T%&'fahD'&dNfKKFikKMC%RJ!"dD'PcPJ2UU'G[G*0
dEj0NEj0hDA4SNfGPEfePG(*TBj0`FQpLB(at-sign)*TE'PdQUabHC(r"9BXNf&ZQ(PhQ''
BH6q1TSf4*i!!5C%%K8GhD(at-sign)aXPJ5&Ef&dG'9YF(56BC0cDj(at-sign)XFQ9dBj0SNhQE")9
[B(at-sign)jcGj0PFLk4"`Nd3fpZFfPNCA+BG*0hNfqBBfpYF'&MG*KMEfk6GT0PH*KcCA4
cQ0G"Q,PKEQ51SBf4*i!!ed+4!*X'Z5k4"ZG19j(r"9CPPJ4k)QPYB(at-sign)GTEQ(at-sign)6ed+
4"48SZA4[Nf+3!&11CC--H'9NNfPZNpGZZ5ecF'&MC5b4"*i!B(at-sign)jNNh4SBA56Gj!
!V(*PNb*NFQp`)T0dD'(at-sign)6FQPRD(at-sign)56Ff9dMU'0N5H!!0G"PJ8E*lPKG*0bB(at-sign)jNEfd
ZQ`M+APGSBA56DA16G'KPNh"bEf*KBQPXDA53!+abHC0dD'&dNpG"NlPYC(at-sign)9dFj2
A3T%!Q`Dj2jKANIm&9Q(at-sign)6B(at-sign)jcGj!!V(*PFT0dD'PcMU'0N5H!!(&eCA0dD(at-sign)pZPJ0
9pfPZNh4SFQ9PNh0dCA"c,T%&"e"'DA*cG#b4!h1dGjUXFQ(at-sign)6FQ9KE'PkCC0dD'&
dNf+BHC0VQ'9PF'PZCj2A3T%$m2fj$(KPC*0KEQ56GT(r(at-sign)14KFRPTEQH1SBf4*i!
!ed'(at-sign)"6NLZ(at-sign)+DV(*jNh4SCC0RFQpeF*0[CT0&G(at-sign)0XD(at-sign)4PB(at-sign)k66(at-sign)pdD(at-sign)pZFbb4"Bc
"GjKPNf4P$'jPNf&ZNfPZQ(D4reMNBA*TB(at-sign)kBG*0YC(at-sign)&cGA*PNfpZMU'0N5H!!'0
[ET(at-sign)XFRD6CAL(at-sign)!f"aFf9dFj2A3T%!Q`Dj,T%&#Xj6C(at-sign)0[EQ3XQ`0m&RH3!+abCC0
KF("XHC0)B(at-sign)4hD(at-sign)GPFLGcNh4SC(at-sign)pbC(at-sign)dXQ'&ZC*0TEQCPFT0dD'&dNh0eBj!!V(*
SNf&ZMU'0N5H!!'PZPDabGT(r(at-sign)14KFQPKET0dPJ168'ePBA0eFQ(at-sign)6CA&eB(at-sign)acNf'
6E'PZC(at-sign)&bNf0[ECUXFQ*TEQ&dD(at-sign)pZNfpQNh4SCC2AETB"q%'j+j-aPJ168'PZQ(4
bD(at-sign)jcD(at-sign)16GTK[E(9YCA-XMU'0N5H!!(GTG'L(at-sign)"%T4BfqD8ijP$Q0TC(at-sign)k3!+abG(1
6C'9`Q'9ZC'PZCj0[ET2A3C1jB(at-sign)jNNfj[G*0[ET2A3T%!Q`Dj,T%'9pY8D'PbC#b
4"')lGj!!V(*PNf4PG'9bE(at-sign)PZCC0dD'9cCBkKMC%RJ!"MEj!!8ijP$Q0TC(at-sign)k9V(*
dFjX&%KKLNhQBG'&VD(at-sign)jRQ(0eDA4KBQaPQ0G#N3#E"VNRFbk4#+m[9'KPQ'9ZC*K
bCA0eE(5BDA1BB(at-sign)kBD(at-sign)4PET0dDA56HCKhD'PMNfLBDA11SBf4*i!!Dfj[QUabGfk
(at-sign)!emeBA16G'KPNfYTEQ9YBA4TBj0QEh*YQ(9XB5b4!hXBGfKTBjKSNfKKFj0LN!"
6MQ9PET0dD'(at-sign)6Ef+4!+FFDQ9MG*0[CT0YQ(9MQ'L6FQ9cC(at-sign)&bBjKSMU'0N5H!!'P
ZPJ2UU(4SDA16Bf9ZQUabG(9bHC(r"9BXNh0dD(at-sign)aXNfG[D(at-sign)jRNfpZNh4[N!"6MQ4
KQ(Q4r`9(at-sign),SkQMC%RJ!"8D'&ZDjB$kUKjQUabEh(at-sign)6CQpbNhQBEh9bNf&dG'9ZQ(4
TEfiZMUD0N5H!!00#D(at-sign)*XD(at-sign)pRFQ&`D*!!S!"jMUD0N5H!!,P%B(at-sign)jTC(at-sign)b(at-sign)!`Cq35k
65faKD(at-sign)k6B(at-sign)jNNdGTB(at-sign)iY3f&bE'q68QpdB5b4!c3J5(at-sign)kDV(*dFQq3!&11C(9MG'P
[ET0dEj0(C(at-sign)pYCA4bD(at-sign)168(*[BQ&LD(at-sign)aTG*KjMU'0N5H!!#K-CATTEfjTPJ2UU%a
TEQ0PC5NXNd0KECUXFQ*bD(at-sign)4RCC09EQPfQ'9bFfPdQ(Q68(*PFh-XNc%j16H1MTm
H!!#0NJ$kEf!b-)k1M)X!!!!9!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!$&XD!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N(at-sign)`cM00*6PD4rS!!39*
*38j8PJ5!!&4)48p5NIlJ!&NXNdp-4*0"6N566N9AMU31J!#0NJ$FS1fjBT!!8ij
PD(at-sign)jRMU'0NJ#BGXa8D'(at-sign)(at-sign)!qUSFf9MEfjNNd0[E'a[N!"6MR&eDA9YNdaPBh4eFQ(at-sign)
1U4U!!)f4-HreC'9XDADDV(*PFQ9NPJ2UU'&dNh4SCC0"EQkBG(at-sign)&XNdePCA4TEQH
6EfD6G'KPNd&YCA*TBf&ZNdeKG'KPE(at-sign)&dD(at-sign)0KE*06Ej!!8ijMD(at-sign)9dQ(Q1SBf5!+-
'$N*KE(4TE(at-sign)pbC5b(at-sign)!qUS5Q&ZN!#XFR9KFRQ61#b6-6Nj0bk1TSf5!,qICNGTB(at-sign)i
Y3f&bE'q4!qUS8QpdBBkKMC)!SHjU4'9`BA*dE(at-sign)9ZN!#XFR5(at-sign)!qUSEfD66(at-sign)&dD'9
YBA4TBh11SBf5!,j(D%e*9#b(at-sign)!qUSFQq3!&11Eff6-Ldc06'1SBf5!+P#I$FhQ`2
UU%eKFh0KBj(at-sign)XFQL6GA0PG(4cQ%&fNf9ZNh9PMU'0NJ#NITC$B(at-sign)f3!+abBR*TC'G
PPJ2UU%e"Nc!b-6-j,63c-$H1Rc1!!)f4*i!!5(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)"%E
&G'KPEh*jNfPcNh4SCC0RFQ9KG*05EfeKETUXFR4TBj0cG'pbHC0[CT0YBA4SC(at-sign)e
KG'PMFbk4"Ndh4T(r"9C[FT0[EQ(at-sign)6D*KeELf1SBf4*i!!C(*PC*B&J5CKEQ56$'C
dQUabHC0jQ'9KFR-XN3AQaQCbEff6DA4cNf+D8ijPCfPZEQPZCh16GfPdD*0#EjK
[E'(at-sign)6G'q6G'KPNh4TE(at-sign)8XN3AQaQ&bEh9ZC)kKMC%RJ!"dD'(at-sign)(at-sign)"8YpE(at-sign)PNC'aPNfp
QNh4SDA16Bf9ZQUabG(9bHC(r"9BXN3(at-sign)MXRGSC(at-sign)k6DA56BR*KEQ1BD'9NNfm,NfP
ZQ(4[Nh0PGTKPFQ&XNfPZC'9`N!"6MQ9ZC'9ZQ(51SBf4*i!!C'PcBfP`E'PZCA-
XN36kefeKG'KPE(at-sign)&dD(at-sign)0TB(at-sign)jcPJ6%D'pQNf&XE*0MEh9ZQUabG(*TCA16GjKPFQ(at-sign)
6BR*[G(at-sign)GSQ(56G'pRCA4SCA+6BTKjNh4SC(at-sign)PbMU'0N5H!!'0[E(at-sign)e[ETB%AiTQB(at-sign)P
dD*0TET0TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G(-kN3BLT'PZN34ILN9ZCfaKEQ3XQ`4m`N0
KNhPXCAQ4r`9(at-sign),*K0B(at-sign)00B(at-sign)K[ELbB8hPXGT0PFh4PFT%%AiTKEQ51SBf4*i!!8f&
XE(at-sign)pZ,*B%&*&KEQ5E"!`[E'&dCA)XNd&XCR*PC*KCNIm&9QpeEQFXNd&TG'ZDV(*
PELb66'PdG'aPGjK[P911Ej0NPJ3-,f&ZC*08NIm&9R9bETKLG(at-sign)aX,T%&RAC*ET0
(CA)YMU'0N5H!!'eKET(at-sign)XFRQ4r`9(at-sign),*X$XE0$E'9LFf16D#bB4fpbC'&Z,*K(FQ&
cFfeKEQiXQ&0[F'L6GA1(at-sign)!k0f6'PP,*K6G(9NH6Z4!lXaD(at-sign)k64T(r"9CbB(at-sign)jMC5b
B5'9bE(at-sign)PdC5b1SBf4*i!!5QpbC'&ZPJ5HV'&ZC*0-B(at-sign)GeCA*bC6Z4"2LZD(at-sign)k65A4
KE(QEr`9(at-sign),*B%bke$BA#3!&11C(at-sign)aXD5b63R*TEh0MN!#XFQKT,*08Q(*eC'Q(at-sign)"*k
XB(at-sign)jNNd0[FR*KC'q68f9RFQ8XMU'0N5H!!'PZQ`1K%N&YCA*TBf%XPJ1[b8GXC(at-sign)j
Z,*0%D(at-sign)13!+abDh0[ELb63f&bGA1B+'pQQ(4SCCK$BA*eFjK0Efj[Ch*KF'Kc+5b
64A*TBjK8NIm&9Q9YF'aPMU'0N5H!!%*PE'b(at-sign)"$jdB(at-sign)jNNfaKG'9bNdKPFQeKEQk
69j(r"9CPH(at-sign)`ZN3Bd490PE'4[EC0TET0SDA0dEh*jNfKKFj0KET0TET!!V(*dCA*
ZBA4TEfjKE*0MEfdYMU'0N5H!!'f9V(*eEQPdNhQ(at-sign)"(P9EfD6Ff1DV(*SEfaKFR1
6CQ9XG*0cEj0eEQPdC(at-sign)56BTKjNf'6BfpYE(at-sign)pZNh0MD(at-sign)9ZQ(4T$'16D(at-sign)4PB(at-sign)b6CQp
bNh0[Nfa[EQH6BBkKMC%RJ!"cG(*PG'1DV(*SPJ6MRfpQNh4TE(at-sign)8ZN3JMa%PZNfp
eFT0MC(at-sign)kBG(9bHC[r"9BXN38Kh%aTCC0dD'9[FRQ6B(at-sign)jNNf&XCf9LFQ&TBj0RC(at-sign)p
YCA4bHCJXN38Kh'4T#f9b,BkKMC%RJ!"PETUXFR4TB(at-sign)b(at-sign)!p'PB(at-sign)aRC(at-sign)*bBC0KEQ5
6B(at-sign)aRC(at-sign)*bB(at-sign)PMNf0[ECKLD(at-sign)jKG'pbD(at-sign)0cNf&bCC0[#h0`FQPZCh16EfD6D(at-sign)kBGT(
r(at-sign)14KFQPKETKdNh4SC(at-sign)pbHC(r"9BZMU'0N5H!!%j[PJ(at-sign)4CQpdD'9bNfeKG'KPE(at-sign)&
dD(at-sign)0KE*0dD'9[FRQ4r`9(at-sign),*%&qa9hDA4SNh4SCC0PH'0PF(4TEfk6EfD6G'KPNh4
SC(at-sign)pbHC0[CT0QG(at-sign)jM,BkKMC%RJ!"dD(at-sign)pZFjB$m54[CT0KNf0[EA"XCAL6GT(r(at-sign)14
KFQPKBQaP,*%$mX0SBA16D'&NNf&cNf4PCA#6B(at-sign)jNNfaKFh4TEQH6B(at-sign)k6D(at-sign)i0G(at-sign)9
ZBf(at-sign)6Efk6G'KPMU'0N5H!!'4PGT(at-sign)XFQ9XEh"YC(at-sign)k6G*B$kUK[CT0YBA4SC(at-sign)eKG'P
MFbk1TSf4*i!!4AD9V(*PET0dG(at-sign)&XE(Q4r`9(at-sign),*%%*4CTET0fNIpBj'&bD(at-sign)&ZNh5
(at-sign)""PQG'KPEh*jNhHDV(*KFj0dEj0LN!"6MQ9MEfePNf'6GQPMG'PYNfpQNfPdFj0
[Q(GZNh0eBf0PFh-kN3(at-sign)(at-sign)A(4SCBkKMC%RJ!"fQUabCA*jPJ2MhR4PFQf6)QPZQ(D
4reMNBA*TB(at-sign)kBG*0dD'9[FRNLNfPcNfj[Q(HBB(at-sign)4KQ(PcNh9ZC'9bFh4[P911Ej0
NPJ2MhQPZNh0eBjKSNf'6GfPNCC0fNIpBj'&bD(at-sign)9dQ(Q1SBf4*i!!EfD(at-sign)"(TGFf9
ZFf9cNh4SBA56DA56D'&cNf+3!&11C(at-sign)0[E(at-sign)(at-sign)6B(at-sign)aXNf*eG*0YC(at-sign)&ZD(at-sign)jRE'9cFbk
4"ZJ!5A56DA16EQq6GjUXFQpZC'9bNh4SBA56HCK[GBkKMC%RJ!"KFQ(at-sign)(at-sign)"!*!BQ%
2C(at-sign)56BT!!V(*jNh4SCC0dDA4XCC0[CT0dD'PcNfaPBh4eFQ8XN33)*Q&ZC*0MGA*
TEh9cNh4[NfKPBA+6GfKKG*0hD(at-sign)aXNf+3!&11CC0cB(at-sign)PNMU'0N5H!!'&LN!"6MQp
eG*X$kUKTET(at-sign)XFRD4reMNBA*TB(at-sign)k6G*KdD'9[FRQBD(at-sign)kBG'KPQ'jPH(5BCQpbG*0
j,(at-sign)9TCfL6G*KYD(at-sign)k6GA4PFbk1TSf4*i!!6'PVQUabCCB%`'pdD'(at-sign)63A*KBQPKET0
`D'q3!&11C(at-sign)jTH*0KFQPcD(at-sign)jRNfCbEff6DA4cNf&cD'9c,*%%pH&ME'&cFfPMB(at-sign)b
6D(at-sign)kBGT(r(at-sign)14KFQPKETKdNh4SC(at-sign)pbHC(r"9BXMU'0N5H!!'pZBf(at-sign)(at-sign)"*(1F(*[EQp
eEQ0PC*0NC(at-sign)&N,*%%ZjGTFj0[EQ0PNf&RB(at-sign)PZNf&dNh4SCC0QEh*PCR*[ET!!V(*
dNfpQNfeKG'KPE(at-sign)&dD(at-sign)0c,T%(,P&8D'(at-sign)1SBf4*i!!EfaNPJ0(%R4bC(at-sign)&dDA0PFj0
KFQ(at-sign)6BT!!8ijPD(at-sign)jRNf4eFh4PC*0[#j0dD'(at-sign)6FfKPE(DDV(*PFj0[CT0XD(at-sign)*bBA*
jNf*KFf9YC(at-sign)kBG(16B(at-sign)jNNh*PFQ9KC#b1MTmH!!#0NJ$kEf!b-Bk1M)X!!!!(at-sign)!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!$1Bk!#S#MeMD$pKYF
,S!*E+2(at-sign)0S2h*e`Z0N5H!!,PbC(at-sign)PZQUabG'9bF(*PG'9NPJ4p6(at-sign)&ZC*0`FQ9cC(at-sign)k
BG'9NNfPZNf'6E'&ZCh9KCf(at-sign)6G'KKG*0YC(at-sign)9dFj0dD'(at-sign)6Fh4KEQ4KFQ56EfD6FQP
REh+1T!k!!)f4*i!!EfD(at-sign)!e`QEh9bNf4KQUabHC(r"9BZN38*Ae4SCC0`FQpRFQ&
YNfpQNf0XBA0cD(at-sign)0KE*0TETKfNIpBj'&bD(at-sign)&ZQ(56G'KPEh*jNIm&9Lb4!hLRG'K
KG*0SB(at-sign)56CQpbNh0[E(at-sign)(at-sign)6G'PYCBkKMC%RJ!"LQP11C(at-sign)9ZPJ4cHQGTGT!!V(*PET0
eF*0KFj0SEh#BC(at-sign)aPFh-XN359VQPcNf&RB(at-sign)PZNf+BC(at-sign)PZCj0`GA*cG(at-sign)9N,*%%PDj
KEQ56Fh9MBf9cFj0YBC!!V(*jNf&dNfaKFh51SBf4*i!!BT!!8ijPPJ2UU(GTG'K
TET0bC(at-sign)&MN!#XFQJZMUNDJ!#0N5H!!&H4r`9(at-sign)CCB%p&ThD(at-sign)aXNh*PGQPPGj0dPDa
bGj0[PJ6d(at-sign)R4eFQjTEQH6F*!!8ij[D(at-sign)kDV(*dFj0TET0dD'(at-sign)6D'PcG'pbHC0[CT0
TETKfNIpBj'&bD(at-sign)&ZQ(56G'KPEh*jNIm&9Lk4#&Ah9'KPMU'0N5H!!!abFh3XQ`0
eYR4SCCB$(at-sign)(SLEQ9h)T0[EQ8XQ'KKF(#3!&11C(at-sign)jPC*0KFQpeEQ56G'KPNh4eFQk
6EfD6G'KPNf0PET!!V(*dGA*jNIm&9LbBB(at-sign)jNNfPdFj0P#f9MG(11SBf4*i!!BA*
PPJ,T+A0dD(at-sign)aXNf+3!&11C(at-sign)PZCj0QC(at-sign)adNf&XE*0[PDabGT0PFTB#k5PYBA4SC(at-sign)e
KG'PMFbk4"1-,9'KPNh0PBfpZC#bE!abTG'KPNb*[E'3LNfpZC5bBD'&`F*!!8ij
PEQ9NMU'0N5H!!(D3!+abCA*jPJ40Xf9KFQajNfPZNh4SCC0RB(at-sign)eP,*%%CRCKEQ5
6E'9NNh4[Nf'6Ff9bD(at-sign)peFj0YDA0eEQ4PFR0dB(at-sign)jND(at-sign)jRNh4SBA56E'&cG(16G'q
1SBf4*i!!G'KTFj%$kUKNBC!!V(*jNIm&9Lk1TSf4*i!!3C%&B1p`N!"6MQ9NCA0
dFQPKETB&B8pNC3aZDA4TEfk6EfD6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)"(at-sign)&2G'KPEh*
jNfeTCfLDV(*dNfG[Nf&cNfC[E'a[Q(Gc1T%)*LpTETKfNIpBj'&bD(at-sign)&ZQ(51SBf
4*i!!G'KPEh*jPJ5(at-sign)d'PcNh4SCC0cG(9NHC0[CT0[FQ*TG(16EfD6Ch*[GA#6B(at-sign)0
dD(at-sign)pZFbk4"ceB8h9MN!#XFQL6BC0NC3aZDA4TEfk6DA16BfpbFQ9MG#b1SBf4*i!
!BR9dPJ1#-QPdNffDV(*eFh56BT!!8ijPNh0eF("XC(at-sign)ePETKdC(at-sign)56BTKjNf'6F(*
[Ch*KE(at-sign)eKG'PMNh0dBA4PE(at-sign)9ZQ(3ZN38(at-sign)$NKPFQeKEQk69j(r"9CPH(at-sign)`XMU'0N5H
!!'PZPJ2,YR4SCC0TET!!V(*dFQqD8ijNG(at-sign)0dD(at-sign)pZNh4[NfKTFj0LQ'qBEfZ6)P4
SCC0$E'&cFfPMB(at-sign)b64h*[GA"c)Lb4!p(QGj!!V(*KFj0dD'(at-sign)6$(*cG*0TET0dD'P
cMU'0N5H!!'0PETUXFR4eFRQ(at-sign)"$#CG'q6CfPfQ'(at-sign)6BC0cGjKPCA"TEQH6EjKfQ'9
bGQPPGj0[CT0dD'(at-sign)6F(*[Ch*KEC0[CT0TETKfNIpBj'&bD(at-sign)&ZQ(56G'KPEh*jNIm
&9Lk4"JUb5'(at-sign)1SBf4*i!!Fh9YE(at-sign)&bDATPC*B%(YGdD'PcNh"bEfGbB(at-sign)f6D(at-sign)k6G*(at-sign)
XFRH6EjB%(YGLBA0TBj0KFh0PFR4TEfjc,T%&e(at-sign)a8D'(at-sign)6$(*cG*0cG'&dCA16G'K
KG*-L3(at-sign)aXMU'0N5H!!'GPEfePG(*TBjB$C0aQB(at-sign)0dFj0KFQ(at-sign)6CAK`FQ9cFf9NNf+
DV(*jNh4SCC0fNIpBj'&ZDA0SD(at-sign)jRNfpQNfPZQ(D4reMNBA*TB(at-sign)kBG(-L,*%$Ijp
KEQ56G'KPNh0PBfpZC)kKMC%RJ!"cG'&dCA1(at-sign)!qUSG'KKG*-LB(at-sign)aXNfPZPDabGT[
r(at-sign)14KFQPKET0dFjB$kUKKFQ(at-sign)6D(at-sign)k9V(*fQ'&bD(at-sign)&ZNh4cPJ2UU'pQNh4PER0[FR-
L,SkQMC%RJ!"-CA5(at-sign)!k34GA16BR*TC3ejNf0[E(at-sign)ePETUXFR56Efk6G'KPFf(at-sign)6E'p
QG*KjNh0dBA4PE(at-sign)9ZQ(4c,T%&)9KAD'&dNfPcNf'6Cf9[E(at-sign)9dFQPMNfCKBh3rMU'
0N5H!!%'4")f-Cf9[E(at-sign)9dFQPMPJ50YQCKBh56DA16BC0QB(at-sign)0dNf&LQP11Eh9dNh0
`B(at-sign)0PNh4SBA56DA16D(at-sign)jNCA#BC(at-sign)jNC(at-sign)kDV(*dNfpQNh4SCC0MQ'K[D(at-sign)0PNfpQMU'
0N5H!!''(at-sign)"*[YBfqD8ij[FQ4TEQ&dCC0cHA0dC(at-sign)dZN3G-VdGPEfePG(*TBj0QB(at-sign)0
dFj0KFQ(at-sign)6C'9cBh*TBTKPC*0LN!#XFRQ6E(at-sign)9KER16EfD6CA&eBA4TEfjcMU'0N5H
!!(GSD(at-sign)1DV(*SPJ-RCh*PFA9TFQ(at-sign)6BC0MQ'K[D(at-sign)0PNfpQNf0[N!"6MQpbC'PZBA4
PFbk4"2I+5(at-sign)k6BC0fQ'9MG'pbNh0`B(at-sign)0PNpG(at-sign)N3A$elP[CT0ND(at-sign)ePER0TEfk6efk
6Z(at-sign)pZCBkKMC%RJ!"MN!#XFQK[QP11Eh0PFjB$3U*KNf0[Q'pbC'PZBA4PNh0jFh4
PEC2AH)fI!Fc-e6'1PJ6!"0FlQ`(rrRL0R`(-c08bMT2A1ifB1TJkQ$U1N4(+R$Z
BH)fI!Fc-f'k1N3MUmVNZN38!hP0TEQ0PPJ0#SN4PFf0KFR4PFbb4!f3pGjUXFQ(at-sign)
6D''BGTKPNfaPBA*ZC(at-sign)51SBf4*i!!G'q(at-sign)"Ha1CAK`FQ9cFj0RC(at-sign)pYCA4bD(at-sign)16CQ&
MG(16BT!!V(*jNf9aG(at-sign)&dD(at-sign)pZFj0TET0dD'(at-sign)6Bfq3!&11Eh*ND(at-sign)jKG'9cNpGiMCm
"c-c9-Bk(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpFlMCJkQ$UB1Sk4%FUF1jKiMCm"c-c
BESk4"DK3Z5k1SBf4*i!!5'q9V(*hNf9fNf9b,*%$GA"KBT!!8ij[GA5(at-sign)!eJLEfj
PNfLDV(*eEQ4bC(at-sign)56HCKPBA*cNf&REbb4!h9`E(at-sign)&dD'9YBA4TBfPKER16B(at-sign)jNNh"
SQ(PcD(at-sign)0TFh4cNfeKC'(at-sign)1SBf4*i!!G'KPPJ5l1A0SEj!!8ijMQUabDfPZCj0NDA0
MEjKfQ'9bHC0dD'&dNh4SCC0eFh9KE*0dQ(P`N!"6MQ(at-sign)6EfD6CA&eBA4TEfjc,*X
%lejdD'&dNfPc,*KPFA9KG'P[ER11SBf4*i!!D(at-sign)k(at-sign)!lL9G'KPNf0[E(at-sign)f9V(*eG'&
dDAD6CCB$Z*9bD(at-sign)jRNfGPEQ9bBA4PC*0LN!#XFRQ6G'KPNhD4reMNBA*TB(at-sign)*XCA1
6ehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ(L
0R`(-c0KZMT%&U&#j,*%$`TPKFQ(at-sign)4!lL9D(at-sign)jKC'8YMU'0N5H!!(&eBA4PPJ2(at-sign)j'C
[FT0dD'(at-sign)6C'9cBh*TF(4TEfk6EfD6BC0XEh56EfD6Cf9[E(at-sign)9dFQPMNf&ZC*0`D*!
!V(*jFfPMB(at-sign)b6CQ&MG(-ZN38b58e[G'PfNIpBj'&dC(at-sign)51SBf4*i!!BTUXFRQ(at-sign)!`b
TG'KTFj0NDA0MEjKfQ'9bHC(r"9BXN3-j$h4SCAQ6D(at-sign)kBG(*[N!"6MQ4eBf9NNf'
6E(at-sign)pbCC0RC(at-sign)jPFQ&XNh*TEQFZN36Zi&4SDA16DA16G'KPNh*TEQH6EfD6EQpZMU'
0N5H!!'0[E(at-sign)f9V(*eG'&dDAD6CCB$9KT`QP11EfajEQpYD(at-sign)&XFj0TET0dD'(at-sign)6Bfq
BEh*ND(at-sign)jKG'9cNpGiMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpFlMCJkQ$U
B1Sk4%FUF1jKiMCm"c-cBESk4"DK3Z5k4"3GE5'pYEfGPEQ9[GA14!eBDC(at-sign)`YMU'
0N5H!!'9YC(at-sign)k3!+abG(1(at-sign)")%DEfD6G'KTFj0bD(at-sign)jR,*X%TVGdD'&dNfPc,*KSEfe
[Cf9ZC(at-sign)peFj0ZEfk6BfpYEC(at-sign)XFR9dBA4TGT0PPJ5"'R#3!&11EfajEQpYD(at-sign)&XFj0
TESkKMC%RJ!"dD'(at-sign)(at-sign)!j4BGT(r(at-sign)14KFQPKBQaPFj2AH)fI!Fc-e6'1PJ6!"0FlQ`(
rrRL0R`(-c08bMT2A1ifB1TJkQ$U1N4(+R$ZBH)fI!Fc-f'k1N3(at-sign)S8,NXN31PQf&
bCCB$P&KMB(at-sign)aXC(at-sign)56G'9ZFfpbFbk4"4`E5(at-sign)D6GjUXFQ(at-sign)6BT!!8ijPE'PPGTKPNdK
PFQeKEQk69j(r"9CPH(at-sign)`RFikKMC%RJ!"`D'PXEh0[F'L9V(*jNIm&9Lb4!iLaG'K
PETX$F$0hNf(at-sign)BGfPXE*KLN!"6MQ(at-sign)BFf&dDA--C(at-sign)5BG'KKG*KPFA9KG'P[ER1BD(at-sign)k
BG'KPQ(4PER0[FTKKE'GPBR*KQ(0eCLf1SBf4*i!!$'0PPJ0BaQC[FT0dD'(at-sign)6C'9
cBh*TF(4TEfk6EfD6B(at-sign)kDV(*jNfGPEfePG(*TBj0QB(at-sign)0dNhHBCC0hD(at-sign)aXNf9fQ'9
bNfePCA3ZN38)2dD4r`9(at-sign)GA*dD'9bE(at-sign)pbC5b1SBf4*i!!D(at-sign)D(at-sign)"&p1G'KPFf(at-sign)6CA&
eBA4TEfjcNf&bCC0dEj0PH("bCA0cNfGPEfePG(*TBj0`FQp`N!"6MQ9bG'PPFbb
4"(aiG'KPET0dD'9jNff3!+abGA0dNfK[E'51SBf4*i!!EQq(at-sign)!rhQE(at-sign)&dG'9bNhG
SBA56Bfq3!&11Eh*ND(at-sign)jKG'(at-sign)6FhPcG'9YNfPcNf1DV(*SEh0PEMZ4"!H&D(at-sign)k6Eh4
SCA+6GjK[FQ4c,*%%!VCPFA9KG'P[ER16G'KKG)kKMC%RJ!"NCA0MFQPLQP11CCB
$!3jRC(at-sign)pYCA4bD(at-sign)16CQ&MG(16EC!!V(*eFh56BTKPNfPZPDabGT(r(at-sign)14KFQPKET0
dPJ-"$R9ZC'9bNf13!+abD'&ZCf9cNfpQNf0[Q'pbC'PZBA4PFbk4"1X#9'KPMU'
0N5H!!("bEfGbB(at-sign)f(at-sign)"!V6EfD6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)"!V6G'KPEh*jQrm
&9Lb4"",HCR*[EC0#Ej!!8ij[E'(at-sign)6G'q6Eh9bNf4KN!#XFRQB,*%%%YjTFj0`FQ9
MDA0PE(Q6G'KPNh4bB(at-sign)jcE'%YMSkI(J!!MC)!qQpJ-M+1MSb,!!!!&`!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!f8UJ!U!SpBfJrBEA#k!#(at-sign)bM
eMD$pbGF,MC%RJ!#jG'P[ETB$qT9[CT0RC(at-sign)pYCA4bD(at-sign)16CQ&MG(16D(at-sign)kDV(*dEj0
TETKfNIpBj'&bD(at-sign)&ZQ(56B(at-sign)aRC(at-sign)*bB(at-sign)PMNf9aG(at-sign)&dD(at-sign)pZFj0PH("bCA0cC(at-sign)56D(at-sign)k
6G'9bEA11T!k!!)f4*i!!EfD4!qUSG'9ZFfpbFbk1U4U!!)f4*i!!9'KTFjB%6N4
`FQpRFQ&YNfpQNh4bB(at-sign)jcE'&dD(at-sign)pZNfpQNfGPEfePG(*jNfPZQUabG'q6B(at-sign)aRC(at-sign)*
bBC0hQ'&cNh4[Nf+3!&11CC0MBA*bD(at-sign)9NNfpeG)kKMC%RJ!"TETX&0'TdPDabGj0
[Q(0dCA"c,T%*&LC8D'(at-sign)B$(*cG*KcG'9`Q'0[ER0TFh4PC*KTETKNC(at-sign)0[EA#3!&1
1Eh0TEQHBG'9ZFfpbQ'&XCf9LFQ'BD(at-sign)k6G'q1SBf4*i!!DA*bC(at-sign)4eBfPLE'(at-sign)(at-sign)"!r
XBfpYF*!!8ij[EQ9ZQUabG(16G(at-sign)jNCA+6BjKSB(at-sign)jRCA16EfD6Bfq3!&11Eh*ND(at-sign)j
KG'9c,T%&U+a8D'(at-sign)6Ff9MEfjNNh0dCA#6BfpZ,BkKMC%RJ!"cDA0dC(at-sign)5(at-sign)"8eZD(at-sign)k
6C'9fDA0TEQH6B(at-sign)k6C3jMD(at-sign)9ZQUabG*0ZEh4KG'P[ET0QEh+6G'KPNf9iF(*PFh0
TEfk6EfD6D(at-sign)kBGT(r(at-sign)14KFQPKETKdFj0QEh+1SBf4*i!!C(at-sign)&MQUabD*B%m60TFR*
PC(9MD(at-sign)*XCC0MEfe`N!"6MQpZC(at-sign)kBG#k4#%b"9'KPN`abFh56Fh4PF*0hQ'&cNh0
eBf0PFh0QG(at-sign)aXHC0MBA*bD(at-sign)9NNfpeG*0TESkKMC%RJ!"dD'PcPJ4(at-sign)p(at-sign)0PETUXFR4
eFRNlN350((4SCC0cC(at-sign)0[EQ56GjKKFj0KBQ&ZC'pZC(at-sign)56FfpYCA4TE(at-sign)(at-sign)6D(at-sign)k6G'K
PNh5BGjKPETKdD(at-sign)9cNf&ZC*0[EQajMU'0N5H!!(*PBf9ZN!#XFR4XHCB$kUKSBA1
6DA56FQ9cGA*QB(at-sign)0PC#k1TSf4*i!!9'KPPJ0#V(at-sign)4PBfpYF*T6MQpcDA4TEfk6EfD
6G'9ZFfpbNf&XCf9LFQ'6D(at-sign)k3!+abG'q6DA*bC(at-sign)4eBfPLE'(at-sign)6BfpYF*K[EQ9ZQUa
bG(16GjKKFj0NDA0MEjKf,BkKMC%RJ!"PFQ9NPJ-C0'&bEh9ZC*0dD'(at-sign)6G(9bET0
[CT0dD'(at-sign)6Bf9ZQUabG(9bHC0KE'e[Fh56FfPYQ(9XG'&ZC(at-sign)peFfajNf+BHC0*Fh0
KDC06BjKSQ(9bNf&ZC)kKMC%RJ!""E'CbC(at-sign)5(at-sign)!kCb(at-sign)C(r"9C[G(at-sign)jR,T%&)L08D'(at-sign)
6CfPcG*0[CT0dD'PcNf4PBfpYF*!!8ij[FfPdD(at-sign)pZNfPcNfpZCC0[CT0dD'(at-sign)6Ch*
PBA56B(at-sign)4fNIpBj'&ZBf9cNfPZMU'0N5H!!'eKG'KPE(at-sign)&dD(at-sign)0cPJ49rQpQNf&XE*0
dD(at-sign)ePFbb4"($8B(at-sign)jNNfPdNfeKQUabHC0LN!"6MQ(at-sign)6GjK[FR4SQ(GSD(at-sign)aPNh4[Nh"
bCA0PETKdNfPdNfPZNf'6CQpbEBkKMC%RJ!"dD'&dPJ2UU'0KET0LN!"6MQ(at-sign)6E(at-sign)&
NCC0KN!#XFRD4reMNB(at-sign)PXB(at-sign)*XCC0dEj0eEQ4PFQGbB(at-sign)4eBA4PFbk1TSf4*i!!6'9
dPJ44*h9cNf0[ER0TC'9bNfCeEQ0dD(at-sign)pZFj0[CT0dD(*PCC0fNIpBj'&bD(at-sign)&LE'9
c,*%%DXGcG(at-sign)13!+abD*0KFj2ACT%"4rqj+0GiMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqH)f
I!Fc-e6+1NpFlQ(L0R`(-c08cMT1j+5k4"QaG9(H9V(*[N344*hH6C(at-sign)aXMU'0N5H
!!'YZEj!!V(*hETB$'G*ME'&cFf9cNfpQNfCeEQ0dD(at-sign)pZFj0[CT0dD(*PCC0fNIp
Bj'&bD(at-sign)&LE'9cNf&bCC0cH(at-sign)eYCA4bD(at-sign)16CR9ZBh4TEfjc,*%$3jGNC3aZC(at-sign)51SBf
4*i!!G'q(at-sign)!qUSFf&dDA0QHC0dD'(at-sign)6CA&eBA4TEfjcMU'KMC)!RMBYefD0R`(-c0K
cMT%%EMbj+0GiMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpFlQ(L0R`(-c08
cMT1j+CB$99)pNpGQMCm"c-cBFik4"'imZ5MAH)fI!Fc-f'Q0R`%FFI-VZ8&KU!!
'!!!!"J!!!!4MEA)feM'1MTB(Mi(A1jKiMCm"c-cBDBfI!4aaeM+1MT2A1jKiMCm
"c-cBDBfI!4aaeM11MT1j+BkT&3!!MC%RJ!"QEh+(at-sign)"Bc%CADDV(*PFRQ6F*!!8ij
PFQfBGA4KG'P[ET0cC(at-sign)jND(at-sign)jRNh4SCC0TEQ4TBf9cNbJaecZ(at-sign)!IrqZ6,A1j1j-bQ
(at-sign)"Bc%G'q6+0GTMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqDBfI!Fc-e6+1NpFlQ'Q0R`(-c08
cMT1j+5b4"I9,B(at-sign)jNN3(at-sign)-a(0VN!#XFQ9hMU'0N5H!!(0jE(at-sign)ePG(*TBjB$kUKQG(at-sign)j
MG'P[ER-XNf4P$'jPC*0LN!#XFRQ6G'KPNf9aG(at-sign)&dD(at-sign)pZFikKSBf5!*G8+0GQMCm
"c-cBBBk4"322Z5MAH)fI!Fc-e6'1PJ6!"0FlQ`(rrRL0R`(-c08bMT2A1jKiMCm
"c-c9-ik6Z5Q(at-sign)!e952C28"YGQMCm"c-cBBBk4"322Z5MAH)fI!Fc-f'Q0R`%FFGB
aMSk(at-sign)"iq"ecZBH)fI!Fc-f'Q0R`%FFGBbMSk6ecZBH)fI!Fc-f'Q0R`%FFGBcMSk
6Z5RA1ikQMC%RJ!#jGfKPFQ(at-sign)(at-sign)!i`pG'KPNh0TCfk6DA16+c'6Eh+6e!#j-C0KBf0
[FQ4TEQH6BA16G'KPNh#3!&11CA*YN!#XFR9dBA4TEfk6Ff9ZC'PZCj0dD'(at-sign)6D(at-sign)j
ND(at-sign)0PFikKMC%RJ!!S-GFlPJ(rrVNbecZ6Z6-TPJ2UU(4[NbMADBfI!Fc-e6'1PJ6
!"0FlQ`(rrQQ0R`(-c08bMT2A1jKTMCm"c-c9-ik6Z5Q(at-sign)!qUSDA16CAD3!+abC(at-sign)k
6Eh+6Ej!!8ijNC#k1RaU!!)f4*i!!5A5(at-sign)"-N%DA16EQpdNh4bG(at-sign)(at-sign)6G'KKG*0KNfC
eEQ0dD(at-sign)pZNfpQNh4SFQ9PNhD4reMNBA*TB(at-sign)*XCA16DA16G'KPNh0eEC0[CT0KNh0
jE(at-sign)ePG(*TBikKMC%RJ!"QG(at-sign)jMG'P[ETB#qUjKEQ56BC0cDjUXFQ9h,A0jE(at-sign)ePG(*
TBj0QG(at-sign)jMG'P[ELk4"1ML3C%#qR&dD'PbC*0dQ(P`N!"6MQ(at-sign)6EfD6CR9ZBh4TEfk
6DA16FQ9aG(at-sign)PbC(at-sign)3XMU'0N5H!!(GSD(at-sign)1DV(*SPJ2UU'PcNf0KE'aPC*0KNf0jBfa
TBj0QG(at-sign)jMG'P[ELb6GfKTBjKSNfPcNf4P$'jPC*0LQ(Q6G'KPNf9aG(at-sign)&dD(at-sign)pZMU'
KMC&[V,$ACSfI!Fc-f'11N33ZHENSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c
9-Sk6ecZBH)fI!Fc-e611NlNTPJ+UU#Z6efD0R`(-c0KMMT%%,RQj+0GiMCm"c-c
9-ik(at-sign)"-!%ecZBH)fI!Fc-e6'1NpFlQ(L0R`(-c08bMT1j+CB#UUJVNpGQMCm"c-c
BBik4"#jjZ5MAH)fI!Fc-e6+1PJ6!"0FlQ(L0R`(-c08cMT2A1jKiMCm"c-c9-Bk
6Z5Q(at-sign)!e952C-`ecU1TSf4*i!!Z89fN!#XFQ9bHCB%kSYQG(at-sign)jMG'P[ET0[CT0dD(*
PCC0fNIpBj'&bD(at-sign)&LE'9cNf0KET0LN!"6MQ(at-sign)6G(at-sign)jTFA9PE(Q6Gh*TG(4PET0KFj0
dD'(at-sign)6Fh9YNfpQNf'1SBf4*i!!FhPYE(at-sign)9dFQPMPJ95NQCeEQ0dD(at-sign)pZ,*X&V)eKNh0
VN!#XFQ9hNh0jE(at-sign)ePG(*TBj0QG(at-sign)jMG'P[ELbBB(at-sign)jNNf'6BhPME'PMNfCeEQ0dD(at-sign)p
Z,*KTESkKMC%RJ!"cH(at-sign)f3!+abBT!!8ij[E(-kMU'KMC&6MhrACT%"4rqj+0GiMCm
"c-c9-Bk(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpFlQ(L0R`(-c08cMT1j+CB$99)pNpG
QMCm"c-cBFik4"'imZ5MAH)fI!Fc-e6'1PJ6!"0FlQ(L0R`(-c08bMT2A1jKiMCm
"c-c9-ik6Z5Q(at-sign)!UUS+j2ACSfI!Fc-f''1N38$clNSehL0R`(-c08aMTB%`!6A1jK
iMCm"c-c9-Sk6ecZBH)fI!Fc-e611NlNTPJ+UU#Z6efD0R`(-c0KMMT%%,RQj+0G
iMCm"c-c9-Bk(at-sign)"-!%ecZBH)fI!Fc-e6+1NpFlQ(L0R`(-c08cMT1j+GFkMSkI(J!
!MC)!qQpJZ6)cMSk-L`!!!"J!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!1KrS!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if4*i!!Z89KBjUXFQL(at-sign)"AL
LEfD6G'KPNh4SFQ9PNh0jE(at-sign)ePG(*jNf0XBA0cCA16DA16D(at-sign)kBGT(r(at-sign)14KFQPKETK
dNh9ZC'9bNh#3!&11CA*YQ(9dBA4TEfjc1j%'2k"dD'PcMU31J!#0N5H!!'CKBh5
(at-sign)"0YqDA16Ef+DV(*fD(at-sign)peFj0QEh+6FhPYE(at-sign)9dFQPMNf&ZC*0cDjKPGj0cH(at-sign)eYCA4
bD(at-sign)16CR9ZBh4TEfjcNf*eG*0ZEh56FA9TG'(at-sign)1SBf4*i!!Ffq(at-sign)!rV$Ef+DV(*fD(at-sign)p
eFj0QEh+6BhPME'PMNfCeEQ0dD(at-sign)pZFbk4"(at-sign)Nb9'KPFf(at-sign)6G'KbC(at-sign)(at-sign)6D(at-sign)kBGT(r(at-sign)14
KFQPKETKdNh0eBR0`B(at-sign)0PFj0`E''BHC0QEh+6G'KPMU'0N5H!!'GbEh9`PJ2(Hfp
QNh#3!&11CA*YQUabGA4KG'P[ER16EfD6BC0cCA56EfD6G'KbC(at-sign)(at-sign)6C(at-sign)aPE(at-sign)9ZQ(4
cNf'6FQpXCC0KEQ&XEfG[GA16G'q6G'KPNh*[E'(at-sign)1SBf4*i!!EfD(at-sign)!qUSG'KPNf9
TCf9ZPDabGT0PBh4[FR1(at-sign)!qUSEfD6BC0cH(at-sign)eYCA4bD(at-sign)16E(at-sign)&dFQPi,SkT'QLpMC%
RJ!"'NIm&9QpbPJ+P`(at-sign)CeEQ0dD(at-sign)pZFj2ACT%"4rqj+0GiMCm"c-c9-Bk(at-sign)"-!%ecZ
E!IrqH)fI!Fc-e6+1NpFlQ(L0R`(-c08cMT2A1jKiMCm"c-c90)k6Z5Q(at-sign)!UA"EfD
6CQpeFT0fNIpBj'&bD(at-sign)&LE'9cNh4SCA*PNf&bCC--GT!!V(*PNh0jE(at-sign)ePG(*jNf0
XBA0cCA-XMU'0N5H!!(GSD(at-sign)1DV(*SPJ2UU'&bCC0NC3aZC(at-sign)56BA16CQpXE'qBGh-
kMUD0N5H!!$%ZN38ii&0jE(at-sign)ePG(*TBj%$kUKQG(at-sign)jMG'P[ER-ZMUD0N5H!!$)ZN38
ii&0VN!#XFQ9hPJ2UU(0jE(at-sign)ePG(*TBj0QG(at-sign)jMG'P[ER-ZMUD0N5H!!$-ZN38ii%0
jBfaTBjB$kUKcH(at-sign)eYCA4bD(at-sign)16CR9ZBh4TEfjc,*0cBA4TFfCjD(at-sign)jRNh4SCC0QEh9
bNf9aG(at-sign)&dD(at-sign)pZFikKSBf4(at-sign)[dAefD4!8IrZ5MAH)fI!Fc-e6'1PJ6!"0FlQ`(rrRL
0R`(-c08bMT2A1jKiMCm"c-c9-ik6ecZBH)fI!Fc-e651NlNTPJ+UU#Z6efD4!8I
rZ5MAH)fI!Fc-e6'1PJ6!"0FlQ(L0R`(-c08dMT2A1jKiMCm"c-c9-Sk6ecZBH)f
I!Fc-e611NlNTPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc-e6'1PJ6!"0FlQ(L0R`(-c08
cMT2A1jKiMCm"c-c90)k6ecZBH)fI!Fc-e6+1NlNTPJ098Mf6-0FlMUN8lSkKMC&
Dr4GQN3&(rlNSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZBH)fI!Fc
-e611NpFlQ(L0R`(-c08dMT1j+CB#UUJVNpGQN3&(rlNSehL0R`(-c08dMTB%`!6
A1jKiMCm"c-c9-Sk6ecZBH)fI!Fc-e6'1NpFlQ(L0R`(-c08cMT1j+CB#UUJVNpG
QN3&(rlNSehL0R`(-c08cMTB%`!6A1jKiMCm"c-c9-Sk6ecZBH)fI!Fc-e651NpF
lQ(L0R`(-c08aMT1j+CB$99)pNc$A1ikQSBf4(at-sign)[dACT%"4rqj+0GiMCm"c-c9-Bk
(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpFlQ(L0R`(-c08cMT2A1jKiMCm"c-c90)k6Z5Q
(at-sign)!UUS+j2ACT%"4rqj+0GiMCm"c-c90)k(at-sign)"-!%ecZBH)fI!Fc-e6'1NpFlQ(L0R`(
-c08cMT2A1jKiMCm"c-c9-Sk6Z5Q(at-sign)!UUS+j2ACT%"4rqj+0GiMCm"c-c9-Sk(at-sign)"-!
%ecZBH)fI!Fc-e651NpFlQ(L0R`(-c08cMT2A1jKiMCm"c-c9-Bk6Z5Q(at-sign)!e952C-
`ecZ1TU'0N9Vp&fD4!8IrZ5MAH)fI!Fc-e6'1PJ6!"0FlQ`(rrRL0R`(-c08bMT2
A1jKiMCm"c-c9-ik6ecZBH)fI!Fc-e651NlNTPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc
-e611PJ6!"0FlQ(L0R`(-c08aMT2A1jKiMCm"c-c9-Sk6ecZBH)fI!Fc-e651NlN
TPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc-e6+1PJ6!"0FlQ(L0R`(-c08cMT2A1jKiMCm
"c-c9-Bk6ecZBH)fI!Fc-e651NlNTPJ098Mf6-0FkMTmJedZ0N5H!!,Nd,T%&11"
'NIm&9R9ZBh4TEfjcPJ2UU(0KG'PcCRPTEQH6G'KPNfC[GA+6CA&eBA4TEfjcMU'
N'PG,MC%XjCMACT%"4rqj+0GiMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpF
lQ(L0R`(-c08cMT2A1jKiMCm"c-c90)k6Z5Q(at-sign)!UUS+j2ACT%"4rqj+0GiMCm"c-c
9-Sk(at-sign)"-!%ecZBH)fI!Fc-e6'1NpFlQ(L0R`(-c08cMT2A1jKiMCm"c-c90)k6Z5Q
(at-sign)!UUS+j2ACT%"4rqj+0GiMCm"c-c9-Bk(at-sign)"-!%ecZBH)fI!Fc-e6+1NpFlQ(L0R`(
-c08dMT2A1jKiMCm"c-c9-ik6Z5Q(at-sign)!UUS+j2ACT%"4rqj+0GiMCm"c-c9-Sk(at-sign)"-!
%ecZBH)fI!Fc-e6'1NpFlQ(L0R`(-c08dMT2A1jKiMCm"c-c9-ik6Z5Q(at-sign)!e952C-
`ecZ1SD'0N5cPQ'D4!8IrZ5MAH)fI!Fc-e6'1PJ6!"0FlQ`(rrRL0R`(-c08bMT2
A1jKiMCm"c-c9-ik6ecZBH)fI!Fc-e651NlNTPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc
-e611PJ6!"0FlQ(L0R`(-c08bMT2A1jKiMCm"c-c9-Bk6ecZBH)fI!Fc-e651NlN
TPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc-e6'1PJ6!"0FlQ(L0R`(-c08dMT2A1jKiMCm
"c-c9-ik6ecZBH)fI!Fc-e6+1NlNTPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc-e611PJ6
!"0FlQ(L0R`(-c08dMT2A1jKiMCm"c-c9-Bk6ecZBH)fI!Fc-e6+1NlNTPJ098Mf
6-0FlMU'KMC%XjCKQN3&(rlNSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk
6ecZBH)fI!Fc-e611NpFlQ(L0R`(-c08dMT1j+CB#UUJVNpGQN3&(rlNSehL0R`(
-c08aMTB%`!6A1jKiMCm"c-c9-ik6ecZBH)fI!Fc-e6+1NpFlQ(L0R`(-c08dMT1
j+CB#UUJVNpGQN3&(rlNSehL0R`(-c08dMTB%`!6A1jKiMCm"c-c9-Sk6ecZBH)f
I!Fc-e611NpFlQ(L0R`(-c08aMT1j+CB#UUJVNpGQN3&(rlNSehL0R`(-c08dMTB
%`!6A1jKiMCm"c-c9-ik6ecZBH)fI!Fc-e6+1NpFlQ(L0R`(-c08aMT1j+CB$99)
pNc$A1ikKR`k!!)f0MBf5!*%kkjrerrbV(at-sign))k1MT)!SDbbeh0TCj9Z1(at-sign)kj+0FENlN
TefD4!8IrZ5MAH)fI!Fc-f"Z3!%Ple6'1PJRP!pFlQ`(rrRL0R`(-c0JEN!"*Hp8
bMT2A1jKiMCm"c-cB'j!!5A[9-ik6ecZBH)fI!Fc-f"Z3!%Ple651NlNTPJ098Mf
6-0FkMTmJedZ0N5H!!,Ne,T%&11"'NIm&9R9ZBh4TEfjcPJ2UU(0KG'PcCRPTEQH
6G'KPNf9aG(at-sign)&dD(at-sign)pZFik1Rai!!)f5!2T[B$)dMSk-L`!!!"N!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!281S!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFR
A#k3CAZk0N5bcbGGQN3&(rlNSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c9-Sk
6ecZBH)fI!Fc-e611NpFlQ(L0R`(-c08dMT1j+CB#UUM8!*2ACT%"4rqj+0GiMCm
"c-c9-Sk(at-sign)"-!%ecZBH)fI!Fc-e6'1NpFlQ(L0R`(-c08cMT2A1jKiMCm"c-c90)k
6Z5Q(at-sign)!UUSe!#6efD4!8IrZ5MAH)fI!Fc-e6'1PJ6!"0FlQ(L0R`(-c08bMT2A1jK
iMCm"c-c90)k6ecZBH)fI!Fc-e611NlNTPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc-e6+
1PJ6!"0FlQ(L0R`(-c08aMT2A1jKiMCm"c-c90)k6ecZBH)fI!Fc-e611NlNTPJ0
98Mf6-0FlMU'KMC%XXmPQN3&(rlNSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm"c-c
9-Sk6ecZBH)fI!Fc-e611NpFlQ(L0R`(-c08dMT1j+CB#UUM8!*2ACT%"4rqj+0G
iMCm"c-c9-ik(at-sign)"-!%ecZBH)fI!Fc-e6+1NpFlQ(L0R`(-c08aMT2A1jKiMCm"c-c
90)k6Z5Q(at-sign)!UUSe!#6efD4!8IrZ5MAH)fI!Fc-e6'1PJ6!"0FlQ(L0R`(-c08dMT2
A1jKiMCm"c-c9-ik6ecZBH)fI!Fc-e6+1NlNTPJ+UU#Z6efD4!8IrZ5MAH)fI!Fc
-e611PJ6!"0FlQ(L0R`(-c08dMT2A1jKiMCm"c-c9-Bk6ecZBH)fI!Fc-e6+1NlN
TPJ098Mf6-0FlMU'KMC%XXmPQN3&(rlNSehL0R`(-c08aMTB%`!6A1jX"rrjiMCm
"c-c9-Sk6ecZBH)fI!Fc-e611NpFlQ(L0R`(-c08dMT1j+CB#UUM8!*2ACT%"4rq
j+0GiMCm"c-c9-Bk(at-sign)"-!%ecZBH)fI!Fc-e611NpFlQ(L0R`(-c08bMT2A1jKiMCm
"c-c90)k6Z5Q(at-sign)!UUSe!#6efD4!8IrZ5MAH)fI!Fc-e651PJ6!"0FlQ(L0R`(-c08
bMT2A1jKiMCm"c-c9-ik6ecZBH)fI!Fc-e6'1NlNTPJ+UU#Z6efD4!8IrZ5MAH)f
I!Fc-e651PJ6!"0FlQ(L0R`(-c08cMT2A1jKiMCm"c-c9-Sk6ecZBH)fI!Fc-e6'
1NlNTPJ098Mf6-0FlMU'N$S!!MBf0MC)!T+NbRrArr+YBMSk1NJ#e'[RACT%"4rq
j+0GiMCm"c-cB'j!!5A[9-Bk(at-sign)#H8$ecZE!IrqH)fI!Fc-f"Z3!%Ple6+1NpFlQ(L
0R`(-c0JEN!"*Hp8cMT2A1jKiMCm"c-cB'j!!5A[90)k6Z5Q(at-sign)!e952C-`ecU1Rar
HlSf4*i!!Z89fQUabCA*jPJ8HJ(at-sign)CeEQ0dD(at-sign)pZNfpQNfC[GA+6GT(r(at-sign)14KFQPKBQa
PFj0TFj0eEQPaG(at-sign)9XHC0PH("bCA0cD(at-sign)*XCC0KFj0dD'(at-sign)6Fh9YNfpQN`afQ'(at-sign)1SBf
4*i!!CR9ZBh4TEfjc,*%%B2jPB(at-sign)1DV(*SPJ4*8fpZCC0LN!"6MQ9XEfjRD(at-sign)jRNh4
[NfpZCC0[CT0dD'9cCC0cH(at-sign)eYCA4bHC0ME'&cFf9c,T%'91&&B(at-sign)1BD*0cH(at-sign)dYMU'
0N5H!!'ePG(*jPJ2UU'0XBA0cNfPcNfPZPDabGT(r(at-sign)14KFQPKET0dPJ2UU(9ZC'9
bNh#3!&11CA*YN!#XFR9dBA4TEfjc,SkT'GV4MC%RJ!"0Eh*PPJ,4)(at-sign)GPEQ9bB(at-sign)a
XHC(r"9BXN3-*Ef9fN!#XFQ9bHC0QG(at-sign)jMG'P[ET0[CT2AET1jGT(r(at-sign)14KFQPKBQa
PFj2ACT%"4rqj+0GiMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqH)fI!Fc-e6+1NpFlMCJkQ$U
B1Sk4%FUF1jKiMCm"c-cBESk4"DK3Z5Q(at-sign)!Y%KBf&ZNf+3!&11CC0eEQPaG(at-sign)9XHBk
KMC%RJ!"hFQPdG'9ZPJ5bP(at-sign)&cNh4SCC0cG(at-sign)f6EfD6eh#0R`(-c0KZMT%+(at-sign)Z(at-sign)jCR9
ZBh4TEfjc,*%%j*&PB(at-sign)1DV(*SNfpZCC0LN!"6MQ9XEfjRD(at-sign)jRNh4[Nf'6C'N,CA*
PETKdNh0jE5f1SBf4*i!!E(at-sign)9dFRQ(at-sign)!m4*BfaKFh-ZQ`8X&NKPFQ8XN32,pGG`MCm
"c-cBESk4#(at-sign)bCZ(at-sign)9aG(at-sign)&XFj0dD'(at-sign)6ET(at-sign)XFR9YNf+3!&11CA+(at-sign)!m4*EfD6F'&bG'P
dD(at-sign)pZFj0[CT0dD'(at-sign)6D(at-sign)k3!+abG'9RCA+6efkj,TK&B(at-sign)13!+abD)kKMC%RJ!"cH(at-sign)e
YCA4bHCB$kUKME'&cFj0TFj0NC3aZC(at-sign)56BTUXFRQ6CA&eBA4TEfjcNhGSD(at-sign)1BD*0
KFQ(at-sign)6EQpdNf4T$Q0eE(56G'q6$'jN,SkQMC%RJ!"8D'PcPJ1e8Q4PBfpYF*!!8ij
[FfPdD(at-sign)pZNfK[E'4cNfC[FT0dC(at-sign)jcEh*cNf&cNhHDV(*PE'`XN31rr(at-sign)&QG'9bNh0
[E(at-sign)(at-sign)6BfpcE(at-sign)9dD(at-sign)16BjKSB(at-sign)jRCA16EfD1SBf4*i!!EQpdBA4TEfiZN36G&&5Er`9
(at-sign)EjB#ed9dD'PcNf4KN!#XFRQB,*%$$PP[EQajNh59V(*hNfq(at-sign)!YG&FhPYE(at-sign)9dFRQ
6BfaKFh0PFj0[CT0dC(at-sign)jcEh*cNfKKPDabGT0PPJ,A4(at-sign)+3!&11C(at-sign)9ZNh0dG(at-sign)4TC(at-sign)5
1SBf4*i!!D(at-sign)k(at-sign)"GS`B(at-sign)kDV(*jNf4PG'&TE#k4#`Gh8hPYE(at-sign)9dFQPMNh4PER0[FR1
6BA*PNfpbC'PZBA*jNf0[E(at-sign)fBGA4KG'PfQ'(at-sign)6F*!!8ij[E(PZEfeTB(at-sign)acMU'0N5H
!!(0eBjUXFQL(at-sign)"(at-sign)TRBA16GjKPNfaPBA*ZC(at-sign)56G'q6GA0PNfPZNf&ZB(at-sign)ajG'PMNfG
PEfePG(*jNIm&9Lk4#EJH8fZBCAH6FhPYE(at-sign)9dFQPMNh4PER0[FR11SBf4*i!!BA*
PPJ29MR#D8ij[E(PZEfeTB(at-sign)acNfPZNh4SCC0MEjK[FQ4TEQ&dCA16ehL0R`(-c08
aMTB%`!6A1jX"rrjiMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ(L0R`(-c0KZMT%
*IGkjF(*[N!#XFRCTC'9NPJ29MR4SBA56G'KPNhD4reMNBA*TB(at-sign)*XCA11SBf4*i!
!BA*PPJ8A['&cFh9YC(at-sign)56G'q6Ff&dDA0QHC0dD'(at-sign)6CA&eBA4TEfjcNpGiMCm"c-c
BDBk(at-sign)!f6DehL0R`(-c0KUMT%*Zp(at-sign)j2C%&9F[8!0GiMCm"c-cBDSk4"'B+ehL0R`(
-c0KTMT1j,T%)`"e8NIm&9Q9ZFfpbFjB&&laLN!"6MQ9XEfjRD(at-sign)jRNh4[MU'0N5H
!!(0jE(at-sign)ePG(*jPJ5KX(at-sign)0XBA0cCA16Eh4SCA+6G'KKET0dD'(at-sign)6BfaKFh0PFj0[CT0
cH(at-sign)eYCA4bD(at-sign)16B(at-sign)jNNh0VN!#XFQ9hNh0jE(at-sign)ePG(*TBikKMC%RJ!"dC(at-sign)jcEh*cPJ-
JTQ&XFfq6Ej!!8ijMBh9bNfPZNfGPEfePG(*jNf&ZC*0`D*(at-sign)XFRPcD(at-sign)0c,T%%pBT
)Ej0hNf9fNf9b,*%$53edD'9cCCB$)+CcH(at-sign)eYCA4bHC0ME'&cFf9cMU'0N5H!!'K
KPDabGT0PPJ-*I(at-sign)+3!&11C(at-sign)9ZNh0dG(at-sign)4TC(at-sign)56GTUXFQ9bHC0XDA4dE'8XN3-fKQ&
ZC*0dD'9jNf&bCC0KNfa[EQH6GjKKQ(Q6CR*[EC0LQP11C(at-sign)PZCj0eEQ4PFR0dEjK
[Q'3ZMUD0N5H!!&0[Q`3+Pff9V(*eBj0SQ'C[FTKdD'(at-sign)BGj0[FQ5B)QjPGb+BD(at-sign)k
BG'KPQ(4TG'aPQ'pQQ(4SDA1BE'9MG(9bC6Z4""U1E'9dQ(9cQ'jPH(5BC'qBFfp
YCBkKMC%RJ!"UGA0dD(at-sign)0PPJ(at-sign)19A4[Nh4SCC0hN!#XFQpbC*-LEfaN)Lk4#L2R9j(
r"9CPNhGTE'b6C'9cBh*TBTT6MQ(at-sign)6G'KPNfe[Fh56F*KPBh9XD(at-sign)&bNfCPBA4eFQ(at-sign)
6EfD1SBf4*i!!BfaKFh0TBf&XQ`0`lfPZPDabGT(r(at-sign)14KFQPKET0dQ(4SC(at-sign)pbHCE
r"9BXQ`1*4fjKE(at-sign)9XHC-XQ(4SCCB$F1pcH(at-sign)fDV(*LN!"6MQpXD(at-sign)16Eh+6G(at-sign)fBBR*
KE*0ZEh4KG'P[ELb4!iP(G'q6GfKTBjKSMU'0N5H!!%9bD(at-sign)1(at-sign)!ViU9*[r"9CPEA"
XCC0#C(at-sign)aXNf4PC'PMBA4PC*0SDA163fpXE'q3!&11FA9TG(at-sign)f66'9MG(9bCA16D(at-sign)k
6-6Nb0bk4"05f9jKPNhGTE'b6BfpZFfPNCA+1SBf4*i!!G'KPPJ,Y[h0TEA"XCA0
dNfGbEh9`,*B$)&4ZB(at-sign)ePE(Q4r`9(at-sign),*0dD'(at-sign)(at-sign)!ZfrCh*[GA#6EfD6G(*KER0XBA4
TEfjcNfpQNh4SCC0XD(at-sign)jP,T%%j**8D'(at-sign)6G(at-sign)k3!+abGA0eB(at-sign)b1SBf4*i!!CQ9KG(9
bCA1(at-sign)!e$4EfD6G'KPNh0jEC!!V(*LQP11EfaTBj0YCA4SEjKNNhGTE'b6B(at-sign)abC(at-sign)&
NHC0LQ'(at-sign)6BA"`BA*PET!!V(*dNfPZNh4SDA16Fh#BC(at-sign)0TB(at-sign)b6Bf&cC5k1SBf4*i!
!6'9dPJ1F(at-sign)GG`Z5MAH,NTNf&ZC*2AFC!!EMQj+0GiZ5Q6BTT6MQ(at-sign)6E(at-sign)pZD(at-sign)16F*K
[E(PZEfeTB(at-sign)acNfPZNh4SCC0fNIpBj'&bD(at-sign)&LE'(at-sign)6ehLj,T%&(XCANIm&9Q(at-sign)6Gh*
TG'(at-sign)6G'KPEC0TESkKMC%RJ!"dD'(at-sign)(at-sign)!qUSCQpXE'qDV(*hD(at-sign)jRNh&eB(at-sign)PZQ(56EQp
dBA4TEfikMSkI(J!!MC)!qQpJ-M(at-sign)1MSb,!!!!'J!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!"!&LJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,T"-!#Bf
43CfQeh#j+0GiZ5Q(at-sign)!e952C2AH)fIq`ZCf'k1N3K5q,NVMBf0N3+UU*rZCPbV))k
0RrIKaBf4#T98efk1Ra"CHBf4#bFEZ6'1MSf4%CMGRqjQA+XKMSk1PKQ$LGGKMCm
"c-c9-BkE"-!%ehL0RrX,QGKZfJ$9-Bk4%bpdZ5Z0MBf4!UUSRqjQA+XJMSfIpq(
&MC%+P96AESkI%&PjMC%,*aZj-Sk1MC%4Q0fIlQCFUb'1MSk6ef'0R`(-c08bMTM
AH)fIq`ZCf'lD!08bMT%6,h5j+if4!UUSecU(at-sign)!Irq1T-kMT%6(r#j+if0MC%#UUL
IlQCFUb#1MCrhiF(at-sign)0N46DTYGZMTm3(at-sign)AQ0N3U99'k(at-sign)!UUSe!#6Z6'1MSf4*L1!Rqj
QA+XKMSk1N5i1,0GKMCm"c-cBEYS!e6'1N4#%c0GiPJ+UU,NVNpGKMCm"c-cBESk
4"DK3ecZ1Ra[9ASf4*i!!Z(at-sign)&ZC)kT$S!!SBf44L+2eh'3!'ijZ5MAH,NTPJ098Mf
6ehL0RrX,QGKVMT%(cMUj+if0MC%#UULIlQCFUb#1MCrhiF(at-sign)0N3U990GVMTm3(at-sign)AQ
0N3VR(,NaMSk0N4%BhjrZCPbV)Bk1MTBC!i[ABSfI!Fc-e6'1Q`6!"0GiMCrl#jR
BDj!!0SAD!08aMT%5UVDj+if0MC%#UULIlQCFUb#1MCrhiF(at-sign)0N3U990GVMTm3(at-sign)AQ
0N3VR(,NbMSk0N4%BhjrZCPbV)Bk1MT2ABSfI!Fc-e6+1Q0GiMCrl#jRBDj!!0SA
D!08bMT%5UVDj+if4!UUSecU(at-sign)!Irq1T-kMT%6(r#j+if0MC%#UULIlQCFUb#1MCr
hiF(at-sign)0N46DTYGVMTm3(at-sign)AQ0N3U99'Z4!a(&e!#4!UUSZ6'1MSf4*D1#RqjQA+XKMSk
1N5f1,YGLMCm"c-cBDj!!0SAD!08aMT%3!!lAH*B#UULj+j2ABSfI!Fc-f'Z1N38
MNYFkMTmFJ!U0N5H!!,PANIm&9Q(at-sign)(at-sign)!r(at-sign)TBA0cG(at-sign)ePNh4SBA56G'KPNh#D8ij[E(P
ZEfeTB(at-sign)b6eh'3!'ijZ5MAH,NTNfPcNfpQNfa[PDabGj0PFTB$pDPNC(at-sign)GbC(at-sign)(at-sign)6G'K
KET0dD'(at-sign)6F*K[E(PZEfeTB(at-sign)b1TSf4*i!!eh#j+0GiZ5NXPJ2UU(4SBA56DA-XNh4
SBA56efZ4!la[e"54!e95efkj,SkI'S!!MC%RJ!"%C3aZCCB$kUKdD'(at-sign)6G(*KER0
XBA4TEfk6Eh#D8ijPFQ&dEh+6ee50N3'KaTrlT6,BBik4#EVRZ(at-sign)pZNf'6F*K[E(P
ZEfeTB(at-sign)b6eh#j+0GiZ5Q6BA16CQpXE'q3!+abGh-kMUDQMC)!Zb0+ee50N3'KaTr
l#jRBBik4"G!reh#j+0GiZ5Q(at-sign)!e952C2AF,NSehL(at-sign)!UUSZ5Z6ef1j+GFkMTm9!!#
0N5H!!,P-CA5(at-sign)!qUSGA16Gh*TG'(at-sign)1TTmI!!Q0N5H!!0G`Z5MAH*B!`!qj+j2ABlN
TPJ098Mf6ehL0RrX,QGKZMT%'D&qj+if0MC%!`!qIlQCFUb#1MCrhiF(at-sign)0N3LUZpG
ZMTm3(at-sign)AQ0N3NmJVNaMSk0N3qZ4*rZCPbV)Bk1MTBAQ2$AF)fI!Fc-e6'1Q`6!",N
Sef1j+GGiMCrl#jRBEYS!e6'1N4&%flNVMBf0N3$!$jrZCPbV))k0RrIKaBf4#+U
lefk1Ra"CHBf4#6b#Z6+1MSf4$kj%RqjQA+XKMSk1NpG`MCm"c-c9-SkBZ5MABlN
TehL0RrX,QGKZfJ$9-Sk4%86EZ5Z0N3$!$pFkPJ(rrMU61Sk4$dUqZ5Z0MBf4!-!
2RqjQA+XJMSfIpq(&MC%5m!hAESkI%&PjMC%)UVYZPJ+UU03!NlNaMSk0N53ijjr
ZCPbV)Bk1MT%X)j2AF)fI!Fc-f'lD!08aMT%3K-bj+0GMZ5RAH*B!`!qj+j2AF)f
I!Fc-f'k1N3(at-sign)S8,NSef1j+GFlMTmL!!U0N5H!!,P8D'(at-sign)(at-sign)!qUSefU4!+c6Z5edD*0
MEjT6MQ81BfPPET!!V(*dNpG`MCm"c-cBDSk4"'B+Z5MABlNTNfpQNh4SCC0`Q'p
XH(at-sign)j[E(at-sign)PKE*2AF,NSehL(at-sign)!UUSZ5Z6ef1j+CB$kUKTFj0MEfe`GA4PC*0dEj0LQ'(at-sign)
1TU'0NA4-rpG`MCm"c-cBDSk4"'B+Z5MABlNTPJ098Mf6ef'0R`(-c0KUMT%(%,+
j+if0MC%#UULIlQCFUb#1MCrhiF(at-sign)0N3V!XGGUMTm3(at-sign)AQ0N3U99,NaMSk0N4"e8*r
ZCPbV)Bk1MTXBArcABBfI!Fc-f'U3!(B)fJ$9-Bk4$d+'ef1(at-sign)!UUSZ5Z0MBf6Rqj
QA+XJMSfIpq(&MC%+`,(ADSkI%&PjMC%+P95j-Sk1MC%3G9#IlQCFUb'1MSkBef'
0R`(-c0KUN!"f#0S!e6+1N3p#KYGMMCrl#jR9-Sk4"fUXZ5Z0NpFkPJ(rrMU61Sk
4%ar`Z5Z6ef10RrX,QGKUMT%%CJVA1SkI()!+MC%RJ!#j3C%&T1p`N!"6MQpXH(at-sign)j
[E(at-sign)PKE*%&T(at-sign)&*+0GKMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqBBfI!Fc-e6+1NpFlMCJkQ$U
B1Sk4%FUF1jKKMCm"c-cBESk4"DK3ecZBBSfI!Fc-e6'1NpFlQ'+0R`(-c08bMT2
A1ifB1TJkQ$U1N4(+R$ZBBSfI!Fc-f'Z1N38MNVNTPJ(at-sign)PB(at-sign)PZNh4SCC0fNIpBj'&
bD(at-sign)&LE'9cNpGKMCm"c-c9-Bk(at-sign)"-!%Z5bE"K32ef'0R`(-c08bMT1j,*JZPJ(e9#k
6,T-Xef'0R`(-c0KZMT%&U&#j,)kQMC%RJ!$ABSfI!Fc-e6'1PJ6!",NXef+0R`(
-c08bMT1j,*%$Ip`ZPJ(e9#k6,T-Xef+0R`(-c0KVMT%)L,ZjDA1(at-sign)!f8TFf&TC*0
dEj0LQP11CC0KET0TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G*B$C5P[CT0dD'(at-sign)6G*(at-sign)XFRH6EjB
$C5P`Q'pXH(at-sign)j[E(at-sign)PKE(16eh#j+0GiZ5NXN30rh0GaN!"Z1ENSehLj+C0hD'9ZMUD
N'S!!MC%RJ!$A5C%!mB1j+0GKMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqBBfI!Fc-e6+1NpF
lMCJkQ$UB1Sk4%FUF1jKKMCm"c-cBESk4"DK3ecZBBSfI!Fc-e6'1NpFlQ'+0R`(
-c08bMT2A1ifB1TJkQ$U1N4(+R$ZBBSfI!Fc-f'Z1N38MNVNTPJ098Mf6edQ4!2'
$Z5MAF)fI!Fc-e6'1PJ6!",NSef1j+GFlQ(#0R`(-c08bMT1j+0GMZ5RA1ifB1TJ
kQ$U1N4(+R$ZBF)fI!Fc-f'k1N3(at-sign)S8,NSef1j+GFlQ('0R`(-c08aMT1j+0GMZ5R
A1jKaMCm"c-c9-Sk6Z5MABlNTecZ0Q$UB1TJkMT%4bT`lQ('0R`(-c0KVMT%&)j+
j+0GMZ5NTMU'0N5H!!'C[FTB$l-0KE'b6BfpYF'aPH*0ZPDabG(at-sign)f6BT!!8ijPFR1
(at-sign)!qc$ef1j,T%&2c&#HC0KBR9cCC0[CT0ZEh4KG'P[ELb4!qe+Gj!!V(*PNhGbDA4
PNpG*N3$aJlNSeh#j+0GiZ5RA1j%"rrjaN!"Z1ENSehLj+5Q6B(at-sign)jNMUD0N5H!!(H
3!+abCCB$I'GcF*T6MQ9KDj0[CT2A5C%%EHUjBA16BTKPD(at-sign)jRNf&ZNfPZPDabGT(
r(at-sign)14KFQPKET0dPJ0mCfpQNh4SCC0`Q'pXH(at-sign)j[E(at-sign)PKE(16eh#j+0GiZ5Q6B(at-sign)jNNpG
aN!"Z1ENSehLj+5k4"43J5(at-sign)k6G'KTFikQMC%RJ!"KBR9cDAD3!+abCCB$+3KZEh4
KG'P[ELb4!dr"BC0`QP11EfajEQpYD(at-sign)&XNpG*N33DLlPTFj0cB(at-sign)PNNh4[Nf+BCC0
KET0TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G*B$+3K[CT0dD'(at-sign)6F*K[E(PZEfeTB(at-sign)acMUD0N5H
!!0G`Z5MAH,NTPJ2UU'&ZC*2AFC!!EMQj+0GiZ5Q6GfKPEQ9fN!#XFQ9bMUDQMC)
!NZb9ed(at-sign)E!,3AZ5MA9)f4!D('RrX,QGKMMTB&d$rAF,NSehLj+GFlN3(rrP50N3'
KaTrl#jRBBik6eh'3!'ijZ5MAH,NT+CB$99)pNpG&Q,NSeh#j+0GiZ5RA1j%"rrj
aN!"Z1ENSehLj+5Q1Ra8!!)f4*i!!CQpbPJ2UU'&XE*0MEfjcG'&ZN!#XFR4cNpG
MZ5k1SBf4*i!!5(at-sign)k9V(*fQrpBj'&bD(at-sign)&ZNh5(at-sign)"!SIG'KPEh*jNfPcNf0[EQ0PFQj
PC*0hDA4SNh4SCC0`FQpLE'9YNfpQN`aZC'PZCj0KE'b6D(at-sign)k9V(*fQ'&bD(at-sign)&ZNh4
cPJ3+(fpQNf'1TSf4*i!!CfPfQUabC(at-sign)k(at-sign)!qUSFf9dNfpQNh#3!&11EfajEQpYD(at-sign)&
XFbb6BA16GjKPE'b6BA16G'KPDA+6FfPREQN-Bf&ZBf8ZMSkI(J!!MC)!qQpJ-MD
1MSb,!!!!'`!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"$N5
J!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j9fKKG*B&MjTTFj0YC(at-sign)&ZQUabG*0
LQ(Q6G'KPNb*cD(at-sign)GZD3aMB(at-sign)jMC5+6EfD6B(at-sign)k6D(at-sign)kBGT(r(at-sign)14KFQPKETKd2j%+*lC
ANIm&9Q(at-sign)6GfPXE*0KF(#3!&11C(at-sign)&XNh4[MU31J!#0N5H!!%KPFQeKEQk(at-sign)"+MA9j(
r"9CPH(at-sign)`ZN3GcEL*&GTUXFQ9bH5+6F(*[F*!!8ijPFR5BHC0[CT0`N!"6MQpXH(at-sign)j
[E(at-sign)PKE(16GfKTBjKSNfPcNfPZQ(D4reMNBA*TB(at-sign)kBG*0eEQ4PFSkKMC%RJ!"dD'(at-sign)
(at-sign)!lqLCh*[GA#6EfD6G(*KER0XBA4TEfjcNfPcNf9iF(*PFh0PC*0LQUabHC0dD'(at-sign)
6GT(r(at-sign)14KEQPcD'PZCj0[CT0KNh0PG*0[CT0TETKfNIpBj'&bD(at-sign)&ZQ(4c,SkKMC%
RJ!"*ETB#fE4[G'KPFT0hPDabEh*NFbb4!a",)Q&ZNhNLPJ,CY(0PG*0[CT0`N!"
6MQpXH(at-sign)j[E(at-sign)PKE(16GfKTBjUXFQL6DA16D(at-sign)kBGT(r(at-sign)14KFQPKETKdNh9ZC'9bNh4
bB(at-sign)jcE'&dD(at-sign)pZFikKMC%RJ!"TFjB%H8&dD'(at-sign)6Ff&YCC0cCA56BA16BC0cCA56EfD
6F*!!8ij[E(PZEfeTB(at-sign)acNfpLG'&TEQ9NNf+3!+abHC0cCA4dD(at-sign)jRNh4[NhTPFQq
6BC0cCA56EfD1SBf4*i!!D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh4cPJ2UU'pQNh0eBj!!V(*
SNh#3!&11EfajEQpYD(at-sign)&XFbk1RaG`,Bf4*i!!5A5(at-sign)!qUSDA16D(at-sign)e`QP11Eh0cD(at-sign)*
XCC0dEj0eEQ4PFR0dB(at-sign)jNNh4SCC0KBTK[PDabGT0PQ`2UU(0dBA4PE(at-sign)9ZNh5BGfP
dD'peG*KPH'&YF'aPFbk1SBf4*i!!6'9dPJ-AjA9cNf0[ER0TC'9bNh4SCC0cD(at-sign)e
`E'9cG*0KEQ56EfaNCA0dNf9iB(at-sign)e`E'8ZN36bRe4SCC0`FQp`N!"6MQ9bG*!!V(*
jNfpQNf'6FA9KC(*KG'PMMU'0N5H!!(#3!&11EfajEQpYD(at-sign)&XMU'KMC)!XJHXeh'
3!'ijZ5MAH,NTPJ098Mf6ehL0RrX,QG8bMT%(DUbj+jB#UUJbef+0R`(-c08aMTX
%`!6AH*1j+j2ABSfI!Fc-e6+1Q0FkMUN5Y#'0N5H!!,P[CTB&,(GSBCUXFRCTEQH
6BC0NEh9LE'(at-sign)6FQq3!&11Eh56DA16D(at-sign)kBGT(r(at-sign)14KFQPKETKdNh9ZC'9bNh4bB(at-sign)j
cE'&dD(at-sign)pZFcZ4"FeHD(at-sign)k6Eh4SCA+6GjK[FQ4c,*%&I1TTCSkKMC%RJ!"dD'(at-sign)(at-sign)"&N
IF*T6MQpXH(at-sign)j[E(at-sign)PKE*2AFC!!EMQj+0GiZ5Q6D'&cNf'6C'peBQaPNh*[Q'pd,*%
%G,acEj0NEjKPFj0dD'(at-sign)6F*K[E(PZEfeTB(at-sign)b6eh'3!'ijZ5MAH*B#pGfj+j2ABlN
TN34C(fC[FSkKMC%RJ!"KET(at-sign)XFRQE")`UBfpZFh4KET0dQ0GMQ,NZN3FGC8D4r`9
(at-sign)EfaXEj0hD(at-sign)jRQ%KPFQeKEQkB9j(r"9CPH(at-sign)`XN35dLRH6CCKXEj!!8ij[DjKQEh+
BB(at-sign)kBD(at-sign)k6GT(r(at-sign)14KFQPKET0dQ(GSEh0PMU'0N5H!!(D4reMNB(at-sign)jTFfKTEQH(at-sign)"%(at-sign)
6CAK`FQ9cFf9cNh4SDA16F(*[F*!!8ijPFR5DV(*jNIm&9Lk4"NQL8h9bCC0PEQp
eCfJXN34F6QPdNfPcNf9KFhQ6G'q6BjKSC(at-sign)1BDj0dD'&dNh4SCBkKMC%RJ!"NDA0
MFQPYD(at-sign)jKET!!V(*dMU'KMC)!ZiHZed53!&11Z5MABSfI!Fc-e6'1PJ6!"0FlN3(
rrQ+0R`(-c08bMT1j+CB$99)pNpGLMCrl#jQ0e6+1R`IVfSdaMSk4"fUXe!#4!UU
Sef+0R`(-c08bMSkQMC%RJ!#jDA1(at-sign)!qUSG'KPNf4PFfPbC(at-sign)56D(at-sign)k9V(*fNIpBj'&
bD(at-sign)&ZNh3ZMU'0N5H!!&4SDA1(at-sign)!q0FCAKKEA"XC5bE!q65C(9PNh4[Nd*[N!"6MQp
XC5bBGjUXFQ&cNh4SCC0cF'&bDj0dD'&dNfaPC*0dEj0dD'(at-sign)6BQPbG'L6EfD6D(at-sign)k
BGT(r(at-sign)14KFQPKETKdMU'0N5H!!(4SC(at-sign)pbHC(r"9BZMTmAF#f0N5H!!%pZCCB%Ira
[CR4PET0SC(at-sign)&bFj0dD'(at-sign)6Ff9ZQUabG'9ZBf(at-sign)6)NKTE'+3!&11CA*dNfYTE'aPC*0
TETKfNIpBj'&bD(at-sign)&ZQ(56G'KPEh*j)Lb4"+94FQ9`N!"6MQ9KG'9NNf&cMU'0N5H
!!'&ZPJ(at-sign)&A(at-sign)9iBh9cCC0dEj0TCfj[FQ(at-sign)6B(at-sign)aXNh4SBA56Gj(at-sign)XFQ9ZNh5(at-sign)"B9GEfk
6D(at-sign)k6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)"B9GG'KPEh*jNf&QG'9bNdKTE'+3!&11CA*
d,T%+#3"*MU'0N5H!!'4[ELGdPJ8Tj(at-sign)YZEjUXFRH6GfK[NfeKC'(at-sign)6GA#6G'KTFj0
TEQCKE(at-sign)peFj0cC(at-sign)kBG'9ZBf8ZQ`MfQ%PdNfPcNfj[G*0dFR9P,TK)D(at-sign)aLN!"6MQ9
bG)kKMC%RJ!"XEj(at-sign)XFRD6C(at-sign)5E!bZFD(at-sign)k6GT(r(at-sign)14KFQPKET0dQ(4SC(at-sign)pbHC(r"9B
XN304dQ&ZC*KSCCKhNf9ZNh5BEfkBF(9LE'PcD'PZCjKcG(*TDfPZCjK`BA#3!&1
1CA*cQ'PZQ'PZNhD4reMNBA*TB(at-sign)k6G)kKMC%RJ!"dD'9[FRQ(at-sign)"!SIGjUXFQ9XE*0
KCR4PFT0SCC0`FQqBGTKPC*0dD'(at-sign)6G'KPEh*PEC0dD'&dNfPcNfj[Q(HBB(at-sign)4KQ(P
cNf0KE'aPC*0dD'(at-sign)65'PXBT!!8ijPFR51SBf4*i!!BQ&cDA1(at-sign)!rM3G'KPEh*PE5b
4!raDG'KPNh4SC(at-sign)pbC(at-sign)f6G'KKG*0TFj0cGA"`N!"6MQpcC(at-sign)56G'q6D''9V(*fNf(at-sign)
(at-sign)!rM3DfPXE'9NNfPZPDabGT(r(at-sign)14KFQPKET0dN32id(4SC(at-sign)pbHC(r"9BZMU'0N5H
!!&0[E(at-sign)(at-sign)(at-sign)")(FEfD6G'KPNfe[Fh56CQ&cBfPZBA4TEQH6FQ9cG(at-sign)adFj0TET0TET(at-sign)
XFRD4reMNBA*TB(at-sign)k6G*B%JGadD'9[FRQ6GjUXFQ9bCC0NDA0MEjKfQ'9bC(at-sign)56D(at-sign)k
1SBf4*i!!G'KPPJ6Yp!abFh56G*(at-sign)XFRH6C(at-sign)k6G*0jQ`6Yp(Q6C(at-sign)&bFjK[CTKdD'P
cQ'0PET0dGA*jNIm&9Lb4"5l(BCKXEfjRQ(4TE(at-sign)(at-sign)BB(at-sign)CdCA+B5'PXBT!!8ijPFR5
BF(*[NhD6C(at-sign)5BD'PcMU'0N5H!!'*KFfPcN32UU(4SC(at-sign)pbC(at-sign)dZMUNAF#f0N5H!!&G
SBA5(at-sign)!iX$G'KPET0TFj0dD'(at-sign)6FQ9KFfpZNfC[FT0dD'(at-sign)6G'9YF*!!8ij[FQ&bHC0
NC(at-sign)eTFf(at-sign)6EfD6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)!iX$G'KPEh*jNfPZNh4SDA11SBf
4*i!!Bf9ZQUabG(9bH6q4"G4M6fjPPJ3HIR*PBA0[ET0TFj0dD'(at-sign)6C(at-sign)jNC(at-sign)eTBj0
eFf(at-sign)6EfD6BC0ZEh4KG'P[ET0dD'&dNfaKBjKVQ'9NNh*TCfpbNf&ZC)kKMC%RJ!"
dD'&dPJ0fp'&YEh9ZQUabG'9NNh4[NfaTG(4XCC0YEh*PNh4SB(at-sign)k6D'&ZC(HBBCK
fD(at-sign)jRNfPZNh"bD(at-sign)kBG#k4"4*29'KTFj0TFj0dD'(at-sign)6FhPYQ'+3!&11EfaTBikKMC%
RJ!"[FTB$kUKeEC!!V(*LFQ&XNfj[G'&dD(at-sign)pZ,SkQMC%RJ!"%D(at-sign)9eC'pZET!!V()
6NITcNQ(at-sign)(at-sign)!kIQGh*[G'(at-sign)6G'KKG*0SB(at-sign)aQNh4SCC0cG(at-sign)0MCA0cNfpQNf'6F'PPBf(at-sign)
6EfD6E(at-sign)&dD'9YBA4TBh16C'9`N!"6MQ9ZC(16Efk1SBf4*i!!BCB%r1Y`FQp`N!"
6MQ9bNf1DV(*SEfPMCC0[CT0ZEh4KG'P[ELk4#'qT5A56GjK[G(at-sign)aNNf+3!&11CC0
TETKdCA*PFh4TEQH6G'q6E(at-sign)&VQ'(at-sign)6BC0XDA0dNfpQNh9Z,BkKMC%RJ!"QEh*dG(at-sign)j
KG'(at-sign)(at-sign)!mP`EQpdBA4TEfjcNh4SBA56DfPXE'9NNhD4reMNBA*TEh9cNf1DV(*SBA"
dCA*cNfpQNfeKG'KPE(at-sign)&dD(at-sign)0c,*%$d"4KFj0hQ'9XE*0KFj0KMSkI(J!!MC)!qQp
J-MH1MSb,!!!!(!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"'fLJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#jE'PcG*B$mqC[CT0QC(at-sign)aTBfP
dEh9cNfj[G'&dD(at-sign)pZFj0dD'&dNh"bEfe[G'9NNh4SCC0NCAD9V(*PE'p`E(at-sign)9ZNh5
(at-sign)!r2QEfD6Eh4SCA+6BR*KEQ13!+abD'9cMU31J!#0N5H!!'pQPJ4&8QeKG'KPE(at-sign)&
dD(at-sign)0c,T%'50j8D'(at-sign)6FhPYQUabBT!!8ij[E'PMNfpbNh9YQ'*bB(at-sign)b6EQpdBA4TEfk
6G'KKG*0hQ'&cNh9cC(at-sign)56BTKjNfPZQ(D4reMNBA*T,BkKMC%RJ!"KETUXFR5(at-sign)"2Q
VG'KPEh*TFh4cNh4SFQpeCfL6G'KPNfjTEQ9dC(at-sign)9ZNh5BGjKPETKdD(at-sign)9cNhHBBA1
6BC0MBA4KFh4bEh"SC5k4#'AS3C%%q(at-sign)9ZQ(9YQ'+3!&11CA+1SBf4*i!!EfD(at-sign)!pV
!E(at-sign)&dD'9YBA4TBfPKER16G(*TC(at-sign)56G'q6E(at-sign)&VQUabCC0cC(at-sign)jcCC0[CT0dD'(at-sign)6FhP
YQ'+D8ij[E'PMNfePG'K[Q'56GfPdD'peG*0cG(at-sign)-YMU'0N5H!!'0PFh-XQ`81Ph4
SCCB%e$4dD(*PCC0YEh0dNfj[G'&LE'(at-sign)6EfjPFj0LN!"6MQ9TEQH65'9bE(at-sign)&ZET0
ANIm&9Q9jE#bB4A*TBj08NIm&9Q9YF'aPNd*PE'`XMU'0N5H!!'&ZC*B%8`C&C(H
3!+abBA*NNdKPCf9XCA+63f&bGA-ZN3Caq89bD(at-sign)169*(r"9CPEA"XCC0#C(at-sign)aXNfC
KD(at-sign)aPC*0dEj0`FQp`N!"6MQ9bE(Q6C'8-EQ(at-sign)6G(at-sign)dYMU'0N5H!!'*bB(at-sign)b(at-sign)""*"EQp
dBA4TEfiXN33F*f&ZC*0SDA16BT96MQq6EfZ(at-sign)""*")N&XCf9LFQ&TBj0"FQPdD'e
PG'PM)T0bC(at-sign)eKD(at-sign)jcNh4[Nh4SDA16C''3!+abHC0dD'(at-sign)1SBf4*i!!BT96MQq6EfZ
(at-sign)"&`0EfD6Ff9fN!#XFQ9ZNh0PB(at-sign)ac,T%'M3p*CT0)CA*YB(at-sign)jZNeHEr`9(at-sign)CAPXNf&
ZC*0&FQPMNe5BC(at-sign)e`E'(at-sign)63Q9XE*0SB(at-sign)56E'PfQUabC(at-sign)56$'CdQ(Q1SBf4*i!!HCU
XFQ9KFR1(at-sign)"()ME'pZCf9b,*%%P!*cEj0KFj0dEj0LN!"6MQ9ZC3adNfpQNh4SCC0
NCADBC(at-sign)a[F'ePETKdNfpQNhGSBA56GjKKFj0TET0dD'9TFT0dD(at-sign)ePMU'0N5H!!'0
KE'aPC*B#hJ)LE(at-sign)q3!&11C'9bEL+6B(at-sign)aRC(at-sign)*bB5b4!a1pG'KPHC0hQUabEh9XC*0
eEQ4[G(at-sign)*dC(at-sign)4XHC0SBCKfQ'(at-sign)6Fh9MBf9PC'9NNfPZNh"bEh#3!&11CA*XHBkKMC%
RJ!"NC3aZD(at-sign)jRPJ2UU(9YN!#XFQ*bB(at-sign)b6EQpdBA4TEfiZMTmDJ!#0N5H!!%PZPJ9
,K'peFT0NBCUXFRQ4r`9(at-sign),*%&SlYTG*0NEj!!8ijPFj0ZEh56G'&VQ'(at-sign)6ECKeBjK
SNhHBEh*VNh4[Nf&MBfpYF'aTFfL6G'KTFj0dBA0V,T%*(at-sign)h0%Ej0ZEh51SBf4*i!
!BT!!8ijPPJ9#EQ&XBA*YC(at-sign)3kN3ISDfPdNhGTE'b6EfjXHC0dB(at-sign)ZDV(*PNf'6CQ9
hNfeTETKeG'9c,T%*3$&#C(at-sign)C[FQ(at-sign)65C%&3K9cG'&bG*0cF*!!8ij[GA4TEQH6Eh9
dMU'0N5H!!'4P$'jTG'P[ER-XN39k,QaPG*B&+NGYCC0cBCUXFRQ6GfKKG*0*N38
Tp'&YNfj[G*0REfPZCj0dEj0cBCKjNIm&9Lk4#2Hm9(at-sign)fBBR*KE*0ZEh4KG'P[ET0
MB(at-sign)k1SBf4*i!!BTT6MQ(at-sign)(at-sign)"++%FfK[N!#XFRGZNh4[Nf+BCC0PFA9TGT(r(at-sign)14KE'9
ZN!#XFR3XQ`63HfpbNb*MFRP`G'pYEh*`D'PM)LbBG'q6GA0PNf'6G'9bEC0TET(at-sign)
XFRD6C(at-sign)k6G'9NN35LK'+6HBkKMC%RJ!"YN!#XFRQ(at-sign)"+Z"E'&dCC0QFQPPEQ564f&
bFQ9dG*0#DA*VD'm,,*%%flGdEj0KEQpdD'9bNfj[G'&dD(at-sign)pZNh4SBA56D'&cNfG
KD(at-sign)jPC*0RFQ9KG)kKMC%RJ!"ZEh4[FQPPG*UXFRQ(at-sign)!j-3D(at-sign)k6Eh9bNf4KQ(NkN38
0&%QE!j,kE(at-sign)9KET0dD'(at-sign)6EQpdBA4TEfk6EfD65'p`CT0KE'GPBR*KFbk4"4ZY5CK
hD(at-sign)aXNfj[G*0UGA0dD(at-sign)CjMU'0N5H!!(4SDA1(at-sign)!dZ$8hPLD(at-sign)aXD(at-sign)jPNh"bEfj[G(at-sign)j
MC(at-sign)ePET!!V(*d,*X$DeKZEh56BT!!8ijPBf&eFf(at-sign)6DA56DA16C'N1Bh9XG*0dEj0
NEj0cEbbBBR9dNf+3!&11C(at-sign)0KGA0PMU'0N5H!!'PdPJ2UU(H3!+abEh9XC*0LQP1
1CC0dEjK[Nf+BEh*TEQH6G'q6C'q6FfmZMU3DJ!#0N5H!!%aPG*B$kUKeFj0REj0
[ET0dEj0dD'(at-sign)6C'8-EQPdD(at-sign)pZNfpQNh9YN!#XFQ*bB(at-sign)b6EQpdBA4TEfiZMU'0N5H
!!&0TC'(at-sign)(at-sign)"#K8BTUXFRQ6FfPNCC0hDA4SNh4SCC0`N!"6MQpXH(at-sign)j[E(at-sign)PKE(16eh#
j+0GiZ5Q6B(at-sign)jNNpGaN!"Z1ENSehLj+5b4"$I!GjKPNf0[ER0TC'9bNf&ZEh4SCA+
6F*!!8ij[E(NYMU31J!#0N5H!!'j[E(at-sign)PKE*B&D'*KE'GPBR*KNp0$Z9[AH$Z(at-sign)!Ir
q#j!!%imlN`bE!+T2Z9f(at-sign)"(at-sign)KLD(at-sign)k6G'KbC(at-sign)(at-sign)6GT(r(at-sign)14KFQPKBQaPFj2AH*1j,*%
&ap$A#j%&Hr'jB(at-sign)jNNpF-Q,NXN3A(d(4[Cf9dD'9bNhGTG'L6BBkKMC%RJ!"XD(at-sign)j
PBA+(at-sign)",T2CR9ZBh4TEfjKE*2A4C%&EQDjC'8-EQ9NNfpZNh4SCC0eEQ4PFQajD(at-sign)j
RNhD3!+abC(at-sign)0dEh+6Fh"KBf(at-sign)6dd1j(at-sign)pGi1jB"rri,N!!6McZ6$*%!UNqjA5k4"kI
99'KPMU'0N5H!!'4P$'jTG'P[ETB$k2j[CT0dD'(at-sign)6E'PZC(at-sign)&bNfCeEQ0dD(at-sign)pZB(at-sign)b
6ed(at-sign)4"*d9Z(at-sign)PcNh4SCC0VQUabCAQ6F*!!8ij[D(at-sign)kBG#k4"6K55A56DA16Bf&bFQP
PC*0[GA56D(at-sign)k6G'KPMU'0N5H!!'C[E'a[N!#XFRGTEQH4!qUSFh4PF(-kMUNDJ!#
0N5H!!&0dCA#4!qUS-5k4"6MJ8f9dMU'KMC)!cN"Xed(at-sign)4!,3AZ5MAH)fIq`ZCf'U
1N34Q#VNTPJ098Mf6ehL0RrX,QGKUMSkI&3!!MC%RJ!#jCQpbPJ4Rc(at-sign)&XE*0ZEfk
6EQ9RBA4TGTUXFQ(at-sign)6D(at-sign)kBG'9RCA*cNpGUN388S,NXN35(&fPZNh"KFR4TBh9XBA+
6ed(at-sign)4!,3AZ5Ja+CB%+P8pNc%ZN3D`8&4SQ(9c,*%%KaGdD'(at-sign)(at-sign)"'I0FQ&ZCf(at-sign)6EfD
1SBf4*i!!G'KPPJ2UU'aTEQ9KFT0QG(at-sign)jMG'P[EQ&XNpG&N35H[lPTFj2A3j%!h*k
j(at-sign)pGiZ9dZMUD0N5H!!&0dCA#4!qUS-Lk4"6MJ8f9dMU'KMC)!c(!8ed(at-sign)4!,3AZ5M
A#if3!"12RrX,QGKUMT%%HCQj+CB$99)pNpGKMCm"c-cBDSk4"'B+ecZ1Ra8!!)f
4*i!!Z(at-sign)PZPJ2UU("KFR4TBh9XBA)XNhHDV(*PNfKKQ(DBCC2A4C%!Y"Hj+0F,MC!
!%iqIqk8bf'U1N34jQENTPJ098Mf6-*B$kUKTCT2ADT%%!L8qN3098Qkj,Sk1Rai
!!)f5!2T[B$)iMSk-L`!!!"d!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!5HSS!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if4*i!!Z90dCA#4!qUS-bk
4"6MJ8f9dMU31J!#KMC)!c5C0ed(at-sign)4!,3AZ5MA$)f4!+T2RrX,QGKUMT%&%&Qj+CB
$99)pNpGLMCm"c-cBDSk4"'B+ecZ1U48!!)f4*i!!Z(at-sign)PZPJ2UU("KFR4TBh9XBA)
XNhHDV(*PNfKKQ(DBCC2A4C%!Y"Hj+0F-MC%!UNqIqk8bf'U1N383(at-sign)ENTPJ098Mf
6-*B$kUKTCT2ADT%%!L8qN3098QZ3!'FGZ5k1RaU!!)f4*i!!8h4PF*B$kUJd,TX
&11"8D'PcNfPcNh4SCC0YB(at-sign)PZNh0dCA!ZQ&0PG)kKSBf5!+2GEGG&Q`#d&lNSe`Z
0N!!6Mjrl#jRBDBk4!hKTe`b0N3#U6jrl#jRBDSk4"4"CehL0RrX,QGKJMT%$q1D
j+CB$99)pNpG&Q,NSe`Z0N!!6Mjrl#jRBDBk4!hKTZ5RA4CLj+0F-MC%!UNqIq`Z
Cf'U1N383(at-sign)ENTehL0RrX,QGKJMT%$q1EA1SkQMC%RJ!#j4T(r"9C[E'a[PDabGfP
ZCjX$+J46H(at-sign)afNf9cG'9b,*%$8)YdD'(at-sign)BGT(r(at-sign)14KFQPKBQaPFjMA#j%$2C1jB(at-sign)j
NQ0F-N3288lPKFQ(at-sign)BBf&XE'9NQ(9YNf*bB(at-sign)8ZN36iU8PZQ'pdD'9bQ(H6Eh*NFbb
1SBf4*i!!G'KPPJ2UU'aTEQ9KFT0QG(at-sign)jMG'P[EQ&XNpG&N35H[lPTFj0YPDabG(at-sign)a
dDA"XD(at-sign)0KG'PfNf(at-sign)(at-sign)!qUSEfk6C'PcG'PZBh56G(at-sign)f3!+abBR*KC5k1U4U!!)f4*i!
!8h4PF*%$kUJe,SkKMC%RJ!"&H(4PEQ5(at-sign)!qUSBTUXFRQ6E'PZC(at-sign)&bDA5BHC(r"9B
ZMUD0N5H!!&4SDA1(at-sign)!qUSBfpYF'aPG'9cNh4SCC0NC3aZDA4TEfk6EfD6G'KPNfa
TEQ9KFT0QG(at-sign)jMG'P[EQ&XNpG&N3#d&lNZMU'0N5H!!&H4r`9(at-sign)CCB''IjZCAKdNf0
[E(at-sign)(at-sign)6G'q6G'KPNfe[Fh56C'PcFA9TCA4TEQH6CQ9KG(9bCC0[CT0eEC!!V(*LFQ&
XNfj[G'&dD(at-sign)pZ,T%,aZ*-CA51SBf4*i!!efD4!8IrZ5MA#jS6McZ(at-sign)!Irq$*%!UNm
lNhLj+CB&TIpKEQ56efH3!'ijZ5MA#jJlPJ(rrJb4!+T21j0iZ5Q(at-sign)"DArBTT6MQ(at-sign)
6G*(at-sign)XFRH6EjB&TIp`Q'pXH(at-sign)j[E(at-sign)PKE(16D(at-sign)k6G'KPNhD4reMNBA*TB(at-sign)*XCA16e`Z
3!"121jB"rri-N3#U6cZ6H,NZN3TUj9H4r`9(at-sign)CBkKMC%RJ!"hFQPdCBkKSBf5!,+
q40GQN3&(rlNSe`ZD%imlPJ(rrJb4!+T21j0iZ5Q0MCrmUe#0MC%$99,8')k1R`2
8X)f0N30Z1VNpMSk1MT%2rr[ACj!!EMQj+0F,Q$Z6$*%!UNmlNhLj+BkT&3!!MC%
RJ!"dEj%$kUKYC(at-sign)&ZMU'KMC)!RUXLed(at-sign)E!,3AZ5MACT%"4rqj+0F,N!!6McZ(at-sign)!Ir
q$*%!UNmlNhLj+5Q(at-sign)!e952C2A4CLj+0GRN!"Z1ENSe`Z3!"121jB"rri-N3#U6cZ
6H,NT+GFkMUD0N5H!!,P5C(at-sign)&NMBfIr+Y3MBf4!qUSe"L1MTm$e,#0MC%%!j!!Z6f
1MSk1N4%UTf&cPJ2UU#*PFA9TGT(r(at-sign)14KE'9ZN!#XFR56G'mL,SkKMC%RJ!"8D'(at-sign)
(at-sign)!qUS)Q0XBA0cD(at-sign)0c)T0hPDabC(at-sign)k6G*B$kUKKNf*TG*0dEj!!8ij[NfCKFLb6G'K
PHC0hFQpdCBkKSBf5!,,A,0GQN3&(rlNSe`ZD%imlPJ(rrJb4!+T21j0iZ5Q(at-sign)!e9
52C2ACj!!EMQj+0F,Q$Z(at-sign)!Irq$*%!UNmlNhLj+BkQMC%RJ!"dD'&dPJ0qR'Pc,*%
$P$KdD'9jNh*PF'aKBf9NNh4SCC0cH(at-sign)fDV(*LN!"6MQpXMBfIr+Y3MBf6e"L1MTm
$e,#0MC%$Pi5j2Bk1MSk4%&+2BTKjNfpbC'PZBA*jNf9aG(at-sign)&XDA5BHC(r"9BZN38
8h&4SDA16GjKKFj0KET0PH'0PFbf1SBf4*i!!FfPfQUabCCB$+6&KBR9cCC0[CT0
ZEh4KG'P[ELk4"2KM9'KPNb*ME'&cFfPMFb+6GjKPFQ(at-sign)6BCKhQ'&bCC0[CT0dD'(at-sign)
6CA*bEh)XN302iQ&ZC*0hD'PXCC0dD'9jMU'0N5H!!''9V(*fNfpTC'9NPJ6*2f0
[EA"eG'&dD(at-sign)pZB(at-sign)b6CA*bEh*cNf+DV(*jNf0XCADBCA+6BA*dDA0dFRQ4r`9(at-sign),*%
&!19dD'9jNhHBCA*PNh9ZB(at-sign)*XCC0dEj0cCA4dE'(at-sign)1SBf4*i!!Efk(at-sign)!qUSBC0MEh*
bC(at-sign)0dNfj[G'&dD(at-sign)pZ,SkI'S!!MC%RJ!"8D'(at-sign)(at-sign)"%8BG(at-sign)fDV(*LFQ&XNfpbNh0jECK
LQP11EfaTBj0YCA4SEjKNNf0[ER0TFh4cNfpQNh*PF'aKBfPZCj0KE'b6EjKMBh9
bFQ9ZBf9cNfpQNh4SCBkKMC%RJ!"MEjT6MQ81BfPPET!!V(*dFjB%pQ*[CT0dD'(at-sign)
6F*K[E(PZEfeTB(at-sign)acNpG`Z5MAH,NTNf&ZC*2AFC!!EMQj+0GiZ5Q6BTUXFRQ6G(at-sign)f
BBR*KCC0KEQ56CA&eDAD4reMNB(at-sign)aPEQ0PFbk1SBf4*i!!4T(r"9C[FT%$kUKPH'&
YF'aP,)kKSBf5!-(P(at-sign)YG`Z5MAH,NTMBfIr+Y3MBf4!e95e"L1MTm$e,#0MC%$EMU
j2Bk1MSk4$rrl+0GiPJ+UU,NVNpF,N!!6MlNTMCrl#jRBESk1TSf4*i!!Z(at-sign)&ZC)k
1Rai!!)f5!2T[B$)jMSk-L`!!!"i!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!6+HS!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#k31J!#0NJ$!a`6AFC!
!EMQj+0GiZ5Q0MCrmUe#0MC%$99,8')k1R`28X)f0N30Z1VNpMSk1MT%2rrXSehL
(at-sign)!UUSZ5Z6e`b4!+T2Z5Q0RrX,QGKVMT%&)j,A1SkT&3!!MC%RJ!#j6'9dPJ2UU(9
cNf0KFQ9QG(at-sign)aXHC0MPDabD'9MNfZ(at-sign)!qUSG'KPN`abFh56CA&eDAD4reMNB(at-sign)aPEQ0
P,SkKMC%RJ!"#HCB$kUKNC3aZDA4TEfiXNh4SCC0PFA9TGT(r(at-sign)14KE'9ZBf(at-sign)6E(at-sign)9
KER16G'KPNh0KE(at-sign)(at-sign)6BA11SD'0NJ#YdMMA4CX!Y"Hj+0G`Z5MAH,NT+CB$99)pNpG
&Q,NS+0GiPJ+UU,NVNpF,N!!6MlNTMCrl#jRBESk4"DK3Z5RA1SkQMC%RJ!#j8fP
ZBf(at-sign)E!ZlRed(at-sign)4!,3AZ5MAH)fIqk8bf'U1N34Q#VNTPJ098Mf6ehL0RrZP-YKUMT%
(92'jCQpbQ'&XE*KZEfkBEQ9RBA4TGT(at-sign)XFQ(at-sign)BD(at-sign)k6G'9RCA*cQ0GUN3#XdlNXN3-
K3(4SDA1BD(at-sign)4PET0dDA56HCKMB(at-sign)kBBT!!8ijPQ(*PGh*TG(4PESkKMC%RJ!"KFik
KSBf5!,FA(at-sign)YG`Z5MAH,NTPJ098Mf6ed(at-sign)4!,3AZ5JSehL(at-sign)!UUSZ5Z6e`Z3!"12Z5Q
0RrX,QGKZMT%&U&#j+GFkMUD0N5H!!,P&H("KEQ4TEQH(at-sign)!qUSG'KPNh*TCfLDV(*
dNfKKEQ56FfPNCC0LQ(Q6G'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)dXNhHBCC0[BR4KD(at-sign)k
1SD'0NJ$&2`cA4C%!Y"Hj+#MAH*B#UULj+j2A#j!!%iqj+BfIq`ZCf'k1N3(at-sign)S8,N
TN3098Mf1Rab!#Bf46&NAed(at-sign)4!,3AZ5MAH)fIq`ZCf'k1N3K5q,NVMBf0N3+UU*r
ZCPbV))k0RrIKaBf4#T98efk1Ra"CHBf4#bFEZ6'1MSf4%CMGRqjQA+XKMSk1PKQ
$LGF,QK12H)fIq`ZCf'lD!08aMT%6,h5j+if0MC%#UULIlQCFUb#1MCrhiF(at-sign)0N3U
990GZMTm3(at-sign)AQ0N3XR'lNbMSk0N4'BhCrZCPbV)Bk1MT2A#ifBRrX,QG8bMT%%dj2
AH)fIq`ZCf'lD!08bMT%6,h5j+if4!UUSecU(at-sign)!Irq1T-kMT%6(r#j+if0MC%#UUL
IlQCFUb#1MCrhiF(at-sign)0N46DTYGZMTm3(at-sign)AQ0N3U99'k(at-sign)!UUSe!#6Z6'1MSf4*L1!Rqj
QA+XKMSk1N5i1,0F,MCLIq`ZCf'lD!08aMT%3Q&[AH*B#UULj+j2A#ifBRrX,QGK
ZMT%&Zpqj+BkI'p9HMC%RJ!"#HCB$kUKXD(at-sign)jPBA*TG*!!V(*jNh4SDA16CA&eB(at-sign)a
cMU'I(`!*MC%Zjq[AH)fIq`ZCf'k1N3K5q,NVMBf0N3+UU*rZCPbV))k0RrIKaBf
4#T98efk1Ra"CHBf4#bFEZ6'1MSf4%CMGRqjQA+XKMSk1PKQ$LGG&Q`#d&lNSe`Z
3!"12Z5RAH)fIq`ZCf'lD!08aMT%6,h5j+if0MC%#UULIlQCFUb#1MCrhiF(at-sign)0N3U
990GZMTm3(at-sign)AQ0N3XR'lNbMSk0N4'BhCrZCPbV)Bk1MT2A4CLj+0F,MC!!%iqIq`Z
Ce6+1N366NlNTehL0RrX,QGKZfJ$9-Sk4%bpdZ5Z0N3+UU0FkPJ(rrMU61Sk4%ar
`Z5Z0MBf4!UUSRqjQA+XJMSfIpq(&MC%8fUEAESkI%&PjMC%+P94ZPJ+UU03!NlN
aMSk0N5BMJ*rZCPbV)Bk1MT%Z$LcA4CLj+0F,MC!!%iqIq`ZCf'lD!08aMT%3Q&Z
j+GGiPJ+UU,NVNpG&Q,NSe`Z0N!!6Mjrl#jRBESk4"E[IZ5RA1SkI)99HMC%RJ!#
j4AD4reMNB(at-sign)aeBA4TEQH(at-sign)!qUSG'KPNfaTEQ9KFT0QG(at-sign)jMG'P[EQ&XNpG&N3#d&lN
XNhH3!+abCC0cC(at-sign)(at-sign)6G'KKG*0dD'PcNfPZNh4eFQk6CA&eB(at-sign)acMU'I%`!*MC&8Arr
AH)fIq`ZCf'k1N3K5q,NVMBf0N3+UU*rZCPbV))k0RrIKaBf4#T98efk1Ra"CHBf
4#bFEZ6'1MSf4%CMGRqjQA+XKMSk1PKQ$LGGKMCm"c-c9-BkE"-!%ehL0RrX,QGK
ZfJ$9-Bk4%bpdZ5Z0MBf4!UUSRqjQA+XJMSfIpq(&MC%+P96AESkI%&PjMC%,*aZ
j-Sk1MC%4Q0fIlQCFUb'1MSk6ef'0R`(-c08bMTMAH)fIq`ZCf'lD!08bMT%6,h5
j+if4!UUSecU(at-sign)!Irq1T-kMT%6(r#j+if0MC%#UULIlQCFUb#1MCrhiF(at-sign)0N46DTYG
ZMTm3(at-sign)AQ0N3U99'k(at-sign)!UUSe!#6Z6'1MSf4*L1!RqjQA+XKMSk1N5i1,0GKMCm"c-c
BEYS!e6'1N4#%c0GiPJ+UU,NVNpGKMCm"c-cBESk4"DK3ecZ1Ra[9ASf4*i!!Z(at-sign)&
cN32UU'4PFfPbC(at-sign)3ZMTmDJ!#0N5H!!&4SCC%$kUKPH("bCA0cD(at-sign)pZMU'KMC)!e-#
E+0GiPJ+UU,NVNpF,N!!6MlNTMCrl#jRBESk1TSf4*i!!Z(at-sign)PcPJ2UU'0KE'aPC*0
KET0eECUXFQ*bB(at-sign)b6FQ9`FQ9cC(at-sign)kBG'&dD(at-sign)pZNfpQNh4SCC0`N!"6MQpXH(at-sign)j[E(at-sign)P
KE*2AF,NSehLj+5k1SBf4*i!!5(at-sign)k(at-sign)!d3VG(at-sign)fDV(*LFQ&XNfj[G'&dD(at-sign)pZ,*%$CAK
KNf0[EA"XCAL6ETKeECKLQP11CA+6eh+4!jHjZ(at-sign)PcNf'6FQqBEh56EfD6G'KPNh#
BEfajEQpYD(at-sign)&XNf9aG(at-sign)&dD(at-sign)pZMU'0N5H!!0G`Z5MAH,NTPJ098Mf6-*B$kUKTCT0
KEQ56EfjXHC0TCSkKSBf5!-LepLMAFT%#rMDj+j%#UUMA#j!!%iqj+BfIq`ZCf'k
1MBfIr+Y3MBf4#2fLe"L1MTm$e,#0MC%*&SUj2Bk1MSk4&DK,-0FkMUD0N5H!!,P
6D(at-sign)eTE'&bE(Q4r`9(at-sign),*%')GKTETB&X'KeECUXFQ*bB(at-sign)b6EQpdBA4TEfk6G'KPNh#
3!&11EfajEQpYD(at-sign)&XNpG8MC%"SFDIqk8bf'11N3A32pG`Z5MAH,NTPJCCU$f6eh#
j+0GiPJ2IPlNVNpGMZ5Q(at-sign)"E"SE(at-sign)'BHC0LN!"6MQ(at-sign)1SBf4*i!!FQ9`FQ9cC(at-sign)kDV(*
dC(at-sign)5(at-sign)!qUSBA16CQpXE'qBGh-kMSkI(J!!MC)!qQpJ-c#1MSb,!!!!(`!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"1[kJ!U!SpBfJrBEA#k!#(at-sign)bM
eMD$pbGF,T!k!!)f5!+bm,GG`Z5MAH*B#UULj+j2ABlNTMBfIr+Y3MBf4!e95e"L
1MTm$e,#0MC%$EMUj2Bk1MSk4$rrl+0GiNlNVNpF,N3+q0lNVNpGMZ5Q0RrX,QGK
ZMT%&U&$A1ikT&-3XMC%RJ!#jB(at-sign)jNPJ1(at-sign),R4SDA16H(at-sign)PPE'4cNh4SCC0eECUXFQ*
bB(at-sign)b6CAK`FQ9cFfP[ET0QEh+6G'KPNf0[N!"6MQ81BfPPETKdFj2AF)fI!Fc-f'U
1N34Q#VNSef1j+C0[CT0dD'(at-sign)6F*!!8ij[E(PZEbf1SBf4*i!!E(at-sign)PKE*X$kUMAF,N
SehL(at-sign)!UUSZ5Z6ef1j+5bBEQ&YC(at-sign)ajMU'KMC)!`&"eeh#0R`(-c0KUMTB%CJUj+0G
MZ5Q0MCrmUe#0MC%$99,8')k1R`28X)f0N30Z1VNpMSk1MT%2rrXSe`Z4!VihZ5Z
4!UUSef1j+BfIq`ZCf'U1NpFkMUD0N5H!!,P-CA5(at-sign)!qf-GA16EQ9iG*0cC(at-sign)(at-sign)6D'q
DV(*hNh9YQ'*bB(at-sign)b6EQpdBA4TEfk6DA16FQ9XBA4PC*0dEj0TETKfNIpBj'&bD(at-sign)&
ZQ(4c,T%&3BY-CA56GA16BA0cG(at-sign)ePMU'0N5H!!(4SBA5(at-sign)"+DVG'KPNh59V(*hNfq
(at-sign)"+DVF*!!8ij[E(PZEfeTB(at-sign)acNpG`Z5MAH,NTNf&ZC*2AFC!!EMQj+0GiZ5Q6D''
9V(*fNf(at-sign)(at-sign)"+DVG'KPNh0KE(at-sign)(at-sign)6C'9RFQ9PNpGZZ5k4"fcU9'KPET0KESkKMC%RJ!"
TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G*B$kUMA3C1jEfD6G'KPNh#D8ij[E(PZEfeTB(at-sign)acNpG
`Z5MAH,NTecZ4!IrqFC!!EMQj+0GiZ5Q6E(at-sign)'3!+abHC0LQ'(at-sign)6C'8-EQ9NNf&cNfC
[E'a[N!#XFRGc1SkKSBf5!+Q&`YG"Z5MAFC!!EMQj+0GiZ5RA1j%"rrj`Z5MAH,N
T+Bf0RrbV8)f0N3098Y3BMSkI!p5`MBf4!fikZ6f1MSk1N3rrqbMA$*%$92I8!*%
#UUMA#j!!%iqj+BfIq`ZCf'k1N3(at-sign)S80FkMUD0N5H!!,P8D'(at-sign)(at-sign)!d84CADEreMNB(at-sign)a
eBA4TEfk6EfD6G'KPNfPZPDabGTKKFQPKET0dPJ0&%GG"NlPTET0dCA*YFj0[CT0
dD'(at-sign)6Bfq3!&11C3jMD(at-sign)9ZN!#XFR4cNfpQNpG`Z5MAH,NTNf&ZC*2AFC!!EMQj+0G
iZ5Q1SBf4*i!!F(*[N!"6MQ0PC(at-sign)4cPJ2UU'&cNfC[E'a[N!#XFRGc1SkKSBf5!*Z
qi0G"Z5MAFC!!EMQj+0GiZ5RA1j%"rrj`Z5MAH,NT+CB$99)pNpG&N3#d&lNS+0F
-N308pp3!N3+UU0F,N!!6MlNTMCrl#jRBESk4"DK3Z5Q62BkQRa-!#Bf44Ji$ed(at-sign)
4!,3AZ5MA$)f4!+T2RrX,QGKZMT%)r8I8!)f0MC%#UULIlQCFUb#1MCrhiF(at-sign)0N3U
990GZMTm3(at-sign)AQ0N3XR'lNaMSk0N4'BhCrZCPbV)Bk1MT%CJiRA$)f4!+T2RrX,QGK
ZfJ$9-Bk4%5mEe`Z4!VihZ5Z0N3+UU0FkPJ(rrMU61Sk4%ar`Z5Z(at-sign)!UUS+03!Z6%
TMCrl#jRBEYS!e6'1MBf0N4#%c*rZCPbV))k0RrIKaBf4)V6+efk1Ra"CHBf4''p
iET28!*1j-Bk1MC%crD5IlQCFUb'1MSk41qK3e`b4!+T2#if3!"12RrX,QGKZfJ$
9-Bk4%d-$Z5Z6+03!Z6%TMCrl#jRBESk4"DK3e`Z0N!!6Mjrl#jRBESk4"E[IZ5Q
4!e952BkQT"jdF)f4+MkGed(at-sign)E!,3AZ5MA$)f4!+T2RrX,QGKZMT%'8Tqj+CB#UUM
8!*2A4CLj+)f0MCrZCPbV))k0RrIKaBf4"qUXefk1Ra"CHBf4#(acZ6'1MSf4$Zi
eRqjQA+XKMSk1N4EBiGF-MC%!UNqIq`ZCf'lD!08aMT%4,a[A#j!!%iqj+C-VMC2
A1TB"rrikNcU1N4-Im,NVNbM8!,Na+BfIq`ZCf'lD!08aMT%3K-cA4CLj+)f0MCr
ZCPbV))k0RrIKaBf4%Lrqefk1Ra"CHBf4"qUXET28!*1j-Bk1MC%MH0LIlQCFUb'
1MSk4+f1%e`b4!+T2#if3!"12RrX,QGKZfJ$9-Bk4%*KEZ5Q6+j-Se!#j-5Q0RrX
,QGKZMT%&U&$A4CLj+0F,MC!!%iqIq`ZCf'k1N3(at-sign)lhlNTN3098Mf1U4RdCk'0N5H
!!0G&PJ#d&lNSe`b0N3#U6jrl#jRBESk4"P+IZ5R8!)f0MCrZCPbV))k0RrIKaBf
4"qUXefk1Ra"CHBf4#(acZ6'1MSf4$ZieRqjQA+XKMSk1N4EBiGG&NlNSe`b0N3#
U6jrl#jRBEYS!e6'1N4%['lNTed(at-sign)6Z5MA#jS6MlNT+ihA1TB"rrikNcU1N3h+S,N
V+03!Z6%TMCrl#jRBEYS!e6'1MBf0N4#%c*rZCPbV))k0RrIKaBf4)V6+efk1Ra"
CHBf4''piETB#UUM8!*1j-Bk1MC%crD5IlQCFUb'1MSk41qK3ed(at-sign)6Z5MA$*%!UNq
j+GG&NlNSe`Z0Q*rl#jRBEYS!e6'1N4#B(at-sign)lNT+bM8!,Na+BfIq`ZCf'k1N3(at-sign)S80G
&NlNSe`Z0Q*rl#jRBESk4"E[IZ5Q4!e952BkQSBf4*i!!ef+0R`(-c0KZMT%(V*c
8!)f0MC%#"%bIlQCFUb#1MCrhiF(at-sign)0N3RZq0GZMTm3(at-sign)AQ0N3U![lNaMSk0N4$bJCr
ZCPbV)Bk1MTBBh5hABSfI!Fc-f'lD!08aMTX3K-cABBfI!Fc-e6'1N3E%8,NVMBf
0N3)%6*rZCPbV))k0RrIKaBf4#Hliefk1Ra"CHBf4#S#rZ6+1MSf4%2+"RqjQA+X
KMSk1NpGLMCm"c-cBEYS!e6+1Q0GKMCm"c-c9-Sk4"X43e!#0N3)%60FkPJ(rrMU
61Sk(at-sign)%G-iZ5Z0N3)%60FkPJ(rrMU61Sk6Z5Z(at-sign)!J4-+03!Z6%TMCrl#jRBEYS!e6'
1MBf0Q*rZCPbV))k0RrIKaBf4)V6+efk1Ra"CHBf4''piETB#UUM8!*1j-Bk1MC%
crD5IlQCFUb'1MSk41qK3ef+0R`(-c08aMT%%`!6ABBfI!Fc-f'lD!08aMT%5L4L
j+j-Se!#j-5Q0RrX,QGKZMTB&U&$ABBfI!Fc-f'k1NpFkMTmJbF(at-sign)0N5H!!,PAD*U
XFRQ(at-sign)!qUSDA16ed'6Z(at-sign)&ZNfPZQ(D4reMNBA*TB(at-sign)kBG$q4"6MJ9'KTFj0TFj0LN!"
6MQ9cG*0cC(at-sign)9ZNfPZNh9YQ'*bB(at-sign)b6EQpdBA4TEfikMU31J!#KMC&V)d2A3ENSee5
0N3'KaTrl#jRBBik(at-sign)"G!reh'3!'ijZ5MAH,NTecZ4!Irq9)f4!D('RrX,QGKMMT2
AF,NSehLj+5Q0MCrmUe#0MC%$99,8')k1R`28X)f0N30Z1VNpMSk1MT%2rrXSe`b
E!e6hZ5Z(at-sign)!UUSef16e!#6e`Z4!Vihe!#6ef1j+BfIq`ZCf'k1N3MpSVNpN3098LM
A$*M8!*2A#j!!%iqj+BfIq`ZCf'k1N3(at-sign)S80FkMTm8a#b0N5H!!,P8D'(at-sign)E!d#CD(at-sign)k
9V(*fNIpBj'&bD(at-sign)&ZNh5Bed'BZ(at-sign)PcQ'0KE'aPC*KdD'(at-sign)BBA#3!&11EfaKFTKTET0
fNIpBj'&bD(at-sign)&ZNh3lN30j5A56Gj0[Q(#3!&11EfajEQpYD(at-sign)&XFjMAF,NSehLj+CK
KEQ5Beh'3!'ijZ5MAH,NTMU'0N5H!!'KKQUabGQPZCjB$p'KdD'(at-sign)6F(*[F*!!8ij
PFR5BHC0dD'&dNpG"Z5MAFC!!EMQj+0GiZ5RA1j%"rrj`Z5MAH,NT+CB$CHXpNc#
(at-sign)!r4SBA*PNh0KD(at-sign)56G'q6BTT6MQ(at-sign)6BA#BEfaKFLk4"9BK5(at-sign)k6G(at-sign)f3!+abBR*KE)k
KMC%RJ!"ZEh4KG'P[ELbE!qUSG*(at-sign)XFRH6EjK`P911EfajEQpYD(at-sign)&XFjKKFQ(at-sign)BBA#
6EfaKFTKhD'9ZCAD3!+abCA+1SD'0NJ$(aQ8Se`b4!e6he!#4!UUSe`Z3!"12Z5Q
0RrX,QGKZMSf0RrbV8)f0N3MpSY3BMSkI!p5`MBf4#4D+Z6f1MSk1N4(at-sign)S5c$A1Sk
1Rai!!)f5!2T[B,Nc-Bk1M)X!!!!J!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!&%NU!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,P8D'(at-sign)(at-sign)!iY
2BfpZBf9`G*0[CT0KF*T6MQpXBA*TG*!!V(*jNfKKFj0KNf4TFh4TEQGeDA0SC(at-sign)5
6F*KPC'PRFQ9PNfG[D(at-sign)jRNf&XE*0dD'(at-sign)6Gj(at-sign)XFQ'6HC%$LdpLB(at-sign)16DikN$S!!MC%
RJ!"dEj%$kUK"F*!!8ij[E'a[EQPeFbk1U4S(at-sign)-)f4*i!!9fKKG*B%KmpTFj0dD'(at-sign)
6)R0TCfjT$'0KEQ0P)T0[CT0dD'(at-sign)6BA#D8ij[E'&bNfPZPDabGT(r(at-sign)14KFQPKET0
d2j%(%&9AD'&dPJ5(cf4[Q'9cNfPdNfePB(at-sign)k6CQpbMU'0N5H!!(59V(*hNfq(at-sign)!qU
SF*T6MQpXH(at-sign)j[E(at-sign)PKE(16G'q6BTKPNf&`Q'pXBA)rN38ii&4SDA16FA9PFh4TEfk
6DA16B(at-sign)jcGjUXFQ9bC(at-sign)56BTKjMUD0N5H!!008D'9[FQ9YN365Xc'j,T%'%-p6GA"
`QP11Eh0PPJ3bSR4SBA56eh+4")B`Z(at-sign)PcNf'6FQqBEh56EfD6G'KPNh#BEfajEQp
YD(at-sign)&XNpGaN!"Z1ENSehLj+5bE"%5KG'KKG*0TFbbBG'KKG)kKMC%RJ!$AFC!!EMQ
j+0GbQP11Z5Q(at-sign)!e952C-`,T%&11"8D'9ZPJ2UU(4SCC0`Q'pXH(at-sign)j[E(at-sign)PKE(16eh'
3!'ijZ5MAH,NTNf&ZC*2AF,NSehLj+CB$99)pNbMAH*B#UUM8!*2AFTLj+BfIqk8
bf'k1N3Q5q,PKFQ(at-sign)4!qUSBA#BEfaKFLk1TSf4*i!!8(*[QP11EfBZN38ii%D4r`9
(at-sign)Eh+4!qUSeh#j+0GiZ5Q(at-sign)!e952C-SehL(at-sign)!UUSe!#6eh+BZ5Q0RrZP-YKZMT%*N[L
jGj(at-sign)XFQ(at-sign)E!qUSD''6GT0PQ0F,MC!!%iqIqk8bf'U1MBfIr+Y3MBf4"mlVe"L1MTm
$e,#0MC%(jp1j2Bk1MSk4&(Q8+03!eh+3!&11Z5Q0RrZP-YKUMT%%CJUj,*KKEQ5
BD'9ZBf(at-sign)1SD'0NAP3rGG"Z5MAFC!!EMQj+0GiZ5RA1j%"rrj`Z5MAH,NT+Bf0Rrb
V8)f0N3098Y3BMSkI!p5`MBf4!fikZ6f1MSk1N3rrqbMA$*B$92I8!*X#UULj+03
!eh+3!&11Z5NTMCrl#jRBESk4#2fLZ6f4!e95+0F-NlNVQ0GbN!"6MVNTMCrl#jR
BESk0MCrmUe#0MC%)rD,8')k1R`28X)f0N3N(at-sign)LVNpMSk1MT%9U%X`ecZ1Ra5`T)f
4*i!!Z(at-sign)&cN32UU'4PFfPbC(at-sign)3ZMUD0N5H!!00$Eh*[E'aKFRQj,T%'0!P*CTB%2Q"
dD'(at-sign)6F*T6MQpXH(at-sign)j[E(at-sign)PKE*2AFC!!EMQj+0GiZ5Q6D'&cNpGZNlPNDA0dD(at-sign)jMG*0
bEjK[G(16eh+0R`(-c08aMTB%`!6A1jX"rrjbMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%
4bT`lQ(+0R`(-c0KZMT%&U&#j,*%%8djKEQ54"$jJD(at-sign)D1SBf4*i!!G'KPPJ0Bh(#
D8ij[E(PZEfeTB(at-sign)b6eh#j+0GiZ5Q6DA16BA#BEfaKFT0dEj2AFC!!EMQj+0GiZ5N
XN30f"A4SC(at-sign)k6G'KPFQ(at-sign)6CAKTFh56BfpZFh4KET!!V(*dFj2ABifI!Fc-e6'1PJ6
!"0FlQ`(rrQ10R`(-c08bMT2A1ifB1TJkQ$U1N4(+R$ZBBifI!Fc-f'k1MU'0N5H
!!,PQEh+4!qUSGfKTBj!!V(*SMU'KMC&NR"6AF,NSehLj+CB$99)pNpGMMCm"c-c
9-BkE"-!%Z5MAH*B#UUM8!*2AFSfI!Fc-e6'1Q,NTMCrl#jRBESk4#&,iZ5Z6ef1
0R`(-c08bMTLj+0GiNp3!NpGbMCm"c-c9-SkBZ5Q0RrX,QGKZMT%)8[Lj+if6ecU
(at-sign)!Irq1T-kMT%6(r#j+j2ABifI!Fc-f'k1Q`(at-sign)S8,NSehL6e!#6eh+0R`(-c0KZMTL
j+BfIq`ZCf'k1Q0FkMTmJ4Y50N5H!!,P3FQqD8ij[CLk4"abB9'KPPJ5,j(at-sign)4TE(at-sign)9
ZFfP[ET0[CT0dD'(at-sign)6B3jZCC0cG(at-sign)*cF'&MCC0[CT0KE'b6E(at-sign)pZD(at-sign)16F*K[E(PZEfe
TB(at-sign)acNpG`Z5MAH,NTMU'0N5H!!(GSD(at-sign)13!+abD*B%p5eKFQ(at-sign)6BA#D8ij[E'&bNh4
[NpGaN!"Z1ENSehLj+C0PFA9KE(16efkj,T%)(at-sign)'j#GA56D(at-sign)D6G'KPNh#BEfajEQp
YD(at-sign)&XNpGaN!"Z1ENSehLj+C0SBA16FfPYF'aPMU'0N5H!!(*[QP11Eh4c,*%$(,T
dD'9ZPJ,T2f+3!+abHC0dD'(at-sign)6B(at-sign)+BEj(at-sign)XFRD6CCB#k6pdD'9[FQ9YNh4SCC0`Q'p
XH(at-sign)j[E(at-sign)PKE(16+0GiPJ#FiY3!NpGbMCm"c-c9-BkE"-!%Z5Q0RrZP-YKZMT%&U&$
A1j%"rrkj+0GiNp3!NpGbMCm"c-c9-SkBZ5Q0RrZP-YKZMT%&U&$A1if(at-sign)!Irq1T-
kNcU1N4(+R$Z4!IrqZ5MAH*28!)kKMC%RJ!$AFSfI!Fc-f'k1N3(at-sign)S8,NTMCrlT6,
BESk4#5IaZ(at-sign)&bCCB$Ik&XD(at-sign)jPBA*XHC0TEQ4PF*T6MQ9ZC'9ZN!#XFR56B(at-sign)jNNf&
`Q'pXBA+6G'q6eh'3!'ijZ5MAH,NT,T%&&60)C(at-sign)jMCC0dD'(at-sign)6F*K[E(PZEfeTB(at-sign)b
6eh#j+0GiZ5Q1SBf4*i!!DA1(at-sign)!qUSBC0XD(at-sign)jPBA+6BfpYN!#XFQ*TEQ&dD(at-sign)pZNfp
QNh4SCA0PNh#D8ij[E(PZEfeTB(at-sign)ac,T%&11"8D'PcNf0[EA"XCA4PFj0dD'(at-sign)6F(*
[Q'pQ,SkQMC%RJ!"8D*(at-sign)XFR9c,*%%,K"hNf(at-sign)(at-sign)"##9Ff9PNh4SBA56BA#3!&11Efa
KFQPdQUabHC0RDADBCA16BC0dFQPfD(at-sign)&XNf&ZFhHBCA+6G'q6G'KPNfC[E'a[Q(G
TEQH6FA9PFh4TEfikMU'0N5H!!(GSC(at-sign)k(at-sign)!VmHBf&ZNf'6F*T6MQpXH(at-sign)j[E(at-sign)PKE*2
AF,NSehLj+C0LQ'(at-sign)6Gh*TG(4PET0KFj0KNfaTEQ9KFT0MEff3!+abBQPZBA4TEfk
6EfD6F*K[E(PZEfeTB(at-sign)acMU'0N5H!!'pQPJ2UU(4SCC0QEh*YNbMAH*B#UUM8!*2
AFSfI!Fc-e6'1Q`6!",NTMCrlT6,BESk4"DK3ecZ4!IrqZ5MAH*28!*2AFSfI!Fc
-e6+1Q,NTMCrlT6,BESkE"DK3ecZ0PJ(rrMU61T-kMT%4bT`lN3(rrVNSehL6e!#
6eh+0R`(-c0KZMTLj+BfIqk8bf'k1Q,NrMUD0N5H!!%'4"$ArBTT6MQ9KGA4TCR9
XPJ3f%h4SC(at-sign)pbC(at-sign)f6Efk6BA#BEfaKFQPdQUabHC0hQ'&cNh"bEjKfQ'9NNf+BHC0
dD'(at-sign)63R*TG'PcD*0YBA4SC(at-sign)eKG'PMD(at-sign)&ZMU'0N5H!!%T[D'k(at-sign)!qUS5'PXG'pZNdG
bB(at-sign)0P,T%&11"ANIm&9Q(at-sign)6Fh4KG'(at-sign)6DA56GfPdD'peG*0`FQq3!&11EfBkMUD0N5H
!!00(FQ&MC5GcN36BXe4SC(at-sign)pbC(at-sign)fj,T%')(T*CTX%0pYdPDabGj0[Q(#98ij[E(P
ZEfeTB(at-sign)acQ0G`Z5MAH,NTQ'&ZC*MAFC!!EMQj+0GiZ5QBEfDBC'9RFQ9PQ0GZQ,P
KFQ(at-sign)BBA#6Ebf1SBf4*i!!E'&b,*%%54KdD'9ZPJ3f0'9fQUabCA*jNf4TFfZ6D(at-sign)k
6G'KPNf0[EA"XCAL6F'aKEQ(at-sign)6BfpZQ(4KD(at-sign)jTEQH6CADBCA*jNhTPFQq6EfD6eh#
j+0GiZ5Q6B(at-sign)acEikKMC%RJ!"MEfk3!+abG'&TER1(at-sign)!qUSBA56E'9KFh56EfjPNhT
PFQq6EfD6eh'3!'ijZ5MAH,NT,SkQMC%RJ!"(FQ&MC5GcPJ5#h&4SC(at-sign)pbC(at-sign)f6DA1
6B(at-sign)k6D(at-sign)jcG'&ZBf(at-sign)6EfD6GfKKG*0YD(at-sign)GSN!#XFR56BT!!8ijPNf0KE'aPC*0KNh0
dGA*NHC0dD'9[FQ9Y,SkKMC%RJ!"'NIm&9QpbPJ3i1Q&XE(at-sign)pcG*0[EQ(at-sign)6D*UXFR9
ZC(*PC*0jQ'9KFR16DA56D'&cNh*PFfPcG'9NNf&XE*0KG(4PEA"dFj0KG*0RC(at-sign)j
PFQ&XDATKG'P[ELk1SBf4*i!!3(at-sign)aYEh0dPJ5p0'&XE*0VEQq3!+abGfk6FQ9cG(at-sign)a
dFj0KBTT6MQpeG*0dD'(at-sign)6C'PcG(*TBR9dD(at-sign)pZNfpQNhTPFQpcNfpQNh#BEfajEQp
YD(at-sign)&XFj0TESkKMC%RJ!"dD'(at-sign)(at-sign)!qUSBfpYF'aPH*0`E'&ZCC0KFQ(at-sign)6BfpbEfaXBA*
TCA16EfD64h*KBf8RFj0dD'9[FQ9Y,Sk1Rai!!)f5!2T[B$-bMSk-L`!!!#%!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!9#rS!+J+2(at-sign)0S2f'e`Z
J!PXSpBfJrFRA#if4*i!!Z9H4r`9(at-sign)CCB%6m"hD(at-sign)aXNfjPH(56Cf9ZCA*KE'PkCC0
dD'(at-sign)6BA#D8ij[E'&bNfPZPDabGT(r(at-sign)14KFQPKET0dPJ42`(4[Nh4SCC0MBA0PNfp
QNh59V(*hNfq4"%r!F*K[E(PZEfeTB(at-sign)acMU31J!#0N5H!!0G`Z5MAH,NTPJ8HH'&
ZC*2AFC!!EMQj+0GiZ5Q6EfD6C'N,CA*PET!!V(*dNf4PCh*PCA16efk6Z(at-sign)&ZC*2
ADj!!Cafj,*X&DfahDA4SNpGVN3A)Ap38N39K3YGZZ5k4#0449*(r"9C[Nh4SDA1
6C(at-sign)jN,*KhN!#XFQ(at-sign)1SBf4*i!!FfaTCfLDV(*dE(Q(at-sign)!qUSCf9ZCA*KE'PkCC0dD'(at-sign)
6C'8-EQPdD(at-sign)pZNfpQNfPZQ(D4reMNBA*TB(at-sign)kBG#b6BA16CQpXE'qBGh-ZMTmDJ!#
0N5H!!%'(at-sign)!qUSF*!!8ij[E(PZEfeTB(at-sign)b6edQ4!2'$Z5MABBfI!Fc-e6'1PJ6!"0F
lQ`(rrQ'0R`(-c08bMT2A1ifB1TJkQ$U1N4(+R$ZBBBfI!Fc-f'k1N3(at-sign)S80FlQ'+
0R`(-c08aMT2A1jKLMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ'+0R`(-c0KVMT%
&)j,A1jKiZ5Q(at-sign)!qUSD(at-sign)k6G'KPNhD4reMNBA*TB(at-sign)*XCA11SBf4*i!!ef'0R`(-c08
aMTB%`!6A1jX"rrjKMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ''0R`(-c0KZMT%
&U&$A1jKLMCm"c-c9-Bk6ecZBBSfI!Fc-e6+1NpFlMCJkQ$UB1Sk4%FUF1jKLMCm
"c-cBDik4"515ecZBH*B&S(1jDA16Ff&TC*0dEj0LQP11CC0KET0TET(at-sign)XFRD4reM
NBA*TB(at-sign)k6G*B&S(0[CT0dD'(at-sign)6F*K[E(PZEfeTB(at-sign)acMU'0N5H!!0G`Z5MAH,NT,*B
$kUMAFC!!EMQj+0GiZ5Q6GfKPESkKSBf5!*60FGG*N3$aJlNSef'0R`(-c08aMTB
%`!6A1jX"rrjKMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ''0R`(-c0KZMT%&U&$
A1jKLMCm"c-c9-Bk6ecZBBSfI!Fc-e6+1NpFlMCJkQ$UB1Sk4%FUF1jKLMCm"c-c
BDik4"515ecZBH,NTN3098Mf1T"8!!)f4CX(at-sign)redQ4!2'$Z5MAF)fI!Fc-e6'1PJ6
!",NSef1j+GFlQ`(rrR#0R`(-c08bMT1j+0GMZ5RA1ifB1TJkQ$U1N4(+R$ZBF)f
I!Fc-f'k1N3(at-sign)S8,NSef1j+GFlQ('0R`(-c08aMT1j+0GMZ5RA1jKaMCm"c-c9-Sk
6Z5MABlNTecZ0Q$UB1TJkMT%4bT`lQ('0R`(-c0KVMT%&)j+j+0GMZ5RA1jKiPJ+
UU,NVNpGMZ5Q1SBf4*i!!CQpbPJ2UU'&XE*0MEfe`E'9iNfk9V(*eEC0LN!"6MQ9
bFj%$kUMABlNZMU3DJ!#0N5H!!&0[E(at-sign)9dD(at-sign)ePFjB$kUKdD'9cCC0YEh*PNfGPEQ9
bB(at-sign)b6D(at-sign)k9V(*fQrpBj'&bD(at-sign)&ZNh4cPJ2UU'&bCC0MB(at-sign)aXC(at-sign)56Bfq9V(*fQ'&bD(at-sign)&
ZNh4c,SkKMC%RJ!"ANIm&9Q(at-sign)(at-sign)!qUSC'8-EQ(at-sign)6BC0YEh*PNfGPEQ9bB(at-sign)b6BA#3!&1
1EfaKFT0TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G*B$kUKKFj0QEfaXEj!!V(*hFcU1T!k!!+'
0NJ#16mrA3ENSeh'3!'ijZ5MAH,NTecZ4!IrqF,NSehLj+5Q0MCrmUe#0MC%$99,
8')k1R`28X)f0N30Z1VNpMSk1MT%2rrXSe`b4!e6he!#(at-sign)!UUSe`ZD%iqj+BfIq`Z
Cf'Z1N38MNVNSehL6e!#6e`ZBZ5Q0RrX,QGKZfJ$BDik4%1KDecU1Ra8!!)f4*i!
!Z8&RB(at-sign)PZ,*%)phThPDabCCX(p1TcBC0jQ(4SBA5BG*0hNfqBF*96MQpXH(at-sign)j[E(at-sign)P
KE(1Beh#j+0GiZ5QBB(at-sign)jNQ0GaN!"Z1ENSehLj+CKKFQ(at-sign)BBA#6EfaKFTKhD'9ZMU'
0N5H!!0G"Z5MAFC!!EMQj+0GiZ5RA1j%"rrj`Z5MAH,NT+CB%L2*TFj0TC'9ZN!#
XFR4TBf&XE(Q6HQ9bEbbE",#&G'KKG*0TFbbBHQ9bEj0QEh+6B(at-sign)aXNpGiZ5k4"a1
r9'KPEh*PEC-aNh*PE(at-sign)&TER11SBf4*i!!GT(r(at-sign)14KE'PNPJ4"Lf&cNh0dBA4PC#k
4"Mf)9'KKG*0TFbb4"&G$D(at-sign)D6eh'3!'ijZ5MAFTT6MVNTPJ2T0Mf6-*B%3BYdD'9
ZNh4SCC0`Q'pXH(at-sign)j[E(at-sign)PKE*2AF,NSehLj+CB$k6BpNbMAH*B#jG$8!*2AFTLj+Bf
Iqk8bf'k1N3RTflPTFikKMC%RJ!"KF*!!8ij[E'&bPJ2UU(4[NpGaN!"Z1ENSehL
j+5k1RaU!!)f4*i!!6'9dPJ3G1h9cNf0[ER0TC'9bNf'6Fh#D8ijPBfPKE*0MBA0
P,T%&d*P6GA"`Q'pcCC0dD'&dNpGaN!"Z1ENSehLj+C0TFj0KNh&eB(at-sign)4bBA4TBj0
`Q'pXH(at-sign)j[E(at-sign)PKE)kKMC%RJ!"KEQ5(at-sign)!qUSeh#j+0GiZ5Q6DA16BC0MG(at-sign)*TBj0`N!"
6MQpXH(at-sign)j[E(at-sign)PKE$U1SD'0NJ#cUA,AFC!!EMQj+0GiZ5Q(at-sign)!e952C2AH)fIq`ZCe6+
1N3GUV,NVPJ+UU$,ABSfI!Fc-e6'1N36!"0GiNlNVNpGLMCm"c-c9-Sk1U48!!)f
4*i!!Z(at-sign)&ZC)kKSBf5!*YL'YG`Z5MAH,NTPJ098Mf6ehL0RrX,QG8cMTB(DUbj+jX
#UUJcef'0R`(-c08aMT%%`!6AH)fIq`ZCe6+1NlNVQ$2ABBfI!Fc-e6+1PJ6!"0G
iQ,NVQ0GKMCm"c-c9-ik6ecU1TSf4*i!!Z94SC(at-sign)k(at-sign)!qUSGjUXFQ(at-sign)6D''BGTKP,*0
TET0eECKLFQ&XNfj[G'&dD(at-sign)pZMU'N'S!!MC%[G*VA3ENSeh'3!'ijZ5MAH,NTecZ
4!IrqF,NSehLj+5Q0MCrmUe#0MC%$99,8')k1R`28X)f0N30Z1VNpMSk1MT%2rrX
Se`b4!e6he!#(at-sign)!UUSe`ZD%iqj+BfIq`ZCe6+1N36!",NSehL6e!#6e`ZBZ5Q(at-sign)!e9
52C-Se`b0N3#U6jrl#jR9-Sk4#"6le!#(at-sign)!UUSZ6,A#jJ-N308plNVNpF,MCLIq`Z
Ce6+1N366NlNTehL6e!#6e`ZB$)f4!+T2RrX,QG8bMT%)&2Zj+j-be`Z0Q*rl#jR
9-Sk4"016e`b4!e6he!#6e`Z0Q*rl#jR9-Sk4"016ecU1SBf4*i!!Z89fNIpBj'&
XG(at-sign)&dD(at-sign)jRPJ+jIh4SCC0XD(at-sign)jPBA+6CR9ZBh4TEfjKE*2A4C%!Y"Hj,*%#pSGhQUa
bCC0[BR4KD(at-sign)k6G'KPNfC[E'a[Q(GTEQH6CAK`E'PMDA56CAK`FQ9cFfP[ESkN$S!
!MC%RJ!"QEh+(at-sign)!qUSG'KPNf&`N!"6MQpXBA+6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh3kMU'
KMC&56+IA3ENSeh'3!'ijZ5MAH,NTecZ4!IrqF,NSehLj+5Q(at-sign)!e952C2A4C%!Y"H
j+#MA$)f4!+T2RrX,QG8bMT%)&2[8!*B#UULj-YF,QK12$*%$92Hj+j2A#ifBRrX
,QG8bMT%%dj1j+GGiNp3!NpF,Q!b0N3#U6jrl#jR9-Sk4#"6lZ5Z6-YF,MCLIq`Z
Ce6+1N366NpF-N308pp3!NpF,MCLIq`ZCe6+1N366NlNTN3098Mf1MTmH!!#0NJ$
kEf!c-ik1M)X!!!!L!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!&GH+!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N8d+Y0G&Q`#d&lNSe`b0N3#U6jr
l#jR9-Sk4"(at-sign)T6Z5Q(at-sign)!UUSe!#6Z6,A4CLj+0F,N!!6M`b4!+T2Z5Q6+j2A4CLj+0F
,MC!!%iqIq`ZCe6+1N366NlNTehL6Z5Z6ed(at-sign)BZ5M8!0F,N!!6M`b0N3#U6jrl#jR
9-Sk4"(at-sign)T6Z5Q6+j-bed(at-sign)BZ5MA#if3!"12RrX,QG8bMT%%dj2A$*%!UNqj+C28!*2
A4CLj+0F,MC!!%iqIq`ZCe6+1N366NlNTN3098Mf1T"8!!)f4-6Y1ed(at-sign)E!,3AZ5M
A$)f4!+T2RrX,QG8bMT%&DP1j+CB#UUM8!*1j-YG&Q,NSe`Z3!"12Z5RA4CLj+0F
-N3#U6lNTNbZ6ed(at-sign)BZ5MA#if3!"12RrX,QG8bMT%%dj1j+GGiNlNVNpG&Q,NSe!$
A#j!!%iqj+GG&Q,NSe`b0N3#U6jrl#jR9-Sk4"(at-sign)T6Z5Q6+j-bed(at-sign)BZ5MA#if3!"1
2RrX,QG8bMT%%dj1j+GG&Q,NSe`b4!+T2Z5Q6e!#6ed(at-sign)BZ5MA#if3!"12RrX,QG8
bMT%%dj1j+C%$99)pMU'0NJ#(djBSef+0R`(-c08bMTB(DUc8!*X#UULj-YGKMCm
"c-c9-Bk4"-!%ef+0R`(-c08aMT1j+jMABBfI!Fc-e6+1PJ6!",NTehLBe!#Bef'
0R`(-c08aMT2ABSfI!Fc-e6+1N3GUV,NVQ$,ABBfI!Fc-e6+1NpGLMCm"c-c9-Bk
4"fUXe!#Bef'0R`(-c08cMT2A1SkKMC%RJ!#j9'L3!+abGA-XN31T,f'(at-sign)!jM4FA9
KC(*KG'PMNh#D8ij[E(PZEfeTB(at-sign)b6eh'3!'ijZ5MAH,NTNf&ZC*0KNf0eBQPMNh#
BEfajEQpYD(at-sign)&XNpG`Z5MAH,NTNf&bCC0KF*K[E'&bNfPQMU31J!#0N5H!!'&ZC*B
$kUK[EQajNfPQNh4SC(at-sign)PbNf0[N!"6MQ81BfPPETUXFR4cNh0KG'PcCRQ6G'KPNh5
BGjK[Nf9aG(at-sign)&dD(at-sign)pZFikKSBf5!,Pp20GLMCm"c-c9-Sk(at-sign)"fUXe!#E!UUSZ6,ABBf
I!Fc-e6'1N36!"0GLMCm"c-c9-Bk6Z5ZBef'0R`(-c08bMT%)&9Dj2C%$99)`MUN
9!!#KMC)!VEY)e!$ABBfI!Fc-e6'1PJ6!"0GLMCm"c-c9-SkE"fUXZ5Z4!UUS-YG
KMCm"c-c9-Sk6ef+0R`(-c08aMTM8!*%#UUMABBfI!Fc-e611N3J99VNpN3098M$
A1SkQMC%RJ!#j9A0TEQH(at-sign)!qUSG'KPFf(at-sign)6CA&eBA4TEfjc,*0hQUabCC0MB(at-sign)k6F(*
[Q(DBCC0dQ(HBEj0TEA#3!&11Eh*dB(at-sign)kBG*0dD'9[FQ9YFcU1U4U!!)f4*i!!de4
SC(at-sign)pbC(at-sign)f4"2%b-VNZN3CJF&4SCA*PPJ40,(at-sign)PcNfPZNfGPEQ9bB(at-sign)b6EfjPNh&eB(at-sign)4
bBA4TBj0`QP11EfajEQpYD(at-sign)&XNhGSD(at-sign)13!+abD*0TFj0KF*K[E'&bMU'0N5H!!(4
[PJ2UU''6CfPfN!#XFQ9ZNf0eBQPMNh#3!&11EfajEQpYD(at-sign)&X,SkQMC%RJ!"3FQq
D8ij[CLk4"6MJ5(at-sign)jNC(at-sign)9N,*B$kUKdD'(at-sign)6B(at-sign)+BEj(at-sign)XFRD6CCB$kUKPFA9KG'P[ER1
6E(at-sign)'3!+abHC0LQ'(at-sign)6FQ9hFQPdG'9ZNf&cMU'KMC)![[lkef+0R`(-c08bMT%(DUc
8!*%#UULj-YGKMCm"c-c9-Bk4"-!%ef+0R`(-c08aMT%)&9Dj2C%$99,8!0GKMCm
"c-c9-Sk1U48!!+'0NJ#i!*R8!0GKMCm"c-c9-Bk(at-sign)"-!%ef+0R`(-c08bMT%(DUb
j+j%#UUJbef'0R`(-c08bMT2ABSfI!Fc-e6'1N3J99VNpN3098YGKMCm"c-c9-ik
6ecU1TSf4*i!!Z94SCCB$kUKcEfaeG'P[ER16ef+0R`(-c08aMTX%`!6A1j%"rrj
LMCm"c-c9-Sk4#+UXZ(at-sign)C[FT0RDAD3!+abC(at-sign)k6ef'0R`(-c08aMTMA1jB"rrjKMCm
"c-c9-SkBecZ6BBfI!Fc-e611N3LUV,PKFQ(at-sign)(at-sign)!qUSD(at-sign)k6Cf9ZCA*KE*0eEQPaG(at-sign)8
ZMUNDJ!#0N5H!!008D'9[FQ9YN352$$1j,T%&B#P8D'9bCCB$pm"TFj0KE(H9V(*
KNhPcPJ2h`''6G*(at-sign)XFRH6EbeND(at-sign)ePER0TEfjKE*B$pm"cF'&MCC0[CT0MG(at-sign)*TBj0
`N!"6MQpXH(at-sign)j[E(at-sign)PKE(11SBf4*i!!GfKTBjUXFQL(at-sign)!qUSBA*PNf&`N!"6MQpXBA+
6G'q6BC0RDADBC(at-sign)k6FA9KC(*KG'PMNh#3!&11EfajEQpYD(at-sign)&X,SkQMC%RJ!"3FQq
3!&11EfBZN38ii%PZC'9PC#b(at-sign)!qUSCfPfQUabC(at-sign)k6ef+0R`(-c08aMT%%`!6A1j%
"rrjLMCm"c-c9-Sk4#+UXZAHBCC0YBCKjNh0[E(DBCC0QEh+6ef'0R`(-c08aMTB
%`!6A1jX"rrjKMCm"c-c9-Sk6ecZBBBfI!Fc-e611N3LUV,PQFQpYPJ2UU(4SCC0
PFA9KG'P[ER11SD'0NJ#kE6E8!,Nbef'0R`(-c08aMT%%`!6ABSfI!Fc-e6'1N3G
UV,NVN3+UU0GKMCm"c-c9-Sk4#"9(at-sign)Z6f4!e95e!$ABSfI!Fc-e6+1MUN9!!#KMC)
!Z!#Ce!$ABBfI!Fc-e6'1PJ6!"0GLMCm"c-c9-Sk4"fUXZ5Z4!UUS-YGKMCm"c-c
9-Sk6ef+0R`(-c08aMT%)&9Dj2C%$99,ABBfI!Fc-e611NpFkMUD0N5H!!,P8D'9
cCCB%GSCPFA9KG'P[ER16B(at-sign)ahPDabBC0jFjX%GSCSBC0fNf(at-sign)BBCKNEh9LE'(at-sign)BD(at-sign)i
-EQPdNhQBEfDBFfpXGA4TEfjc,*%%QAjKFjKdD'9jQ(9cC(at-sign)5BG'q1SBf4*i!!Ff'
DV(*jPJ2UU'PZNh4SCC0[E'56C''BHA-ZMUNDJ!#0N5H!!&4SC(at-sign)pbC(at-sign)ecPJ3j`$+
6B(at-sign)jNNc16F(*[N!#XFRCTC'(at-sign)6BC0cD(at-sign)e`E'(at-sign)6B(at-sign)jNNf9iF'aTBfPdNfePG'K[N!"
6MQ56CQpbNh0[E(CTEQH6BC0MG(at-sign)*TBikKMC%RJ!"PFA9KG'P[ELk4"6MJ5A5(at-sign)!qU
SCfq3!&11CA16BA16CQpXE'q3!+abGh-ZMUD0N5H!!%GTGT!!V(*PETB$kUKdD'(at-sign)
6Bh9LD(at-sign)16F*!!8ij[E(PZEfeTB(at-sign)b1MTmH!!#0NJ$kEf!c0)k1M)X!!!!M!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!&SXU!#S#MeMD$pKYF,S!*
E+2(at-sign)0S2h*e`ZN$S!!MC)!Qf)Deh#j+0GiZ5Q(at-sign)!e952C2AH)fIq`ZCe611PJGUV,N
VQ`+UU$2ABBfI!Fc-e6'1N36!"0GiMCrl#jR9-Sk6Z5ZB-pGKMCm"c-c9-Sk(at-sign)"-!
%ehLBZ5ZBef'0R`(-c08cMT2A1ikI&13`MC%RJ!#j$(*cG#b4"EfEBTUXFRQ(at-sign)"(at-sign)!
h9'KPEh*PEC-bNhHBCC--EQ56BC0eEQPaG(at-sign)(at-sign)6FA9KC(*KG'PMNh#3!&11EfajEQp
YD(at-sign)&XNpGaN!"Z1ENSehLj+C0hD'PMQ'L6DA11SBf4*i!!BA#D8ij[E'&bPJ0Mqh4
[NpG`Z5MAH,NT,T%&#ra*ET0RC(at-sign)jPFQ&X,*%$IZTcG(at-sign)13!+abD*0KNh&eB(at-sign)4bBA4
TBj0`Q'pXH(at-sign)j[E(at-sign)PKE*2AFC!!EMQj+0GiZ5Q6D'&cNh59V(*hNfq4!f2lC'PcG'P
ZBh51SBf4*i!!FQqD8ij[G(1(at-sign)!fMBeh+0R`(-c08aMT%)+0bjB(at-sign)jNNpGbMCm"c-c
9-Sk4"-!%Z5k4"3fE3RQ69'KPEh*PEC-a,*%$JXjdD'(at-sign)6Bh9LD(at-sign)16F*K[E(PZEfe
TB(at-sign)acNbMAH*B"SB,8!*2AFSfI!Fc-e6'1Q`6!",NTMCrlT6,9-ik4##MFZ(at-sign)&ZC*%
$D0JSehL6e!#6eh+0R`(-c08bMTLj+BfIqk8be611MU'0N5H!!,PKFQ(at-sign)(at-sign)",([BA#
3!&11EfaKFT0dEj2AFC!!EMQj+0GiZ5NZN3H1YP0PBfpZC#bE"12"BT!!V(*jNe4
SC(at-sign)pbC(at-sign)f6-bbBG'KPNf%1EQ(at-sign)6E'PZC(at-sign)&bNh0`B(at-sign)0PNfpQNf0eBQPMMU'0N5H!!(#
98ij[E(PZEfeTB(at-sign)acQ`32&(at-sign)&`NfpXBA+BG'qBeh'3!'ijZ5MAH,NTQ'KKFjKND(at-sign)e
PER0TEfkBG*(at-sign)XFRH6Ebk4"DBS8fPZBf(at-sign)Beh#j+0GiZ5QBDA1BBA#3!&11EfaKFTK
dEjMAFC!!EMQj+0GiZ5NXMU'0N5H!!(HDV(*PPJ3-(at-sign)Q0[EQ0XG(at-sign)4PNh4SBA56eh#
j+0GiZ5Q6DA16BC0XD(at-sign)jPBA+6BfpYQ'*TEQ&dD(at-sign)pZNfpQNbMAH*B#`CR8!*2AFSf
I!Fc-e6'1Q`6!",NTMCrlT6,9-ik4#-aHZ(at-sign)&ZC*%%$&SSehL6e!#6eh+0R`(-c08
bMTLj+BfIqk8be611Q,NZN3(at-sign)GpdPZMU'0N5H!!(0jEC!!V(*LN!"6MQpXFcU1SD'
0NJ#29),AF,NSehLj+CB$99)pNpGMZ5MAH*B#UUM8!*2AFSfI!Fc-e6'1Q`6!",N
TMCrl#jR9-ik4"fUXZ5Z6+$'6e!#6ef1j+5MAH*28!*2AFSfI!Fc-e6+1Q,NTMCr
l#jR9-ik1U46N-)f4*i!!Z(at-sign)C[FTB$)UTcEfePNf0[ER0dB(at-sign)kDV(*dNpGMZ5k4"2B
f6f*cCA*fQ'(at-sign)6G'KKG*2ABlNXN30+UYGbMCm"c-c9-BkE"q+ZZ(at-sign)&ZC*2AFSfI!Fc
-e6+1Q,PKFQ(at-sign)6BfpYF(9dC(at-sign)56BT!!V(*jNh0[E(CTEQH6E'PZC(at-sign)&bMU'0N5H!!'&
ZC*B$kUKaG(at-sign)&NFQ&dD(at-sign)16CA&eBA4TEfjc,SkI'PVUMC%RJ!"*ETB%h9&dD'PcNhH
9V(*KNhQ4r`9(at-sign),*%&'IadD'(at-sign)(at-sign)"0e4FfpXGA4TEfk6EfD6G'KPNf0eBQPMNf9aG(at-sign)&
dD(at-sign)pZNpG`Z5MAH,NTPJ6bA$f6-*B%h9&TFj0bC(at-sign)4eBf9NNh4[Nh4SCBkKMC%RJ!"
cEfaeG'P[ETB$kUK[CT0dD'(at-sign)6CA&eBA4TEfk1SD'0NJ#E(m$ABlNSehL(at-sign)!UUSe!#
6eh+0R`(-c08aMTX%`!5j+BfIq`ZCe611N3J99VNpN3098Y3!Z5JaNp3!NpGMZ5N
SehL6e!#6eh+0R`(-c08bMTLj+BfIq`ZCe611Q0FlMUD0N5H!!,PKEQ5(at-sign)!qUSG'K
TFj0PFA9KG'P[ET0TFj0PBA0TE(Q6FfpXGTUXFQ9NNf+BHC0dB(at-sign)YTEQH6BC0MG(at-sign)+
D8ijPNh*[Q'pd,SkN'PVUMC%RJ!"8D'PcPJ2UU'ePG'K[QP11C*0[CT0cEfafD(at-sign)j
RNf'6Bh9LD(at-sign)16CA&eBA4TEfk6DA16G'KPNfpZE(Q6EfjPNdQ6Bf&ZNh*PE(at-sign)9YN!#
XFQ+BCA)ZMU'0N5H!!%aPG*B%NA0YCC0ND(at-sign)GbCA0cNhGTG'L6BC0`N!"6MQ9bFfp
ZB(at-sign)b6B(at-sign)jPBf4[G'8ZN3FY3%'E"*&)CQ9hNhQ3!+abC(at-sign)&bFj0KCfmXN35l*8QBGj!
!V(*KFj0XC(at-sign)0dGA*TEQH1T!k!!)f4*i!!Efk(at-sign)"9aYG'KTFj0YBA4PFQPKE*0KG*0
KNh0jEA#D8ij[FfPeEC0TET0MEff3!+abBQPZBA4[FQPMFj0dD'&dNh4[Q'pVNh"
XB(at-sign)0PNf&dNh4SCBkKMC%RJ!"9EQPfPDabCA*cDA56HCB%q$4KG*00D(at-sign)jZCA0[G'%
ZN3KKJe#DV(*PFR0TNd4TB(at-sign)0[EQPcNhHBBA16FfPdG'PZCj0TET0dD'(at-sign)6CR*[ETK
dNh*[Q(FXN38lPf&ZC)kKMC%RJ!"*Q`5B)f0[G(at-sign)aNPJ5B6h4PE'b6BA165CKcG'&
bG'9NNh4[NfaPBh4eFQ(at-sign)6G'KKG*0SCC0hQUabBA16CQ&XE'PZCj0KFfaPCA!lN36
[)fKPNf9fQ'9ZQ(4eB(at-sign)aXHBkKMC%RJ!"LN!"6MQ9RB(at-sign)k(at-sign)!qV-G'q6C'pkCC0[#bk
4"6P-3R9dNh4SCC0YEfePETUXFR565C0YC(at-sign)kBG'P[EQ9NNh4SCC0YB(at-sign)GTBj0hQ'p
bC(16)R0[E(CTEQH6BBkKMC%RJ!"MG(at-sign)*TBjB$kUKPFA9KG'P[EL+6D'(at-sign)6Gj(at-sign)XFQp
VNf(at-sign)(at-sign)!qUSGA#6GfPdD*0KNh0dBA*dNf&ZC*0cB(at-sign)PN1TB&11!L8Q9KE'aj)C0)Ej!
!V(*h2b+1T"TDkSf4*i!!9'KPPJ2UU("bC(at-sign)0PC'PZCj0dPDabGj0[PJ2UU(4SC(at-sign)p
bC(at-sign)ecNf&bCC0PBA0TE(Q6Cf9ZCA*KE'PkC(at-sign)3ZMU'0N5H!!008D'9[FQ9YN300-M5
j,T%%hpe8D'(at-sign)(at-sign)!YqJC'PYC(at-sign)jcD(at-sign)pZNfpQNh4SCC0cF'&MCC0[CT0KE'b6+'e[EQP
M+C0`N!"6MQpXH(at-sign)j[E(at-sign)PKE(16EfD6C'9RFQ9PMU31J!#0N5H!!0GVN32(at-sign)(at-sign),PdD'&
dPJ0[1f&bCC0KF*T6MQpXBA+6G'q6BC0`Q'pXH(at-sign)j[E(at-sign)PKE*0[CT0NC(at-sign)GbC(at-sign)(at-sign)6efk
6Z(at-sign)9aG(at-sign)&XFj-befZ4!K(at-sign)Ve!#4!Dk1efkj,*X$KqTTET0RC(at-sign)jPFQ&X,*KhD'9ZMU'
0N5H!!0GVN31mEp38N3098YGZZ5k1U4TDkSf4*i!!de4SC(at-sign)pbC(at-sign)f4!ddb0ENZN36
Ih94SCCB#hk"ND(at-sign)ePER0TEfk6EfD6G'KPNh0`B(at-sign)0PNfpQNf&XE*-SE(at-sign)pZD(at-sign)-TNh#
3!&11EfajEQpYD(at-sign)&XFj0[CT0NC(at-sign)GbC(at-sign)(at-sign)1SBf4*i!!efk(at-sign)!qUSZA4SBA56BA*PNf&
`QP11EfaKFT0dEj0KNh#BEfajEQpYD(at-sign)&XNfpQNf4PCh*PCC2ADj%%8F(at-sign)jCA&eB(at-sign)a
cNpGVN!"R(ENXNfPQNpGVN31mEp38N3098YGZZ5k1TSf4*i!!6'9dPJ-8EA9cNh4
bHC0dEj0cEfafQUabCC0KET0PFA9KG'P[ET0[CT0NC(at-sign)GbC(at-sign)(at-sign)60C0TET0YQ(9MQ'L
6G'KPNh0KE(at-sign)(at-sign)6GjKKQ(Q6BA16GjKPNh0[E(DBC(at-sign)51SBf4*i!!BCB$kUKMG(at-sign)*TBj0
PFA9KG'P[ELk4"6MJ4fPfQUabC(at-sign)k6G'KPNh&eD(at-sign)kBG'PMNh#3!&11EfajEQpYD(at-sign)&
XMSkI(J!!MC)!qQpJ-c(at-sign)1MSb,!!!!*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!"FN1J!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,T!k!!)f4ArREeh#
j+0GiZ5Q(at-sign)!e952C2AH)fIq`ZCe6(at-sign)1PJGUV,NVQ`+UU$AABBfI!Fc-e6'1N36!"0G
iMCrl#jR90)k6Z5ZB-6$ABBfI!Fc-e6+1N36!"0GiMCrl#jR9-ik6Z5ZB-6$ABBf
I!Fc-e611N36!"0GiMCrl#jR9-Sk6Z5ZB0GGKMCm"c-c90)k4"-!%ehLBZ5ZBef'
0R`(-c08eMT%)&9Dj2C%$99)`ecZ1Ra8!!)f4*i!!Z94SC(at-sign)pbC(at-sign)f(at-sign)!f6$0*0KFh0
eFQ9cNh9cNh4SBA56G'KPFQ(at-sign)6DA16D(at-sign)k6Cf9ZCA*KE*0KNh9ZDA&eCC0MG(at-sign)*TBj0
`N!"6MQpXH(at-sign)j[E(at-sign)PKE*2AFC!!EMQj+0GiZ5Q1SBf4*i!!GfKTBj!!V(*SPJ3&PfP
cNf&`QP11EfaKFT0dEj2AF,NSehLj+5k4"BQX5(at-sign)k6Cf9ZCA*KE#b4"!a5G'KTFj0
MG(at-sign)*TBj0`Q'pXH(at-sign)j[E(at-sign)PKE*0SBA16G'KbC(at-sign)(at-sign)6C'PcG'PZBh51SBf4*i!!FQq3!&1
1Eh4cN322BYGbMCm"c-c9-Bk(at-sign)"-!%ecZE!IrqFSfI!Fc-e6+1NpFlQ(+0R`(-c08
cMT1j,T%&,mP#HCB$cf*8D'9[FQ9YNc%XN328eR4SCC0`N!"6MQpXH(at-sign)j[E(at-sign)PKE(1
6+0GiPJ*bmp3!NpGbMCm"c-c9-Bk(at-sign)"-!%Z5Q0RrZP-Y8eMT2A1jLj+0GiPJ*bmp3
!NpGbMCm"c-c9-Sk(at-sign)"-!%Z5Q0RrZP-Y8eMT2A1jLj+0GiPJ*bmp3!NpGbMCm"c-c
9-ik4"-!%Z5Q0RrZP-Y8eMSkKMC%RJ!#jBA*PPJ3%j'aTEQ9KFQajNfPZC'9`QP1
1C(at-sign)jNC(at-sign)k3!+abG*0KEQ56BA#BEfaKFT0dEj2AFC!!EMQj+0GiZ5Q6,T%&Kj0#HC0
8D'9[FQ9YNc8XN33,Fh4SCC0ND(at-sign)ePER0TEfk1SBf4*i!!EfD(at-sign)"#KBG'KPNh0`B(at-sign)0
PNfpQNf&XE*0`QP11EfajEQpYD(at-sign)&XFj0KF*K[E'&bNh4[NpGaN!"Z1ENSehLj+C0
PFA9KE(16-bk4"I([3R9dNh4SCC0`Q'pXH(at-sign)j[E(at-sign)PKE)kKMC%RJ!$AF,NSehLj+CB
$kUKTFj0KF*T6MQpXBA+6G'q6eh'3!'ijZ5MAH,NT,T%&11")C(at-sign)jMC5b6eh#j+0G
iZ5Q6Bf&ZNf+BCC0hFQPdG'9ZNfPZNh4SCC0QEh*YMU'KMC&fT&6AF,NSehLj+CB
$99)pNpGMMCm"c-c9-BkE"-!%Z5MAH*B#UUM8!*2AFSfI!Fc-e6'1Q,NTMCrl#jR
90Bk4"fUXZ5Z6ef10R`(-c08bMTLj+0GiNp3!NpGbMCm"c-c9-SkBZ5Q0RrX,QG8
eMT%(DUbj+j2ABifI!Fc-e611Q,NSehL6e!#6eh+0R`(-c08cMTLj+BfIq`ZCe6(at-sign)
1MTm9!!#0N5H!!,PQEh+(at-sign)!mK5Fh9TG'&LE'(at-sign)6BfpZFh4KETUXFR4cNpGMMCm"c-c
BDBk4!f6DZ5k4"5eZ9'LBGA-XN322-(HBCC0cC(at-sign)(at-sign)6G'KKG*0KNfGPEQ9bD(at-sign)16F*!
!8ij[E(PZEfeTB(at-sign)b6EfD6C'9RFQ9PNc(at-sign)1SBf4*i!!Bf&ZPJ16l(at-sign)+D8ijPNhGbDA4
dC(at-sign)k6BA16BC0XD(at-sign)jPBA+6BfpYN!#XFQ*TEQ&dD(at-sign)pZNfpQNh4SFQ9PN`aQG'L6F*K
[PDabGj0PFR1(at-sign)!j2YEfD6E'PZC(at-sign)&bNh#BEfajEQmYMU'0N5H!!'eTB(at-sign)ac,T%&(F&
8D'9cCCB$Q8YKFQ(at-sign)6BfpYF(9dC(at-sign)56BT!!V(*jNh0[E(CTEQH6E'PZC(at-sign)&b,*%$UC!
!FA9KC(*KG'PMNf&ZC*0MG(at-sign)*TBj0PFA9KG'P[ER-ZMU'0N5H!!&4SDA1(at-sign)!kcFFQ9
NG(at-sign)0dD(at-sign)pZNh4[Nf0KEQpZD(at-sign)0KE*0QEh*YNfpQNh4SCC0aG(at-sign)PZQUabG'PMNfPcNf&
cNf0XEh0PNf&cNhHBCC0MB(at-sign)k6BfpYCC0dEikKMC%RJ!"cEfafD(at-sign)jRPJ2UU''6FA9
TETUXFR4TBj0PFA9KG'P[ET0LQ(Q6FQ&ND(at-sign)0KE(-ZMUNDJ!#0N5H!!%'DV(*dPJ6
)rR4SDA16F*!!8ij[D(at-sign)kBG#b4"3#8FfpYC(at-sign)pZCC0TET0dD'(at-sign)6BA9ND(at-sign)9ZBf(at-sign)6GfP
XE*0bB(at-sign)PcCC0SDA16Eh+6D'9bNfKKEQ56B(at-sign)jNNh0KQ(NkMU'0N5H!!#*&H'0eFf(at-sign)
(at-sign)"&IJE(at-sign)8XN34c,(at-sign)*eG*0dD'(at-sign)6G(at-sign)fDV(*LFQ&XNfePG'K[N!"6MQ56HCK[GC0SBCK
fQ'(at-sign)6D(at-sign)kBG(*[N!"6MQ4eBf9NNfPcNfj[G*0PGTKPET0REj96MQq6C)kKMC%RJ!"
PEQpeCfL(at-sign)!qUSG'q6CAK`FQ9cFj0dD'(at-sign)6C'PcBh*TE(at-sign)PZB(at-sign)k3!+abG*0[CT0KNh&
eB(at-sign)4bBA4TBj0PFA9KG'P[EL'4"6MJ)SkQMC%RJ!"4G(at-sign)PdCC%$kUKbD(at-sign)GSN!#XFR3
ZMUD0N5H!!&4SCCB&%raNC3aZDA4TEfjcNfpQNh9YQUabBR*KCC0KEQ56EfD6G'K
PNfaTEQ9KFT0QG(at-sign)jMG'P[EQ&XNpG&N3A)%lPSBCKfQ'(at-sign)6B(at-sign)k6Ef+BGQP[GA11SBf
4*i!!Cf9ZCA*KE'PkBA4TEfk(at-sign)"(aeG'q6B(at-sign)kDV(*jNf&bFQ'BHC0[CT0`N!"6MQp
XH(at-sign)j[E(at-sign)PKE(-XN35Jk(0KQ(Q6eh#0R`(-c08aMTB%`!5j+0GiZ5RA1jX"rrj`MCm
"c-c9-Sk6Z5MAH,NTecZ0Q$UB1TJkMT%4bT`lQ(#0R`(-c0KJMT%$q1Dj+0GiZ5N
ZN3EZ4NpZCBkKMC%RJ!"cD(at-sign)e`E(Q(at-sign)!qUSBfpZFfPNCA*cNh4SCC0cF'&MCC0[CT0
`N!"6MQpXH(at-sign)j[E(at-sign)PKE(11SD'0NJ#kTac63lPEehJlPJ(rrJZ0R`(-c08aMTX%`!6
A1j-,MCm"c-c9-SkBecZ0NcU61T-kMT%4bT`lN`Z0R`(-c0KJMT%$q1DjABkT&3!
!MC%RJ!"KEQ5(at-sign)!qUSEfjPNh0PG(11SD'0NJ$E3AAA4C%!Y"Hj+0F,MCrk6kD0N!!
6MpKUMTm)%hb0G)k1N34jQENTMUD0N5H!!(4[PJ6d2Q9aG(at-sign)&XNh4SCC2ADT%!V01
j,A4SNf0[QP11C3jMD(at-sign)9ZN!#XFR56EfD6G'KPNh#BEfajEQpYD(at-sign)&XNpG`MCm"c-c
BG)k4!j(+Z5MAH,NT,T%)9D&AD'&dNfPcNf0bG(at-sign)0TB(at-sign)`XN38fSh4SCBkKMC%RJ!"
XD(at-sign)jPBA+(at-sign)!qUSCR9ZBh4TEfjKE*2A4C%%RVqjDA16B(at-sign)GKD(at-sign)k6EC(at-sign)XFR9XG'P`E'P
MBA4TGT0PPJ2UU'pZNf4TFh4TEQ0dNh9YN!#XFQ*bB(at-sign)8kMU'KMC&k6[6A4CX!Y"H
j+0F,MCrl#jQ0N!!6MpKTMTm(kpU0e6'1MTB%`!6A#ifIqNqQMC!!%irBDSkI#%q
JMG8bMSk6e`Z0RrX,QBf3!"12f'Z1R`IVfSh9-ik1MC%(0arA1TB"rrikNcU1N4F
"[AL0RrX,QGKJMT%$q1Dj+CB$99)pNpG&Q,NSe`Z0RrX,QBf3!"12f'Q1R`IVfSh
9-Bk1PJ6!",NTed(at-sign)BZ5MA#ifIqNqQMC!!%irBDSkI#%qJMG8bMSk6Z5RA4CLj+0F
,MCrl#jQ0N!!6MpKVMTm(kpU0e611MT%&0b'j+Bf(at-sign)!IrqecU61T-kMT%4bTaiMCr
l#jRBB)k4!rMQecU1TSf4*i!!Z8j[QUabGjB&L&eMEfePFj0dD'(at-sign)6Bf&dBjKSNh4
SBA56D(at-sign)k6G'KPNfpXC*0NBCKjFj0hQ'&cNh4[Nh4eFQk6D(at-sign)kBG'q6BC0ZEh4KG'P
[EQ&XMU'0N5H!!'jTCfLDV(*dE(at-sign)&bC6U4"Ip+G'KPPJ40hA#3!&11EfajEQpYD(at-sign)&
XFj0cBCKjNpG`MCm"c-c9-Bk(at-sign)"-!%Z5MAH,NTecZE!IrqF)fI!Fc-e6+1NlNSehL
j+GFlMCJkQ$UB1Sk4%FUF1jK`MCm"c-cBB)k4!rMQZ5MAH,NTPJ40h(at-sign)jPC(at-sign)56EQp
dNf+3!&11CC0NDA0dD(at-sign)jMG#k1SBf4*i!!5(at-sign)k(at-sign)"UdGCQ&MG#b4"efkG'KPNfe[Fh5
6D(at-sign)e`QP11Eh*dB(at-sign)k3!+abG*0MBA0PNfqBBf0eFR16GfKPET0PB(at-sign)13!+abD*0[CT0
dD'(at-sign)6F*K[E(PZEfeTB(at-sign)acMSkI(J!!MC)!qQpJ-cD1MSb,!!!!*3!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"IBHJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$
pbGF,MC%RJ!$AF)fI!Fc-e6'1PJ6!",NSehLj+GFlQ`(rrR#0R`(-c08bMT1j+0G
iZ5RA1ifB1TJkQ$U1N4(+R$ZBF)fI!Fc-f'#1N32ijVNSehLj+CB$km9TFj0PFA9
KE*0dEj0[EQ(at-sign)6B(at-sign)jNNh4SCC0cB(at-sign)ePNh#3!&11EfajEQpYD(at-sign)&XNpG`Z5MAH,NT,T%
&2$C*ET0dD'PcMU31J!#0N5H!!'0KFf8XPJ2UU(4SCC0NC3aZDA4TEfk6EfD6G'K
PNfaTEQ9KFT0QG(at-sign)jMG'P[EQ&XNpG&N35H[lPYBCUXFRQ6BT!!8ijPNh0TEA"XD3a
PC*0KFj0QEfaXEjKhFcU1SBf4*i!!-5k1SBf5!-i4fYG&N3#d&lNSe`Z0RrT2TSf
3!"12f'U1R`KaYBeTMSk4"(QCZ5Q(at-sign)!e952C2ABBfI!Fc-f'U1MUN9!!#0N5H!!,P
QEh+(at-sign)!qUSCAD3!+abCA*jNpGTNlNXNf&ZC)kKMC%RJ!!b,SkKMC)!QTR1ed(at-sign)4!,3
AZ5MA#ifIq`ZCMC!!%irBDBkI"q[DMG8aMSk(at-sign)"-!%e`Z0RrT2TSf3!"12f'U1R`K
2S)h9-Sk1NpF,MCrl#jQ0N!!6MpKVMTm(kpU0e611MSf4"cFIecU(at-sign)!Irq1T-kMT%
A!EeiMCrl#jRBB)k4!rMQZ5Q(at-sign)!e952C2ABBfI!Fc-f'Q1N30NfYGKMCm"c-cBDSk
4"'B+ef'0R`(-c0KVMSf4"b13!0FkPJ(rrMU61Sk4&ZiZH)fIq`ZCf'#1MUD0N5H
!!,PQEh+(at-sign)!qUSB(at-sign)aXNfj[ET0ZC(at-sign)GKG'PfQUabCC0TETKdC(at-sign)GPFR16efNlPJ(rrQU
3!!(at-sign)h1j0VN!"R(6Z0NcU61T-kMT%4bTaJZ5k1RaU!!)f4*i!!9(at-sign)f3!+abBR*KCC%
$miVA#ifI!Fc-e6'1PJ6!"0FlQ`(rrJZ0R`(-c08bMT2A1ifB1TJkQ$U1N4(+R$Z
B#ifI!Fc-f'#1N3IXF,PcBA4TFfCjD(at-sign)jRPJ2cLM'6B(at-sign)jNNc+6BA*PNh0KD(at-sign)56G'q
6BT!!8ijPNf9iBj(at-sign)XFQKKEQGPB(at-sign)*XC5k4"91'9'L6GA-XMU'0N5H!!'C[FTB$kUK
PH'1DV(*SB(at-sign)jRC(at-sign)&LE'(at-sign)6G(at-sign)fBBR*KCC0hQ'(at-sign)6D''BGTKPMU'KMC)!X,,q+0GiPJ+
UU,NVNpF,MCm"c-c9-BkE"-!%Z5Q0RrX,QGKZMSf0RrbV8)f0N3MpSY3BMSkI!p5
`MBf4#4D+Z6f1MSk1N4(at-sign)S5bMAH*1j+j2A#ifI!Fc-e6+1Q,NTMCrl#jRBESk4"DK
3ecU1TSf4*i!!Z89bD(at-sign)1(at-sign)"*I49*(r"9CPEA"XCC0#C(at-sign)aX,*X%`aahD'q6Gh*[G'(at-sign)
62C0TET0`E'&MCC0[CSf0RrbV8)f0Np3BMSkI!p5`MBf4",#jZ6f1MSk1N3hY+#b
BGjUXFQ&cNf*K$f9NNf+BHC0dD'(at-sign)6CQ&MG*0dD'&dMU'0N5H!!(59V(*hNfqE!qU
SG(at-sign)f6BR*KCCKMEh9XC*KLN!"6MQ(at-sign)BCAKMNfKKEQGPB(at-sign)*XCCKhDA4SEh9dQ'+3!&1
1C(at-sign)PZCjKPFA9KE#k1U4U!!)f4*i!!9j(r"9CPPJ38R'0KET0ZEjUXFRH6Fh4KG'(at-sign)
6G'KPNfeKD(at-sign)k6G'KPEh*PEC0[CT0TETKfNIpBj'&bD(at-sign)&ZQ(56G'KPEh*jP[m&9Lk
4"EDm9j0PPJ38R(GTE'b6BfpZFfPNCA+6BBkKMC%RJ!"cD(at-sign)jRE'(at-sign)4!qUSF*!!8ij
[E(PZEfeTB(at-sign)`ZMUD0N5H!!008D'9[FQ9YN3320cDj,T%&'#G&GT(at-sign)XFQ9bHCX$L(j
TET0fNIpBj'&bD(at-sign)&ZNh5BEfDBBCK`N!"6MQpXH(at-sign)j[E(at-sign)PKE*MAF,NSehLj+CKTFjK
[BR4KD(at-sign)jPC*KLNhQBCAD4reMNB(at-sign)aeBA4TEQH1SBf4*i!!FfpYCCB%$m&`N!"6MQp
XH(at-sign)j[E(at-sign)PKE*0TET0dD'(at-sign)6C'N,CA*PEQ0PFj2A#ifI!Fc-f'Q1Q`BSa03!N3,$kYF
,MCm"c-cBDSk4#(A,Z(at-sign)&ZC*2A#ifI!Fc-f'Q1Q03!N3,$kYGi1j1jGfKPFQ(at-sign)6e`Z
0R`(-c0KTMT%(G*ZjB(at-sign)jNNpF,MCm"c-cBDSk4#(A,Z(at-sign)&bCBkKMC%RJ!"PH'19V(*
SB(at-sign)jRC(at-sign)&LE'(at-sign)E"6QpG(at-sign)f6BR*KC5k4#5BJ3fpZNhD6CA*cC(at-sign)ajNIm&9Lb4"Bf$CAD
6CA*jQ(#3!&11EfajEQpYD(at-sign)&XQ'PZQ(0eBj0SQ'4T#f9bC(at-sign)jMCA1BDA11SBf4*i!
!CA&eDAD4reMNB(at-sign)aPETUXFR5(at-sign)!qUSG'q6B(at-sign)k6D(at-sign)kBGT(r(at-sign)14KFQPKETKdNfpQNh4
SCC0`N!"6MQpXH(at-sign)j[E(at-sign)PKE*2AF,NSehLj+5k1TSf4*i!!9'KPPJ2UU("bEjT6MQp
QNfPcNf9iG(*PE(at-sign)9XHC0cD(at-sign)e`E'8XNf*eG*0hD(at-sign)aXNf+BCC0[E(at-sign)PdG'9N,SkQMC%
RJ!"-CA5(at-sign)!qUSGA16FQ9fD(at-sign)9hNh0[E(at-sign)(at-sign)6BfaKFh0TBf&XNf9iB(at-sign)e`E'9c,SkQMC%
RJ!"8D'(at-sign)(at-sign)"(c#C'PcBh*TE(at-sign)PZB(at-sign)kDV(*dNfpQNf'6FA9KC(*KG'PMNh#3!&11Efa
jEQpYD(at-sign)&XNpG`Z5MAH,NTPJ41!$f6ehL0RrZP-Y8bMT%(cL(at-sign)j+jB$$L%bef'0R`(
-c08aMT%%`!6AH*1j+j2ABBfI!Fc-e6+1N3NmaVPYBCKjN34m`Q+3!&11CBkKMC%
RJ!"eEC(at-sign)XFQ*bB(at-sign)aXHCX$kUKbCA"bCA0PET0dC(at-sign)5BBA1BCQpXE'q6Gh-kMU'KMC)
!V!CIed53!&11Z5MAF,NSehLj+5Q0MCrmUe#0MC%$99,8')k1R`28X)f0N30Z1VN
pMSk1MT%2rrXSe`Z0R`(-c08aMT%(DUc8!*%#UUMA#ifI!Fc-e6+1PJ6!",NTMCr
l#jR9-Sk6ecfj-YFlMTm9!!#0N5H!!,PhD'9bCCB$kUMA#ifI!Fc-e6'1Q`LUV,P
KEQ56e`Z0R`(-c08bMTLjBA*PNf9iBjUXFQKKEQGPB(at-sign)*XCC0eECKLFQ&P,T%&11"
*EQ4PC(at-sign)3kMU'QMC%Vjr6A4CX!Y"Hj+#MA#ifI!Fc-e6'1N3GUV03!N3+UU0F,MCm
"c-c9-Sk(at-sign)"-!%Z5Q0RrX,QG8bMT1j+CB$99)pNpG&Q,NSe`Z0RrX,QBf3!"12e6+
1R`IVfSdaMSk4"016Z5Q(at-sign)!UUSe!#6Z6,A4CLj+0F,MCm"c-c9-Bk(at-sign)"-!%e`Z0R`(
-c08bMT1j+CB#UUJVNpG&Q,NSe`Z0RrX,QBf3!"12e6+1R`IVfSdbMSk4"016Z5Q
(at-sign)!e952C2ABBfI!Fc-e6+1PJGUV03!Q`+UU,Nbef'0RrX,QBh9-SkI"q[DM6'1MT1
j+jMABBfI!Fc-e6+1N3J99VNpN3098M)Sef'0R`(-c08bMT28!*MABBfIq`ZCMG8
bMTm(kpU0-Bk1N36!",NTecZ1TSf4*i!!Z(at-sign)&cN32UU'4PFfPbC(at-sign)3ZMSkI(J!!MC)
!qQpJ-cH1MSb,!!!!*J!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!"LG(at-sign)J!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j6'9dPJ2!iR9cNfjPH(5
6BfpZFfPNCA+6BC0MG(at-sign)*TBj0`N!"6MQpXH(at-sign)j[E(at-sign)PKE*2AF,NSehLj+CB$99)pNpG
iMCrlT6,9-ik(at-sign)"a9CZ5ZE!P99-pGKMCm"c-c9-Bk4"-!%ehL0RrZP-Y8bMT1j+jJ
cef'0R`(-c08bMTB%`!6AH*Lj+jMABBfI!Fc-e611NlNZN38Ume4SCBkN$S!!MC%
RJ!"NDA0MFQPYD(at-sign)jKET!!V(*dPJ3Sf(at-sign)pQNh4SDA16F*T6MQpXH(at-sign)j[E(at-sign)PKE#b4"$K
PE'9dNh9cNf0KE'b6DA56ed5BZ5MAF,NSehLj+5NXPJ3iC(at-sign)9aG(at-sign)&XFbb6BA1(at-sign)"#M
CHCUXFQpeNfYZEjKh,)kKMC%RJ!"dD'(at-sign)4!qUSCAK`FQ9cFfP[ESkKRa+Jh)f4E1(at-sign)
fed53!&11Z5MAF,NSehLj+5Q4!e952Bf0MCrhiF(at-sign)0N35)K63Sef'0R`(-c08bMTX
(DUc8!*%#UUMABBfI!Fc-e6'1PJ6!",NTMCrlT6,9-Sk6Z5MABBfI!Fc-e6'1NpG
KMCm"c-c9-ikBe!#E!UUSef'0RrZP-Sh9-SkI"e*"M6+1MT1j+CM8!*Lj+0GKMCm
"c-c9-ik4"fUXe!#Bef'0R`(-c08aMT2ABBfI!Fc-e6+1NlNTMCrlT6,9-Sk1N35
)KCm&8(at-sign)f*!!"QC3#mc(at-sign)#I#`J,MC&EGV+j-Sk1MSk5!-+*'0FkMTmB%*+0N5H!!,P
8D'(at-sign)(at-sign)!qUSG(at-sign)fDV(*LFQ&XNf9iF(*PFh0TEfk6EfD6G'KPNf4TFf0bD(at-sign)eTEQ&ZQ(5
6DA16C(at-sign)&cD(at-sign)9bNh4[Nh*PE(at-sign)9YQ'+3!&11CA)ZMU'KMC&R*9cA4*!!8ikj+0G`Z5M
AH,NT+Bf0RrbV8)f0N3098Y3BMSkI!p5`MBf4!fikZ6f1MSk1N3rrqbMA#ifI!Fc
-e6'1Q`GUV03!N3+UU0F,MCm"c-c9-Sk(at-sign)"-!%Z5Q0RrX,QG8bMT1j+0F,MCm"c-c
9-ikBe!#4!UUSe`Z0R`(-c08dMT1j+BfIq`ZCe6+1NlNSe`Z0R`(-c08aMTM8!*%
#UUMA#ifI!Fc-e651NlNT+0F,MCm"c-c9-SkBe!#4!UUSe`Z0R`(-c08cMT1j+GF
kMUN9!!#0N5H!!,P"FjB&92KjQUabEh(at-sign)6Dfj[Q(FXN3(at-sign)[M(4SCC0NDA0MFQPYD(at-sign)j
KETKdNhD4reMNB(at-sign)jTFfKPFj0TCT0KEQ56EfjXHC0TCT0dD'(at-sign)6Bh9LD(at-sign)16CA&eBA4
TEfk1SBf4*i!!eh#j+0GiZ5Q(at-sign)!e952C-`PJ2UU'KKFj0KNf4[G(at-sign)*XCC0bEj!!8ij
[G#k1RaU!!)f4*i!!9'KPPJ2J(at-sign)%KPFh0TB(at-sign)k6EfD6BC0MG(at-sign)*TBj0`QP11EfajEQp
YD(at-sign)&XNf0KET0LQ'(at-sign)6C(at-sign)aPCf&ZQUabG'ajNhGbDA4dC(at-sign)k6D(at-sign)k6G(at-sign)fBBR*KE*0ZEh4
K,BkKMC%RJ!"dD(at-sign)pZPJ2UU'&cNfC[E'a[N!#XFRGc1SkKSBf5!)ER9YG)N3$Y9VN
Seh#j+0GiZ5NTMBfIr+Y3MBf4!e95e"L1MTm$e,#0MC%$EMUj2Bk1MSk4$rrl+0F
,MCm"c-c9-BkE"fUXe!#4!UUSe`Z0R`(-c08bMTB%`!5j+BfIq`ZCe6+1NlNSe`Z
0R`(-c08aMTM8!*B#UUMAH,NT+0F,MCm"c-c9-SkBe!#6ehLj+GFkMUD0N5H!!,P
8D'(at-sign)(at-sign)"#TS5'9cFfPKET0fNIpBj'&ZDA0SCA16D(at-sign)D6B(at-sign)jNNfpZE(Q6D(at-sign)D6G'KPNf0
eBQPMNh#D8ij[E(PZEfeTB(at-sign)b6DA16G'KPNh4SDA*NNh#BEj(at-sign)XFRH6CA+1SBf4*i!
!EfD(at-sign)!qUSBC0`N!"6MQpXH(at-sign)j[E(at-sign)PKE*0[CT0NC(at-sign)GbC(at-sign)(at-sign)6EfjP,SkI'S!!MC%RJ!"
"E'a[N!#XFRH(at-sign)!iN5E(at-sign)(at-sign)6B(at-sign)j[G'KPFT0ND(at-sign)GbCA0cD(at-sign)pZ,T%&'&P2ET0SC(at-sign)&bD(at-sign)j
RNf&LN!"6MQpeG*0dD'(at-sign)6GT(r(at-sign)14KEQPcD'PZCj0[CT0dD'(at-sign)65'9cFfPKESkKMC%
RJ!"KFjB%H*4dD'(at-sign)6BfpZC'PdD(at-sign)pZNh4SBA56BC0MG(at-sign)*TBj0`QP11EfajEQpYD(at-sign)&
XNf+BCC0KNh#BCA*QC(at-sign)0dNf0eBTKP,*%%R!pTG*0MEfePFj0ZBA4e,BkKMC%RJ!"
bB(at-sign)aXHCB$5PKdEj0KFfZ6G'KPNfGPEQ9bB(at-sign)b6FA9PFh4TEfikN36SZ(GSD(at-sign)1DV(*
SNfPZQ(D4reMNBA*TB(at-sign)kBG*0[CT0KNh#3!&11EfajEQpYD(at-sign)&XNfpQNf4PCh*PCC2
AESkKMC%RJ!#jGT(r(at-sign)14KEQPcD'9cPJ1lc'PQNf&ZC*0[EQajNfPQNh4SCC0`QP1
1EfajEQpYD(at-sign)&XNfPcNh4SCC2ADj!!Cafj,C0dD*0`Q'q9V(*hNf9bPJ1lc'pQNh0
[E(at-sign)(at-sign)6F*K[E(PZEfeTB(at-sign)b1SBf4*i!!EfD(at-sign)!fqdC'9RFQ9PNpGZ2(at-sign)Z3!'FGZ6qE"3r
N5'9bCC2ADj%$eY'jDA16BC0NDACTFfpbNfpQNpGZZ5kB4T(r"9C[FT0KNfa[EQH
6G'PYCC0*N30[P(4SEh9RD*UXFR56G'KPNf&ZFhHBCA+1SBf4*i!!G'q(at-sign)!bTKG'K
TFj0aG(at-sign)9cG'P[ET0dEj0LQP11CC0LQ'9jQUabEfjNNh*PB(at-sign)1BD#b4!e$(at-sign)G(at-sign)kBG'P
XNfpZCC0NBCKjNIm&9Lb4!e$(at-sign)GfKTE'(at-sign)6E'9K$'jRNf4PFh#3!&11EfjNC(at-sign)kBG'a
jMU'0N5H!!(4SFQpeCfL(at-sign)"'B%G'KPNh0PBfpZC*0fN!#XFQpXG(at-sign)ePNfpQNdKTE'+
D8ijPFR3RFj0MEfaXC(at-sign)0dC(at-sign)56F'&`Q'9bFbb4")6E5C%%CH4KBf0TC'9ZN!#XFR4
KE'ajNf4TFbf1SBf4*i!!Bfq9V(*fNf9bC(at-sign)5(at-sign)!jASG'KKG*0)D(at-sign)aLQP11CA*dNfK
KC*0MEfe`E'9dC(at-sign)ajNh0[E(D3!+abC(at-sign)56DA3ZN38FS&4SCC0cEfaeG'P[ET0MB(at-sign)k
6BTKPNf9XC(at-sign)GKET!!V(*dE(Q1SBf4*i!!CAK`FQ9cFf9NPJ3k%QPZNh9YQUabBR*
KE*0ZEh4KG'P[ELk4"LFG9'KTFj0TFj0[EQajNfpZCC0[CT0cCADBCA*KE*0cG(*
TDfPZCj0bCA0eE(4cNfpQMU'0N5H!!%KTE'+D8ijPFR3RFjB$kUK[ET0TET(at-sign)XFRD
4reMNBA*TB(at-sign)k6G*B$kUKdD'9[FRQ6G'KKG*0SBC(at-sign)XFRD6CCB$kUKLQ'9PET0QEh*
REh4dC(at-sign)iZMTmDJ!#0N5H!!%aPG*B%Dc"eFj0MEfjcD(at-sign)4PFT0ZCAKdNf&ZNfPZPDa
bGT(r(at-sign)14KFQPKET0dPJ4V-'pQNh4SCC0aG(at-sign)PZN!#XFR4TBbk4"VTh9'KPEh*PEC-
cNh4PE'acNh9cNh4SBA56BBkKMC%RJ!"aG(at-sign)PZN!#XFR4TBj%%AMlAF,NSehLj+CB
%'JmpNpGiMCrlT6,90Bk(at-sign)"lPHZ5ZE![PD0GGKMCm"c-c9-Bk4"-!%ehL0RrZP-Y8
dMT1j+jJa-0GKMCm"c-c9-Sk4"-!%ehL0RrZP-Y8cMT1j+jJeef'0R`(-c08cMT%
%`!6AH)fIqk8be651NlNVQ0GKMCm"c-c90Bk4#4j#Z(at-sign)KKFjB%AMjKNh9ZDA&eCC0
KF*!!8ij[E'&bNf0eBQPMMU'0N5H!!(#D8ij[E(PZEfeTB(at-sign)b(at-sign)"#kCeh'3!'ijZ5M
AH,NT,T%'",*8D'(at-sign)6F*K[E(PZEfeTB(at-sign)b6eh'3!'ijZ5MAH,NTNfPcNf&ZNfPZPDa
bGT(r(at-sign)14KFQPKET0dPJ3ZQ(at-sign)pQNpG`Z5MAH,NT,T%'",*%EjKPFj0TG*0SBC(at-sign)XFRD
6CBkKMC%RJ!"KPJ52&R0TEA"XCC0PH("bCA0cD(at-sign)pZNfPZNh9YN!#XFQ*bB(at-sign)b6EQp
dBA4TEfirQ`FQ+8PZC'9PC*0TG*0NEj!!8ijPFbkB9'KPNf9iF(*PFh0TEfk6DA1
1SBf4*i!!G'KPN32UU'C[E'a[N!#XFRGTEQFkMU'KMC&'NlcAFC!!EMQj+0GiZ5Q
0MCrmUe#0MC%$99,8')k1R`28X)f0N30Z1VNpMSk1MT%2rrXSe`Z0R`(-c08bMTX
(DUc8!*%#UUMA#ifI!Fc-e611PJ6!",NTMCrl#jR9-Sk6Z5MA#ifI!Fc-e611Q03
!N3+UU0F,MCm"c-c9-Bk6Z5Q0RrX,QG8bMT1j+0F,MCm"c-c9-BkBe!#4!UUSe`Z
0R`(-c08bMT1j+BfIq`ZCe6+1NlNSe`Z0R`(-c08aMTM8!*B#UUMAH,NT+0F,MCm
"c-c9-SkBe!#6ehLj+5MA#ifI!Fc-e611Q03!NpGiZ5RA1SkQMC%RJ!#j5(at-sign)k(at-sign)!qU
SG'KPNf0XBA0cD(at-sign)0KE*0XDA4PFQ&dGA*PNh4SDA16D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh5
(at-sign)!qUSDA16C'9ZEh4PC*0LN!#XFRQ6G'KPNfaPG(4PFT2ADT%!V01j,Sk1Rai!!)f
5!2T[B$-iMSk-L`!!!#F!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!C3PS!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if4*i!!Z9GSBA5(at-sign)"+[8F(*[F*!
!8ijPFR5DV(*jNhGTE'b6G'KPNh&eD(at-sign)kBG'PMNh#3!&11EfajEQpYD(at-sign)&XNpG`Z5M
AH,NTNfKKQ(DBCC0hD'9ZNh4SCC0TETKfNIpBj'&bD(at-sign)&ZQ(56efU1T!k!!)f4*i!
!ZAD4reMNB(at-sign)jTFfKPFcq1SBf4*i!!9'KPPJ1kff&ZFhHDV(*PFT0dEj0dD'PcNh&
eCA0dD(at-sign)pZNfPcNh"XC(at-sign)&cD(at-sign)jR,T%&+2&8D'(at-sign)6D(at-sign)kBGT(r(at-sign)14KFQPKETKdNpGUN34
RVVP[CT0KNh&eD(at-sign)kBG'PMNh#3!&11EfajEQmYMU'0N5H!!'eTB(at-sign)b(at-sign)"'akDA16D(at-sign)4
PETUXFR4TBf&XE(Q6CA&eB(at-sign)b6G'q6HQ9bEj0TCT0KEQ56EfjXHC0TCT0dD'(at-sign)6FA9
TETKdD(at-sign)16DA16BA#3!&11EfaKFT0dEj0cEfePMU'0N5H!!'j[ETB%+!"dFQPfD(at-sign)&
XNh#3!&11EfajEQpYD(at-sign)&XNfpQNf4PCh*PCC0dPDabGj0[,T%&m1G#GA5(at-sign)"#J!G'K
PET08D'9[FQ9YNc(at-sign)6G'9XE(16GA16G'KKG*0dD'(at-sign)1SBf4*i!!FA9TET(at-sign)XFR4TBjX
$kUKYBC0jQ'+3!&11CCKhFQPdG'9ZQ'PZQ(4SCCKQEh*YMU'KMC)!ME+meh#j+0G
iZ5Q(at-sign)!e952C2ABlNSehL(at-sign)!UUSe!#6eh+0R`(-c08aMTX%`!5j+BfIq`ZCe6(at-sign)1N3G
UV,NVNbJaNp3!NpGMZ5NSehL6e!#6eh+0R`(-c08bMTLj+BfIq`ZCe6(at-sign)1Q0FlMTm
9!!#0N5H!!,PhD'9bCCB$E"lAFSfI!Fc-e6'1N36!"0FlN3(rrR+0R`(-c08bMT%
),#+jBA*PNh4SCC0bEj!!8ij[G(16EfD6BC0aG(at-sign)&NFQ&dD(at-sign)16CA&eBA4TEfiZN38
1XP4SN!#XFR9c,*%$K(at-sign)adD'(at-sign)6GT(r(at-sign)14KEQPcD'PZCj0[CT0dD'(at-sign)1SBf4*i!!D(at-sign)k
9V(*fNIpBj'&bD(at-sign)&ZNh5(at-sign)!c9cefU4!q*'Z(at-sign)PcNf'6EQ9MCA0cBA*jNf&ZC*0cG3j
MD(at-sign)9ZQUabG*0MEfjNDA4TEfk6G'KKG*0dD'(at-sign)6FA9TETKdD(at-sign)16F*!!8ij[E(PZEfe
TB(at-sign)b1SBf4*i!!eh#j+0GiZ5Q(at-sign)!aY9E(at-sign)'DV(*jNf+3!&11CC0hFQPdG'9ZNf&cNh4
SCC0cG(at-sign)f6EfD6G*KhQ'q6FQ&dD'9bNh4SB(at-sign)k6G'KbC(at-sign)(at-sign)6$'CdD*0`N!"6MQqBGjK
PFR16EfD6E'PZC(at-sign)&bMU'0N5H!!(#3!&11EfajEQpYD(at-sign)&XFbk4"U5J9fKPETB%BqK
dD'PcNfPcNh4SCC0MBA0P,*%%JMKdD'(at-sign)6$'CdD*0NC(at-sign)GbC(at-sign)(at-sign)6CA&eBA4TEfk6eh#
j+0GiZ5Q(at-sign)"#1d2C-`N34Mk'0KESkKMC%RJ!"LN!"6MQ(at-sign)(at-sign)!qUSFfpXGTUXFQ9NNf+
BHC0bB(at-sign)4TBf&XFbk1SBf4*i!!3RQ(at-sign)",'SFfPYD(at-sign)aKFT0KFQGeE(at-sign)9ZQUabG(-XN36
MD(at-sign)pZCC0MB(at-sign)k6BfpYF(9dCC0KE'b6D(at-sign)kBGT(r(at-sign)14KFQPKETKdFj0hD'pcCC0fNIp
Bj'&ZDA0SD(at-sign)jRNfPY,BkKMC%RJ!"`E'PPFjB%+$YdD'&dNh4SCC0PFA9KG'P[ET0
[CT0NC(at-sign)GbC(at-sign)(at-sign)6$(DDV(*PNfPcNf&XCfpbDA4SE(at-sign)PMB(at-sign)aXHC0cEfafNIpBj'&LE'(at-sign)
6BTKjNh*KC'PMB(at-sign)ac,SkKMC%RJ!"8Gj(at-sign)XFQ9ZNh56H5edD(*PCCX$kUKTET0fNIp
Bj'&bD(at-sign)&ZNh4cQ("XBC0jQ''BFQ9XCAD4reMNB(at-sign)k6G*KbEfaP,*KKFjK$BC0jE'9
jQ(H6BA1B$(*cG*KdEjKcD'q6Gbk1U4U!!)f4*i!!5'PXBTT6MQ9bG#GcPJ2Hmh4
SC(at-sign)pbC(at-sign)f6Efk6$'jTG'(at-sign)6Cf9ZCA*KG'P[ET0[CT0dD'(at-sign)6FQPZCj0[CT0TET(at-sign)XFRD
4reMNBA*TB(at-sign)k6G(1(at-sign)!plcBf&ZNf+BCC0bC(at-sign)0KFh51SBf4*i!!D(at-sign)k(at-sign)",`BG'KPNfa
KEQGeB(at-sign)GPNfpQNh9YQUabBR*KC5b4"2"eB(at-sign)jNNf0KET0LN!"6MQ(at-sign)6CfPfQ'9ZNf'
6FfPYF'aPNf0[ECKLD(at-sign)jKG'pbD(at-sign)&XNh"bEj!!8ij[CSkKMC%RJ!"dD'&dPJ2UU'4
TFh#D8ijPER0PFj0hDA4SNh4SCC0)D(at-sign)aLQ'9bG*0LBA0TFj0dD'9[FQ9Y,SkQMC%
RJ!"*ETB%3QjME'pcD(at-sign)jR,*%%(at-sign)&pXCA56GA16G'peBjUXFQL6GA#3!&11Efk6B(at-sign)j
[G'KPFT0bC(at-sign)&cEfk6CQpbNh4SCC0NC(at-sign)eTFf(at-sign)6EfD6G'KPNh0jECKLN!"6MQpXD(at-sign)1
1SBf4*i!!E(at-sign)9dD'q3!&11C*B$kUKTET0TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G*%$kUKdD'9
[FRQ4r`9(at-sign),SkQMC%RJ!"*ETB$G2KYBA4SC(at-sign)eKG'PMFbb4!ib#DA56DA16CAKdFQ9
YC(at-sign)ajNf4T$Q0eE(56G'q6G'9XE*0dD'(at-sign)6G(*eG'JZN384T94SCC0QEh*YB(at-sign)b6CAK
`N!"6MQpcD5f1SBf4*i!!G'P[ETB%!cY[CT0KNfeKG'KPE(at-sign)&dD(at-sign)0KE*0dD'9[FRQ
6C'q3!&11CA16EQpdNh4PE'b6G'KPNhGSEfaPNh4bGA4S,T%&JTT8D'(at-sign)6G(*eG'L
6EfD6BBkKMC%RJ!"YBA4SC(at-sign)eKG'PMB(at-sign)b(at-sign)!mXdG'KPEh*jNfPcNfe[FQ(at-sign)6E'PVN!#
XFQ9XHC0dEj0LQP11CC0RFQ&cF*KPC*0hD'PXCC0hN!#XFQ(at-sign)6E'PcG'9ZNh4[Nf'
6Bf&cG(at-sign)&XMU'0N5H!!(*PE(at-sign)&bDjB$J,CYB(at-sign)4PNf+DV(*jNh0[E(at-sign)(at-sign)6CAK`N!"6MQ9
bG*0dD'&dNfGTGTKPFj0KQ(HBBCKjNh0[E(at-sign)(at-sign)6D'PNC'9ZNfe[G'PfNIpBj'&dD(at-sign)p
Z,*%$PHGhD'9ZMU'0N5H!!(HDV(*PPJ1j[3aZB(at-sign)aXHC0`D(at-sign)k6C'qBGfk6G'KPNh5
BHA"TBf&XNf9iB(at-sign)e`E'9c,*%$`iC[FT0hD'9ZNhHBCC0NDA0MEjKfQ'9bNhGSBA5
6G'KPNh*PB(at-sign)b1SBf4*i!!F(*[BQaPEA1(at-sign)!qUSBA*PNh4SBA56GjUXFQ9bCC0cG'p
bC(at-sign)56BT!!8ijPD'PZC*0dD'(at-sign)6FfK[Q(HBBf&cCC0`FQpLE'9YFbk1TSf4*i!!8'K
TE'pcEh"SCA*cPJ38L(at-sign)&ZC*0`FhPMQUabD'PKG(*TFh4cNh0SEh9XC*0PH("XB(at-sign)P
ZNhGSQ(Q6DA56DA16G'KKG*0hQ'(at-sign)6E(at-sign)&dD'9YBA4T,BkKMC%RJ!"MD(at-sign)&ZFjB$M"P
KFQ(at-sign)6D(at-sign)k6G'KPNfKKBQPdNfpQNh0jFh4PE(at-sign)&dD(at-sign)0KE'ajNf9bBA0TEQH6Eh9bNfC
[N!"6MQpdFh4PF(-ZN38C(at-sign)e0MD(at-sign)9ZQUabG'PcG(16D''BGTKPMU'0N5H!!'&XGj(at-sign)
XFQ'6HA1E"3(at-sign)5E'q3!&11EfZ6C(at-sign)5BBA0VNIpBj'&ZBf(at-sign)BBA5BG'KTFjKcG(*KEQG
PQ'KKBQPdQ'pQQ'eKG'KPE(at-sign)&dD(at-sign)0TB(at-sign)jc,*%&6%ahD'PMNfLBD'&cMU'0N5H!!'1
DV(*SB(at-sign)jRC(at-sign)5(at-sign)!qUSE'PdG'aPNfCbEff68(PdD'&REh*KFj0dEj0[GA+6C''BHC(
r"9BZMUD0N5H!!&4SCCB&RI*SD(at-sign)4NC(at-sign)k6F(9bF*T6MQpcCC0[CT0dD'(at-sign)6FhPYN!#
XFQ+BEfaTBj0YCA4SEjKNNfPZNfPZPDabGT(r(at-sign)14KFQPKET0dPJ(at-sign)GmR4SC(at-sign)pbHC0
hN!#XFQ&cNfj[G)kKMC%RJ!"cD(at-sign)e`E(Q(at-sign)"3*MG'KKG*0[CT--EQ4TEQH6C(at-sign)&cHC0
PH("bCA0cD(at-sign)pZFj0QEh+6D(at-sign)k9V(*fNIpBj'&bD(at-sign)&ZNh4c,T%)J"""N38#'f4PCA#
3!&11CA+(at-sign)"3*MCQ&TG'L6Gj!!V(*KFikKMC%RJ!"RG(at-sign)PND(at-sign)jRPJ0)5(4SDA16E(at-sign)9
dD'qD8ijN,T%&!X"*G*0hN!#XFQ&cNh4SCC0PH(#BC(at-sign)0dBA4TEfk6G'KKG*0dD'(at-sign)
6CAK`FQ9cFfP[ET0[CT0TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G(11SBf4*i!!BTUXFRQ(at-sign)!U(at-sign)
)G'KPNh0jECKLQP11EfaTBj0YCA4SEjKNNhHDV(*[G(at-sign)aNNf9fQ'9ZQ(4eB(at-sign)aXHC0
RG(at-sign)PNCC0eFj0dEj0cD(at-sign)jRE'(at-sign)6Eh9dNh4SCC-LFQ9XCAD4reMNB(at-sign)kBG#+1MTmH!!#
0NJ$kEf!c1Bk1M)X!!!!S!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!'K(at-sign)U!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`Z0N5H!!,P[FTB$2BXLD(at-sign)e`N!"
6MQpbG'&ZQUabG#+6D(at-sign)kBGT(r(at-sign)14KFQPKETKdFj0KE(at-sign)pZCj0KET0TEJaZDA4PNhD
4reMNBA*TCA5BHC(r"9BZN36r,%PdNhHBBA16G'KPNfK[F*!!8ijPNh4SBA56G'K
PMU31J!#0N5H!!(0TCfjT$'0KEQ0PPJ0DN(at-sign)pQNh4SCC0fQrpBj'&ZDA0SD(at-sign)jRNfp
QNf&ZNfPZPDabGTKKFQPKET0dPJ0DN(at-sign)0[G(at-sign)aNNf+3!&11CC0RE'9KEQ9NNfCbEff
6DA4cNh9YN!#XFQ*bB(at-sign)b1SBf4*i!!CAK`FQ9cFfP[ELk4"8&99'KPPJ2YHRD4reM
NB(at-sign)jTFfKTEQH6EfD6G'KTFj0QB(at-sign)PdD*0hN!#XFQ&cNh4SCC0bC(at-sign)&XNh*PBA0[ET0
QEh+6G'KPNf4PE(at-sign)PcCC0[CSkKMC%RJ!"ME'&cFfPMB(at-sign)bE"0KZD(at-sign)k9V(*fNIpBj'&
bD(at-sign)&ZNh5BG'KPEh*jNIm&9Lb4"42JB(at-sign)jNQ(4SCCKbCACTGT(r(at-sign)14KE*K[CTKdD'P
cQ'CKDA4SQ'PcQ(4SCCKbC(at-sign)&cEfkBCQpbQ'PdFikKMC%RJ!"`FQ9cC(at-sign)k3!+abG*%
$kUKbC(at-sign)*TFR4S,SkT'S!!MC%RJ!"AD'9dD'9bPJ38MQpbNfj[G*0hN!#XFQ(at-sign)6GfP
XE*0cG(at-sign)0MC(at-sign)9NNh4SDA16Ff9MEfjNNh4TE(at-sign)(at-sign)6GfKPFQ(at-sign)6G'KPNf0XBA0cD(at-sign)0cNfC
KD(at-sign)aPC*0TFikKMC%RJ!"KPJ5D2(at-sign)0XD3YSB(at-sign)jRCA+6G'KKG*0hD(at-sign)aXNh"bEf*KBQa
jNf+3!&11CC0bCA0[E(DDV(*PC*0TET0dD'(at-sign)6EQ9iG*0QCAH6HCKPBA*c,T%(4jj
*N35D%(HBEh9XC)kKMC%RJ!"ZEh5(at-sign)!qUSBTT6MQ(at-sign)6Fh#BC(at-sign)&VD(at-sign)jRNh4[NhQDV(*
[GC0ZEjKhNfPQNdQ6C'PNNfj[G*0LN!"6MQ9XD(at-sign)9fQ'(at-sign)6D(at-sign)k6Fh9MBf9cFbk1TSf
4*i!!9'KKEQZ(at-sign)!qUSHCUXFQpeNfC[FT0jQ'peFT0KG(4PETKdD(at-sign)pZ,SkQMC%RJ!$
63QPLE'P[Ch*KF'L3!+!!HBkQMC%RJ!#j5Lj3NIm&9Lj6,TB&LP*,G(at-sign)jRNf&ZC*0
(D(at-sign)&Z,80KFQa[Ne*[G'%XN3Ab2&4SCC0TET(at-sign)XFRD4reMNBA*TB(at-sign)k6G*B&LP*dD'9
[FRQ6EfD6BQPZBA*jNfC[FQec,)kKMC%RJ!"#G(at-sign)aXCA4TETB$kUK[CT0dD'(at-sign)63(at-sign)e
PFQPMB(at-sign)k66(at-sign)&dD'9YBA4TBf&XNe0[N!"6MQ0TCA5DV(*jNIm&9Lb6GTK[E#ia-*-
S-6Ni0#NXNh"`,T%&11!b0bdi05k1TSf4*i!!4T(r"9CbB(at-sign)jVPJ53!+j%,T0(FQp
cFfKKER-XQ`5k,dGTB(at-sign)iY3f&bE'q68QpdBC0KEQ565Qq3!&11C(at-sign)b635k68h4PD(at-sign)i
XQ%PZPDabGT(r(at-sign)14KFQPKET0dN353!+j8D'9[FRQ1SBf4*i!!B(at-sign)jNPJ6lqe0eF*!
!8ijPFQ&XCf9LFQ&c,*X&3&"$3Ne6N36lY9*PCfP[EQ&XNd0[EQCPFQ9ZBf9cNfP
ZNdeKG'KPE(at-sign)&dD(at-sign)0c,*KfN!#XFQpX,T%)E0Nf15b1SBf4*i!!3(at-sign)ePFQPMB(at-sign)k(at-sign)!qU
S6(at-sign)&dD'9YBA4TBf&XNe0[N!"6MQ0TCA53!+abHC(r"9BX-6Ni0bk1TSf4*i!!6Lk
(at-sign)"2%46(at-sign)9dFQp`N!"6MQpXDA16B(at-sign)jNNdGTB(at-sign)iY3f&bE'q68QpdB5b4"6+X8hPYE(at-sign)9
dFRQ6BfaKFh0PFcU4"d(at-sign)cCR9ZBh4TEfjcNfpQNh4SFQ9PMU'0N5H!!(D4reMNBA*
TB(at-sign)*XCA-XPJ2UU%&YCA*TBf&ZNdeKG'KPE(at-sign)&dD(at-sign)0KE*00EfkDV(*dD'ajNIm&9Lb
6GTK[E#kE"6MJ16L6+$%j16%T,*0`F#kB-c)i,6-c-Lk1TSf4*i!!6Lk(at-sign)"+9e6(at-sign)9
dFQp`QP11EfaTFbb4"03S4biY3bk68QpdBC0KEQ565QqBC(at-sign)b635k68h4PD(at-sign)iXN36
8+&4SC(at-sign)pbHC0[CT0cH(at-sign)eYCA4bHC0ME'&cFf9c,)kKMC%RJ!"3FQq3!&11Bf9PC'P
ZCh1(at-sign)"!A3EfD6G'KPNdjKG'P[EQ&XNd&MB(at-sign)4PECUXFRQ6EfD68f0TC(at-sign)jMCA-XN33
-QRDBEf`ZQ`(at-sign)+(at-sign)6JiNbJa16Na+5b4"!bDF(!ZQ$Jd-68YMU'0N5H!!$Jd-6NZMUD
0N5H!!&*TBj(at-sign)XFQKKFQ5E"Hm(at-sign)4(at-sign)KbC(at-sign)k6BT96MQpbCjKKEQ5B4fPKELe$BA*XEjK
5Eh4K,*%'F$&"F*0[E'&bDA53!+abHCKKEQ5BBf&ZEfjTBf&XQ'C[FQecMU'0N5H
!!'C[FTB%a!aSEfe[Cf9ZC(at-sign)peFj0`P911EfajEQpYD(at-sign)&XFbbE"2TP4A9bEh#6C(at-sign)&
ZPJ6%$%T[GA*ZB(at-sign)b6EfD63fpYPDabBQPZBA4[FQPMFbbBGT0[,T%(a3da0)kKMC%
RJ!!S-6Nj-bNXN32UU("`,T%&11!a06FY-6Ja,SkQMC%RJ!"(D(at-sign)&Z,80KFQa[PJ(at-sign)
P1e*[G''6B(at-sign)jNNd)ZNd3ZNe54r`9(at-sign)BCUXFRPXEh)XN3B6i&4SCC0ME'&cFfPMB(at-sign)b
6G(at-sign)fBBR*KE*0MB(at-sign)aMG(at-sign)aeFbb4"K2J8dP"6BkKMC%RJ!"+Eh9bEQ&XPJ2UU'pQNde
KG'KPE(at-sign)&dD(at-sign)0KE*0"EQ&XHA0TFbb6GT!!V(*[E#k4"6MJ-M(at-sign)6+$%j163T,*0`F#i
f163Y0c%a,SkQMC%RJ!""ETUXFR4[EQP[PJ5X%d4TNd0bCA0MC(at-sign)jkEj0KEQ564fP
KELe$BA*XEj05Eh4K,*%%h'e6G(at-sign)b6Bf&XBfpXEj0eECKLFQ&XC5b4"0aY8QPMCA*
MQ'KPMU'0N5H!!'4TPJ2UU%eKG'9YBA4TBf%XNhD3!+abEfb6(at-sign)%a*P9115C0*PJ2
UU#Ja16Nd+5b6F(!ZN38ii$%b15da0M)ZMUD0N5H!!%iZPJ(at-sign)2,%ePG(*[F*T6MQp
XDA-XN3Ai68GTB(at-sign)iY3f&bE'q68QpdBC0KEQ565QqBC5k4#LCX35k68h4PD(at-sign)iXN3A
i690jE(at-sign)ePG(*jNf0XBA0cCA16EfD1SBf4*i!!CR9ZBh4TEfjc,*B$kUK+Eh9bEQ&
XNfpQNd&XCf9LFQ%XNhD3!+abEbkE"6MJ-6FaNbJa16Ne+5b6F(!ZQ$Jd05di0MB
ZMSkI(J!!MC)!qQpJ0$#1MSb,!!!!+3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!"VFbJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC)!J`MMdd0268*
*6N'4rZ!!9%p558&-N35!!&0139"65%p88ikN$S!!MC)!fRZ1BT!!B!"PD(at-sign)jRMU'
0NA8Df94SCCB%J!"dD'PbC*0[CT0dD(*PCC0$EfaXEj!!B!"aG(at-sign)PeEC0-C(at-sign)0dGA*
PFikKMC%Xp(0NC(at-sign)aTGTUJ!'9bC(at-sign)5(at-sign)")!!BA56G'KPNd&ZETKeB(at-sign)b66(at-sign)9PG'PZCj0
[CT0dD'(at-sign)63(at-sign)ePFQPMB(at-sign)k66(at-sign)&dD'9YBA4TBf&XMU'0NJ$91ij6Ej!!B!"MD(at-sign)9dN!#
J!(Q1SBf5!*SR(%*KE(4TE(at-sign)pbC5b(at-sign)")!!5Q&ZN!#J!(9KFRQ615b6-6Nj1)kKMC)
!Z8mF4fPKELe$BA*XEj%%J!"5Eh4KMU'0NJ#(at-sign)blY%CA"KFR4YC(at-sign)k3!+!!G*B%J!"
[CT00BA4SC(at-sign)eKG'PMFikKMC)!h11168P8MU'0NJ#CXFG$B(at-sign)f3!+!!BR*TC'GPPJ5
!!%e"Nc!b-6-j,63c-$H1Rc1!!)f4*i!!Z9GSC(at-sign)k(at-sign)!q(A5C%$iG4hQUabBA16D(at-sign)k
6D'PRD*0cBjKSEj!!8ij[E#b4!q1DECKjNd9ZCfaTFfL6G'9KBjKSCA+6Cf'BGTK
PNfePNh4[Nh*PB(at-sign)56B(at-sign)k6CA0cBCKjNf+BHBkKMC%RJ!"+B(at-sign)ePFjB$+mG8D*UXFR9
bBT!!8ijPFLb4!e(dBf&XE'9NNb*8D'(at-sign)6Ff9MFQ9dNfaTCQ(at-sign)6EfD69j(r"9CKE(4
PFT00DA4dQ(NL,T%%q8""CR4PFT0bCA*PB(at-sign)4TEQH6G'KTFikKMC%RJ!"PFh0KPDa
bHCX%'f4PGT0PFRQBCQ9hQ(Q6C(at-sign)&bFbb4"#H65C%%'eGNC(at-sign)0TC'9NQ(4SBA5BCAD
6CA*jNfpZCCKSBA1BBCKANIm&9Q&XG'9bQ%eTG(56HCKMEfe`E'9i,SkKMC%RJ!"
2EQ(at-sign)E"$fPGj(at-sign)XFQ'6HCKdEjKeEQ4PFR0dB(at-sign)jNQ''BF*!!8ijPFR0[ETKYD(at-sign)GSNh5
BBT!!8ijPQ(4[Q'4TFf0[NhD6CA+BG'KKG*K`N!"6MQ9bFfpZ*h1B9j(r"9CKE(4
PFSkKMC%RJ!"0DA4dPDabHC%$kUKQB(at-sign)k6G'&cD(at-sign)9c,SkKMC%RJ!"0Eh0dPJ6CY'p
QNfpeFT0YBA4SC(at-sign)eKG'PMB(at-sign)b6G'K[G(at-sign)GSQUabG(16D(at-sign)k6D'PRD*0cBjKSEj!!8ij
[E*0[FT0TET0MEfaXC(at-sign)GPNhHBCA*PNeH4r`9(at-sign)B(at-sign)`YMU'0N5H!!(4PFTB%XY"0DA4
dQUabHC0QB(at-sign)kBG'&cD(at-sign)9c,T%(N9KAD'9ZNhHBCC0XC(at-sign)&bEQ9NNf'6EQ9hNh"TC(at-sign)0
PNfpQNfeKG'JXN36NfRHBCC0hQ'peE'56$'jNMU'0N5H!!'peFR0PE(D9V(*PFjX
%cpjQB(at-sign)k6G'&cDATTEQHBEfkBDA4cQ(#98ij[Fh0TBQaPQ'GPEQ9bB(at-sign)aTHQ&dD(at-sign)p
ZFbk4"qL"3A1BFfq6EfkBBA1BGj!!V(*PQ(9ZC'9b,BkKMC%RJ!"cG'q98ij[Nf5
(at-sign)",3PBQPZEfeTB(at-sign)b6Bfq3!&11C3jMD(at-sign)9ZPDabG(-XN36QK(H6CCX%Y#9QB(at-sign)k6G'&
cDATPC*KKBT!!8ij[GA5BG'KPDA+BCf9ZCA*KE'PkBA4TEfkBG'qBG'KPMU'0N5H
!!'0KFf(at-sign)(at-sign)!b,eGfKPET0dD'(at-sign)6C'9ZEfeTEQ&dEh+6DA16EQ9RBA4TGTUXFQ8lN30
PKR4SCC0YEfePETKdNhHBCC0XC(at-sign)&bEQ9NNf&LN!"6MQpeG*0NCA*TGT(r(at-sign)14K,Bk
KMC%RJ!"dDAD9V(*PFbb4!`KXGj0PQ`,2h(at-sign)aKG(at-sign)jMNfKPC*KTET0dEjKNCA*TGT(
r(at-sign)14KG'PfNf9cQ'pQQ'CbB(at-sign)0dD(at-sign)pZB(at-sign)bBEh*NCA)ZN36DR%PQQ(H6CCKhNf9bCCK
PGT0PFTKPH(#3!&11Eh0PC)kKMC%RJ!"dEjB$+&pdD'(at-sign)68QPPE(at-sign)&ZET0kCA4KNfC
eEQ0dD(at-sign)pZ,*%$6cThQUabCC0hQ'peE'56FQpYB(at-sign)kBG'PMDATPNh0[E(at-sign)(at-sign)6EQ9hNfP
ZQ(4PFR"bCA4KG'P[ESkKMC%RJ!"[CTB$kUKdD'PcNfCeEQ0dD(at-sign)pZNh4SBA56GjU
XFQpeE'56CfPfQ'(at-sign)6BCKhQ''BHC0TG(16Ff9MFQ9d,SkI'S!!MC%RJ!"8D'PcPJ3
d,(at-sign)aPBh4eFQ(at-sign)6FfK[G(at-sign)aNNfKKPDabGT0PPJ3d,(at-sign)+D8ijPC(at-sign)k6CfPfN!#XFQ9ZNf&
ZEh4SCA+6G'PdE'8ZN3B9F%PdNh0SEh9XC*0LQ'(at-sign)6Bf&XE'9NNb*8D'(at-sign)1SBf4*i!
!E'&dCA+(at-sign)!hQ#E'PQCC0[CT0ANIm&9Q&XG'9bNdeTG(5DV(*j)Lk4"4-T5A56GfP
XE*0MEfjcDA0dNfpQNf'6Ff9aG(at-sign)9ZBf(at-sign)6EfD6C'PcF'aKQ(PcNfpQNb*MQ'LBGA4
k,BkKMC%RJ!"`B(at-sign)JLPJ5,5f+DV(*jNf'69j(r"9CKE(4PFT00DA4dQ(Q6GfK[NfK
KFj0XEh0dNfKTFj0cD*KjEQ9cFbk4"aV*4(at-sign)&MQ'L6FfjKF(0SEh56GfPXE*0NC(at-sign)&
XMU'0N5H!!(GTG'L(at-sign)!qUSFfpYCC0jQUabEh9dD'CeE*0QB(at-sign)kBG'&cHC0dD'&dNfK
KFj0`BA*dD(at-sign)&XE(Q6GjK[FQZBC(at-sign)56Eh9d,Sk1Rai!!)f5!2T[B$3aMSk-L`!!!#S
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!EG,S!+J+2(at-sign)0S2f
'e`ZJ!PXSpBfJrFRA#if4*i!!Z8C*8P08PJ-FX&0139"65%p81T%$(14"6T0&(at-sign)%&
08%a&Ndp'Ne"5N!#XFNp'58j*9%(at-sign)63dp03NP13C(r"9C86e**3e-ZMU3DJ!#0N5H
!!%aPG*B#r#TeFj0LN!"6MQ9RD(at-sign)k6GfPdD*0KNh"TC(at-sign)0PNfpQNfKTFh4[FRNY$'0
dD(at-sign)pZ,*%$+paKEQ56CQ&ZQUabG'&cDATPNfK[Q(H68QPPE(at-sign)&ZET0YD(at-sign)GSQ(51U3k
!!)f4*i!!D''9V(*fNf(at-sign)E!qUSC'PcBfq6GT0PFQ9NQ(4SCCK5D(at-sign)9YB(at-sign)jZQ(TPG''
BCR9ZBh4TEfiZMU'0N5H!!&"bEfCPFh0[FTB$qV"5D(at-sign)9YB(at-sign)jZNhHDV(*KFj0KQ(H
BBA*PNh4SBA56BA*TG'KYCA4TBj0NC(at-sign)jcDA5BHC0TFj0[CT0QG(at-sign)jNB(at-sign)ePETKdB(at-sign)b
6D(at-sign)dYMUD0N5H!!(#D8ij[FR4KEQ0PPJ1P%fPZNfk9V(*eEC0LQ'9bPJ1P%h4SC(at-sign)p
bHC(r"9BZN38KVNPQNpG"NlPTFj0KNh0eBR0PG*0[CT0dD'(at-sign)6Ff9dNfpQNh#BEh0
TG'PfQUabCC0TETKdC(at-sign)GPFR16edk4!8$NZ5b1TSf4*i!!G'KPETB$kUKdD'(at-sign)6BA*
TG'KYCA4TBj0NC(at-sign)jcDA53!+abHC0[CT0dD'(at-sign)6Ff9dNpG"NlPTFj0NC3aZC(at-sign)56G'q
6BT!!8ijPMUDI%0RfMC)!LZQ#ef4PER1j+0G"Z5Q(at-sign)!e952Bf0N3C!T'aTEBkI"J!
!MC2BEYSK-Bk1MBf0RrIKaBf4(8+UZ6'1N4b`ijm&8(at-sign)f*!!"QC3!(!iQI#`J,MGG
ZMSk1MT%Njjr8DihA3CB#UUM8A*0QZ6(A1jB"rrkj-YFlMC-kNcU61Sk4%FUF1j0
Ze'H1N94mq(at-sign)[A1SkI*,XqMC%RJ!#jGfKPEQ9fN!#XFQ9bPJ8B(at-sign)h4SCC0XD(at-sign)eTG*0
PH'PcG(-ZQ`M"q%D4r`9(at-sign)Eh+6CAKKEA"XCC2AC'9ZFlNSedk4!8$NZ5Q(at-sign)"9EB2C-
a,TK*CTB&'&[A3BfI!Fc-f(#1N3RIZ,PTFj0dD'(at-sign)6Ff9dNfpQMUD0N5H!!'f3!+a
bG(at-sign)adDA"XCA1(at-sign)"+McEfD6G'KPNh"bD(at-sign)ePNpG`Z5bE"0L'G'KPET2AC'9ZFlNSed'
0R`(-c0K`MT%%aefj+C%%Q6NpMBf0RrY'GBf4"G!Ce6'1N3A-E*m"l,f*!!"QC3!
%4efI"[!RMGK`MSk1MT%,4[bj1j%&#"KhD'&dNfPcNfe[FQ(at-sign)6BA"`N!"6MQ9KE'P
ZCbbBEfjPMTm3N['0N5H!!'9KFfPXHCB&',eMEfe`GA4PFj0dD'&dNpGNC(at-sign)jcZ5M
A3BfI!Fc-f(#1N3JrXG4FN30i90G"MCm"c-cBFBk4"*6qZ5Q4"9H!2Bf0MCrl4R(at-sign)
0N3LBhp8aMT%'LV1I!HbpL3!!CQ8!#&aER`E`*ihBF('1MSk1N48brVPQEh+6B(at-sign)k
DV(*jNh5BGjK[Nh"bD(at-sign)ePFj2AF*1jB(at-sign)jNNpGaN!"Z1ENZN3M$(dPQMTm2XC10N5H
!!'4PER0TG*(at-sign)XFRQE!fD1Gj0PFQ(at-sign)BBCJSBfpeET0dB(at-sign)*XHCKKC'4TG'PfNf8TQ("
bEf*KBQPXDA56HCKYC(at-sign)&cGA*P,*%$J2ThNf(at-sign)BGj0[G(at-sign)aNQ'PZCQ9bQ(4SBA51TSf
4*i!!G'KPQ`-2KQ9fPDabC(at-sign)k6G(1BG'KKG*KKQ(*KEQ4[E(at-sign)ajQ'16D'pcC(at-sign)kBET0
eEC0LN!"6MQ9bQ'PcQ'4TGQPcD(at-sign)*XCCKLNhQBC(at-sign)PdD'9bQ'pQQ(56Gj0[Q("bD(at-sign)e
PFikQMC%RJ!"KFQ(at-sign)(at-sign)"%5UD(at-sign)jNCA#3!&11C(at-sign)jNC(at-sign)kDV(*d,T%'4ZC9EQC[FR4eEQ&
dC(at-sign)ajNIm&9Lb4"&XUBA*TG'KYCA4TBj0NC(at-sign)jcDA5BHC0cD'&bCA16FfpYCC0LGA5
6EQpdNf&XE)kQMC%RJ!"`FQp`N!"6MQ9bG'PPFjB%K+Y[CT0KNh"bEf*KBQPXDA5
DV(*jNfePBA0eFQ8ZN3F'k%PdNfPcNfe[Fh56C(at-sign)e`D'&dD(at-sign)0KE'ajNfj[G*0MEh9
ZQ(4KBQajMUD0N5H!!'&NC'PdDAD3!+abC5k1SBf4*i!!3(at-sign)CdCA+(at-sign)![)SBC0`P91
1CA*TEj0NPJ,b+'pQNh0[G(at-sign)b6Ff9KFQ1DV(*SD(at-sign)jR,*%$)pY3FQpQCA0cEh+68QP
PE(at-sign)&ZET0hQ'&cNf&LE'(at-sign)6G'q6$'jNNf'6FQ9YC(at-sign)4jMUD0N5H!!(4[PJ35pR0[E(at-sign)(at-sign)
6C'8-BfPPEQ0TCA16EfD6BA*TG'KYCA4TBj0NC(at-sign)jcDA5DV(*jNf+BHC0KNf*bD(at-sign)a
XD(at-sign)&ZQ(56E'9KF*0[CT0TE(at-sign)&RD(at-sign)jKG'P[ELk1TSf4*i!!5'(at-sign)(at-sign)!qpDBjUXFQK[Ff(at-sign)
6BC0bC(at-sign)&XNfkBG(at-sign)fBBTT6MQ9bNpGcPJ0G8$k6Z6%XN32`KQ&ZC*B$leTNC3aZC(at-sign)5
6G'KPNfePBA0eFQ(at-sign)6EfD6BC0`Q'pcDA4TGTUXFQ(at-sign)6D(at-sign)kBG'9RCA+1TSf4*i!!efk
(at-sign)!qUSZA4[Nf9aG(at-sign)&XMBf0RrY'GBf4"j,Pe6'1N38Gfjm"l,f*!!"QC3!*+KHI"[!
RMGKZMCrpZ16CFik1MSk1N3pl*ENlNfPZNh4SDA16Gj(at-sign)XFQ'6HC(r"9BXPJ2UU(4
SCC0YC(at-sign)&cGA*PNfpQNh4SCC0cCA56edk4"5Z-ZA4eFQjPC*0[GA56G'q6CA&eB(at-sign)b
1TTm5FFU0NJ$'VC2A%*%!k!'j+0GcZ5Q(at-sign)!e952BfIm[rpMC%(&l6D-BkI$3!$MBf
4""l8RrArr+YBMSkI#p+"MC2BEY8p-Bk1MBf0RrIKaBf4'9BdZ6'1N4D06jm&8(at-sign)f
*!!"QC3!,FF(at-sign)I#`J,MGGZMCrmL)RBFik1MSk1N5-b4pFkMTmT8S'0N5H!!,P8D'9
bC(at-sign)C[FQ8XN32d'QKPPJ2b0Q0[G(at-sign)aNNf4P$'jPNf'6+'0[G(at-sign)kDV(*dB(at-sign)*XHC0KC'4
TG'PfQ'8TNh"bEf*KBQPXDA5BHC0YC(at-sign)&cGA*PNpG3MCm"c-cBFik4#'"bZ(at-sign)pZMUD
0N5H!!(4SCCB$kUKcCA56edk4"5Z-Z(at-sign)pQNh#3!&11Eh0TG'PfQUabCC0TETKdC(at-sign)G
PFR16BTKjNh0PG(4TEQH1TU33fID0NJ#dkT6A8)fI!Fc-f(11N34Z2,NSed'j+C%
$99)pMBf0RrIKaBf4#r1H-Bk4")L&R`94EBN!!'CP!"5f,Tm,#!Z0ea#4!1J"Z5M
AFlNTMSk1MSf0MC%GdA+IpIrmUeL1MTm-)L(at-sign)0N4aaj0KZfM,B3Bk1MBf0RrIKaBf
4-jlHZ6'1N6$9qCm&8(at-sign)f*!!"QC3!,FF(at-sign)I#`J,MGGZMCrmL)RBFik1MSk1N6ekmGF
kMTmGH-'0N5H!!,P5D(at-sign)9YB(at-sign)jZPJ1EJh4SC(at-sign)k6F(*[N!"6MQ0PC(at-sign)4PC*0dEj0fN!#
XFQ9bD(at-sign)CjNhGSBA56D'(at-sign)6D'&NNh0PER0PC*0KE'b6B(at-sign)a[EQFXPJ1V9fjKE(at-sign)9XHC(
r"9BXNh4SCBkQMC%RJ!"QG(at-sign)jNB(at-sign)ePET(at-sign)XFR4KE*%$kUK`FQp`N!"6MQ9bG*0jMUD
KMC)!N!#m9pG3MCm"c-cBFikE"'imZ5MA3BfI!Fc-f(#1N3Gb"G4FN3+UU0G"MCm
"c-cBFBk4"*6qZ5Q(at-sign)!e952C2A8)fI!Fc-f(11Q,NSed'0R`(-c0K`MT%%aefj+GG
3MCm"c-cBFikBZ5MA3BfI!Fc-f('1N358rVNTNcf0MBfIpq(&MC%(A9%aMT%%L)(at-sign)
I"9&YL3!!CQ8!#iQ6R`X)#ihAF('1MSk1N4&&5cU1MTmH!!#0NJ$kEf#j0$+1MSb
,!!!!+`!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"[c'J!U!
SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j5(at-sign)k(at-sign)"'E4Eh4SCA+6GjUXFQpbC(-XN35
&fh4SCC0PGTKPETKdFj2A3BfI!Fc-f(#1N3NZ,VPKEQ56ed'0R`(-c0KaMT%)qmq
jG'KKG*0KNh*KEQ4[E(at-sign)ajNf1BD'pcC(at-sign)k6D(at-sign)kBG'9RCA+6efk6Z(at-sign)+3!&11CBkN$S!
!MC%RJ!"NDACTFfPLE'(at-sign)(at-sign)"9HRBTUXFRQ6EfjPNfpQNh4SCC0dQ(HBEj0`FQPYCA1
6eh#6Z(at-sign)pbNpGaN3A&i,PKFQ(at-sign)6D(at-sign)jNCA#3!&11C(at-sign)jNC(at-sign)kBG*0bC(at-sign)aKG'PfQ'(at-sign)6G'q
6G'KPMU'0N5H!!("bEf*KBQPXDA53!+abHC%$kUMA8)fI!Fc-f(11N34Z2,NZMU'
0N5H!!&4SCCB$kUK5D(at-sign)9YB(at-sign)jZNhTPG''6CR9ZBh4TEfk6Gj!!V(*KFj0REj96MQq
6C*B$kUKQEh+6FfpYCA4SD(at-sign)jR,*0KCR4PFT0KE'`ZMUNDJ!#0N5H!!%Q4"3L(GfP
XE*B&#0"ZEjUXFRH6GA0PNf'6FQKPG'pbD(at-sign)0KE*0NCACTBf(at-sign)6G'KKG*0hQ'&cNf8
,C(at-sign)0dDADBC(at-sign)ajNf9YF'a[Q(QBC(at-sign)56BTKjNfpZCC0[CSkKMC%RJ!"YQUabHCB%+Tj
eEQ4PFQGbB(at-sign)4eBA4PNh4PB(at-sign)1BD'9bFbb4"$UF8(*[CQ9cFfpbNd*[N!"6MQ1BD'j
PFLk4"IM$5(at-sign)k6G'KPNf0XBA0cFQq3!&11EfdXN33kR&"bEfCPFh0[FSkKMC%RJ!"
#Ej!!8ijMPDabD'jPFTX&#$PhNfpeE'5BF(*P$(LBG'KPQ(0dBA4PE(at-sign)9ZNh5BEfD
BBCKdD'9[FQ9YQ'+6HCKdD'(at-sign)BGj0[FQ4c1T%(G!%L8h9LN3#R('TPBh51SBf4*i!
!G'q(at-sign)"EC`G'9MQUabD'jTBf&XNf&cFh9YF(4TEfjc,*%'+(at-sign)*dD'(at-sign)6CQpXE'qBGfP
ZCj0TFj0dFR9P)MZ4"Ta8GfPdD'peG#bE"LPLEfD6BfpeFR0P,*KPGT!!V(*PFSk
KMC%RJ!"NDA0ME'pcD(at-sign)jRPJ2UU(GSBA56D'PcNh4PBjUXFQKZD(at-sign)0KE*0KFh0eEA"
dD(at-sign)pZFj0hQ'9bC5k1TSf4*i!!8(*[CQ9cFfpbPJ(at-sign)"P&*TC(at-sign)eKEQk6G'KPET0`FQq
3!&11Bf9PC'9NNh4[Nh0SEjUXFRH6G'KKG#b4"HG2Fh9LN3#R('TPBh56G'q6G'9
MQ'KZD(at-sign)0KE*0KFbf1SBf4*i!!Fh9YF(4TEfjcPJ2UU'pZNh4SCC0cCA56ed'j,)k
KSBf0MC)!Xmk)E'PYMTm'dRf0NJ#cS%,BFpSKe6'1MT)!aNk)ee#0R`(-c0KcMT%
%EMbj+0G"Z5Q(at-sign)!e952C2AC'9ZFlNSed'j+GFkMTmMTp'0N5H!!,P8D*(at-sign)XFR9c,*%
$9%YPGT0PETB$,V4dD'peCfL6BA*TG'KYCA4TBj0NC(at-sign)jcDA5DV(*jNfPcNfj[G*0
KNh"bEf*KBQPXDA5BHC(r"9BXN3085fPdNfPcNh9ZC'9bNh0eDA4KBQaPMU'0N5H
!!'0[EQ4TG'P[ER1(at-sign)!qUSG'KPNfaTE(at-sign)PdNfpQNh"bEf*KBQPXDA4TCA-ZMUD0N5H
!!%a[EQH(at-sign)"'NHB(at-sign)CdCA+68QPPE(at-sign)&ZET0hQUabBA16CfpZC5b4")LlDA56GjKKFj0
cD'qBGfiXN35)Zf&RB(at-sign)PZNh0eBT%!TaaUC(at-sign)0dNh4[Nh4PBjKSEQPMB(at-sign)b6BA-YMU'
0N5H!!(0eEA"dD(at-sign)pZFbb4"GFAG'KKG*B&G*TdD'(at-sign)6F(*[BQ&LD(at-sign)aTG'PPFj2A8)f
I!Fc-f(11N3RLeVPKFQ(at-sign)6G'KPNfpZE(Q6F(*[BQ&LD(at-sign)aTG'PPFj0NC3aZC(at-sign)56Efk
1SBf4*i!!G'KPPJ3(,R0PG*2A6T%&5"+jEfD6EQ&dGA*KE*0TETUXFR4PCf9bFj0
QEh+6GfKTBjKSNh4SCC0PGTKPETKdFj0[CT0NDACTFfPLD(at-sign)aTG*KjNf+BHC0ND3Y
PFQ9ZQ(51SBf4*i!!F(*TE(at-sign)9cPJ3V[f&bCC0TEQ4PF*T6MQ9ZC'9ZN!#XFR3ZN3A
m*94SDA16CQ&MG*0cC(at-sign)9YFj0dEj0XC(at-sign)jNNh0eF(#BEh*dNh4[Nh4SCC0`FQpRFQ&
YNfpQMU'0N5H!!("bEjUXFRCTEQH(at-sign)"+S-FQ9cG(at-sign)adFj0[CT0ZQ(9YQ'+3!&11CA+
6G'KPEh*jNf+BHC0`FQpLB(at-sign)*TE'PcG'PMNfePG'K[QP11C(16BQ&cC(at-sign)56GA#BEfk
6G'KPMU'0N5H!!&*TC(at-sign)eKEQk(at-sign)!qUSHQ9dBC0QG(at-sign)jMG'P[ELk1TSf4*i!!9fLDV(*
jPJ2L"'4TC'iRG*03FQpQCA0cEh+68QPPE(at-sign)&ZET0PGTKPFT0`G(at-sign)*XDA0SNh4SDA1
6GjK[EQ4PFQCeE*0TC'9KNfpQNfKTFcq4"6Ar9'KPMU'0N5H!!'&ZFhHDV(*PFTB
%TBCTFj0ZEh56D'&bC*0dEj--EQ3ZN3GTHP54r`9(at-sign)FR9P,*%%e$ecEfePNh4SC(at-sign)p
bC(at-sign)ecNfpQNfkBG(at-sign)fBBTT6MQ9bNh4SC(at-sign)pbHC0MB(at-sign)k6BTKPMU'0N5H!!("bEj(at-sign)XFRD
6C(at-sign)5(at-sign)!l(at-sign)DF(*[BQ&LD(at-sign)aTFh4TBf&XE(Q6BTUXFRQ6G'KTFj0XD(at-sign)eTG'PZCj0`FQq
3!&11Bf9cFbb4!m!hCQpbNf9iB(at-sign)e`E'(at-sign)64'PbD(at-sign)1BD'aPG#GcNh4SC5f1SBf4*i!
!Eh*PECB$+Vj[ET0`FQPYCA16D(at-sign)k6BA*TG'KYCA4TBj0`FQpRFQ9cFfP[ELk4"2M
R5'q9V(*hNf9fNf9b,*%$85"NC(at-sign)9`QP11CA+4!bUqET0eEC0LQ'9bN3-U[R4SC(at-sign)p
bCA4TBikKMC%RJ!"bCA0eE(4cQ`4B2QKKPDabGT0PQ(4[Q(4SDA1BC''6HCKPE(9
NC(at-sign)5BG'KTFjKKF("bEf&MNfJXPJ4cT'C[FTKPH'&YF'aP,*0ZEjK[EQ(at-sign)BD'&cQ(0
eBbf1SBf4*i!!Bf9PC'9NPJ(at-sign)J!fPZNh"bEjUXFRCTEQH6G'KPNh"bD(at-sign)ePNfkBG(at-sign)f
BBT!!8ijPFT0dD'9[FQ9YNf+BHC0dD'PcNfePG'K[N!"6MQ3ZN3TBm9"bEfCPFh0
[FSkKMC%RJ!"5D(at-sign)9YB(at-sign)jZ,*X%!*&KPDabGj0KFQ(at-sign)(at-sign)!r`[EfD6G'KTFj0NC3aMD(at-sign)9
ZBhQ4r`9(at-sign),*KdD(*PGj0SDA16EQpdCA16D(at-sign)kDV(*dEj0dD'(at-sign)6GjKKFh4PBQ&cDjK
PG*0KEQ51SBf4*i!!F(*[N!"6MQ0PC(at-sign)4PC*B$T4edEj0XD(at-sign)jVNh4SCC05D(at-sign)9YB(at-sign)j
ZNhTPG''6CR9ZBh4TEfk6G'q6G'KPNf4TFh4bD(at-sign)*eG'P[ET0[CT0`FQPYCA16D(at-sign)k
1SBf4*i!!B(at-sign)k(at-sign)"-M&B(at-sign)adEfGPG'KPFT0ND3YPFQ9ZQUabG*0hQ''BHC(r"9BXN38
!6(at-sign)+BHC0cG'&dD(at-sign)jRNh4SCC0SQ(P`QP11Eh4SCA0TFj0dD'&dNf+BC(at-sign)&bFj0SDA1
6EQ&YCBkKMC%RJ!"KEQ5(at-sign)!qUSG'KKG*0bC(at-sign)eKD(at-sign)jcNh9ZF(*[PDabGT0PC*B$kUK
dEj0dD'PcNf4KN!#XFRQ4r`9(at-sign),SkQMC%RJ!"AD*UXFRQ(at-sign)"#"XB(at-sign)f65C%%)&jdC(at-sign)a
XD(at-sign)jRNhQBEh(at-sign)6G'KTFj0LDA56EfD6D'PcG'pbH5d-Bh4TEfirN3AD+d*PBf&eFf(at-sign)
65C%%)&jhQ'&ZQ(56G'q6F(*[F*!!8ij[Ff(at-sign)1SBf4*i!!B(at-sign)j[G'KPFTB#V8"`FQp
LB(at-sign)*TE'PcG'PMNfPZN!#XFR4PFR"bCA4KG'P[ET0[CT0dD'(at-sign)68QPPE(at-sign)&ZET0kCA4
KNfCeEQ0dD(at-sign)pZNh4SBA56DA16FA9TG'(at-sign)1SBf4*i!!C'N,CA*PETUXFR5(at-sign)!qUSCR*
[EC0dD'(at-sign)6D(at-sign)kBG'9bF(*PG'&dD(at-sign)pZNfTeFh56Eh9dE'PZC(at-sign)3ZMSkI(J!!MC)!qQp
J0$11MSb,!!!!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"bQ(at-sign)J!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j6'9dPJ0+d(9cNf0[ER0TC'9
bNf'6F(*[BQaPEC0TET0MEffDV(*LD(at-sign)jKG'pbD(at-sign)&XNf9ZQ(9YCA*KG'P[ELk4"31
B6'9dNh9cNh4KDjKPNf'6BhPME'PMMU31J!#0N5H!!'GbEh9`PJ6J4fpQNfpbC'9
bNpGbN!"6MVNXN38GVR0KQUabHC2A3ifI!Fc-f(+1N355BVNZN3JC[%9fQ'9bHC0
MQ'KKFQ&MG'9bNpFINlP[CT0dD'(at-sign)6Ch*[GA#6ed10R`(-c0KbMT%*FUQjD'&cNf'
6DjKPFQjPE)kKMC%RJ!"hD'PMQUabD*B&H9&TFj0KNh0eBQGbEh9`NfpQNpG$MCm
"c-cBFSk4"**LZ5k4#H6E6(at-sign)pbCC0RC(at-sign)jPFQ&XE(Q4r`9(at-sign),*%&h2YPGTKPFRQ6Ff9
aG(at-sign)9ZBf(at-sign)6eaq0R`(-c08aMTB%`!6A1jX"rriIMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%
4bT`lQ"q0R`(-c0KcMSkKMC%RJ!#jEfD(at-sign)"48[BjUXFQKKFQ&MG'9bFj0[CT2A3if
I!Fc-f(+1N3QRNEPSBA16BC0UEfPZQ(56DjKPFQjPE*0hD'PMQ'L6DA16B(at-sign)acEj0
KNh0eBQGbEh9`NfpQNpG$MCm"c-cBFSk4"**LZ6Z4"DTbG'KPMU'0N5H!!'T[D(at-sign)k
9V(*dQ`2p9QZ6CA*ZC(at-sign)bBEfDBBCKcCA&eC(at-sign)jMCCK[CTKMNfKKFQ&MG'9bFjKTFjK
cD(at-sign)e`E(QBG'KPQ'PZNh4PFR0PBh4TEfkBEfDBG'KPQ'Z6CA)YMU'0N5H!!'jPE(1
(at-sign)"(AkEfD6C(at-sign)&MQUabD*0[CT0dD'(at-sign)6BjKSBA*KBh4PFR16D(at-sign)k6G'KPNh0PFA9PEQ0
PNpFIMCm"c-c9-Bk(at-sign)"-!%ecZE!Irq(ifI!Fc-e6+1NpFlMCJkQ$UB1Sk4%FUF1jJ
IMCm"c-cBFik4#13fZ5k4"YVA5(at-sign)D(at-sign)"(AkBC0cCA&eC(at-sign)jMCBkKMC%RJ!$A(ifI!Fc
-e6'1PJ6!"0FlQ`(rrKq0R`(-c08bMT2A1ifB1TJkQ$U1N4(+R$ZB(ifI!Fc-f(1
1N3HqHVP[CTB$8$lAFj1jBjUXFQKKFQ&MG'9bFj0TFj0MQ'K[Ff9ZNfPZC'9`N!"
6MQ9ZC'9ZQ(4XHC0KEQ56BA56FQ&ZC'pY,*%$Eb"hD'&dNfPcMU'0N5H!!(4SCCB
$S$*`FQpLB(at-sign)*TE'PdQUabHC0dD'&dNh4SCC0UEfPZQ(56DjKPFQjPE*0[CT0dD'(at-sign)
6Ff9aG(at-sign)9ZBf(at-sign)6CA&eB(at-sign)acNf'6CfPfQ'9ZNh0eBQGbEh9`MU'0N5H!!0G$MCm"c-c
BESk4#C,iZ(at-sign)pQN32UU0G$MCm"c-cBFSk4"**LZ6q1RaU!!)f4*i!!9'KPPJ2[NR"
bEf*KBQPXDA5DV(*jNfpQNh4SCC0PGTKPETKdNh4SBA56G'KPNfZBCA*ZC(at-sign)b6EfD
6BC0bB(at-sign)jNEfeXHC0MQ'K[Ff9ZNf1BD'&bB(at-sign)0dCA+1SBf4*i!!GfPXE*B%kZYMEfk
DV(*dB(at-sign)PZNh4SCC0cG(at-sign)*RFQpeF*2A3ifI!Fc-f'k1N3U61lPPFA9KE(10MBfIqdC
eMC%'NN69-Bk4"KiHR`(X[BN!!'CP!!8S8*m'm#H0f'k1MSk1N3ajSENXN38Ur(0
TEQ0PNh4SCA*PNf&bCC2AFT%&2RQjBjKSBA*KBh4PFR16EfD6G'KPMU'0N5H!!'G
bEh9`PJ0(b0G$MCm"c-cBFSk4"pSUZ(at-sign)&ZC)f0MCrl4R(at-sign)0N38&mYKbMT%%H[ZI!Hb
pL3!!CQ8!"5K3R`E`*ieZMSk1MT%1(NDjFh9MQUabD*0MQ'KKFQ&MG'9bFj0hD(at-sign)a
XNhD4reMNB(at-sign)jTFfL6Efk6ed10R`(-c0KZMT%&U&#j,T%&!T98D'9bC(at-sign)C[FQ8XN30
S(at-sign)h4SCC0`FQpLB(at-sign)*TE'PdQ(Q1SBf4*i!!G'KKG*B$lETdD'(at-sign)6DQpTETUXFR56DjK
PFQjPE*0[CT0KNh*KEQ4[E(at-sign)ajNf1BD'pcC(at-sign)k6Ff9aG(at-sign)9ZBf(at-sign)6eaq0R`(-c08aMTB
%`!6A1jX"rriIMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ"q0R`(-c0KcMT%)(at-sign)rD
jEfD(at-sign)!qfkeh16Z(at-sign)13!+abD'&b,BkKMC%RJ!"KBh4PFR1(at-sign)""UNFfKKE'b6BfpZQUa
bG'&TET0dD'(at-sign)6Fh9LCh*[GA#6ed10R`(-c0KZMT%*`[5jCA&eB(at-sign)acNbL0MBfIqdC
eMC%"TeR9-Bk4!6-cR`(X[BN!!'CP!!8S8*m'm#H0f'k1MSk1N3H1YVNTMCrlT6,
BFik4"'imZ5k4"FM86'9dNh9cNf4PEQpdCC0LQ(Q6ee#0R`(-c0K$MCm"!!$CESk
1N3r([EPdD'(at-sign)1SBf4*i!!F(*[BQ&LD(at-sign)aTG*UXFRQ(at-sign)"*8CG'KKG*0dD'(at-sign)6DQpTETK
dNfZBCA*ZC(at-sign)b6EfD6G'KPNf1BD'&bB(at-sign)0dCA*cNpFIMCm"c-c9-Bk(at-sign)"-!%ecZE!Ir
q(ifI!Fc-e6+1NpFlMCJkQ$UB1Sk4%FUF1jJIMCm"c-cBFik4#309ZA0SB(at-sign)aXN35
9'(at-sign)9aG(at-sign)&XMU'0N5H!!(4SCCB$kUKcG(at-sign)*RFQpeF*2A3ifI!Fc-f'k1N3(at-sign)S8,NZN38
ii&4SC(at-sign)k6GjUXFQ(at-sign)6D''BGTKPNh4SCC0TC'9ZQ(4TG*KjMU'T%0RfMBf0MCrhiF(at-sign)
0NJ$+Rd!aMT)!apCER`94EBN!!'CP!!YaaCm,#!Z0efk0Rrb)LGKcMSk1MSk5!0I
3TENpMBf0N39'$jrerrbV(at-sign))k1R`bUVSf4!e95f'lDDYKNfQVBFSk1N4HSNYG3MCm
"c-cB3ifI!(at-sign)Nrf(at-sign)51MT%+kERA1SkI(i!#MC%RJ!#j5'9bC5b4"$`QGj!!V(*PPJ3
VfR9cCC0dD'(at-sign)6CQ&MG*0dD'&dNh4SCC0`BA*dD(at-sign)&XE(Q6Eh*NCA*PC*0cCA56EfD
6Fh9LCh*[GA"cNfpQNf'6BhPME'PMMU'0N5H!!'GbEh9`PJ2qfpG$MCm"c-cBFSk
4#*%pZ(at-sign)PcNfPcEfe[FR"SD(at-sign)16G'q6G'KPNh"KFR4TB(at-sign)aXHC0[FQ4PFQ9NNh0PG*0
[CT0NDACTFfpbFj0[CT0dD'(at-sign)6D(at-sign)k3!+abG'9RCA+1SBf4*i!!eh+3!&11Z5k1RaU
!!)f4*i!!9j(r"9CPPJ1B(fj[QUabGj0eFf(at-sign)6G'KPNderNISJ"'pLDA9cNfPZQ(D
BCA*cD(at-sign)pZNfC[FQfBG(at-sign)aKNfpQNfkBG(at-sign)fBBT!!8ijPFT0dD'9[FRQ4r`9(at-sign),*%$U+"
dD'9bC(at-sign)+BHC0[BR4KD(at-sign)iYMU'0N5H!!'PZCikKTSf5!,8EdYG3MCm"c-cB3ifI!3!
!f(at-sign)k1MT%2!QZj2Bf0MC%&4JqIpIrmUeL1MTm-UUk0N3098YKZfQVBC0TUf(+1MT%
AU*,A&VNSef3pEVNTMBf0RrIKaBf4!iI8-Bk4!6-cR`94EBN!!'CP!!U*2Tm,#!Z
0ef50Rrb)LGKcMSk1MSk4$1qNecU1Rb!UVSf4*i!!Z8KPFQ8XPJ2UU0F(at-sign)Z5MADT%
!V01j+C0TFj0dD'(at-sign)66Aq4qL!%Ef*TGA16CR9ZBh4TEfk6EfD6ET(at-sign)XFR9YNf+3!&1
1CA+4!qUSG'KPEh*jNIm&9Lk1RaU!!)f4*i!!3(at-sign)CdCA+(at-sign)"4VVG'KPNf1DV(*SB(at-sign)j
RCC0[CT0fNIpBj'&bD(at-sign)&LE'(at-sign)6ef5(at-sign)"9XfZ6f6efjUN3A([VPhQ'(at-sign)(at-sign)"4VVBf&ZNh*
PBf&cG*0dD'(at-sign)6FQPRD*KdNfKKEQ56FfPNCC0KFikKMC%RJ!"QEfaXEj!!V(*hFcU
1SDD0NJ#f)rcA8)fI!Fc-f%10R`%!!0PZMSk4$`*VZ6f0MBfIpq(&MC%(8(at-sign)SaMT%
%L)(at-sign)I"9&YL3!!CQ8!#h(&R`X)#ihAESfIr)L*f(11MSk1MSf0MC%6,AZIpIrmUeL
1MTm,p*D0N4Kc(at-sign)pKUMSk4)jp#eaDj+0GUPJ#XdlNTMBf0RrIKaBf4!cle-Bk4!6-
cR`94EBN!!'CP!!RhIjm,#!Z0efU0NjrmL)RBFik1MSk1N3aGjGFkMTmH(at-sign)#10N5H
!!,P8D'(at-sign)(at-sign)!j(at-sign)6GT(r(at-sign)14KFQPKBQaPNpGUN34#CVP[ET0dD'(at-sign)6FQPRD*UXFR56FQ&
ZCf9cNfqBGTKPFT0cEfePNh0eBR0PG*0[CT0NDACTFfpbFj0[CT0dD'(at-sign)6D(at-sign)kBG'9
RCA+1SBf4*i!!eh+3!&11Z5b(at-sign)!qUSGfKTBjUXFQL6GjKPNfjPC(at-sign)56EQpdNhHBEh*
bHC0KBT!!8ij[GA3ZMSkI(J!!MC)!qQpJ0$51MSb,!!!!,3!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"eIQJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF
,MC%RJ!#j6QqDV(*hPJ082(at-sign)PQNh4SCC0cG(at-sign)f6Efk6G'KPNh*TCfLBG*0bB(at-sign)jRC(at-sign)5
6EjKfQ'9bNf&XE*0`N!"6MQpcDA4TGTKPNfPZQ(4PCf9bFj2ADT%!V01j,*%$FP0
dD'9ZNh4SCC0bD(at-sign)GSQ(51T!k!!)f4*i!!D'&ZC*B$kUKcD(at-sign)4PNhH3!+abEh9XC*0
PFA9KE)kKRa$CpSf0MBfIpq(&MC)!fPrc-Bk5!0HA$Tm&8(at-sign)f*!!"QC3!,FF(at-sign)I#`J
,MGGZMCrmL)RBFik1MSk1MBf0RrIKaBf5!1cD8VNaMT)!j(at-sign)mjR`94EBN!!'CP!"5
f,Tm,#!Z0ea#4!1J"Z5MAFlNTMSk1MT)!qeLDecZ1RbF3NSf4*i!!ZA4SBA5(at-sign)"1m
4DA-XN38`,'PdNf0[G(at-sign)aNNf+3!&11CC0PH("bCA0cC(at-sign)56D(at-sign)k6G'9bEA16EfD6G'K
PNfPZPDabGT0PFR0PPJ6[%(at-sign)pQNh4SCC05D(at-sign)9YB(at-sign)jZNhTPG''1SBf4*i!!CR9ZBh4
TEfiZN368C%PQPJ+p0(HDV(*PNf0[G(at-sign)aNNf1BD'&ZCf(at-sign)6Eh9bNf0[ECKLD(at-sign)jKG'p
bD(at-sign)&XNh"bEf*XC(at-sign)f6G'q6Cf9dNf&ZNh9ZFQ9cG(*TBh4PC)kKMC%RJ!"cG(at-sign)f(at-sign)!ap
kEfk6G'KPNh*TCfLDV(*dNfKKEQ56FfPNC5b4!dJGG'KPET0hQ'(at-sign)6GjK[G(at-sign)aNNfK
KQ(DBCC0KNh"bEf*KBQPXDA0dD(at-sign)16D(at-sign)kBG'9bF(*PG'&dD(at-sign)pZMU'0N5H!!'pQPJ2
UU(4SCC05D(at-sign)9YB(at-sign)jZNhTPG''6CR9ZBh4TEfiZMU3DJ!#0N5H!!&4SDA1(at-sign)!Y5MDA1
6C'pZCC0LQUabHC0bCA"XB(at-sign)0TEQH6G'KPN`aZDA4PNf0jBfaTBj0RFQpeF*2A3if
I!Fc-f'k1N3KmmlPLQ(Q6BC0`FQm-EQPdCC0MH(at-sign)0XD(at-sign)16Ch*[GA!ZMU'0N5H!!%0
[ER0TC'9bPJ30iR4SCC0RFQpeF*2A3ifI!Fc-fM'1N3d0kVP[CT0bBA4TEfjKE*0
ZPDabG(at-sign)f6BT96MQ9bFjX%$H*YEj0NG(at-sign)a[Q'pZC5k4"D+24T(r"9C[FTKPGT!!V(*
PFRQBF*0[FfPdDAD3!+abCBkN$S!!MC%RJ!"TETUXFR4PCf9bPJ5DSYGZZ5b4"-D
JG'KPNfGbEh9`NpG$MCm"c-cD-Bk4$CUUZ(at-sign)KKFj0KNh9ZDA&eCC--EQPdCC0cG(at-sign)*
RFQpeF*2A3ifI!Fc-f'k1N3T#mVPhDA4SNpGZNlPPE'9YC(at-sign)kBG(-ZMU'0N5H!!&4
SCCB%BDpMN!#XFQKKFQ&MG'9bNfGbEh9`NpG$MCrlT6+0N3$FRYS$MTm(8N'0-Bk
1Q`eKYlP[CT2A3ifI!Fc-fM'1Q,PTFj0KNf0[EA"KBh56Ch*[GA!lN35G-fPdNfK
KFj0KNdKKBA+6E(at-sign)9KFh9bCBkKMC%RJ!"hD'PMQUabD*B&+Q"TFj0KNh"bEf*KBQP
XDA5BHC0YC(at-sign)&cGA*PNpG3N3'KaVNZN3Mi#&4SCC0RFQpeF*2A3ifIqk8bMC%!h*l
D!ikI"e*"M6'1MT%1+QLjDA16G'KPNf4PFfPbC(at-sign)56F(*[$'jTG'(at-sign)1SBf4*i!!Ch*
[GA#(at-sign)!qUSEfk6GfKTBjUXFQL6GjKPNf0KET0RC(at-sign)jPFQ&XDATPNh4SCC0`FQ9MC(at-sign)4
TEQH6BfpYF(9dBA4TEfiZMTmDJ!#0N5H!!&4SCCB%qNGcCA56EfD6B(at-sign)aXNf1DV(*
SBA*KBh4PFR16EfD6G'KPNfGbEh9`NpG$MCm"c-cD-Bk4$IT2Z5KdD'&dNfPc,*%
&2LjdD'(at-sign)6Ff9dNfpQNf&XE*0PE'9YC(at-sign)kBG(11SBf4*i!!EfD(at-sign)"&*0G'KPNfGbEh9
`NpG$MCrlT6+0N3$FRYS$MTm(8N'0-Bk1N3N!#,NTNhGSD(at-sign)13!+abD*0fNIpBj'&
ZDA0SNfpZNf'6Fh9LCh*[GA#6ed10R`(-c0KZMT%*qTfjEfD6ed10R`(-c0SaMT%
08P(at-sign)jD'&cNdKKBA+6E(at-sign)9KFh9bCBkKMC%RJ!"PFA9KE*B%Y-&dEif0MCrl4R(at-sign)0N3C
F'Y8aMT%&jr5I!HbpL3!!CQ8!"5K3R`E`*ihBESk1MSk4$%0hZ5k4"jFX9'LDV(*
eFbb4"1G)D(at-sign)D6GjKPNf1BD'q3!&11Eh0PNf'6Ff9aG(at-sign)9ZBf(at-sign)6eaq0R`(-c08aMTB
%`!6A1jX"rriIMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ"q0R`(-c0KcMT%*)[f
jEfD(at-sign)",6"eh16Z(at-sign)13!+abD'&bB(at-sign)0dCA*cNfpQMU'0N5H!!0G$MCm"c-cD-Bk4$(0
CZ(at-sign)PZC'9`N!"6MQ9ZC'9ZQUabG'ajPJ0c8(at-sign)&ZC*0KG*0bB(at-sign)jNEfdXN31,,h4SCC0
`FQpLB(at-sign)*TE'PdQ(Q6G'KKG*0dD'9TFT0UEfPZQ(56DjKPFQjPE*0hD(at-sign)aXMU'0N5H
!!'0[ETUXFR4KD(at-sign)k(at-sign)!h-BG'KPNfGbEh9`NpG$MCm"c-cBESk4#4YSZ(at-sign)9aG(at-sign)&XFj-
SMBf0RrY'GBf4!DGCe6'1N3%c-jm"l,f*!!"QC3!&+&#I"[!RMGKZMSk1MT%(MVD
j+BfIqk8bf(11N3IK9,NZN384"8PQNhHBCC0KCf&TET0NC(at-sign)j[G'(at-sign)6BTKjNpG3MCm
"c-cB3ifI!3!!f(at-sign)k1MT%2)$'jG'KPNh"bEf*KBQPXDA5BHBkKMC%RJ!"dD'&dPJ5
DD(4SCC0UEfPZQUabG*0VQ'9bEQ9XNfpQNf'6Ff9aG(at-sign)9ZBf(at-sign)6eaq0R`(-c08aMTB
%`!6A1jX"rriIMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ"q0R`(-c0KcMT%*#+5
jEfD(at-sign)"*TSeh16Z(at-sign)13!+abD'&bB(at-sign)0dCA*cNf9aG(at-sign)&XFj0dD'(at-sign)1SBf4*i!!Ch*[GA#
(at-sign)!qUSed10R`(-c0KZMT%&U&#j,*0dD'9ZNhHDV(*PNfKKQ(DBCC0dD'(at-sign)6D(at-sign)4PETK
dDA5BHBkKU4$CpSf0MBfIpq(&MC)!c)rm-Bk5!-R(&jm&8(at-sign)f*!!"QC3!,FF(at-sign)I#`J
,MGGZMCrmL)RBFik1MSk1NJ$C`(at-sign)'j2Bf0MC%$99+IpIrmUeL1MTm-UUk0N35F!GK
ZfQVBC)k1N42('GG3MCm"c-cB3ifI!(at-sign)Nrf(at-sign)51MT%+kERA1ikI(i!#MC%RJ!#jGfK
PFQ(at-sign)(at-sign)"'Z#G'KPNh0eEC0[ET0dD'(at-sign)6FQPRD*UXFR56DA16EQqBGj0TEJaZDA4P,T%
'Zfe"Cf&TET0LQ(Q6G'KPNderNISJ"'pLDA9cNfPZQ(DBCA*cD(at-sign)pZMU'0N5H!!'C
[FQf9V(*eE''E!qUSGj0PQ'pLG'&TESkKTSf5!*bV'GG3MCm"c-cB3ifI!3!!f(at-sign)k
1MT%2!QZj2Bf0MC%$99+IpIrmUeL1MTm-UUk0N35F!GKZfQVBC)k1N42('GF(at-sign)Z5M
AC$eZZ5Q0MBfIpq(&MC%$Kp3aMT%"-c1I"9&YL3!!CQ8!#SNqR`X)#ihAC)fIr)L
*f(11MSk1MT%342Dj2Bf0MCrhiF(at-sign)0N3G4DM'1N35)KCm&8(at-sign)f*!!"QC3!,FF(at-sign)I#`J
,MGGZMCrmL)RBFik1MSk1MBf0RrIKaBf4'F[*Z6'1N4*JX*m&8(at-sign)f*!!"QC3!8YLk
I#`J,MGF3N3$S!ENSeh1j+Bk1MSk4+%S4ecU1Raq!!Sf4*i!!Z94SDA1(at-sign)"'6!DA1
6G'KPNh"bEfeTFf9NNh"bEf*KBQPXDA0dD(at-sign)16D(at-sign)k3!+abG'9bF(*PG'&dD(at-sign)pZNfp
QNh4SCC05D(at-sign)9YB(at-sign)jZNhTPG''6CR9ZBbf1SBf4*i!!G'P[ELk4"pr(at-sign)8fpYCCB%c2T
`FQp`QP11CA*dD(at-sign)9cNfpQNh4SCC05D(at-sign)9YB(at-sign)jZNhTPG''6CR9ZBh4TEfk6Bf&ZNf+
BCC0`FQq9V(*fNf9NN36-qR"bEf*K,BkKMC%RJ!"LD(at-sign)aTFh4TBf&XE(Q(at-sign)"4iKGA0
TEQH6G'KTFj0TET!!V(*dCA*`FQ9dBA4TEfiXQ`9UrfC[FT0PH'&YF'aP,*KdD'(at-sign)
6F(*[N!"6MQ4eBh56CQpbEC!!V(*eE'%ZN3M65dPdMU'0N5H!!(*PE(at-sign)&TER1(at-sign)!p4
fG'q6BTT6MQ(at-sign)6Ff9PET0hD'PMN!#XFQL6Eh4SCA+6F(*[F*KPFR4TCA16EfD6G'K
PNe*TC(at-sign)eKEQk6HQ9dBC0QG(at-sign)jMG'P[ET0MB(at-sign)k1SBf4*i!!BT!!8ijPQ`2UU("bEj(at-sign)
XFRD6C(at-sign)5BD(at-sign)kBG'KTFjKhNf'6HC(r"9BZMSkI(J!!MC)!qQpJ0$(at-sign)1MSb,!!!!,J!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"iTDJ!U!SpBfJrBE
A#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j9'KPPJ6``h"bC(at-sign)0PC'PZCj0KFQGeE(at-sign)9ZQUabG*0
TFj0KET0TER0dB(at-sign)jMCC0[CT0KNfGPEQ9bB(at-sign)aTHQ&dD(at-sign)pZNfpQNf&ZNf9ZQ(9YCA*
K,BkN$S!!MC%RJ!"dD(at-sign)pZPJ6mfR"bEf*XC(at-sign)f6Efk6BC--EQPdCC0cCA56G'q6B(at-sign)k
6C(at-sign)kDV(*eE(at-sign)9bBA4TEfk6Efk6BC0`FQm-EQPdCC0cCA3ZN3K[GP0eBjKSNf'1SBf
4*i!!FQ9`E'&MC(at-sign)ePETUXFR5(at-sign)"!qDEfD6BC--EQPdCC0cCA56BTKjNf'6F(*[$'j
TG'(at-sign)6)R0PG#+6GjK[FQYcNfPZNfpdD'9bNf0[ECKLD(at-sign)jKG'pbD(at-sign)&XMU'0N5H!!("
bEf*XC(at-sign)ec,T%&dC!!9fPXE*B%(BehQUabCC0PGTKPFT0SBCKfQ'(at-sign)6BC0`FQm-EQP
dCC0MEffBBQPZBA4[FQPMFj0[ET0`FQm-EQPdCC0cCA4cNh0TC'(at-sign)1SBf4*i!!BTU
XFRQ(at-sign)!qUSFfPNCC0hDA4SNf0[ECKLD(at-sign)jKG'pbD(at-sign)0cNfpZN`aZDA4PNh0PG(-rMSk
I(J!!MC)!qQpJ0$D1MSb,!!!!,`!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!"lb'J!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j8d9$6dj%PJ2
UU&0139"65%p81T085%(at-sign)63eP$6%P$Nd4&8NP(at-sign)NIkab%'4r`9(at-sign)9%P(at-sign)45k1RaR(VSf
4*i!!9'KPPJ104QpbC'PZBA*jNf4PFQPfQrpBj'&dDAD3!+abCC0[CT0KNh#3!&1
1EfajEQpYD(at-sign)&XNfPZNfpZCC0fQ'&bD(at-sign)&LE'(at-sign)6D'&cNf+3!&11C(at-sign)9ZNfGPEQ9bB(at-sign)a
THQ9NMU31J!#0N5H!!'+3!+abHCB&)-e)BA9cC'pb#j0dEj0`QP11EfajEQpYD(at-sign)&
XFj0KEQ56CQpbE(at-sign)&XNh#BEj(at-sign)XFRH6CA+(at-sign)"5$0Ff9bD(at-sign)9cNfPZNfj[ET0MEfeYPDa
bGA4KG'PfNf(at-sign)1SBf4*i!!GT(r(at-sign)14KFQPKBQaPFjB$r2*KFj0QEfaXEjUXFRGc,T%
&Ele$EfjcD(at-sign)4PFT0dD'(at-sign)6BA0cEj!!8ijMD(at-sign)&dDADBCC0KE'GPBR*KNp0$e'MAB6Z
(at-sign)!IrqBMZ0NcU61T-kMT%4bT`lNf-lNhM8DC%$r2+jCf9ZCA)YMU'0N5H!!'&dC(at-sign)5
(at-sign)"(V0BT!!V(*jNf'6Ff9dNfpQNfaPG(4PFR16e'EAB6Z(at-sign)!IrqBMZ0NcU61T-kMT%
4bT`lNf-lNhM8ClNZN3ET6P4SCCB%HXeXCA4dCA+6ehL6Z(at-sign)PcNf0KE'aPC*0KNhD
4reMNBA*TB(at-sign)*XC5b4"*l(at-sign)B(at-sign)aXMU'0N5H!!'pdD'9bPJ3KDQaPG(4PFR16BA*PNf0
KE'aPC*0MEfjcG'&ZQUabG(-ZN3AG*8'4"#&FE(at-sign)pZEfeTB(at-sign)b6D(at-sign)k6G'KTFj0KFh0
[N!"6MQ0TBA4TGTKPNf&XCf9LFQ'6DA11SBf4*i!!GfKKG*B$kUKjQUabEh(at-sign)6G'K
TEQZ6DA56FfK[G(at-sign)aNNf+3!&11C6U4"6MJDA56DA16BC0hQ'pbC*0XD(at-sign)ZBCBkKSBf
5!,k`kGGYPJ098VNpNpGKH'*KH)fIq`ZCe611N36!"0GLBhKN1SkI&(A#MC%RJ!#
j3C%#mh&`QP11EfajEQpYD(at-sign)&XPJ,cX'PcNf'6E'PZC(at-sign)&bNf0[EC!!V(*LD(at-sign)jKG'P
[ET0[CT0YEfj[E(at-sign)PKE(-XN3-P&(at-sign)&ZC*0KNfC[FQeKE*0`Q'q9V(*hNf9bN3,cX(0
PFQPPFikKMC%RJ!"TFjB%*+eNC3aZC(at-sign)56BA16B(at-sign)k6D(at-sign)i-EQPdCC0cG(at-sign)f6EfD6E(at-sign)p
ZEfeTB(at-sign)ac,*%%-bjhDA4SNh0eDA4KBQaPNh*PFh4bD(at-sign)0dD(at-sign)pZFj0[ET0dD'(at-sign)1SBf
4*i!!Ch*[QUabGh4SPJ-,PQpQNf4PCh*PCA16EfD6G'KPNh0eE(at-sign)eKEQ4c,T%%lS9
'NIm&9QpbE(at-sign)&XNh#3!&11EjKhQ'9bNh0PFQPPFj0TET0ZEfk6BfpYECKeG'&dDAD
BCBkKMC%RJ!"fNIpBj'&bD(at-sign)&LE'9cPJ3c*'C[FQf6B(at-sign)k6B(at-sign)aRC(at-sign)*bBC263p4SD0G
K1jB"rrjL1if61T-kNcU1N4(+R$Z6BcZ6H04TDENZN3B59&H4r`9(at-sign)CCB%-b4hD(at-sign)a
XNf4PEQpdCC0LQUabHC2ACT%"4rqj+0GiZ5Q6Fh9MQ'L6BBkKMC%RJ!"QEh*YB(at-sign)b
E!qUSF*!!8ij[PDabGj0PFTKcCA*TCA-ZMUNCakk0N5H!!&4SCCB$kUK)BA9cC'p
b#j0NCA*TGT(r(at-sign)14KG'PfQUabCC0[CT0dD'(at-sign)6E(at-sign)pZEfeTB(at-sign)b6eff6Z(at-sign)PcNf0[EA"
eG'9NNf&cNfC[E'a[Q(Gc1SkKSBf430NJedLE!1e(at-sign)Z5MAEENTPJ098Mf6edLBZ5M
ABAKLBAL0RrX,QG8cMTX%`!6ABQ0iC,NTNcf6ef&LBAL0RrX,QG8cMTMABQ0iC*B
#UULj+j-cef&iBQ&iMCrl#jR9-SkBef*MH'56Z5Z6ef&iBQ&iMCrl#jR9-ikBef*
MC$U1Ra4e`Sf4*i!!Z94SDA1(at-sign)"#)RC'8-EQPdD(at-sign)pZNfPcNf9iG'9ZC'9NNf+DV(*
jNfaTEQ9KFQPdQ(Q6G'q6F*T6MQpXH(at-sign)j[E(at-sign)PKE(16B(at-sign)jNNh4[NfC[FQeKE*0`Q'q
9V(*hNf9bMU'0N5H!!(0PFQPPFbk1TSf4*i!!5(at-sign)D(at-sign)!VV&eff0RrZP-YS`MT%&L2k
jDA16B(at-sign)j[G'KPFT0YEfj[E(at-sign)PKE#b4![H-GjUXFQ(at-sign)6D''BGTKPNh4SCC0PH(#3!&1
1C(at-sign)0dC(at-sign)56FR9XCC0QEh+6$'jND(at-sign)jRNh4SCC0)BA9cC'pb#ikKMC%RJ!"NCA*TGT(
r(at-sign)14KG'PfN!#XFQ(at-sign)(at-sign)!qUSEfD6BC0`FQq3!&11C(9MG$U4"6MJedLE!1e(at-sign)Z5MAE(at-sign)f
0RrZP-YS`MT%#cMQj+CB$99)pNpG)Q,NSeffj+GGYMCrlT6,D-)k4"AMKZ5Z4!UU
Sefe)Q,NSeff0RrZP-YS`MT%#cMQj+GFkMUD0N5H!!,P8D'(at-sign)(at-sign)"!Hc5'&eFf4[FJZ
6C'9bDAD4reMNBA4TGTUXFQ(at-sign)6Fh8,CA*cNfCbEff6BC0YBC%!TaaUEh+6GjKPB(at-sign)Y
ZCA0c,T%&N!!"9'KPFQ(at-sign)6Ff9PEA16G'q6BT!!8ijPMU'0N5H!!'j[PJ1ZDQ&ZB(at-sign)a
[Cj0[CT0dD'(at-sign)6BjUXFQKKD(at-sign)k6FR9XCC0QEh+6G'KPNf4T#f9bC(at-sign)kBG'PKG'P[ET0
[CT0KNfCeEQ0dD(at-sign)pZNfpQNf'6CR9ZBh4TEfiZMU'0N5H!!%D4r`9(at-sign)Eh+(at-sign)"(at-sign)cMCAK
KEA"XC5bE"FebG'KPNdKKGA0NEh),Nf4PFQPfNIpBj'&dDAD3!+abCC0[CT0dD'(at-sign)
6F*!!8ij[E(PZEfeTB(at-sign)b6+0GKH,NTMCrlT6,BESk4"DK3Z5bBGfKPET0dD'(at-sign)1SBf
4*i!!E'9dG'9bFjB$kUMABC1jB(at-sign)jNNpGiNlPNEj0ZEh56BfpYEC!!V(*eG'8XNfP
cNfj[G*0PFA9KE*0dEj2AEVNSef&iZ5Q0RrZP-YKZfJ$9-Bk4%)6-ef'j,T%&11"
*G*0TFj0KNfePFh-ZMUD0N5H!!&4SCA*PPJ6NUQPcNf&ZEh4SCA+6EQpdD(at-sign)pZNfp
QNf4PFQPfNIpBj'&dDADDV(*PNh4SBA56C'q3!&11CA16Ff&dDA0QHC0KNh0TEA"
XCC0MQ'KKD(at-sign)k6FR9XCBkKMC%RJ!"eEQ4PFTB&MN0QG(at-sign)jMG'P[EQ&XNf0[EA#3!&1
1Eh0TG'P[ELk4#L1b5A56DA16G'KPNf0jBfaTBj0NCA*TGT(r(at-sign)14KG'PfQUabC5b
4"IFUC'9ZEh4PC*0LQ(Q6G'KPMU'0N5H!!'aPG(4PFT%$kUMA4*!!8ikj,SkQMC%
RJ!"8D'(at-sign)(at-sign)"#K`BhPME'PMNf4PFQPfNIpBj'&dDADDV(*PNfPcNf4P$'jPC*0KFj0
QEfaXEjKhFbk4"I)i4QPbFh56C'8-EQ(at-sign)6G'KPNh4bG(at-sign)jMBA4TEfk6Eh#3!&11CA)
YMU'0N5H!!'&dEh+(at-sign)!qUSee54"BaZZ(at-sign)&cNfC[E'a[N!#XFRGc1SkQMC%RJ!"K,T%
&11"TCTB$kUKdD'(at-sign)6$(*cG*0XCA4dCA+6EfD6BC0YEfj[E(at-sign)PKE*2AEC1jDA16EQp
dNh4SCC0fNIpBj'&bD(at-sign)&LE'(at-sign)6ehLj,*0cCA56ee54!D('Z5MAEENTPJ098Mf6-$Z
1TSf4*i!!BLk4"6CdD(at-sign)D(at-sign)!q0NG'KPN`abFh56E'9dG'9bNfpQNf'6E(at-sign)pZEfeTB(at-sign)b
6eff6Z(at-sign)PcNh4SCC0fNIpBj'&bD(at-sign)&LE'(at-sign)6ehLj,*X$j0KcEj0dD'&dNpGYPJ098VN
pNpGiEBfIqk8bfM#1N3,11ENXQ(0PG)kKMC%RJ!$A9*%"SFDj+0GYZ5Q(at-sign)!e952C2
AEBfIqk8bfM#1N3,11GFkMSkI(J!!MC)!qQpJZ63hMSk-L`!!!$!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!I%9S!+J+2(at-sign)0S2f'e`ZJ!PXSpBf
JrFRA#if4*i!!Z(at-sign)-ZN38ii%9iG'9ZC*B$kUKLQUabHC0XD(at-sign)jPBA*TG*KjNh4[Np0
$e'KSef%lPJ(rrQ)lMC-kNcU61Sk4%FUF1j0M1j0ie'PTZ5k1T"U!!)f4*i!!9'K
PPJ3A`(at-sign)0jBfaTBj0NCA*TGT(r(at-sign)14KG'PfN!#XFQ(at-sign)6EfD6BC0YEfj[E(at-sign)PKE*2AEC1
jDA16C'8-EQ9NNfPZNh4PFQecNfpQNh4SCC0dFR9ZBf&dD(at-sign)pZMUN1J!#0N5H!!'p
`N!"6MQ9bBA4[FTB$kUKKFj0QEfaXEj!!V(*hFcU1SBf4*i!!B5k4"4,36'9dPJ0
iGpG`NlPLQP11CC0dD'(at-sign)6F*K[E(PZEfeTB(at-sign)b6Ef*dB(at-sign)PZC(at-sign)56BT!!V(*jNf&NC'P
ZCj0KE'b6BhPME'PMNh#BCA*YN!#XFR9dBA4TEfjcNfpQNh4SCBkQMC%RJ!"YEfj
[E(at-sign)PKE*%$kUMAEENZMU'0N5H!!')ZN38ii&0PG*%$kUMA4*!!8ikj+0GYZ5Q(at-sign)!e9
52C2A9*%"SFDj+0G`Z5RA1SkKMC%RJ!#jBbk4"6MJ4AKdC(at-sign)jNPJ2UU'+DV(*jNfa
TEQ9KFQPdQ(Q6G'q6B(at-sign)aXNfC[FQeKE*0`N!"6MQqBGjKPFT0cCA*TCA-ZMU'0N5H
!!%D4r`9(at-sign)Eh+(at-sign)""L$CAKKEA"XC5b4"#2kG'KPNf0jBfaTBj0NCA*TGT(r(at-sign)14KG'P
fQUabCC0[CT0dD'(at-sign)6B(at-sign)+3!&11EjKfQ'(at-sign)6E(at-sign)pZEfeTB(at-sign)b6eff6Z(at-sign)PcNf0[EA"eG'9
NNfPZMUD0N5H!!(4SCCB$kUKQEfaXEj!!V(*hD(at-sign)jRNh0dCA"c1SkKMC%RJ!"6G'9
`PJ6dL$%ZN3K(at-sign)IeH4r`9(at-sign)FQPdCC0NEjUXFRGZNf&XE*0MH(at-sign)0XD(at-sign)16F*!!8ijPFQf
BGA4KG'P[ER16Efk6G'KPNfe[EQpYD(at-sign)&XNpGKH'*KH)fIqk8be611N36!"0GLBhK
NZ5k1TSf4*i!!9'KPFf(at-sign)4!qUSBA*PMUDKMC%RJ!$AH'*KH)fIq`ZCe611PJ6!"0G
LBhKNB6ZE!IrqBQ&iMCrl#jR9-ik6ef*MH'4KH$ZBBAL0RrX,QG8cMT2ABQ0iC'&
iBMZBH)fIq`ZCe611NpGLBhKNBAKLB6ZBH)fIq`ZCe6+1NpGLBhKNBAKLBAJlQ(K
LBhKNBAKLBAL0RrX,QG8bMT2A1ikKMC&XELPLBhKNBAKLBAL0RrX,QG8cMTB%`!6
A1jX"rrjMH'4KH'*KH)fIq`ZCe611NpGL1jKiC'&iBQ&iMCrl#jR9-ik6ef*M1jK
NBAKLBAL0RrX,QG8cMT2ABQ0i1SkI)3!!MC%RJ!#j8h4PF*B$kUJb,T%&11"*ET0
dD'(at-sign)6B(at-sign)+D8ij[PDabGT0PPJ2UU'aTFh3XNh#BCA*QEh*YNfpZCC0[CT0dD'(at-sign)6CQp
XE'q3!+abGfPZCj0[F*KPFQ&dD(at-sign)pZFcU1SBf4*i!!B5k4"P`eD(at-sign)D(at-sign)"%[%G'KPN`a
bFh56E'9dG'9bNfpQNf'6E(at-sign)pZEfeTB(at-sign)b6DA16EQpdNpGiZ5b4"'3,FQ9YEj(at-sign)XFRD
6CCB%5m4dD'(at-sign)6E(at-sign)pZEfeTB(at-sign)b6CR*[EC0dD'(at-sign)1TSf4*i!!E'PcG$Z1SBf4*i!!BLk
4"6MJD(at-sign)D(at-sign)!qUSG'KPN`abFh56E'9dG'9bNfpQNf'6E(at-sign)pZEfeTB(at-sign)b6DA16ehLj,*0
bC(at-sign)e[PDabGT0PPJ2UU(4SCC--FR0dNfaPG(4PFLk1SBf4*i!!9fKPETB$fNThN!#
XFQ(at-sign)6F*T6MQ9bCQpbEC0[F*KPFQ&dD(at-sign)pZFj0K,TX&-fYKEQ56BLkBEfk6G'KPNf9
KBj!!V(*SNfpQNh4SCC0YEfj[E(at-sign)PKE(16D(at-sign)k6G'KPMUD0N5H!!'&LN!"6MQq9V(*
fNf(at-sign)(at-sign)!qUSE'PcG#b6Gj!!V(*PNfpLG'&TET0KNh0SEh*dCA+6E'PcG#b6EQ&YC(at-sign)a
j1SkQTSf48Q"Ief*KH)fIq`ZCe611PJ6!"0GLBhKNB6ZE!IrqH)fIq`ZCe6+1NpG
LBhKNBAKLB6ZBH'*MH'4KH'*KH$ZBBQ0iC'&iBQ&iMCrl#jR9-Sk6ecZBC'&iBQ&
iMCrl#jR9-ik6ef*M1SkI&3!!MC%RJ!#j8h4PF*B$kUJc,T%&11""C'56G'KPNfe
[EQpYD(at-sign)&XFj0dD*UXFR9cNfpLG'&TEQ9NNh4[NfGPG*0dD'(at-sign)6BhPME'PMNf4PFQP
fNIpBj'&dDADBC6U1TU'0MBf4*i!!ed53!&11Z5MAEENTMSk0MC&1rk3pMSk0MC&
L)bcA4*!!8ikj+0GKH'*KH)fIq`ZCe611N36!"0GLBhKNZ5Q1MSkI%B!!MBf0N8l
rT$f1MSf0N(at-sign))M,0GLBAL0RrX,QG8cMTX%`!6ABQ0iC''(at-sign)!UUSZ5Z6ehL0RrX,QG8
bMTMABQ0iC'&iBQ'6Z5Z6ehKLBhKNBAKLBAL6Z5Z6ef*MH'4KH'*KH)fIq`ZCe6+
1N3GUV,NVNpGNBAKLBAL0RrX,QG8cMTMABQ-kMSk1SBf4*i!!Z8&ZEh4SCA+(at-sign)!qU
SCAKKEA"XC6U4"6MJG'KPNf0jBfaTBj0NCA*TGT(r(at-sign)14KG'PfN!#XFQ(at-sign)6EfD6G'K
PNfe[EQpYD(at-sign)&XNpGKH'*iBhKNH*1jCA&eB(at-sign)acMSkI(J!!MC)!qQpJ0$L1MSb,!!!
!-3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"r&#J!U!SpBf
JrBEA#k!#(at-sign)bMeMD$pbGF,T!k!!)f4841Ued53!&11Z5MABAKLH'0iC(Lj+CB$99)
pNpGLH'0iC(KKPJ+UU,NVNpGMH'4iBAKLNlNVNpGNH'&iBRKMH*1j+j2ABAKLH'0
iC$U1Ra8!!)f4*i!!Z94SCCB$kUKMH(at-sign)0XD(at-sign)16C'9bDAD4reMNBA4TGTUXFQ(at-sign)6EfD
6G'KPNfe[EQpYD(at-sign)&XNbMABALj+BfIqk8bf'k1N3Q5q,PTFj0dD'(at-sign)6CQpXE'qBGfP
ZCcU1SDNDJ!#0MBf48'2Aed53!&11Z5JSef&iZ5Q0RrX,QGKZMT%&U&#j+Bk1MBf
5!)P!N6f1MSf0NJ#FC"RA4*!!8ikj+0GKH'&iMCB"rrikNcU61Sk4%FUFBALj+Bk
1MU34J!#0MBf5!)P!N6f1MSf0NJ#FC"RABAKKH)f(at-sign)!Irq1T-kNcU1Qa(+R'&iBCB
#UULj+j2ABAKKH)f(at-sign)!Irq1T-kNcU1Q'&iBC1j+if6ecU(at-sign)!Irq1T-kMT%5G8CKH'&
iMCB"rrikNcU61SkBBAKKMSk1SBf0MC)!L8#4Z6f1MSf0NJ#FC"RAEVNSef&iZ5Q
0RrX,QGKZfJ$9-Bk4%)6-ef%kMSk1TSf4*i!!Z90TE(at-sign)PXBA*XHC(r"9BXPJ2UU'p
ZCC0MEfe`GA4PFikN$S!!SBf5!+U"BGG%N!"6MVNSehL(at-sign)!UUSZ5Z6ef'j+BfIq`Z
Cf'k1N3MpSVNpN3098YGZZ5MAH*1j+j2ABENTMCrl#jRBEYS!e6'1MUN9!!#0N5H
!!,PKEQ3XPJ2UU'C[FT0QEh*YB(at-sign)b6F*!!8ij[PDabGj0PFT%$kUKcCA*TCA-XMU'
KMC)!`b&(at-sign)ed53!&11Z5MACBfIq`ZCf(M9+pKKMT%3D5kj+CB$99)pNpGPMCrl#jR
BH08Vf''1MUD0N5H!!,PKEQ51SD'0NJ$&"SIA4*!!8ikj+0GPMCrl#jRBBAL1Q`R
-YVNTPJ098Mf6ef(at-sign)0RrX,QGKKH)kBef%kMUD0N5H!!,P5C(at-sign)ePECUXFQ+3!&11CA)
XN3(at-sign)ce(4SCCB&(at-sign)'9XCA4dCA*cNpGKNlPKEQ56ehL6Z(at-sign)4[Nfj[G*0MEfeYQ(9dC5'
4#B)A5(at-sign)k6G'KPFf(at-sign)6CAKKEA"XCA-XN3(at-sign)ce(4SCBkKMC%RJ!"MEh*bCA0`N!"6MQp
ZC'PZCjB$kUK)BA9cC'pb#j0NCA*TGT(r(at-sign)14KG'PfN!#XFQ(at-sign)6DA16BC0YCA0c,Sk
KMC%RJ!"8D'(at-sign)(at-sign)!YXRBhPME'PMNf4PFQPfNIpBj'&dDADDV(*PNf9ZDQqBHA16B(at-sign)a
XNh"bEh#D8ijPFR4TCA16CAK`Q'9MG'9NNfpQNh4SCC0[FQ4TEQ&bHC0NCA*TGT(
r(at-sign)14KG'PfN!#XFQ8lMU'0N5H!!'PZPJ9(2A"KFR4TBh9XBA)XN3(at-sign)HBfPdNh0KG'P
c$'9cNh4SCC0MQUabD'&TET0bG(at-sign)aPNfC[FT0dD'(at-sign)6BfpYF*!!8ij[FfPdD(at-sign)pZNfp
QNh5BGjK[NfC[FQeKE)kKMC%RJ!"`N!"6MQq9V(*hNf9bN32UU(0PFQPPFbk1U4U
!!)f4*i!!9*(r"9C[PJ46$R0dBA4PNh4SCC0bG(at-sign)aPFj0QEh+6G'&VD(at-sign)jRNf0jBfa
TBj0NCA*TGT(r(at-sign)14KG'PfPDabCA-XN34Y+(H6CCB%8`jZC(at-sign)9NNfpZCC0YEh*PNfp
`N!"6MQ9bBA4[FLb1SBf4*i!!Bf&XE'9NPJ2UU(4SCC0hFQ&`F'PZCj0[F*!!8ij
PFQ&dEh)ZMUD0N5H!!&4SCCB$Q6&hFQ&`F'PZCj0[F*!!8ijPFQ&dEh+6DA16C'8
-EQ9NNf&cNfC[E'a[N!#XFRGc,T%&(EK-CA56ef10R`(-c08aMTB%`!6A1jX"rrj
MMCm"c-c9-Sk6ecZ0Q$UB1TJkMT%4bT`lQ'10R`(-c0KZMT%*3B'jBT!!8ijPPJ1
C-(at-sign)&ZN!#XFRQ6E'9dG'9bFbk1SBf4*i!!5(at-sign)D(at-sign)!qUSefH3!'ijZ5MAH,NTNfPcNf&
ZQUabHC0QEh*YB(at-sign)b6F*!!8ij[Q(HBCA+6Ff9bD(at-sign)9c,*0cCA51SD'0NJ#f61,8D0G
$N3$FRQ10R`(-c08aMT%%`!6ABifI!Fc-e6+1MC%'`!,A1TB"rrikNcU1N4D+S'1
0R`(-c0KZMT%&U&$8DYGRN!"Z1ENSehLj+G4TN3098VNpMUN9!!#0N5PD,GGMMCm
"c-c9-Bk4"-!%ef10R`(-c08bMSf4"X!#ecU(at-sign)!Irq1T-kMTX(at-sign)LU"MMCm"c-cBESk
4"DK3efH3!'ijZ5MAH,NTPJ+UU#Z6ef10R`(-c08bMSf4"X!#ecU(at-sign)!Irq1T-kMTK
MMCm"c-cBESk4"DK3efH3!'ijZ5MAH,NTef10R`(-c08aMT%(DUbj+j2ABifI!Fc
-e611MC%'`!,A1TB"rrikNcU1Q'10R`(-c0KZMTX&U&$ACj!!EMQj+0GiZ5RABif
I!Fc-e6'1N36!"0GMMCm"c-c9-Sk4"fUXZ5Z0NpFkPJ(rrMU61Sk4%ar`Z5Z6ef1
0R`(-c0KZMTMACj!!EMQj+0GiZ5RABifI!Fc-e6'1N36!"0GMMCm"c-c9-Sk0N3E
!!YFkPJ(rrMU61Sk4&SUJBifI!Fc-f'lD!08aMT%3K-cA1SkI)3!!MC%RJ!#j5(at-sign)D
(at-sign)!qUSefD4!8IrZ5MAH,NTNfPcNf&ZQUabHC0QEh*YB(at-sign)b6F*!!8ij[Q(HBCA+6Ff9
bD(at-sign)9c,*0dD'(at-sign)6Gh*KF("TEQH6Eh#3!&11CA*KG'pbMU'KMC)!b5EIe'MA3j%!h*j
QN3&(rlNSehLj+G4UefH3!'ijZ5MAH,NTe'Q1TSf4*i!!Z(at-sign)PcPJ2UU'4P$'jPC*0
LQUabHC0XD(at-sign)jPBA*TG*KjNIm&9Lk1MTmH!!#0NJ$kEf!d1Bk1M)X!!!!b!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)%+D!#S#MeMD$pKYF,S!*
E+2(at-sign)0S2h*e`Z0N5H!!,P%C3aZCBkN$S!!SBf5!*8)MG4Sed53!&11Z5MACTX"4rq
j+0GiZ5NTe'VACj!!EMQj+0GiZ5R8DCB$99+j2C2A9*%"SFE8D0G$N3$FRQDBZ5M
AH,NTe'VACj!!EMQj+0GiZ5R8DGFkMUN89$q0N5H!!,P'NIm&9QpbN32UU'9iB(at-sign)e
`E'8kMU'KMC)!VCA!e'MA4*T6MVNSefD4!8IrZ5MAH,NT+G4UZ6(8DCB$99+j2C2
A4*Lj+0GQN3&(rlNSehLj+5RA1SkI(fmqMC%RJ!#j9'KPPJ51Pf0jBfaTBj0NCA*
TGT(r(at-sign)14KG'PfQUabCC0[CT0dD'(at-sign)6F(*[N!"6MQ4eBh56EfD6G*KhQ'q6)QCeEQ0
dD(at-sign)pZFb+6DA16CfPfQ'9ZNf+BHC0dD'(at-sign)6CQpX,BkKMC%RJ!"XEj(at-sign)XFRGTEQH4!qU
SD(at-sign)4PET0dDA56H6U1SD'0N(at-sign)d(at-sign)eGG%QP11Z5MACT%"4rqj+0GiZ5RACj!!EMQj+0G
iZ5NTPJ098Mf6e'MA4*Lj+0GQN3&(rlNSehLj+5R8DYGRN!"Z1ENSehLj+G4TPJ+
UU,NVNp4Sed5BZ5MACj!!EMQj+0GiZ5NTe'VACT%"4rqj+0GiZ5R8DGFkMUD0N5H
!!,P'NIm&9QpbPJ2UU'9iB(at-sign)e`E'8XNfpZCC0[BR4KD(at-sign)jcMTmCQ[qKMC&3(at-sign)mMA4*!
!8ikj+#JaPJ+UU03!NpGKH,NTMCrl#jRD!08aMTX,A(bj+$'6e!#6ef*iZ5Q0RrX
,QGS!e6'1Q,NTPJ098Mf6+$'(at-sign)!UUSe!#6ef&iZ5Q0RrX,QGS!e6'1Q,NS-C28!*2
ABRLj+BfIq`ZCfJ$9-BkBZ5JaNp3!NpGKH,NTMCrl#jRD!08aMTMABBkQSBf5!*)
NSENV+$'(at-sign)!UUSe!#6ef*iZ5Q0RrX,QGS!e6'1Q`YFI,NS-C28!*2ABALj+BfIq`Z
CfJ$9-BkBZ5JaNp3!NpGLH,NTMCrl#jRD!08aMTMABMU1Rap[2Sf4*i!!Z8j[PJ2
UU(0eBjUXFQL6D(at-sign)4PETKdDA5BHC0SEfaNFj0QEh+6G'KPNdKKGA0NEh),Nf4PFQP
fNIpBj'&dDADBC5k1RaQDrif4*i!!9'KPPJ4M+'0jBfaTBj0NCA*TGT(r(at-sign)14KG'P
fQUabCC0[CT0dD'(at-sign)6F(*[N!"6MQ4eBh56EfD6B(at-sign)kBHC0cCA&eC(at-sign)jMCC0[CT0QEh*
YB(at-sign)b6F*!!8ij[Q(HBCA+6Ff9bD(at-sign)9cMU'0N5H!!'PcPJ2UU(0TE(at-sign)PXBA*XHC0MEfe
`GA4PC*0LN!#XFRQ6G'KPNhGbBA"`D(at-sign)jRNfp`N!"6MQ9bBA4[FMU1SD'0NJ#Ta)r
A4*!!8ikj+0GQMCm"c-c9-Bk(at-sign)"-!%Z5MAH,NTefD0R`(-c08bMT1j+0GiZ5Q0PJ(
rrYFkNcU61Sk4%FUFCSfI!Fc-f'k1N3(at-sign)S8,NSehLj+5Q4!e952BkQMC&2feM8D0G
%QP11Z5MACSfI!Fc-e6'1PJ6!",NSehLj+5R8DYGQMCm"c-c9-Sk6Z5MAH,NTMCB
"rrlA1T-kNcU1N4(+R'D0R`(-c0KZMT%&U&#j+0GiZ5R8DCB#UULj+j28D0G%Q,N
SefD0R`(-c08bMTB%`!5j+0GiZ5NTe'VACSfI!Fc-e611NlNSehLj+Bf(at-sign)!IrqecU
61T-kMT%4bTaQMCm"c-cBESk4"DK3Z5MAH,NTefD0R`(-c08aMT1j+0GiZ5R8DEN
VMUD0MC)!KGIqecU(at-sign)!Irq1T-kMT)!Q%e%Z5Z4!UUSe'MA4*!!8ikj+0GQMCm"c-c
BESk4"DK3Z5MAH,NT+G4UefD0R`(-c08aMTB%`!5j+0GiZ5RACSfI!Fc-e6+1NlN
SehLj+Bf(at-sign)!IrqecU61T-kMT%4bTaQMCm"c-cBEYS!e6'1N4#%c,NSehLj+G4TecU
1Rap[2Sf4*i!!Z9H4r`9(at-sign)CCB%3l*MEfePNfj[QUabGj0dEj0dD'(at-sign)6E(at-sign)&TET0`FQp
`N!"6MQ9bG*KjNfpQNh4SCC0MH(at-sign)0XD(at-sign)16C'9bDAD4reMNBA4TGTKP1T%&k[4dD'(at-sign)
6BjKSB(at-sign)PZNh*eE'8ZMU'0N5H!!%GTGT(at-sign)XFQ9ZQ`6N(h56Gj0[Q'C[FQeKE*K`N!"
6MQq6Gj0PFTKcCA*TCA1BefD4!8IrZ5MAH,NTQ'&ZC*MACj!!EMQj+0GiZ5QBD(at-sign)k
Bdd28D'MAB6Z(at-sign)!IrqBMZ0NcU61T-kMT%4bT`lNf-lNhM8D(at-sign)Qj,*%&)ReKFh0eE(at-sign)(at-sign)
1SBf4*i!!G'KKG*B$&redD'(at-sign)6CQpbE(at-sign)&XNh#D8ij[PDabGj0PFTB$&recCA*TCA1
6efH3!'ijZ5MAH,NTNf4[Q'9cNfj[G*0SBC(at-sign)XFRD6CCB$&reKNf0[ER0dB(at-sign)k3!+a
bG*0dCA*Y,T%%mUG9EQ4PFT0dD'9cCBkKMC%RJ!"MDA*MG(at-sign)ecG'&ZBf9c,*%%$9a
dD'(at-sign)(at-sign)"!CVBfpYF*!!8ij[FfPdD(at-sign)pZNpGQN3&(rlNSefH3!'ijZ5MAH,NT+C0TFj0
hQUabC(at-sign)aXNf4P$'jPC*0LQ(Q6FQ9`E'&MD(at-sign)jRNpGRN!"Z1ENSehLj+C0QEh+1SBf
4*i!!CAD3!+abCA*jPJ2UU'qD8ijMBh9bFQ9ZBf(at-sign)6EfD6G'KPNhD4reMNBA*TB(at-sign)*
XCC2AH*1jD(at-sign)k6G'KPNfC[FQeKE*0`Q'q9V(*hNf9bPJ2UU(0PFQPPFj2ACT%"4rq
j+0GiZ5NZMU'0N5H!!%aPG*B$i0CeFj0hFQPdCC2A4)fI!Fc-f'H1N368rVNSefD
4!8IrZ5MAH,NT+C0dEj0NC(at-sign)j[G'(at-sign)6G'KPNfC[FQeKE*0`N!"6MQq9V(*hNf9bPJ2
JeR0PFQPPFj0[BR4KD(at-sign)jPC*0LN!#XFRQ6Fh9LFh4T,BkKMC%RJ!"dGA4TEQH(at-sign)!k!
cefH3!'ijZ5MAH,NTNfPZNh"XB(at-sign)0PNfpQNf9fQUabCA*jNfq3!&11Bf0eFR*PEQ0
PNfpQNpGiNlPTET0dD'(at-sign)6BhPME'PMNf4PFQPfNIpBj'&dDADBCC2A4*!!8ikj+0G
QN3&(rlNSehLj+5Q1SBf4*i!!EfD(at-sign)!k+bG'KPNfC[FQeKE*0`N!"6MQq9V(*hNf9
bPJ1LXR0PFQPPFj2ACT%"4rqj+0GiZ5NZN38Jie4SC(at-sign)k6G'KPNf1DV(*SB(at-sign)PZNh*
eE'(at-sign)6CQpbNh4SCC0MH(at-sign)0XD(at-sign)16C'9bDAD4reMNBA4TGTKPMU'0N5H!!'G[N!"6MQ9
cPJ2UU'&cNfC[E'a[N!#XFRGc1SkKSBf5!*Da`YG%QP11Z5MACT%"4rqj+0GRN!"
Z1ENSehLj+5NTPJ098Mf6e'MA4*KRN!"Z1ENSehLj+G4Ued50R`(-c0KRMT%%e2k
j+0GQN3&(rlNSehLj+5R8DGFkMSkI(J!!MC)!qQpJZ68`MSk-L`!!!$-!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!JbPS!+J+2(at-sign)0S2f'e`ZJ!PX
SpBfJrFRA#if4*i!!Z8D4r`9(at-sign)Eh+(at-sign)!qUSCAKKEA"XC5b6GjUXFQ(at-sign)6D''BGTKPMU3
1J!#KMC)!QXG3ed53!&11Z5MACBfIq`ZCf'&iBRL1Qa)iHENTPJ098Mf6ef*iCBf
Iq`ZCf'&iBRL1Q0GKPJ+UU,NVNpGPMCrl#jRBBAKLH)kBef&iBMU1U48!!)f4*i!
!Z8'(at-sign)!qUSE(at-sign)pbCC0PE'9RB(at-sign)kDV(*dNf9iB(at-sign)e`E'(at-sign)6DA16G'KPNfC[E'a[Q(GTEQF
kMU'KMC&cZ0RA4*!!8ikj+0GPMCrl#jR9+$(D!0KKH08TMCrp,6Vc-('K*3X!"J!
!!!B!!!!&BfecH6EE!0BaMSkE*6GUZ5Q(at-sign)!e952C-S-CB#UUM8!*2ABALj+BfIq`Z
CfJ$9-Bk4#eamef(at-sign)0RrX,QG8S-GS!f'&ie5Q0RrdY1YX!eM'1MTLj+$'6e!#6ef&
iZ5Q0RrX,QGS!e6'1N3YFI0GK1SkQMC%RJ!#j6fjPPJ2k`f0KET0`FQq9V(*fNf(at-sign)
(at-sign)!rV$G'KKG*0dD'(at-sign)6BhPME'PMNf4PFQPfNIpBj'&dDADDV(*PNfpQNf'6FQ&dD(at-sign)p
ZB(at-sign)b6CQpbE(at-sign)&XNh#3!&11EjKhQ'9bNh0PFQPPFj0TESkKMC%RJ!"ZEfkE")T-Bfp
YEC(at-sign)XFR9dBA4TGT0PQ'aPG(4PFR1BDA1BB(at-sign)GKD(at-sign)kBBCKbBA4TEfjKE*KZEfkBBfp
YEC0eG'&dDAD6CCK`N!"6MQq6Gj0PFTKcCA*TCA-XMU'0N5H!!'&ZC*B&T0KdD'&
dNh4SCC0MH(at-sign)0XD(at-sign)16C'9bDAD4reMNBA4TGTUXFQ(at-sign)6EfD6B(at-sign)k6B(at-sign)aRC(at-sign)*bB(at-sign)PMNfC
[FQeKE*0`N!"6MQqBGjKPFT0cCA*TCA16D(at-sign)k6EQpZMU'0N5H!!'0[E(at-sign)f9V(*eG'&
dDAD6CCB$kUKXCA4dCA*cNfPcNf&RB(at-sign)PZNf&ZNf&XCf9LFQ&TBj0QEh*YB(at-sign)b6F*!
!8ij[PDabGj0PFT%$kUKcCA*TCA-ZMTmDJ!#0N5H!!%4PFh"TG'(at-sign)(at-sign)!aREG'KPNf9
fD(at-sign)4PEQ0PNh4SBA56G'KPNf0jBfaTBj0NCA*TGT[r(at-sign)14KG'PfN!#XFQ(at-sign)6DA16G'K
PNfjKG(9bB(at-sign)b6EQpdD(at-sign)pZNfpQNf4PFQPfQ'%YMU'0N5H!!(4TGTUXFQ(at-sign)(at-sign)!`'(at-sign)CQp
bNfj[ET0MEfeYQ(9dBA4TGTKPNf&XCf9LFQ&c,*%$-$0dD'(at-sign)6G'KPEh*jNf&cNfP
dNfPcNf&dNh"bCA0PETKdNfPcNfj[G*0cBA4TFfCj,BkKMC%RJ!"TEQFZN38A2P4
SCCB$KF&MH(at-sign)0XD(at-sign)16C'9bDAD4reMNBA4TGTUXFQ(at-sign)6DA16B(at-sign)k6C(at-sign)e`DA*TBf&XNf4
TFf0[Q(DBCA*jNIm&9Lk4"4Fq5A56EQ9PC(16G'q6BT!!8ijPNf9ZFf0[EQ0PC)k
KMC%RJ!"TETB#pBjcEfePNf*bEf&NCA+6B(at-sign)aRC(at-sign)*bB(at-sign)PMNh4SC(at-sign)pbHC(r"9BXN3-
QNff9V(*eBj0SQ`,eMQaTDj0PQ(4SCCK)BA9cC'pb#jKNCA*TGT(r(at-sign)14KG'PfNf(at-sign)
BD'&cQ'+3!&11C(at-sign)9ZMU'0N5H!!'9ZFf0[EQ0PC*B$kUKTET0dD'(at-sign)6G'KPEh*jNfp
QNdK[F'D6B(at-sign)aRC(at-sign)*bBA-ZMSkI(J!!MC)!qQpJ06'1MSb,!!!!0!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#&U1J!U!SpBfJrBEA#k!#(at-sign)bMeMD$
pbGF,MC%RJ!#j9%K*8N5(at-sign)!k5f8dj"8&0)6e3kN31Nada24d&5594)691638j%Ne4
)4C0#58j268P"6*085%928N90,SkI'S!!MC%RJ!"8D'(at-sign)(at-sign)!j'%4A9XCA)Y6(at-sign)&M6'&
eFQPZNh0eE(at-sign)eKG'P[ET0QEh*YN!#XFR9XBC0TFj0[EQ(at-sign)6EfD6G'KPNfe[Fh56FQ9
YBA*VNIpBj'&LE'(at-sign)6CQpb,BkN$S!!MC%RJ!"YN!#XFR9XBA1(at-sign)"8MQEfD6E(at-sign)&dD'9
YBA4TBh-ZN3P6Q8D4r`9(at-sign)Eh+6BC0cG(at-sign)PdB(at-sign)*XCC0QG(at-sign)jMG'P[ET2ACT%"4rqj+0G
iZ5Q6EfD6BC0bC(at-sign)&XNfpbNf0[EA"XCAL1SBf4*i!!GT(r(at-sign)14KFQPKBQaP,*B$kUK
TG*0TFj0cG'&dC(at-sign)56BA16CQpXE'q3!+abGh-kMU'KMC&cMShACTX"4rqj+0GiZ5Q
(at-sign)!UUS+j2ACTLj+0GiNlNVNc%TNbZ6efDBZ5MAH*1j+j-b+C-VMC2A1TB"rrikNcU
1N4-Im,NVNpGQQ,NSehL6Z5Z6efkj+C%$99)pMTmDQD'0NAHDX0G#MCm"c-c9-)k
0N3E!!TraiiHV(at-sign)Sk0Rr81-Sf4%-!$f(M9+pKZe5XaMTm6MNL0N3a120KiMSk4,US
XefD(at-sign)!8IrZ5MAHC!!EMQj+GGNHC%$'1'j+jX#UUMA3SfI!Fc-e6'1N36!",NSefD
6Z5MAH*Lj+jMAETLj+jJa+CM8!*MACT1j+0GiZ5NTMTmJEHL0N8(l9#Z0MBfIpq(
&MC%"-c2A3SfI!Fc-e6+1MT%"-c1I"9&YL3!!CQ8!$DdQR`X)#if4!N62Z6)KMSk
1MT%3%icA4*!!8ikj+0GQQ`&(rlNSehL(at-sign)!UUSZ5Z6efk6Z5Z6-5Q6e!#6efDBZ5M
AH,NT+C-VMBf0RrIKaBf4!phEed+0R`(-c08cMSk4!phER`94EBN!!'CP!!fY*Tm
,#!Z0N3*%clNc)Bk1MSk4%Vided50N!"6MTrl#jR9-Sk4"415Z5MACTLj+0GiNlN
VNpGZNlNVNc%TNp3!NpGQQ,NSehLj+5Q6+if6ecU(at-sign)!Irq1T-kMT%5G8BkMTmB%*+
0N5H!!,P8D'(at-sign)(at-sign)!df5ed+0R`(-c0KZMT%)pH+jBA*PNh4SCC0#CA*ZEh9XE'Q6ET(at-sign)
XFR9YNf+D8ijPFR1(at-sign)!df5B(at-sign)jNNpG%N31K),PTFj0dD'(at-sign)6Eh*ND(at-sign)jKFRQ6C'9bDAD
4reMNBA4TGT!!V(*PNfp`Q'9bBA4[FLk1U4U!!)f4*i!!9'KPPJ4db89eE'9b,8e
KBdaKGA*TET0QEh*YQUabG(at-sign)aKNfKKFj0`FQqBGTKPC*0fQ'9bHC0eFf9QG(at-sign)b6CQp
bNfqBGTKPFT0dQ(HBEj0SQ(9ZC(*PC)kKMC%RJ!"jQUabC(at-sign)&bFbk4"5Ba6QpZCA4
SC(at-sign)aPFh-XN31pdR4SCCB$XTa&G(at-sign)aPFLe0B(at-sign)0-BA9bD(at-sign)k6CQpbECKeE''6Fh8,CA*
cNfCbEff6BC0cCA*TEh9cNf4P$#f1SBf4*i!!BfPPEQ0jNIm&9Lk4"I'Z9'KPPJ3
S3R0PFQPPFj0[ET0dD'(at-sign)6FQPRD*UXFR56D'&ZC*0cD(at-sign)4PNfPcNf&XE(at-sign)pcG*0ZCAD
BCA+6BfpZQ(DBCA*RC(at-sign)kBG#b4"$HTG(at-sign)jXCA0cMU'0N5H!!'PdPJ2UU(*PC(9MCA1
6G'q6BC--EQPdCC0cG(at-sign)dZMUD0N5H!!%peFTB&K(at-sign)CaG(at-sign)9cG'P[ET0TFj0dD'(at-sign)6CQp
XE'qDV(*hD(at-sign)jR1T%)EPaTFj0dD'9bCC0KNhDBC(at-sign)0dEh+6Fh"KBf(at-sign)6EfD6CR9ZBh4
TEfjcNhGSD(at-sign)1BD)kKMC%RJ!"MEfkDV(*dB(at-sign)PZFjB&,d4KFj0YB(at-sign)kBHC0[CT0dD'(at-sign)
6C(at-sign)aPE(at-sign)9ZQ(4KFRQ6CR9ZBh4TEfjcNf&cNh#D8ij[Fh0TBQaP,*%&J'YKEQ56BC0
dEh#BEfa[ChQ1SBf4*i!!Efk(at-sign)",$UFh9MQUabD*0KNhDBC(at-sign)0dEh+6Fh"KBf8XN36
LHh*PE'&dDADBCC0dEj0hD'PMQ'L6G'KPNh*TCfLBG*0SB(at-sign)jNNh0TC'(at-sign)6EfD6G'K
PNd9eE'9b,BkKMC%RJ!"0B(at-sign)0-BA9bD(at-sign)k(at-sign)!qUSCQpbECUXFR9XBC0TFj0KNf0[ETK
fQ'9bCf9ZQ(56Ff9bD(at-sign)9c2ikQMC%RJ!"8D'(at-sign)(at-sign)!ekqB(at-sign)jcGjUXFQ9bNh4[Nh4SDA1
6FA9PFh4TEfk6DA16G(at-sign)jPH(#3!&11C(at-sign)0dC(at-sign)4XHC0bC(at-sign)aKG'9NNh4[Nh4SCC0KER0
hQ'9bNh4[Nf&ZEh4SCA+1SBf4*i!!FA9PFh4TEfiZN3VbM&GSBA5(at-sign)"G-hDA16G'K
PNb*bD(at-sign)GSQUabG#+6Cf9ZCA*KE'PkBA4TEfk6EfD6G'KPNf*TEQpYD(at-sign)&XNf0[N!"
6MQ81BfPPETKdFikKMBf0MC%RJ!#Ip'CJUa#1MCrkV0b0N5eij0KZMTm*GRk0N5f
l3fZ1MSf4-U%dRr4QB+X4MSk1N6jiX,PhD'9ZPJAHQ0GVN3C&YEPTFj0KE'a[PDa
bGj0PC*B&hTKdEj0LN!"6MQ(at-sign)6BC0ZC(at-sign)GKG'PfQUabCC0TETKdC(at-sign)GPFMq4#a5a9'K
TFj0aG(at-sign)9cG'P[ET0XC(at-sign)&NFj0TESkKMC%RJ!"dGA*ZPJ2JV(4[Nf'6G'KTFQ56FA9
PFh4TEfikN38ciQK[QUabGj0cD'&XE*0hQ'(at-sign)6Dfj[Q(H6GfKPG'KPFT0KNfGPEQ9
bB(at-sign)aTHQ&dD(at-sign)pZNfpQNh4SCBkKMC%RJ!"LD(at-sign)j[E(at-sign)PKE*B$cl9MEj!!8ijP$Q0TC(at-sign)k
DV(*dFj0TFj-LFQPRD*Kd)Mq4"5rN9'KPNf&ZFhHBCA+6G'q6G'KTFj0dD'PbC*0
aG(at-sign)9cG'P[ET0TFj0PBA0j1T%&+fGKMU'0N5H!!'GPEQ9bB(at-sign)aTHQ&dD(at-sign)pZPJ4Z6'p
QNh4SCC0LD(at-sign)j[E(at-sign)PKE*0MEj!!8ijP$Q0TC(at-sign)kDV(*dFj0TFj-LFQPRD*Kd)T0TCT0
TG*0XC(at-sign)&NFj0dEj0KNh0PER0TBQaPMU'0N5H!!'GPEQ9bB(at-sign)aTHQ&dD(at-sign)pZPJ2UU'p
QNh4SCC0LD(at-sign)j[E(at-sign)PKE*0dD'9[FQ9Y1SkQRa-!#Bf5!+C+$5MABCB#UULj+j2AH,N
TMCrl#jRBESk4#2fLZ6f0Rr,rrBf4#)%af'k1R`d!!if0N32FGCrerrbV(at-sign))k1R``
ijif4!e95f'Z3!$D&e6d`MSk0MBf4&09HRqjQA+XJMSfIpq(&MC%F`!VAESkI%&P
jMC%G!!PVMSk0N52$NjrZCPbV)Bk1MT%VVMrABBfIq`ZCf'Z1N38MNYGiMCrl#jR
BEYS!f'Z1N4$S(at-sign)YFkMTmT$MZ0N5H!!,PAD'9ZPJ-eL8Q4!c9EGjUXFQ&cNhQBEh9
ZCbb4!eR$5CX$09YeFf9NNh4[Nh4SD(at-sign)jVNfpQNh4SCC0LD(at-sign)j[E(at-sign)PKE*0dD'9[FQ9
YNf&cNh4bDACTB(at-sign)`ZN36mJ%QBG'KTEQZ1SBf4*i!!5CX%PE*SBC(at-sign)XFRD6CCB%PGj
XC(at-sign)&bEQ9NNff3!+abHC0XCA0cEfiZN3FkJN'BGj(at-sign)XFQ9XE#eVEQq6Gfk(at-sign)"*AHF'K
TE'pcEh"SCA)XN36!V%QBBf&Z*h56FQ9YC(at-sign)f3!+abBT!!8ijPFT0SDA11SBf4*i!
!EQ&YC5b4"-YEGh*[G'(at-sign)(at-sign)"*jUG'KKG*0dD'(at-sign)6GfK[E'(at-sign)6G(at-sign)jTGT!!V(*PFR0PNf0
KET0LN!"6MQ(at-sign)6D(at-sign)jQCA*bC(at-sign)56CR*[EC0KNfGbB(at-sign)PZNfpQNh0KEQ3ZMU'0N5H!!%K
PPJ5Kp(0SEh9XC*0SBC(at-sign)XFRD6CCB%SI4KC'4PC*0dD'&dNf'6Ch*PBA56C'9KE*0
[CT0YBA4SC(at-sign)eKG'PMFj0MB(at-sign)k6BT!!8ijPNf4PFQPfQUabC(at-sign)56BTKjMU'0N5H!!'e
PC'PdBA4TEQH(at-sign)!qUSGA#3!&11Efk6G'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)dZMSkI(J!
!MC)!qQpJ06+1MSb,!!!!03!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!#'kDJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j6'9dPJ8NaR9cNh4
KDjUXFQ(at-sign)6G'KPNf*eE'b6BTKjNh4SCC0SEh*ZFbb4"A01B(at-sign)jNNh0dBA4PNh4SCC-
LFQPRD*Kd)T0RC(at-sign)jPFQ&XDATKG'P[ET0[CSkN$S!!MC%RJ!"dD'(at-sign)(at-sign)"-(2BQPZEfe
TB(at-sign)b6BfqD8ijP$Q0TC(at-sign)k3!+abG(-ZN3Hq9PH4r`9(at-sign)CC0`FQqBBf9PC*0TET0dD'(at-sign)
6E(at-sign)pcG*0`Q'9NCA0dFQPKET0hPDabBC0jNIm&9Lb4"2HCBT0jN36"c`abFh51SBf
4*i!!Cf9ZCA*KE'PkD(at-sign)jRPJ,3"R4SCC0NC3aZDA4TEfk6EfD6G'KPNfCKBh4[FQP
KE#k4"0UU9'LDV(*eFbb4!`L0E'9dNpGZNlPLN!"6MQ(at-sign)6B(at-sign)kBHC0TETKdC(at-sign)GPFLb
4!`L0F*!!8ij[FfPdDADBCBkKMC%RJ!"[FTB$kUKZC(at-sign)GKG'PfQUabC5k4"6MJ9j(
r"9CPNf4P$'jPNh4SCC05EfeKET0QB(at-sign)0dEh*TB(at-sign)b6(at-sign)pGZZ9dKNf&cNfC[E'a[Q(G
c1SkKSBf5!08159[AEVPG)CB$99)pNpGZZ5'1Ra8!!)f4*i!!D(at-sign)DE!qUSefk(at-sign)!e9
5e"(at-sign)6Z6!XQ'&ZC)kKRa+Jh)f5!-"C99[AEVPG)C%$99)pMBf0RrIKaBf4#,lp+03
!Z6%TMCrlT6,BEY8V-Bk1N35)KCm&8(at-sign)f*!!"QC3!a5THI#`J,MENSe!$AETB#UUM
8!*1j-5NKMSk1MSkI'a#5MC%RJ!"TCT%$kUMAETB$99)mNlN`,SkI'S!!MC%RJ!"
AD'9bCCB$D[eNEj!!8ijPFj0dD'PcNf4P$'jTG'P[ET0MEfePNfCbEff62j%&$P*
*N30Uh(at-sign)0[G(at-sign)aNNh0TEA"XHC0cBCUXFRQ6G'KKG*0TG*0hQ'pbDh-XN31%KQ*eG)k
KMC%RJ!"dD'&dPJ1-,AH3!+abEh9XC*0ZEh56BT!!8ijPNh4SCC0hD'pXCC0dFR9
dD#k4"4PL9'KPNhD4reMNB(at-sign)aeCC0[CT0dD'(at-sign)68QpYB(at-sign)k6CQ&MG'pbD(at-sign)&XNe[AEVP
G)C0QEh+1SBf4*i!!efk(at-sign)!qUSZ(at-sign)jPCf&dDADDV(*PNf9aG(at-sign)&XFj0dD'(at-sign)6FQ9cD(at-sign)4
eCC0[CT0dD'(at-sign)6Cf&YE(at-sign)'6CR9ZBh4TEfk6BA56G'KPNfPZQ(4PCf9bNpGZPJ+UU03
!NlNa,SkKMC%RJ!"9FfPZCjB$kUKdD'(at-sign)68QpYB(at-sign)k6CQ&MG'pbD(at-sign)&X,*0hQUabCC0
NC3aZCC0dD'(at-sign)68QpYB(at-sign)k6Bfq3!&11C3jMD(at-sign)9ZQ(4cNf&cNfC[E'a[Q(Gc1SkKRa-
!#Bf0MBf5!,Y(MjrZCPbV)Sk0RrIKaBf5!-%FjYGZMTm3(at-sign)AQ0NJ$"A19VMSk0NJ$
))'qIlQCFUb11MSk5!0&,',NpMBf0RrIKaBf4&dDf(at-sign)pGZZ9dKMT%%L)(at-sign)I"9&YL3!
!CQ8!0NU1R`X)#ieEefZDCafjA5&Eefk(at-sign)!UUSe!#6efZBZ9dKMSk1MT%m"NEA1Sk
I'p9HMC%RJ!#j9fKPETX%6#2AETB$qcr8&C2ADj%%BPc8&C1j-#b4"'5#G'KPQ&*
[E(at-sign)&ZQ'0[P911C3jMD(at-sign)9ZN!#XFR4cQ'0[D(at-sign)jMD(at-sign)4PQ(GTG'LBG'KPQ'*TEQpYD(at-sign)&
XQ'0[Nf81,BkKMC%RJ!"MD(at-sign)9ZQUabG(-ZN39(at-sign)58D4r`9(at-sign)Eh+(at-sign)!r4fB(at-sign)aXNfPZQ(4
PCf9bFj2AET1jB(at-sign)jNNpGVN!"R(ENXN32fkA4SCC05EfeKET0MEj!!8ijP$Q0TC(at-sign)k
BG(16FfKKFQ(at-sign)6B(at-sign)aXNf9XC(at-sign)ePETKdBA*jMU'0N5H!!("bEh#D8ijPFR4TCA1(at-sign)"+N
cEfD6BQPZEfeTB(at-sign)b6BfqBC3jMD(at-sign)9ZPDabG(-XQ`6BeA0eBj0SPJ5T-f&cNe#3!+a
bBA0MB(at-sign)`RFj0dFQPKEQGXC5bBCA4M,T%(G)")Ej(at-sign)XFRH6CAD6CA)XMU'0N5H!!(4
SCA*PPJ2UU'&bCC0cEfePNh0eFR"bDA0PFj0TET0cG'pbC5b6CQpbNf9iB(at-sign)e`E'8
XNfC[FT2ADj%%8F(at-sign)jF*!!8ij[FfPdDADDV(*PNhHBCC--EQ51SCm6"d+0MBf0NJ#
Xei+IlQCFUb+1MCrhiF(at-sign)0NJ#hU8bj-)kI%&PjMC)!XUcCe!$ADik1MC)!`S(at-sign)lRqj
QA+XMMSk1NJ$,X'5j2Bf0MC%$99+IlQCFUb+1MCrhiF(at-sign)0N3PmFEN`MTm3(at-sign)AQ0N3N
UUGGVMSk0N3qZ0*rZCPbV)ik1MT%Bf0fj2Bf0MCrhiF(at-sign)0N35)K5M8!,Na+BfIqk8
bf'Z3!$D&e5XaMSk4")L&R`94EBN!!'CP!#KBkCm,#!Z0N4$UVpGVMSk1MT%Z&+%
kMTmRe9k0N5H!!,P%Ej!!8ijPFjB%JFGdD'PcNfeKDjUXFQ(at-sign)6B(at-sign)kBHC0cC(at-sign)jcC6q
4"[im9j(r"9CPE'`XPJ5RMRQBCA-XNf+3!&11C(at-sign)0KGA0PPJ5"ahHBCC0MB(at-sign)k6$'j
NNf'6Cf9ZCA*KE'PkBA4TEfk1SBf4*i!!EfD(at-sign)"%FZG'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)p
bC(at-sign)f6G'KKG*0REj!!8ijPFj0hDA4SNh4SDA-ZQ`C1FdPdNfPcNh4SCC0QEfaXEj!
!V(*hD(at-sign)jR,TK5C(at-sign)0KE'b6G'KPMU'0N5H!!(#3!&11Ej(at-sign)XFRH6CA+(at-sign)!qUSFf9bD(at-sign)9
cNf9iF'&ZFfP[ET0[CT0dD'(at-sign)6E'pRBA*TG'KY1SkKU4-(3Sf5!)pf30GXQN12EfH
3!'ijZ5MAH*B#UULj+j2ABENTPJ098Mf6efbBEfH3!'ijH*B#UULj+ifIm[rpMC%
'+U[D-BkI$3!$MBf4!c(,RrArr+YBMSkI$$MRMC2BDj!!0SA926'1MSf0MCrhiF(at-sign)
0N49GjlNSe!#j-5Q0RrZP-YKVN!!fKG8V-Bk1N49Gjjm&8(at-sign)f*!!"QC3!S(at-sign)1QI#`J
,MC%3kUrADik1MSk0MBfIpq(&MC&!APpKMCrlT6,BDik1N8!G0Tm&8(at-sign)f*!!"QC3!
,d15I#`J,MGGiMCrmL)RBDik1MSk1N8dK6GFkMTmG$MZ0N5H!!,PANIm&9Q(at-sign)(at-sign)!hY
YBf&ZNh*PBf&cG*0dD'PcNh#D8ij[PDabGj0PFTB$HfecCA*TCA16CAK`B(at-sign)jcD(at-sign)p
ZNfPZNh4PFQecNfpQNh4SCC05EfeKET0MEjKP$Q0TC(at-sign)k3!+abG(11SBf4*i!!BA1
4!qUSCQpXE'q3!+abGh-kMU'QMC)!Qll,efbD3ip[Cj!!EMQj+0GiPJ+UU,NVNpG
KZ5Q(at-sign)!e952C2AE*K[Cj!!EMPiPJ+UU,NVMCrbrrf0N3BUUpSaMTm0!!10MC%$-FZ
IpIrmUeL1MTm-11H0NpKVN!!fKG8p-Bk1MBf0N43UY*rZCPbV)Sk0RrIKaBf4'P(
6Z6#1Ra"CHBf4'J!,efZ1MSf4))1(at-sign)RqjQA+XMMSk1MBf0RrIKaBf4*me*ef'0RrZ
P-YKVMSk4*i`JR`94EBN!!'CP!![3j*m,#!Z0ehL0Rrb)LGKVMSk1MSk40*!!0pF
kMSkI(J!!MC)!qQpJZ68cMSk-L`!!!$B!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!LG0S!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if4*i!!Z94SDA1
(at-sign)!l(at-sign)BDA16BTT6MQ9RD(at-sign)jZD(at-sign)jRNh4[Nfa[Q'pVNfaTDj!!V(*PNf'6Cf9ZCA*KE'P
kBA4TEfk6EfD6G'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)dXN32!0(GTG'L1T!k!!)f4*i!
!G'KPPJ50Bfa[Cf&bDA4SEC0`E''DV(*jD(at-sign)jRNh4SCC0bEfaPFj0[CT0kCA*[,A4
SNh#3!&11EjKhQ'9b,T%()4&"EQpdD'9bNh#3!&11EjKhQ'9bNh0PFQPPFj0PH#f
1SBf4*i!!F'&ZFfP[ETB$"hjhD'9bCC0dD'(at-sign)68QpYB(at-sign)k6Bfq3!&11C3jMD(at-sign)9ZQUa
bG(16E(at-sign)&VQ'(at-sign)6G'KPDA+6BA"`N!"6MQ9KFQ&ZBf(at-sign)6DA16G'KPNfC[E'a[Q(GTEQF
kMU'I%`!*MC&*1+)SehL(at-sign)!UUSZ5Z6ef'j+5MAE*T$MfpRN!"Z1ENSehL6Z5Z6ef'
j+C28!*1j-5Q(at-sign)!e952C2AH,NSefbBEfH3!'ijH*B#UUM8!*1j-5Q6+j2AB(at-sign)bBEfH
3!'ijH*1j+ifIm[rpMC%'+U[D-BkI$3!$MBf4!c(,RrArr+YBMSkI$$MRMC2BDj!
!0SA926+1MSf0MC%8+V5IlQCFUb+1MCrhiF(at-sign)0N4T4dlNaMTm3(at-sign)AQ0N4S!#pGVMSk
0N5#$PTrZCPbV)ik1MT%Q(at-sign)1hABBfIq`ZCf'Z1N38MNYGiMCrl#jR9-GS!f'Z1N4!
!$YFkMTmG$MZ0N5H!!,P%EjB$kUKhQUabCC0cC(at-sign)(at-sign)6BC0`BA4dCA*Z2j%&11"ANIm
&9Q9XE#b6E'9dNh9cNh4bHC0jQ'9dNf&ZEh4SCA+6F*!!8ij[Q(HBCA+6Ff9bD(at-sign)9
cNf9iF'&ZFfP[EMU1SCm3fID0NJ#EBc-SehL(at-sign)!UUSZ5Z6ef'j+BfIq`ZCe6+1N36
!",NSefb3!%12EfH3!'ijZ5MAH*1j+j2ABENTNp3!NlNaNp3!MBf0RrIKaBf4!ph
EZ6'1N32Gfjm&8(at-sign)f*!!"QC3!&hrbI#`J,M6+1MSk1N3Va#LQ4!e952BkI)EY(MC&
&-IcAH,NSefbD3ip[Cj!!EMPiPJ+UU03!NlNaNp3!MBf0RrIKaBf4!phEZ6'1N32
Gfjm&8(at-sign)f*!!"QC3!&hrbI#`J,M6+1MSk1N3Va#LQ6+if0MC1IlQCFUb+1MCrhiF(at-sign)
0N3KrrlNbMTm3(at-sign)AQ0N3Krrc'1MSf4$PrlRqjQA+XMMSk1N43e8YGKH,NSefbBEfH
3!'ijH*28!*1j-5Q6+if0MC1IlQCFUb+1MCrhiF(at-sign)0N3KrrlNbMTm3(at-sign)AQ0N3Krrc+
1MSf4$PrlRqjQA+XMMSk1N43e8YGKMCrl#jR9-Sk4"-!%efbBEfH3!'ijH*1j+if
Im[rpMC%'+U[D-BkI$3!$MBf4!c(,RrArr+YBMSkI$$MRMC2BDj!!0SA92611MSf
0MC%8+V5IlQCFUb+1MCrhiF(at-sign)0N4T4dlNbMTm3(at-sign)AQ0N4S!#pGVMSk0N5#$PTrZCPb
V)ik1MT%Q(at-sign)1hABBfIq`ZCf'Z1N38MNYGiMCrl#jR9-YS!f'Z1N4!!$YFkMTmGZ1H
0N5H!!,P1Ej(at-sign)XFRHE!qUSGj0PQ'0KETKXC(at-sign)&`Q(4[Q''BCf9ZCA*KE'PkBA4TEfi
ZN38ii%D4r`9(at-sign)Eh+BFh9TG'&LE'(at-sign)BCR9ZBh4TEfjcQ0GQN3&(rlNSehLj+5bBFf9
dMU'KMC)!e&9qed50N!"6MTrl#jRD!08aMT%,X!VACT%"4rqj+0GiZ5Q1U48!!)f
4*i!!G'q(at-sign)""i)BT!!8ijPNh4SCC0eEQPaG(at-sign)(at-sign)6D(at-sign)jNC3aZDA4PNfPZQUabG'9RFQ&
XNfpQNh4SCC0QG(at-sign)jMG'P[ET2ACT%"4rqj+0GiZ5Q6GfKTBjKSNfKKFj0MEfjcG'&
ZQ(51SBf4*i!!G'9bECB$kUKPFA9KE*0dEj0kCA*[,T%&11"%Ej0ZEh56GjUXFQp
bFRQ4r`9(at-sign),*0dD'PcNhGTE'b6E(at-sign)&VQ'(at-sign)6Ff9ZFf(at-sign)6D(at-sign)k6BC0YEfePETKd,SkI'S!
!MC%RJ!"%C3aZCBkKSBf5!,&-EYF9MCrl#jQ0e5Ja+BkI"q[DMGKZMSk4#eamZ5M
AH,NTPJ098Mf6(at-sign)pGZZ9dKed50N!"6MTrl#jRD!0KZMT%-Q&EAE*!!3ip[Cj!!EMP
i1SkQMC%RJ!#j5'9bC5b4"!Z&efk(at-sign)"!6cZ(at-sign)PcNf&ZQUabHC0TETKdC(at-sign)GPFLb4"!Z
&F*!!8ij[FfPdDADBCC0[FT0ZC(at-sign)GKG'PfQ'8ZN3(at-sign)(`&4SCC0QG(at-sign)jMG'P[ER16ea(at-sign)
0RrZP-Sh9+$%TMTm(8N'0f'k1MT%,A(bj+0GiZ5Q6BA*PNf0KE'aPC)kKMC%RJ!"
dD'(at-sign)(at-sign)!qUSD'&bE(at-sign)pZD(at-sign)16E'pRBA*TG'KYFj0[CT0[FQ4PFT0[EQ8ZN38ii%D4r`9
(at-sign)Eh+6efk6ZA#3!&11Eh0TG'PfQUabCC0hQ'(at-sign)6D''BGTKPMU'T%0RfMC)!J%T8ea(at-sign)
0RrX,QBh9+$%TMTm(kpU0f'k1MT%,A(bj+0GiZ5Q(at-sign)!e952C2AH)fIq`ZCf'k1N3(at-sign)
S8,NSefb3!%12EfH3!'ijH*B#UUM8!*1j-C28!)f0MCrhiF(at-sign)0N32GflNaMT%$hGZ
I"9&YL3!!CQ8!"GrmR`X)#idbMSk1MTB0Ql,8!)f0MCrhiF(at-sign)0N32GflNaMT%$hGZ
I"9&YL3!!CQ8!"GrmR`X)#idcMSk1MT28!)f4!UUSecU(at-sign)!Irq1T-kMT%6(r$8!)f
0MCrhiF(at-sign)0N34[SVNaMT%$hGZI"9&YL3!!CQ8!"`1*R`X)#ihAESk1MSk4$"5AZ5Q
1RaJ3NSf4*i!!B(at-sign)jNMU'QMC)!b#$mea(at-sign)0RrRGhih9+$%TMTm)EZb0fJ$BESk1N3a
%b,NSehLj+C%$99)pMBf0RrIKaBf4"m0B-Bk4")L&R`94EBN!!'CP!!a9STm,#!Z
0ehL0Rrb)LGKZMSk1MSk4%K&DecU1RaJ3NSf4*i!!Z8pQPJ2UU'0[GA*cCC0hQUa
bCC0KE(0[NfKKQ(DBCBkKSBf5!-6JE0F9MCrjhGq0e5Ja+BkI#-&RM6#1MT%,A(b
j+0GiZ5Q(at-sign)!e952C2AE*!!3ip[Cj!!EMPi1SkI&3!!MC%RJ!#j9j(r"9CPPJ10P(at-sign)&
bCC0ZEj!!V(*hNfPZNf'6F*!!8ij[FfPdD(at-sign)pZNh4[Nh0dBA4PNh4SCC0RC(at-sign)jPFQ&
XDATKG'P[ET0[CT0dD'(at-sign)6BQPZEfeTB(at-sign)b6G'KPEh*PEBkKMC%RJ!"dD'&dPJ2UU'P
cNf&cFfqD8ijMD(at-sign)&dC(at-sign)56GfPdD*0dD'(at-sign)6D'&bE(at-sign)pZD(at-sign)16E'pRBA*TG'KYFbk4"6M
J5A56CfqBCA16BA16CQpXE'q3!+abGh-kMU'I%`!*MC)!QKMLea(at-sign)0RrX,QBh9+$%
TMTm(kpU0f'k1MT%,A(bj+0GiPJ+UU,NVNpGKZ5Q(at-sign)!e952BfIm[rpMC%'e9AD-Bk
I$3!$MBf4!paeRrArr+YBMSkI$$MRMC2BDj!!0SA926#1MSf0MC%8e9kIlQCFUb+
1MCrhiF(at-sign)0N4UUYGGZMTm3(at-sign)AQ0N4VUY'Z1MSf4)DiqRqjQA+XMMSk1N5H$PGGKMCr
l#jRBDik4"515ea(at-sign)0RrRGhih9+$%TMTm**mf0f'lD!0KVMSk4%1KDZ5MAH,NTecU
1MTmH!!#0NJ$kEf#j0651MSb,!!!!0`!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!#-ELJ!U!SpBfJrBEA#k!#(at-sign)bMeMD$pbGF,MC%RJ!#j9'KPPJ2
UU(4SFQ9PNfPNC(at-sign)kDV(*dDA4TCA16B(at-sign)+3!&11EjKfQ'(at-sign)6BA*PNh0`N!"6MQ9MD(at-sign)&
XNf0KFf9cNfpQNh4SDA16D(at-sign)4PETKdDA5BHC(r"9BXNfC[FT2AETB$99+j2C-`ecZ
(at-sign)!IrqZ6(A1j1j-Lk1RaU!!)f4*i!!9'KPPJ9qMfGPEQ9bB(at-sign)aTHQ&dD(at-sign)pZNfpQNh4
SCC0LD(at-sign)j[E(at-sign)PKE*0dD'9[FQ9YNh4[NfKKFQe[EQPMNfa[Cf&bDA4SEA16CfPfN!#
XFQ9cMU31J!#0N5H!!'j[G'KTEQH(at-sign)!qUSEQ9hNfC[FT0ZC(at-sign)GKG'PfQUabCC0PH(#
3!&11EfjPETKdFbb6GfKPFQ(at-sign)6DA56FQ9NG(at-sign)0PFj0dEj0dD'(at-sign)6D(at-sign)4PETKdDA5BHBk
KRa-!#Bf5!*X#k5MAH*B#UULj+j2ABENTMCrl#jRD!0KZMT%2QKUj2BfIm[rpMC%
'e9AD-BkI$3!$MBf4!paeRrArr+YBMSkI$$MRMC%$99,BDj!!0SA926#1MSf0MC%
8e9kIlQCFUb#1MCrhiF(at-sign)0N4c!#Y3!efk1Ra"CHBf4)DUeDik1MC%Y'1UIlQCFUb'
1MSk4031(at-sign)ef'0RrX,QGKVMT%&)j,AH)fIq`ZCfJ$BEYS!f'Z1N4H%dYFkMTmG$MZ
0N5H!!,P)Ej(at-sign)XFRH6CAD6CA)XN39+C'C[FTX&"!Y`N!"6MQpcDA4TGT0PQ'9iF*!
!8ij[EQ9ZNh4cQ(H6CCK[BR4KD(at-sign)kBBCKRC(at-sign)k6G(at-sign)PZCCKKEQ5BBQ%2D(at-sign)jRQ'GPEQ9
bB(at-sign)`YMU'0N5H!!'PkBA4TEfk(at-sign)"+Q(at-sign)EfD6G'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)dZN3G
eUdPdNh0dBA4PFj0dD'&dNh4SCC0QG(at-sign)jMG'P[ER16ea(at-sign)0RrZP-Sh9+$%TMTm(8N'
0f'k1MT%,A(bj+0GiZ5NXN36C8QC[FT2AESkKMC%RJ!#jF*!!8ij[FfPdDAD3!+a
bC5bE"*!![(0KG'PcCRQ(at-sign)"'q%G'KPNfpbC'PZBA*jNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)d
XQ'e[QP11C(9XEj0ZC(at-sign)GKG'PfN!#XFQ(at-sign)6F*K[PDabGj0PFR14"'q%EfD1SBf4*i!
!ehLj,T%&11"*ETB$kUK[G'KPFT0hQUabEh*NFbb6GjKPNfKKQ(DBCC0dD'(at-sign)6CQp
XE'qBGfPZCj0TC'9ZQ(4TG*Kj1SkKRa$CpSf4G4Lb+0GiPJ+UU,NVNpGKZ5Q0RrX
,QGKZMT%&U&#j+0GXN!"$MfpRN!"Z1ENSehL6Z5Z6ef'j+C28!*1j-C28!)f0MCr
hiF(at-sign)0N32GflNaMT%$hGZI"9&YL3!!CQ8!"GrmR`X)#idbMSk1MTB0Ql,8!)f0MCr
hiF(at-sign)0N32GflNaMT%$hGZI"9&YL3!!CQ8!"GrmR`X)#idcMSk1MT28!)f4!UUSecU
(at-sign)!Irq1T-kMT%6(r$8!)f0MCrhiF(at-sign)0N34[SVNaMT%$hGZI"9&YL3!!CQ8!"`1*R`X
)#ihAESk1MSk4$"5AZ5Q0MCrmUe#0MC%$99,8')k1R`28X)f0N30Z1VNpMSk1MSk
I&3!!Ra-!#Bf0Rr,rrBf4EL,If'k1R`d!!if0N(at-sign)Pq)jrerrbV(at-sign))k1R``ijif4D2F
!f'Z3!$D&e6d`MSk0MBf4HRF-RqjQA+XJMSfIpq(&MC)!JQ'iefk1Ra"CHBf5!)+
KYfZ1MSf5!)PP3CrZCPbV)Bk1MT)!N8rYef'0RrX,QGKVMT%&)j,AH)fIq`ZCf'l
D!0KVMT%3k&Uj+0GXN!"$MfpRN!"Z1ENSehLj+CB#UUM8!*1j-C28!)f0MCrhiF(at-sign)
0N32GflNaMT%$hGZI"9&YL3!!CQ8!"GrmR`X)#idbMSk1MTB0Ql,8!)f0MCrhiF(at-sign)
0N32GflNaMT%$hGZI"9&YL3!!CQ8!"GrmR`X)#idcMSk1MT28!)f4!UUSecU(at-sign)!Ir
q1T-kMT%6(r$8!)f0MCrhiF(at-sign)0N3m'ZlNaMT%$hGZI"9&YL3!!CQ8!($'lR`X)#ih
AETB#UUM8!*2ADik1MSk4)8,*Z5RA1SkI(3ilMC%RJ!#j9'KPQ`4A"fPNC(at-sign)k9V(*
dDA56HCKTFjKfNIpBj'&XD(at-sign)5BE(at-sign)q3!&11C(9XEjKZC(at-sign)GKG'PfNf(at-sign)BF*!!8ij[NhH
6CA*cQ'pQQ0GiZ5k4"Rhq6(at-sign)PbB(at-sign)0XCA1BEfDBBf&ZBf9XE'&dD(at-sign)pZMU'0N5H!!'&
bCCB$jEY[N!"6MQ0MGA*bD(at-sign)jRNfPZNh4SDA16D(at-sign)4PET(at-sign)XFR4TG*0jNIm&9Lk4"6F
m5CB$jEPhDA0SQ`2PZdQ6DfjPGjKKQ'0[EC!!V(*LD(at-sign)jKG'pbD(at-sign)&XQ'pbQ("bEf*
KBQPXDA0dD(at-sign)11SBf4*i!!D(at-sign)k3!+abG'9bF(*PG'&dD(at-sign)pZPJ2UU'pQNh4SDA16E'p
RBA*TG'KYD(at-sign)16Cf9ZCA*KE'PkBA4TEfk6EfD6G'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)d
ZMUNDJ!#0N5H!!&0[PJ2C)QCKFLb4!pbMGjUXFQ(at-sign)6D''BGTKPNf&cFh9YC(at-sign)56G'K
KG*0KE'b6Ff9bD(at-sign)9cNf0[ETKfQ'9bCf(at-sign)6D(at-sign)k6G'KPNh4[F*!!8ij[E'pRHC0[CT0
dD'(at-sign)6BfpY,BkKMC%RJ!"`E'9iQ`22QQk9V(*eEC0LN!"6MQ9bFbk4"5rE9j(r"9C
PQ(GTE'bBEQq6GjKMNfKKEQGPQ(4SCCKdEh#3!&11Efa[ChQ4r`9(at-sign),*%$e30hD'P
XCCKbCA4KD(at-sign)jTEQHBG'KPQ'0[ET0fNf9b,BkKMC%RJ!"RC(at-sign)jMC5k1TSf4*i!!9'K
PPJ6F8'e[G'PfNIpBj'&dD(at-sign)pZNfC[FT0dD'(at-sign)6E'pRBA*TG'KYD(at-sign)16G'p`QP11Efa
[ChQ6Gj!!V(*PNf&bCC0KBTK[GA56G'q6C'8-EQ(at-sign)6DA16G'KPMU'0N5H!!'&XCf9
LFQ'(at-sign)"2MREfD6CQpbE(at-sign)&XNdaKGA*PETUXFR56Ff9bD(at-sign)9c,T%)Bje8D'PcNh4[F*!
!8ij[E'pRD(at-sign)0KE*0KE'GPBR*KNfeKQ(Q6BT!!8ijPNf4P$'jPC)kKMC%RJ!"LN!#
XFRQ(at-sign)!lp$C'8-EQPZCj0KNh4[F*!!8ij[E'pRHC0[ET0dD'(at-sign)6B(at-sign)aRC(at-sign)*bBC0[CT0
bBA4TEfjKE*0QG(at-sign)jMG'P[ER16D(at-sign)k6G'KPNhD4reMNBA*TB(at-sign)*XCC2AH,NXMU'0N5H
!!'&ZC*B$8KGdD'9ZNf0[EA"XCA4TEQH6G'KTFj0KE'GPBR*KNh*PE'&dDAD3!+a
bCC0dEj0dD'(at-sign)6G'p`QP11Efa[ChQ4r`9(at-sign),T%&"J98D'(at-sign)6G'p`Q'pXEfGjNfPcNh0
[MU'0N5H!!'1DV(*SEh0PETB%&3pKFj0dEj0SBCKfQ'(at-sign)6efb3!%12D(at-sign)f0R`(-c0K
ZfL%aMT%(at-sign)U'$AH)fIqk8bfJ$BESk4$q*'Z6f4!jeq-#k4"EJ94ADBCA*jNf9XC(at-sign)e
PETKdNfpQNh4SCC0MEfe`E'9dC(at-sign)56B(at-sign)aRC(at-sign)*bBBkKMC%RJ!"dGA*ZFjB$kUK[GA5
6G'q6BT!!8ijPNf'6CQpbE(at-sign)&XNdaKGA*PET!!V(*dNh0PFQPPFbb6G'KKG*0TFbb
6BC0cCA*TCA16EfD6G'KPNfC[FQf1SD'0MBf0NJ$8N!$HRrArr+YBMSkI$$MRMC)
!dlH+f'imC)k1NJ$PfrMABBfI!Fc-f'k1PJ(at-sign)S80GiMCrl#jRBESk6ecU1Raf2Jif
4*i!!Z9H4r`9(at-sign)CCX&!3ehPDabB(at-sign)k6G*KdEjK`P911CA*QEh*YQ'&ZQ'&ZB(at-sign)a[Cfp
eFjKMEfe`E'9dD(at-sign)pZQ("bEj0MCA0cQ'pZQ'&ZEh4SCA+BB(at-sign)aRC(at-sign)*bB6U1SBf4*i!
!G'KPPJ3!8f&XCf9LFQ'6Cf9ZCA*KG'9NNf+DV(*jNf&XE*0QG(at-sign)jMG'P[ER16EfD
6G'KPNfC[FQf6ehL0RrZP-YKZMT%&U&#j+0GXN!"$MfpRN!"Z1ALj+BfIqk8bf(5
1N314bVNXN33&[AGSCA*PNpGZNlPTFj0KETKjMU'0N5H!!'PZPDabG'9RCA)XQ`2
UU(#3!&11Eh0TG'PfNf(at-sign)BEh+BEQ9RBA4TGT0P,*KKEQ5BGfKPFQ(at-sign)Beh5BZ(at-sign)PcQ''
BEQpZQ'jPCf&dDAD6CCKTET0dC(at-sign)GPFLk1SBf4*i!!5(at-sign)k(at-sign)!i(at-sign)SEh*NCA+6G'q6Fh#
3!&11C(at-sign)0TCRQ6GfKTBjUXFQL6C(at-sign)aPE(at-sign)9ZQ(4cNfpQNh4SDA16B(at-sign)aRC(at-sign)*bBC0KFQ(at-sign)
6G'q6BfpZQ(DBCA*RCC0dEj0kCA*[,*%$QGYhQ'(at-sign)1SBf4*i!!EQ9PC*B%aVKKNf+
D8ijPG(4PFT0LQ'9SBC(at-sign)XFRD6C(at-sign)5(at-sign)"-DiBQ&cDA16EfD6G'KTFj0KE'GPBR*K,T%
(c3p8D'PcNf*KFfPcNfPcNh"bEjUXFRCTC'9NNf+BHC0dD'(at-sign)1SBf4*i!!D'&bE(at-sign)p
ZD(at-sign)1(at-sign)!qUSE'pRBA*TG'KYFj0[CT0KFQ*TG(*KFRQ6Eh*NCA+6eh5j,T%&11"8D'9
jNf&bCC0NC3aZC(at-sign)56BA16CQpXE'q3!+abGh-kMSkI(J!!MC)!qQpJ06(at-sign)1MSb,!!!
!1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#2(LJ!U!SpBf
JrBEA#k!#(at-sign)bMeMD$pbGF,T!k!!)f5!,"ij0F9MCrl#jQ0e5MBG08TMTm(kpU0f'k
1MT%+,N+j+0GiZ5Q(at-sign)!e952C0EefkjA5(A4)f3!&11RrX,QGKZMT%&qpkj+0GXN!"
$MfpRN!"Z1ALj+BfIq`ZCf(51MUN9!!#0N5H!!,PQEh+(at-sign)"1C&CADDV(*PFRQ6EQp
ZEQ9RBA4TGTKPNfPZQ(4PCf9bNpGdNlPKEQ56CQpbNf9fQ'9bHC0TETKdC(at-sign)GPFT2
AET1j,T%)+lC'NIm&9QpbNf9iB(at-sign)e`E'8XN38P,(HBCBkKMC%RJ!"SBC(at-sign)XFRD6CBk
KSBf5!-YU'pF9MCrl#jQ0e5J`+BkI"q[DMGKZMSk4#eamZ5MAH,NTPJ098Mf6ehL
0RrX,QGKZMSkQMC%RJ!#jCQpbPJ2UU'9fQUabCA*jNfj[EQjPCf&dDADBCC0TETK
dC(at-sign)GPFT2AEVNXNf&ZC)kKSBf5!-kNlYF9MCrl#jQ0e5J`+BkI"q[DMGKZMSk4#ea
mZ5MAH,NTPJ098Mf6-)kQMC%RJ!"QEh+(at-sign)!qUSEQ9RBA4TGT!!V(*PNpGZZ5k1RaU
!!)f4*i!!4AK`E'PMDA5(at-sign)"3)!CAK`FQ9cFfP[ER16BA*PNfYZEj!!V(*hET0QEh+
6G'KPNfKKFQe[EQPMNfa[Cf&bDA4SEA-ZN3KqjdD4r`9(at-sign)Eh+6G'KPNfKKFLf1SBf
4*i!!E(at-sign)pZD(at-sign)1(at-sign)!qUSE'pRBA*TG'KYFj0[CT0[FQ4PFT-bNhHDV(*PNfKKQ(DBCC2
A&BfIqGhIMG8S-LQ1R`M"Cid`MSk4#eamZ5MAH,NTPJ098Mf6+0GXN!"$MfpRN!"
Z1ALj+BfIqk8be6+1N3LUV,PKEQ5(at-sign)!qUSCQpbNpGZNlP`N!"6MQpcDA4TGTKPMU'
I(0RfMC%RJ!$A&BfIq`ZCMG8S-LQ1R`IVfShBESk1N3YFI,NSehLj+CB$99)pNpG
iMCrl#jRBESk0MBf4"DK3Z5L1MSk4#MS8+0GXP812EfHDEMPiZ5Q0RrX,QG8bMT%
%`!68!,NS-LZ0MBfIpq(&MC%"-c-bMT%"-c1I"9&YL3!!CQ8!"GrmR`X)#idbMSk
1MT%)4Q)VMGFkPJ(rrMU61Sk4$FUJZ5Z0MBfIpq(&MC%"a2SbMT%"-c1I"9&YL3!
!CQ8!"`1*R`X)#ihAESk1MSk4#(at-sign)R[Z5RAE*0[CjKiZ5Xb+if0MCrhiF(at-sign)0N3%c-c+
1N3%c-jm&8(at-sign)f*!!"QC3!&hrbI#`J,M6+1MSk1PJK'BLJa+if0MCrhiF(at-sign)0N3%c-c'
1N3%c-jm&8(at-sign)f*!!"QC3!&hrbI#`J,M6+1MSk1NbNVMGFkPJ(rrMU61SkE$FUJZ5Z
0MBfIpq(&MC%"a2SbMT%"-c1I"9&YL3!!CQ8!"`1*R`X)#ihAESk1MSk4#(at-sign)R[Z5J
a+if0MCrhiF(at-sign)0N3%c-c'1N3%c-jm&8(at-sign)f*!!"QC3!&hrbI#`J,M6+1MSk1NbZ0ecU
(at-sign)!Irq1T-kMTLj+if0MCrhiF(at-sign)0N3(%qM'1N3%c-jm&8(at-sign)f*!!"QC3!(!iQI#`J,MGG
ZMSk1MT%*DHqj+Bf0M5Q1MSk4"*(%ecZ1Raf3!*+0N5H!!,PKEQ51SCm3fID0NAN
TBGF9MCrjhGq0e5Jb+BkI#'lXMGS!f'k1MTX-4-Lj+0GiZ5Q(at-sign)!e952C-behL0RrX
,QGS!f'k1MBf0Q,NSMSk1N4$(at-sign)M0GXN!"$MfpRN!"Z1AL(at-sign)!UUSe!#6Z6'6e!#0MBf
Ipq(&MC%$hGZj-Bk4!phER`94EBN!!'CP!!AIr*m,#!Z0-Sk1MSk4$CZbe!#0NpF
kPJ(rrMU61Sk4%ar`e!#0MBfIpq(&MC%1Y21j-Bk4!phER`94EBN!!'CP!"Z1,*m
,#!Z0efk6e!#6Z6'1MSk1MBf0N5#I1LQ1MSk4*6$qecU1RaN3NSf4*i!!Z8D4r`9
(at-sign)Eh+(at-sign)"(Z5CADDV(*PFRQ6EQpZNfjPCf&dDADBCC0TETKdC(at-sign)GPFT2AG*1jG'KPNfK
KFQe[EQPMNfa[Cf&bDA4SEA16EfD6Eh*NCA+6eh56ZA0KG'PcCRQ1SBf4*i!!G'K
PPJ4Bf(0KE(at-sign)(at-sign)6Cf9ZCA*KE'PkBA4TEfk6EfD6G'KPNf*TEQpYD(at-sign)&XNh4SC(at-sign)pbC(at-sign)f
6G'KKG*0hQUabCC0SBCKfQ'(at-sign)6B(at-sign)abC(at-sign)&NHC0cC(at-sign)9ZMU'0N5H!!'C[FTB$kUKdD'(at-sign)
6D'&bE(at-sign)pZD(at-sign)16E'pRBA*TG'KYFj0[CT0[FQ4PFT-a1SkKRa-!#Bf5!*U[rpF9MCr
l#jQ0e5MBG08TMTm(kpU0f'k1MT%+,N+j+0GiPJ+UU,NVNpGKZ5Q(at-sign)!e952BfIm[r
pMC%'e9AD-BkI$3!$MBf4!paeRrArr+YBMSkI$$MRMC2BDj!!0SA926#1MSf0MC%
8e9kIlQCFUb+1MCrhiF(at-sign)0N4UUYGGZMTm3(at-sign)AQ0N4VUY'Z1MSf4)DiqRqjQA+XMMSk
1N5H$PGGKMCrl#jRBDik4"515ea(at-sign)0RrRGhih9+0Kde5Q1R`NRcBhBEYS!f'Z1MT%
3k&Uj+0GiZ5RA1SkI(3ilMC%RJ!#j9'KPPJ-DJQKKFQe[EQPMNfa[Cf&bDA4SEA1
6BA*PNf'6BQ&cDA16EfD6G'KPNf&XCf9LFQ'6Cf9ZCA*KG'9NNf+3!+abHC0KE'b
6CR9ZBh4TEfjcMU'0N5H!!0GiMCrlT6,BESk4"DK3Z5MAE*!!3ip[Cj!!EMPiZ5Q
0RrZP-YKdMT%$NFUj,T%&11"ANIm&9Q(at-sign)(at-sign)!qUSC'8-EQ(at-sign)6BC0dEh#3!&11Efa[ChQ
6Efk6G'KTFj0KE'GPBR*KNf+3!+abHC0bCA&eDA*TEQH6G'KKG)kKSBf5!,H61pG
XN!"$MfPYMCm"c-cBEYSK!$'1N4e%f0F9MCrl#jQ0e5MBG08TMTm(kpU0f'k1MT%
+,N+j+0GiZ5Q(at-sign)!e952C-`MUD0N5H!!'C[FTB&PKaPGTUXFQ9bHC0ZEfk6EQ9RBA4
TGTKPNfPZQ(4PCf9bNpGdZ5k4#MXm9'KTFj0dEh#3!&11Efa[ChQ6DA16Bf&XE'9
NNh4SCC0XEfGKFQPdD'eTBikKMC%RJ!"dEh#D8ij[E'pRHC(r"9BZN3FS%&4SCCB
%MlKMEfe`E'9dD(at-sign)pZNfpQNh4SDA16B(at-sign)aRC(at-sign)*bBC0bC(at-sign)aKG'PfN!#XFQ(at-sign)6G'q6G'K
PNfa[Cf&bDA4SE(at-sign)PMNh4[F*K[E#f1SBf4*i!!EfGjPJ3)"fPcNh4SCC0KE'GPBR*
KNfpQNfC[FQeKE*0`QP11Ej(at-sign)XFRH6CA+(at-sign)"!J(Ff9bD(at-sign)9cNfpQNfa[Cf&bDA4SE(at-sign)P
MNh53!+abHA#BC5b4"!pIEh+6E'pRBA*TG'KYD(at-sign)11SBf4*i!!B(at-sign)aRC(at-sign)*bB5k1RaU
!!)f4*i!!4AD9V(*PFRQE!V6AC(at-sign)aPE(at-sign)9ZNh5BEfDBG'KPQ'a[Cf&bDA4SE(at-sign)PMQ'&
XCf9LFQ'BDA1BBCKXD(at-sign)jPBA+BBfpYNf*TEQ&dD(at-sign)pZQ'pQQ'0[ET0fNf9bCf9ZNh5
1SBf4*i!!F*!!8ij[PDabGj0PFTB$kUKcCA*TCA16EfD6G'KPNfC[FQf1MTmH!!#
0NJ$kEf!e0Sk1M)X!!!!j!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!**[+!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*e`ZT$S!!MC)!XZBhefD4!8IrZ5M
AH,NTPJ098Mf0MBf4"ZA&RrArr+YBMSkI$$MRMC2BG$YZfK6BC)k1N4VRrYGLMCm
"c-cBEMYdMT%,&SlA&BfIq`ZCMG8Sf(69+BkI"q[DMGKZMSk4#Lj#Z5MAH,NTMTm
HR(50N5H!!(*KEQGTEQHE!qUSEj(at-sign)XFRD6CA+BBCJ-EQPdCCKcCA5BEfDBGT(r(at-sign)14
KE(9PFjK[CTMAG,NZMU3DJ!#0N5H!!&H4r`9(at-sign)CCB$kUKMB(at-sign)k6EQqDV(*hNh*PG(9
bET0dEj0dD'(at-sign)64A9XCA)Y6(at-sign)&M6'&eFQPZNh0eE(at-sign)eKG'P[ET0QEh*YQ(9XB6U1SBf
4*i!!de4SC(at-sign)pbC(at-sign)fj,T%&&5T'NIm&9QpbPJ0rKf9fQUabCA*jNf9XC(at-sign)ePETKdNpG
QN3&(rlNSehLj+C0[CT0dD'(at-sign)6E'pRBA*TG'KYD(at-sign)16B(at-sign)aRC(at-sign)*bBC0dD'(at-sign)6FQPRD*K
dNfKKEQ51TSf4*i!!FfPNCCB$kUK[CT0dD'(at-sign)64A9XCA)Y6(at-sign)&M6'&eFQPZNh0PFQP
PFj0MEfk9V(*fNf9bCf9cPJ2UU'PZNh4SCC0XEfGKFQPdD'eTBj0dEh#3!&11Efa
[ChQ4r`9(at-sign),SkKMC%RJ!"'NIm&9QpbPJ9iMf9iB(at-sign)e`E'8XN3AF#A4SCC0QEfaXEjU
XFRGTEQH6D(at-sign)i-EQPdCC0cCA*TCA16DA16BfpZQ(DBCA*RC(at-sign)kBG*0TET0dD'(at-sign)6E'p
RBA*TG'KYD(at-sign)11TSf4*i!!G'p`N!"6MQpXEfGj1SkQTSf4Ce9CefbD3ip[Cj!!EMP
iPJ+UU,NVNpGXQ'pRN!"Z1ENSehL6Z5Z6-5Q6+j2AE*K[Cj!!EMQj+0GiNlNVNc)
TNbZ0NpFkPJ(rrMU61Sk4%ar`Z5Z6efbBEfH3!'ijZ5MAH*1j+j2AEVNTN3098Mf
1Ra8!!)f4*i!!ed+0R`(-c08`MT%%`!5j+#MAH*B#D1Hj+j2AET1j+j-a+GGXQN1
2EfH3!'ijZ5MAH*1j+j2AET1j+j-a+C28!*2AH'bBEfH3!'ijH*28!*2AET28!*1
j-5Q6+j2A3SfI!Fc-e6'1N36!",NSefbBEfH3!'ijZ5MAH*1j+j2AET1j+j-a+C2
8!*2AE*K[Cj!!EMPiZ5NVMTmDd(at-sign)k0MBf0RrIKaBf5!+HhRpG#MCm"c-c9-Sk1NJ#
RYjqI"9&YL3!!CQ8!$DdQR`X)#if4!N62Z6)KMSk1MT)!YTIi+)f0MCrhiF(at-sign)0N4D
%H6'1N3%c-jm&8(at-sign)f*!!"QC3!`JSHI#`J,MGGiPJ+UU,NVNpGZNlNVNc'1MSk1N6(at-sign)
6PG3!MBf0RrIKaBf4"%5'Z6'1N32Gfjm&8(at-sign)f*!!"QC3!'V9+I#`J,MGGiMSk1MT%
,[Q#j+CB#UUJVMC2A1TB"rrikNcU1N4"e5,NTecU1Rb6l1Sf4*i!!Z8&ZEh4SCA+
(at-sign)!qUSCAKKEA"XCC0TFj0dD'(at-sign)6CQpXE'qDV(*hD(at-sign)jR,T%&11""Fj0jQ'peNfYZEjK
h,*0dD'(at-sign)6Fh9YMUDQMC)!JMVSehL0RrX,QGKVMTX(cMUj+jB#UUJSehL6Z5Z6-5Q
0RrX,QGKVMTLj+j-SehL6Z5Z6-LQ0RrX,QGKVMTLj+if6ecU(at-sign)!Irq1T-kMT%6(r#
j+j-SehL6Z5Z6efkj+BfIq`ZCf'Z1MTm9!!#0N5H!!,PMB(at-sign)k(at-sign)")2+BT!!8ijPNf9
iF(*PFh0PC*0TET0ME'pcC(at-sign)56CQpbEC0LQUabHC0dD'(at-sign)64A9XCA)Y6(at-sign)&ME'&eFQP
ZNfC[FQfBG(at-sign)aK,T%("%98D'(at-sign)6F(*P,BkQMC%RJ!"MC(at-sign)4TEQH(at-sign)",S5G'KPEh*PEC0
XC(at-sign)&NFj0dEj0KEQ&XEfG[GA16Bfa[Ff9NNfC[FQf6CAK`FQ9cFfP[ER16CQpbNh0
eEA16EfD6G'KPMUD0N5H!!'C[FQf1TU'0N5H"jpGiMCrl#jRBDikE"515efb3!%1
2EfH3!'ijH*B#UULj+j-SehL6Z5Z6-5Q0RrX,QGKVMTMAE*!!3ip[Cj!!EMQj+0G
iNlNVNc%TNbZ6+0GiNlNVNc)TMCrl#jRBDikBefb3!%12EfH3!'ijZ5MAH*1j+j-
b+C-VMC2A1TB"rrikNcU1N4-Im,NVNbMAH*1j+j2AEVNTMCrl#jRBDikBefb3!%1
2EfH3!'ijZ5MAH*1j+j2AEVNTecU1SBf4*i!!Z94SCCB$B$KSBA*YEfjTBj0XEfG
KFQPdD'ecNfKKPDabGT0PPJ0J1'pdD'9bNf&`F'aTBf&dD(at-sign)pZFbb4!h[SE'9dNh9
cNfePET!!V(*dD(at-sign)pZNfpZCC0TET0ME'pc,BkQMC%RJ!"TEQFZMU'0N5H!!&*PBf&
XE*B$SlPdD'(at-sign)6C'8-EQPdD(at-sign)pZNfpQNh4SCC0cD'PQG*0[F*!!8ijPFQ&dEh+6EfD
6G'KPNf0KE'0eE(9cNfpQN`aZDA4PNf4T#f9bC(at-sign)jMCA-kMUDQMC)!Z)aaed(at-sign)0N3#
d&jrl#jRBBBk4"EIQefDE!8IrZ5MAH,NTPJ098Mf6efDBZ5MAH*B#UULj+j2ABEN
TecU1T"8!!)f4*i!!Z8D4r`9(at-sign)Eh+(at-sign)!qUSefk6Z(at-sign)'6EQpZNfjPCf&dDADDV(*PNfP
ZQ(4PCf9b,*0NC3aZCC0dD'(at-sign)6Eh#3!&11CA*KG'pbNpG&MCm"c-c9-Bk4#+UXZ(at-sign)&
cNfC[E'a[Q(Gc1SkQTSf5!,Jee0G&MCm"c-c9-Bk4"-!%ea(at-sign)0RrX,QBh9+$!TMTm
(kpU0f'k1MTX,A(bj+0GiZ5Q(at-sign)!e952C2A&BfIq`ZCMG8S-5Q1R`IVfShBESk1Q,N
SehLj+GFkMU'0N5H!!,P*ETB$kUK[FQ4TEQ&bHC0ZEh4KG'P[ELb6G'KTFj0TFj0
dD'(at-sign)6Ff&YCC0KFj0cBC!!V(*jD(at-sign)jRMSkI(J!!MC)!qQpJ06H1MSb,!!!!1J!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#9$5J!U!SpBfJrBEA#k!
#(at-sign)bMeMD$pbGF,U3k!!)f4G-),ed(at-sign)0R`(-c08aMT%%`!6AH)fIq`ZCf'k1N3MpSVN
pN3098YGiMCrl#jRBESk4"DK3Z5MAE*!!3ip[Cj!!EMPiPJ+UU03!NlNaNp3!NlN
aecfj-T28!*1j-GFpZ616e!#0NpFkPJ(rrMU61Sk4%ar`e!#6Z6(A2(at-sign)kj+GFkMTm
8kh10N5H!!,P2EQ(at-sign)(at-sign)!qUSBf&ZNh"bEj(at-sign)XFRD6CCB$kUKdD'(at-sign)6CQpXE'qDV(*hD(at-sign)j
RNh5BGjK[Nh"bEh#3!&11Eh0TG'P[ER-kMU3DC*Q0N5H!!003FQp`N!"J!'pcDA4
TEfkj,T%&11"8D'(at-sign)(at-sign)!qUSEh#3!&11CA*KG'pbFj2A4Bf4!,3ARrZP-YKKMT%*SSk
jB(at-sign)jNNpG&MCm"c-c9-Bk4#+UXZ(at-sign)0[E(at-sign)f3!+abGA4P,SkKMC%RJ!$68(*[F*!!B!"
[FfPdD(at-sign)pZZ5k4"3Ae9'KPPJ04jh*PFh4bD(at-sign)0dD(at-sign)pZNfpQNh4SCC0NCA*TGT(r(at-sign)14
KG'PfN!#XFQ(at-sign)6Eh#3!&11CA*KG'pbNpG%N31PGEPdEj0dD'(at-sign)6Fh9LB(at-sign)aRC(at-sign)*bBBk
QMC%RJ!"[CTB%,IKdD'(at-sign)6E'pRBA*TG'KYD(at-sign)16B(at-sign)aRC(at-sign)*bBC0RC(at-sign)jPFQ&dC(at-sign)56BT!
!V(*jNh4SCC0SBA*YEfjTBj0XEfGKFQPdD'ecNpF9MCrlT6+0e5MBG08TMTm(8N'
0f'k1MT%+,N+j+0GiZ5Q6CQpbMUD0N5H!!(#3!&11Eh0TG'PfQUabCCB$kUMAG*1
j+(4SBA56DA-XNf9iBfaeC'PZCj0dD'(at-sign)6EQpZNfjPCf&dDADBCC0`N!"6MQqBGjK
PFR16EfD6ehLj+C0TFj0TETKfQ'9bG'PLE'8ZMU'0N5H!!&4SCA0PQ`8PYA59V(*
hNfqBF(*[F*96MQpcDA4TEfjcQ'0KETKLNf(at-sign)BGA0PC*KdEjK[BR4KD(at-sign)kB)Qa[Cf&
bDA4SE(at-sign)PMQ'9iG'9ZFfP[ER-LQ'pQMUD0N5H!!(0`N!"6MQ9MD(at-sign)&XPJ-aKQCeEQ0
dD(at-sign)pZFbk4"2XU6'9dNh9cNf0[EQ0XG(at-sign)4PNhGTG'L6G'KPNh0TEA"XCA0dNf9iB(at-sign)e
`E'8kN36F6faPG*0eFj0MEfe`GA4PMUD0N5H!!(4SCCB&F`0XEfGKFQPdD'eTBj0
PH(4PER0TEfk6EfD6G'KPNh0PFA9PEQ0PNfpQNfa[PDabGj0PFT%&F`0QB(at-sign)0dEh*
TB(at-sign)ac,*B&e4TZB(at-sign)ePE(Q4r`9(at-sign),*0dD'(at-sign)1TSf4*i!!F*!!8ij[E(PZEfeTB(at-sign)acQ`+
Ll#MAH,NTMCm"c-cBESk4#2fLZ6f4!e95ehLj+0GiP3dre!#6Z6%T+0GiNp3!NlN
b+Bf(at-sign)!IrqecU61T-kMT%4bTbj+0GiNp3!NpGZNlNVNc%T,T%%bk&8D'PcQ(0PFA9
PEQ0PQ'pQQ(#3!&11EfajEQpYD(at-sign)&XFikQMC%RJ!"cBA4TF`aPFjB$kUKdD'(at-sign)6C'N
,CA*PEQ0PNf9aG(at-sign)&dD(at-sign)pZMUDQMC)![DLQ!5MAH,NTMCm"c-cBESk4#2fLZ6f4!e9
5efkj+0GiZ5Q0R`(-c0KZfJ$9-Bk4%)6-ecZ1T"6VFif4*i!!ZAGSCA*PPJ2UU!'
6DA16G'KPNf4T#f9bC(at-sign)jMCC0[F*!!8ijPFQ&dEh)kN38ii!(ACTX"4rqj+0GiZ5Q
(at-sign)!e952C2ACTLj+0GiPJ+UU,NVNc%TNp3!NpGQQ,NSehLj+GFkMUD0N5H!!,P8D'P
cPJ2UU(0PFA9PEQ0PNf0KET0LN!"6MQ(at-sign)6CAKdC(at-sign)jNC(at-sign)56G'q6EQ9RBA4TGTUXFQ(at-sign)
6efk6Z(at-sign)+BHC0cCA4dD(at-sign)jRMUDI%0RfMC)!PJ1$+0GiZ5Q0R`(-c0S!f'k1N3qD'VN
pMBf0RrIKaBf428iP-Bk4")L&R`94EBN!!'CP!(GV1jm,#!Z0+0GiPJ+UU,NVNc%
T+0GiNlNVNc)TecSk1VNSehL6Z5Z6efkj+Bk1MSk4I5EcecZ1RaVm"Bf4*i!!Z(at-sign)&
ZC*B$kUKhQUabCC0SBCKfQ'(at-sign)1TUD0NJ#bBB)"+0GiZ5Q0R`(-c0S!f'k1N3qD'VN
pN3098Y3!efkj+0GiZ5Q0R`(-c0S!f'lD!08aMT%A)86A1SkKMC%RJ!#j4T(r"9C
[FTB%+1*`N!"6MQpcDA4TGTUXFQ(at-sign)6efkj,*%%1("hQ'(at-sign)6E(at-sign)'BHC0NC3aZCC0dD'(at-sign)
6E'pRBA*TG'KYD(at-sign)16CAKdC(at-sign)jcD(at-sign)pZNfpQNh4SDA16Ff9aG(at-sign)9ZBf(at-sign)6BTKjMUD0N5H
!!(0PG(4TEQH1TU33fID0NJ#!%C`SehLj+BfIqGhIMG8S-5Q1R`KZl)hD!0KZMSk
(at-sign)$jSDZ6f4!e95+0GiZ5Q0R`(-c0S!f'k1NlNpMBf0RrIKaBf428iP-Bk4")L&R`9
4EBN!!'CP!(GV1jm,#!Z0+0GiPJ+UU,NVNc%T+0GiNlNVNc)TecSk1VNSehL6Z5Z
6efkj+Bk1MSk4I5EcecU1Ral)dSf4*i!!Z8D4r`9(at-sign)Eh+(at-sign)!qUSCAKKEA"XC5b6+0G
iZ5Q0RrRGhih9+$%TMTm)`(at-sign)H0fJ$9-Bk1N3kacVNpMBf0RrY'GBf4#MXee6'1N35
)KCm"l,f*!!"QC3!2T(at-sign)1I"[!RMGKie5XaMSk1MT%9B4Zj,SkI(IpXMC%RJ!"8D'(at-sign)
(at-sign)"KAdC(at-sign)aPE(at-sign)9ZN!#XFR4cNbMAH,NTMCrjhGq0e5Ja+BkI#'lXMGS!f'k1MT%5(at-sign)Vb
jBTT6MQ9XEfjRNh4[Nh4SCC0cG(at-sign)*YEjKNG(at-sign)aPNfpQNh4SCC0XEfGKFQPdD'eTBj0
KE'GPBR*KMUD0N5H!!(0`B(at-sign)jZC(at-sign)5(at-sign)"1)QBTUXFRQ6ea(at-sign)0RrZP-Sh9+$%TMTm(8N'
0f'k1MT%,A(bj+0GiZ5NXN38J"Q&cNpGZNlPbB(at-sign)jRCA16EjKfQ'9bNf&XE*0TETK
dC(at-sign)GPFR-ZN3JI(at-sign)dpZNh4SDA16Fh9LE(at-sign)q3!&11C(9XC5b4"5!'G'KPMUD0N5H!!'p
`N!"6MQ9bBA4[FTB$kUJ"NfPcNfPZPDabGT0PFR4TBQaP,*B$kUKKEQ56Gj!!V(*
PNf0KET0dD'9bC(at-sign)C[FQ(at-sign)6Ff9dMUDKMC)!YGB&+0GiZ5Q0RrX,QBh9+$%TMTm(kpU
0f'k1MT%1XFkj2C%$99)"MCrl#jRD!0KZfJ$9-Bk0MBfIpq(&MC%Ljibj-Bk4'&4
hR`94EBN!!'CP!"X'*Tm,#!Z0ehL(at-sign)!UUSZ5Z6-Bk1MSk1RaMQVBf4*i!!CQpbPJ2
UU'&XE*0ZEfk6EQ9RBA4TGTUXFQ(at-sign)6D(at-sign)kBG'9RCA*cNpGZZ5k1MTmH!!#0NJ$kEf!
e1)k1M)X!!!!l!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*
FfD!#S#MeMD$pKYF,S!*E+2(at-sign)0S2h*q5b0N5H!!,P*G*B#kaadGA*ZFj0[GA56G'K
KG*0dD'(at-sign)6C(at-sign)aPE(at-sign)9ZQUabG*-SehLj+BfIqGhIMG8S-5Q1R`M"Cid`MSk4$NHBZ(at-sign)P
cNfGTGTKPET0LQ(Q6G'KPNfC[E'a[Q(GTEQH6Ff9bD(at-sign)9c,*%$(MKMEfkBGTKPFQG
PETKdMU31J!#0N5H!!'PZPJ2UU(4SCC0XEfGKFQPdD'eTBj0dEh#3!&11Efa[ChN
kMU'I%9&ZMC&8S'NSehLj+BfIqGhIMG8S-5Q1R`M"Cid`MSk4$V(1Z6f4!e95efb
3!%12EfH3!'ijZ5MAH*B#UULj+j-a+C-VMBf0RrIKaBf4#STEed+0R`(-c08aMSk
4!phER`94EBN!!'CP!"X'*Tm,#!Z0Z6'6+j2AH)k1MSk4)X(Fe!#0MBfIpq(&MC%
8E"rA3SfI!Fc-e6+1MT%$hGZI"9&YL3!!CQ8!,XQZR`X)#ifj-LJaNbZ6ehLj+Bf
Ir)L*e6+1MSk1MTXfK(at-sign)5j+if0MCrhiF(at-sign)0N44X(pG#MCm"c-c9-ik1N32Gfjm&8(at-sign)f
*!!"QC3!ZbDkI#`J,MENc+$'6+j2AH,NTMCrmL)R9-ik1MSk1Q03!MC2A1TB"rri
kNcU1N4*e4MU1RaX3NSf4*i!!Z8*eG*B%H8"dD'PcNfPcNf'6CQ&YD(at-sign)aTBA+6Ef+
4!+FFDQ9MG$U4"PB3DA56DA16G'KPN`NYNfCeEQ0dD(at-sign)pZ,*%%R1CSCA9bDA0dD(at-sign)0
KE'ajNfPZN!#XFR4bEj!!8ijNG(at-sign)0PC)kKMC%RJ!"LQUabHCB&*[9(BA9cFbk4#1h
)4f&eFh16E(at-sign)pdDAD4reMNBA4PC*0dD'(at-sign)6#5eQG(at-sign)jMG'P[ET0KFj0dD'(at-sign)6)R*TCfL
BG#+6FfpXGA4TEfk6EfD6G'KPMU'0N5H!!'4T#f9bC(at-sign)jMCC%$kUKPFA9KG'P[ESk
KRa$CpSf5!,Ddm3%*+0GiPJ+UU,NVNc%TN3098Mf0MBfIpq(&MC%2'jSaMT%%L)(at-sign)
I"9&YL3!!CQ8!'`BQR`X)#ihAH*1j+j-aMSk1MT%J`GlA1SkI'2XkMC%RJ!#j9j(
r"9CPQ`3l!QKKPDabGT0PQ'j[NhHBFQPREh*[GA0XHCKfNf9bD3aPC*K(BA9cFbG
cQ'GeCA0c,T%'+Hj'NIm&9R9bG'KPFTKMEfe`GA4KG'P[ER1BFfK[NhH1SBf4*i!
!G'KKG*B&(YGdD'(at-sign)6C(at-sign)aPE(at-sign)9ZN!#XFR4cNbMAH,NTMCrjhGq0e5Ja+BkI#-&RM6'
1MTX3He1jB(at-sign)jNNbMAH,NTMCrjhGq0e5Ja+BkI#-&RM6+1MTLjB(at-sign)acEj0MEfPZBfP
NCC0hDA4SNh0`N!"6MQ9MD(at-sign)&XNfCeEQ0dD(at-sign)pZFj0TELf1SBf4*i!!G(*[N!"6MQ4
eBf9NPJ8S2Q+DV(*jNdGKGA0c,*B&Gk4ZB(at-sign)ePE(Q4r`9(at-sign),*0dD'(at-sign)(at-sign)"5JqC'PRB(at-sign)e
YBC0KEQ56G(*TCf&YE(at-sign)'6CR9ZBh4TEfjc,*%&Gk4hD'PMQ'L1SBf4*i!!BA*PPJ2
c$f&dNfaKFh56FQPREh*[GA0XHC0NC3aZC(at-sign)56BTUXFRQ6D(at-sign)i-EQPdCC0cCA*TCA1
6BfpZQ(DBCA*RC(at-sign)kBG*0TET0dD'(at-sign)6E'pRBA*TG'KYD(at-sign)11SBf4*i!!G'p`N!"6MQp
XEfGjNIm&9Lk1U4U!!)f4*i!!5(at-sign)k(at-sign)"-D[BC0cD(at-sign)eTE'&bNhD3!+abC(at-sign)PZ,*%%rE&
[EQ(at-sign)6C'8-EQ9cNfa[Cf&bDA4SE(at-sign)PMNf9iG'9ZFfP[ER16EfD6G'KPNd*PFQj[G(at-sign)a
XDC0`N!"6MQpXH5f1SBf4*i!!EQpYD(at-sign)&XFbbE"2[CG'KPPJ6&0NKPFQeTG'(at-sign)6F*!
!8ij[E(PZEfeTB(at-sign)ac,*KPG'-Z,*KKEQ56EfjPN`aZC(16G'KKG*0dD'(at-sign)6BA0jEA"
dEh4TBikKMC%RJ!"PH("KER0TEfjcPJ5'p'pQNh4SCA0PNh#D8ij[E(PZEfeTB(at-sign)a
cNh*PBA"`Q'9KFT0ZBA4eFQ&XE(Q6BA16E(at-sign)9YN!#XFQ+BCA*cNfpQNh4SCC0XEfF
YMU'0N5H!!'&bDA4SE(at-sign)PMPJ3ed(at-sign)9iG'9ZFfP[ER16EfD6G'KPFf(at-sign)6CR9ZBh4TEfj
c,T%''PY"Fj0KNfeKG(4PFT0[CT0QB(at-sign)0d,*%%5*YdD'(at-sign)6E'pRBA*TG'KYD(at-sign)11SBf
4*i!!G'p`N!"6MQpXEfGjPJ0mN!"KE'a[QUabGh16GA16G'q6FQ9`E'&MCC0KFhP
YF(4[G'PMNf9iF'&ZFfP[ER16BTKjNh0PFQPPFj0hD'PMQ'L6BA*PNf0[ELf1SBf
4*i!!GT(at-sign)XFQ9bCf9ZNh5(at-sign)!qUSD(at-sign)k6G'KPNfa[Cf&bDA4SE(at-sign)PMNh4[F*!!8ij[E'p
RHC(r"9BZMUD0N5H!!%PZPJ2UU'0XEh0TEQFXNh59V(*hNfq(at-sign)!qUSEh#D8ijPET0
`FQpLE'9YFj0YBC!!V(*jNf+BCC0YC(at-sign)k3!+abG'P[EQ9N,SkQMC%RJ!"'DA*cG#b
4!edFEQq(at-sign)!cQjBfa[Ff9NNfC[FQf6CAK`FQ9cFfP[ET0TFj0VEQqDV(*hET0QEh+
6G'KPNf0[N!"6MQ81BfPPETKdFj0[CT0dD'(at-sign)6CAK`B(at-sign)jcD(at-sign)pZMU'0N5H!!'pQPJ2
UU''6F(*[N!"6MQ4eBh51SD'0NJ$*1!AA&BfIq`ZCMG8Sf(69+BkI"q[DMGKZMSk
4#Lj#Z5MAH,NTea(at-sign)0RrRGhih9+0Kce5Q1R`NRcBhBDik1N3X+Y,NSehLj+BkI&3!
!MC%RJ!"TETUXFR4[PJ3&&''6E'pRBA*TG'KYD(at-sign)16F*!!8ij[Q(HBCA+6Ff9bD(at-sign)9
c,T%&L#06C(at-sign)0[EQ3XN33,VhHBCC0NEj0ZEh56Dfj[Q(H6BC0MEffBBQPZBA4[FQP
KE*0[FSkKMC%RJ!"`FQpLB(at-sign)*TE'PcG'PMPJ2UU'PZQUabG'9bF(*PG'&dD(at-sign)pZNfp
QNh4SCC05EfeKET0MEj!!8ijP$Q0TC(at-sign)kBG(10MBf6Rr4QB+YSMSfIqUcFMC%)Sic
BESkI#ACqMC%)jHYVMSk0N3h,h*rdCQ#VDBk1MT%(at-sign)EfLjD(at-sign)k6Cf9ZCA*KE#k1TSf
4*i!!9'KKEQZ(at-sign)!qUSHC!!V(*[GC0QEh+6E'PcG'9ZD(at-sign)jR,SkQMC%RJ!$63QPLE'P
[Ch*KF'L3!+!!HBkQMC%RJ!#j5Lj3NIm&9Lk(at-sign)!UpJ8bk65h9ZCbbE!Zj[65k68Q&
YNdeeFR4SN!#XFRQ6B(at-sign)jNNdGTB(at-sign)iY3f&bE'q68QpdB5bB9'KPNe*PC'9TNhTPG''
6CR9ZBh4TEfiXMU'0N5H!!%T[GA*ZB(at-sign)b(at-sign)!qUSEfD66R9YQUabBT!!8ijPFT08D'9
[FRQ4r`9(at-sign),*0fQ'pX,TX&11!a-T-S-6Ni-#NXNh"`,TJd-M%Y0$-f,Sk1Rai!!)f
5!2T[B$8jMSk-L`!!!$`!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!QD4S!+J+2(at-sign)0S2f'e`ZJ!PXSpBfJrFRA#if4*i!!Z8GTB(at-sign)iY3f&bE'q(at-sign)"EP
r8QpdB5bE"Lde3R*eBf(at-sign)68f&RB(at-sign)k6B(at-sign)jNNe#3!+abBA9XNe)ZNe0dC(at-sign)PZ,*K"N3(at-sign)
j#'0jBfaTBj0NCA*TGT(r(at-sign)14KG'PfN!#XFQ(at-sign)6D(at-sign)k1T!k!!)f4*i!!EQpZBfpYEC(at-sign)
XFR9dBA4TGT0PPJ2UU'&XCf9LFQ%XNdT[GA*ZB(at-sign)b6EfD63(at-sign)aRC(at-sign)*bB5b6GT!!V(*
[E#k4"6MJ0M56+$%j1$!T,*0`F#ie0#dh05q1U4U!!)f4*i!!4'&ZD(at-sign)9XPJ9`2%8
ZNda[QP11C(at-sign)+6B(at-sign)jNNdGTB(at-sign)iY3f&bE'q68QpdB5b4"G'K4T(r"9C[FQeKE*0`Q'q
9V(*hNf9bPJ9`2(0PFQPPFj0[CT0XEfGKFQPdD'eTBikKMC%RJ!"dQUabHA#3!&1
1C5b(at-sign)!qUS3(at-sign)4fNIpBj'&ZBf9cNfPZNdeKG'KPE(at-sign)&dD(at-sign)0c,*0fQ'pX,MFeNbJa16J
j+5b6F(!ZN38ii$%Y-6%i,SkQMC%RJ!",C(at-sign)jZCA4SPJ18De-ZNd&XCAKKEQ4PFLb
E!k(at-sign)V5f9ZEQ9dD*0#B(at-sign)0XBC!!V(*hFfYTNf&ZC*0(D(at-sign)&Z,80KFQa[Ne*[G'%XQ%'
4!j4(at-sign)Fh4[N!"6MQ13!+abD'&c,BkKMC%RJ!"dD(at-sign)1(at-sign)!l8fD(at-sign)k3!+abG'9bF(*PG'&
dD(at-sign)pZNfpQNh4SCC05D(at-sign)9YB(at-sign)jZNhTPG''6CR9ZBh4TEfiXN31rjP"bEj!!8ijMC(at-sign)9
ND(at-sign)jRFj0[CT0dD'(at-sign)66Q&dD(at-sign)pZB(at-sign)b1SBf4*i!!3(at-sign)0KC'9YQUabHCB$kUK[CT06BfP
PEQ0PFbb6GTK[,TX&11!j-*-S-6Nj-bNXNh"`,TJf16FY0MNj,SkQMC%RJ!"+,P#
4r`9(at-sign),P-ZQ`9ZBdYeEQFXPJA28(at-sign)9N,Lb64fPKELe$BA*XEjK5Eh4KQ'pZQ%0[EC!
!V(*LD(at-sign)jKG'pbD(at-sign)0c,*0#DA*VD(q4qL!%BA9cCA+B3QpcG'pZ,)kKMC%RJ!!a16N
f,*B$kUK*8d*1Nc!Y1$%h0Ldc0c%c,616Eh+6590#6T-c,6Ff0$-Y-cFa-bdcMSk
I(J!!MC)!qQpJ0M#1MSci!!*`R!'$NX!F1`!!!!!$k!+J+28"f4ki!!N!22-!qV&
e%J!+!!!!#J!!!!CME(at-sign)9i-6$c$PLV83X!$!!!!!`!!!!&Bfeb-6,c$d66lA3!&,e
a!"&(VJ!&Bfeb-6Ic%&LV83X!$QCQ!!`!!!!&Bfeb-6,c+-,(at-sign)6U!!$!!!!!`!!!!
'BfeLH$%bmbNK)LbD!!`!!!!+!!!!"Q0YFhNa-2-UI(YC"`!)!!!!#!!!!!4MEA)
imbZj3(at-sign)'S!!B!!!!'!!!!"'0YFMEc,,IKCk-!$!!!!!`!!!!'BfeYD6%bmbhA!4F
b!!J!!!!)!!!!"(at-sign)0YE(at-sign)Nimbi3cEl1!!B!!!!'!!!!"(at-sign)0YE(at-sign)Nfmbqq5mJ,!!J!!!!
)!!!!"(at-sign)0YFhNimc"aS58,!!B!!!!'!!!!"(at-sign)0YFhNfmc%bad$*!!J!!!!)!!!!"(at-sign)0
YBRJimcQEZiK!!!`!!!!-!!!!"Q0YG'Na-[N!!R2h!YrIhprIhpp+q3!!:
--============_-1322142191==_============--

Date:           Sun, 15 Mar 1998 19:45:35 -0800 (PST)
To:             combinatorics(at-sign)math.mit.edu
From:           Gian-Carlo Rota <gcrota(at-sign)earthlink.net>
Subject:        Colloquium Lectures

--============_-1322120415==_============
Content-Type: text/plain; charset="us-ascii"

I am sending you a new version, since I have received complaints that the
one sent out earlier cannot be printed except with a Macintosh.

G.-C. R.

--============_-1322120415==_============
Content-Type: text/plain; name="colloquium_3=14=98_2"; charset="us-ascii"
Content-Disposition: attachment; filename="colloquium_3=14=98_2"

\documentstyle[12pt]{article}

\def\bphiz#1{\overline\phi_{\hat0}^{#1}}
\def\bPhi{\overline\Phi}
\def\phiz#1{\phi_{\hat0}^{#1}}
\def\sigalg#1{\Sigma^{\otimes{#1}}}
\def\sigalgpi#1{\Sigma^{\otimes{#1}}_\pi}
\def\finalg#1{\Sigma^{\otimes{#1}}_{\rm fin}}
\def\finalgpi#1{\Sigma^{\otimes{#1}}_{{\rm fin},\pi}}
\def\ltimes{{\rm X}}
\def\D{{\cal D}}
\def\F{{\cal F}}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\E{{\cal E}}
\def\C{{\cal C}}
\def\G{{\cal G}}
\def\R{{\bf R}}
\def\S{{\bf S}}
\def\Inpi{{I^{[n]}_\pi}}
\def\bs{\backslash}
\def\zpione{{\hat 0 \le \pi \le \hat 1}}
\def\zpilone{{\hat 0 \le \pi < \hat 1}}
\def\halmos{{Q.E.D.}} % This is temporary
\def\St{{\rm St}}

\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}

\begin{document}

\setlength{\textwidth}{6in}
\setlength{\textheight}{8in}
\parindent=0pt
\tolerance=6000
\title{The American Mathematical Society Colloquium Lectures, 1998}

\author{Gian-Carlo Rota}

\date{Baltimore, January 7, 8, and 9, 1998}

\maketitle

\newpage

\begin{center}

\bigskip

{\bf Contents:}

\end{center}

\begin{align}

1. Introduction to geometric probability.
\bigskip

2. Invariant theory, old and new.

\bigskip

3. Combinatorial snapshots:

\bigskip

\begin{itemize}

\item First snapshot: an example of profinite combinatorics.

\item Second snapshot: the cyclic derivative.

\item Third snapshot: logarithms and the binomial theorem.

\end{itemize}

\end{align}

\newpage

\newpage

\begin{center} {\bf INTRODUCTION TO GEOMETRIC PROBABILITY}\\

being\\

The first of three colloquium Lectures\\

delivered at the Annual Meeting of the American
Mathematical Society\\

Baltimore, January 7, 1997\\

Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\
Cambridge MA 02139-4307\\

\end{center}

\bigskip

\bigskip

I am  very happy to be here before you as the
Colloquium Lecturer for this year, and I feel deeply honored to be
given this great opportunity to share with you some of the
mathematics we love.

\bigskip

When I received from Bob Fossum the invitation to be the
Colloquium Lecturer for this year, I rushed to the library
to read in an old issue of the Notices the list of all
previous Colloquium Lecturers, going all the way  back to
James Pierpont in 1896. It is a list of distinguished
mathematicians, and I wondered how my name could ever
belong in such lofty company.  My immediate temptation was
to decline the invitation outright; but Bob Fossum assured
me that no one in the history of the Society has ever
declined the invitation to be the Colloquium Lecturer.  So I
went back to the list of previous colloquium speakers, in
search for a justification of my presence in that list. As
often happens in such situations, I soon enough found such
a justification. I computed the average age of colloquium
lecturers and discovered that this average is  somewhat
lower than my age, as a matter of fact my age exceeds by
approximately  one standard deviation the mean age of
previous speakers.  As I came to this realization, I began
to fantasize on the probable topics that my younger
predecessors might have chosen. I imagined a brilliant
young mathematician, eager to establish himself as a leader
in his field,  delivering one single  dazzling proof
beginning with the first colloquium lecture and lasting all
the way to the end of the third.  Or else, some middle aged
mathematician, anxious to have his latest theory accepted
by the mathematical world, delivering to a thrilled
audience a three-hour condensation of material that would
normally take an entire term in an advanced graduate
course. How could I, a mathematician one standard deviation
older, ever hope to match such enviable feats?

\bigskip

These fantasies came to an abrupt end when  Bob Fossum
informed me in no uncertain terms that the Council of the
Society had decided that the three colloquium lectures must
deal with three independent and unrelated topics, thereby
allowing any member of the audience  to skip one or more
lectures, without missing anything. Bob Fossum's command
deprived me of all possible role models among previous
colloquium lecturers. In a state of temporary panic, I
again scanned the list  of previous colloquium speakers,
this time looking for names of mathematicians who had not
been chosen for this honor. Sure enough, one name was
conspicuously missing: that of Hermann Weyl.

\bigskip

I hope you will forgive me if I  digress with some personal
reminiscences.

\bigskip

In the fall of 1950 I enrolled as a freshman at Princeton,
having graduated a few months before from the American High
School of Quito, Ecuador.  The principal of the American
High School of Quito was a Princeton graduate,  and he
steered me towards Princeton University.

\bigskip

In November 1950 I listened to my first mathematics
lectures. These were the three Vanuxem lectures, delivered
by Hermann Weyl and bearing the generic title "Symmetry".
These lectures were an unforgettable experience. The lectures took
place in the old chemistry auditorium, packed with an
expectant public. As I shamelessly sat in the first row
trying to guess which of the other persons sitting in the
same row was to be the speaker, a hush fell upon the
audience:  Einstein was entering the lecture room. To my
disappointment, he sat somewhere in the middle of the
auditorium.

\bigskip

The first lecture began with an impressive and lengthy
quotation in Greek, which no one in the audience understood
except Luther Pfahler Eisenhart.  This brilliant start was
followed by a display  of slides portraying charming women
wearing the long brimmed hats fashionable at the time, and
later  by more slides showing the Alhambra and the
Pentagon. Not a word of mathematics.  The audience was left
wondering where such a sparkling display of  "Kultur" was
leading up to. Not much more mathematics was mentioned in
the second lecture, when more slides were shown of physics
experiments, for  which the lecturer provided a learned oral
commentary. Only in the last lecture did some group theory
make a modest appearance.  By that time the audience, which
had not dwindled, was enthralled with the subject, and did
not mind the fact that the speaker had said very little
about mathematics, actually he had said very little about
anything at all. What is more remarkable, the audience
seemed to be thankful to the speaker for making the
contents of the three lectures independent of one another,
thereby minimizing all memory requirements. I hazard to
guess that the success of Hermann Weyl's lectures may be in
part attributed to the speaker's astute foresight in making
his lectures  self contained,  independent and lightweight.
As I recall this distant episode, I realize that  Bob
Fossum's injunction about the independence of the present
colloquium lectures is a wise one, all the more so when the
speaker is not Hermann Weyl.

\bigskip

You may wonder why I cited my age as an aid to delivering
these colloquium lectures. What difference does one standard
deviation make? I think it makes some difference. It is a
relief, both to you and to me, to know right at the start
that the speaker does not feel the need to impress you by
stating the results of his latest research. Nor do you or I
suffer from any lack of exposure to the latest fashions in
mathematics; we'll hear enough about them in other lectures
scheduled to be delivered at this meeting.  We can therefore
afford to spend these three hours on leisurely discussion
of  some mathematics that may matter to  both
you and me.

\bigskip

We will  cover in these lectures a few items that
are not widely known, that  should be better known, and that
I vouch can be understood by anyone with a B.A. in
mathematics. I solemnly promise that I will not state  any
big theorems, I will not subject you to   any ingenious
arguments, and that  I will not announce   any
revolutionary developments.

\bigskip

The title of this lecture is "Geometric Probability".
 A definition
of geometric probability might run as follows: geometric
probability is the study of invariant measures. Like all
definitions, this does not  tell us anything until  we are
shown some typical examples, and these examples are the
content of this lecture.

\bigskip

About one  hundred years ago, the properties that underlie
such notions as length, area, volume, as well as   the
probability of events were abstracted  under the banner of
the word  "measure".  Let us review the definition of
measure, since we will be using this definition in an
unusual way.

\bigskip

A measure $\mu$ is a function defined on a family of
subsets of a set $S$, which takes real values, not
necessarily positive.  The family of sets on which a
measure is defined is closed under unions and
intersections, and contains the empty set.

\bigskip

A measure is characterized by two simple axioms. Let us
take a minute to review these axioms.

\bigskip

Axiom 1.

$$ \mu ( \emptyset ) = 0,$$

where $\emptyset  $ is the empty set.

\bigskip

Axiom 2.   If $A$ and $B$ are two measurable sets, then

$$\mu(A\cup B) = \mu (A) +\mu(B) - \mu (A\cap B) .$$

\bigskip

The meaning of this  second axiom is clear.  The axiom states that
measure is additive. In particular,  if we have two disjoint sets $A$ and
$B$, then

$$\mu(A\cup B) = \mu (A) + \mu (B) .$$

\bigskip

More generally, for any finite family $F$ whose members are
sets, and for which any two members are disjoint, we
have :

$$\mu( \bigcup_{A\in F} A ) =\sum_{A\in F}\mu (A).$$

We most emphatically do not assume that a measure is
countably additive.

\bigskip

The best known  example of a measure is the volume
$\mu_n(A)$ of a solid $A$ in ordinary $n$-dimensional
Euclidean space. The volume  $\mu_n (A)$  of a solid $A$ satisfies axioms
1 and 2  above, but axioms 1 and 2 do not characterize volume
among all possible measures. What additional axioms must we
add to the definition of a measure, in order to
characterize volume? It is possible  to characterize volume among all
measures by adding to axioms 1 and 2 two additional intuitive  axioms,
namely, the following:

\bigskip

Axiom 3.

\bigskip

The volume of  a set $A$  is independent of the position of
$A$.  If a set $A$ in $n$-dimensional
Euclidean space can be rigidly moved onto a set
$B$, then $A$ and $B$ have the same volume.

\bigskip

In  other words,  volume is invariant under the group of
Euclidean motions.

Lastly, we must prescribe a normalization, as physicists
say. This is done by taking a parallelotope $P$ with orthogonal
sides of lengths $ x_1, x_2, \dots ,  x_n $, and setting

\bigskip

Axiom 4.

$$ \mu_n (P) = x_1x_2 \dots x_n.$$

\bigskip

These axioms, together with suitable continuity conditions,
uniquely determine the volume of solids in Euclidean $n$- space.
For example,  starting from these four axioms, by a
limiting process such as one finds in an advanced calculus
textbook, one establishes the fact that  the volume of
a ball $S_r$ of radius $r$ in
$n$-dimensional space  is given by the following formulas:

$$\mu_n(S_r) = \frac{ \pi^{n/2} r^n}{(n/2)!} $$

if the dimension $n$ is even

and

$$\mu_n (S_r) = \frac{2^n \pi^{(n-1)/2}((n-1)/2)! r^n}{n!} $$

if the dimension $n$ is odd.

\bigskip

It is still widely believed  that volume is the only
invariant measure in Euclidean $n$-space. But in point of fact there are
other invariant
measures, defined on all reasonable subsets of Euclidean $n$-space, which
have a notable geometric significance. Our objective is to
describe all such invariant measures.

\bigskip

What happens if we keep the first three axioms, but tamper
with the fourth axiom, the normalization axiom ?  Will we
get something interesting, or will we get nothing new?  To
answer this question, we will appeal  to  the basic tools
of combinatorial mathematics.

\bigskip

The basic tools of combinatorial mathematics are the
elementary symmetric functions, to wit, the following
polynomials in $n$ variables:

\begin{align}

$$e_1(x_1,x_2,...,x_n) = x_1 + x_2 + ... +x_n.,$$

$$e_2(x_1,x_2,...,x_n) = x_1x_2+x_1x_3+...+x_{n-1}x_n,$$

$$ \dots $$

$$e_{n-1}(x_1, x_2, \cdots, x_n) = x_2x_3 \cdots x_n +
x_1x_3x_4 \cdots x_n+...+ x_1x_2 \cdots x_{n-1},$$

$$ e_n(x_1, x_2, \cdots , x_n) = x_1 x_2\cdots  x_n .$$

\end{align}

Observe an interesting coincidence.  The last of these
$n$ symmetric functions is also the formula for the
volume of a parallelotope. Axiom 4 can be rewritten as

Axiom 4.

$$\mu_n (P)  = e_n(x_1,x_2,\cdots , x_n).$$

\bigskip

Let us try an experiment, and replace the $n$-th symmetric
function by the
$n-1$-st symmetric function. Let us first take $n = 3$,
that is, three-dimensional space,  so that we can better
visualize what is going on. Let  us see whether we
can  define a measure on subsets of $3$-dimensional space
by keeping three of the above axioms, but by replacing the
normalization Axiom 4 by using another symmetric function
instead of the symmetric function $e_3(x_1,x_2,x_3)$ which
gives the volume.  Let us first replace the symmetric
function $e_3$ by the symmetric function $e_2$, thereby
changing Axiom 4 to

\bigskip

 Axiom 4':

$$\mu_2(P) = x_1x_2 + x_1x_3 + x_2x_3 . $$

Does  this axiom define a measure? Of course it does. The
right hand side is the formula for the surface area of  the
parallelotope $P$, divided by $2$.  Again we will find in
any advanced calculus textbook the explanation of the fact
that axioms 1, 2, 3, and 4', together with some continuity
considerations, completely determine an invariant  measure
which is the surface area of solids in ordinary space.  For
example, the  following well known formula for the surface
area of a ball $S_r$ of radius $r$ in $3$ dimensions is
obtained from these axioms:

$$\mu_2(S_r)= 4\pi r^2 .$$

Let  us take the next step.

\bigskip

Emboldened by our success with  two symmetric functions, we
now replace axiom 4 by yet another axiom, using another
symmetric function.  Let us set

\bigskip

Axiom 4".

$$ \mu_1(P) =  e_1(x_1, x_2, x_3) = x_1 + x_2 + x_3 $$

The new measure $\mu_1$ will satisfy Axioms 1, 2, and 3,
and in addition it will satisfy axiom 4". The symmetric
function of degree one plays the role that in the previous
two examples was played by the other two symmetric
functions.

\bigskip

But wait a minute: is  this  definition
consistent?

\bigskip

To realize that the definition of the new measure $\mu_1$ is
consistent, that is, that
$\mu_1$ as defined by axioms 1, 2, 3, and 4" really exists
and is not a dream of  reason,   look at two parallelotopes
$P_1$ and $P_2$  that have a face in common. The first
 parallelotope has sides
equal to $x_1, x_2, x_3$ , and the second parallelotope has sides equal to
$x_1, x_2, y $.   The two parallelotopes have a common face
with sides equal to $x_1, x_2$.   The measure
$\mu_1(P_1\cup P_2)$ of the parallelotope $P_1\cup P_2 $
can be computed in two ways: using the left  side of axiom
2, or using  the right side,  and the two computations had
better yield the same answer, in symbols:

$$\mu_1(P_1 \cup P_2) = \mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1
\cap P_2).$$

  Let us check this.

\bigskip

The  left side  is computed by observing that the
parallelotope $P_1\cup P_2$ has sides equal to $x_1$,$
x_2$ and $x_3 + y$. Therefore, Axiom 4" tells us that

$$\mu_1(P_1 \cup P_2) = x_1 + x_2 + x_3 + y.$$
Now let us compute the right side.  We have

$$\mu_1(P_1) = x_1+x_2+x_3$$

$$\mu_1(P_2) = x_1 + x_2 + y$$

$$\mu_1(P_1 \cap P_2) = x_1 + x_2 ,$$

again by Axiom 4" applied to $P = P_1 \cap P_2$, since
one side equals zero when the parallelotope is a flat, that
is, a rectangle. Therefore, the  right side of Axiom 2
equals

$$\mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1 \cap P_2 )=
x_1+x_2+x_3 + x_1+x_2+  y - (x_1+x_2)  = $$
$$ x_1+x_2+x_3+y ,$$

and the two sides  of our equations agree,  thereby
convincing us that the definition may well be  consistent.

\bigskip

The preceding argument is convincing, even though it proves
nothing.

\bigskip

Actually, the   definition of $\mu_1(P) $ for a
parallelotope $P$  has a simple geometric interpretation.
When multiplied by  4, it  equals the perimeter of the
parallelotope $P$, that is, the sum of the lengths of all
the edges of the parallelotope $P$.

Just as happens for volume and area, it  can be shown by
continuity considerations that the measure $\mu_1$ can be
extended to all reasonable solids in ordinary space, for
example, to  all convex sets and to all polyhedra, convex
or nonconvex.

\bigskip

But, one may object, $\mu_1(P)$  makes sense for a
parallelotope $P$, because a parallelotope has a well
defined   perimeter.  What if $A$ is a  solid that does not
have a well defined perimeter, a sphere for example?   The
definition of the measure $\mu_1(A) $ for solids $A$ that
may  not have a well defined perimeter   flies in the face
of common sense.

\bigskip

Einstein wrote: "Common sense is the residue of those
prejudices that were instilled into  us before the age of
seventeen".  Common sense  must constantly  readjust to
reality.

\bigskip

The new  measure $\mu_1$  that we obtain in this way is
called  the mean width, a misnomer that has been kept for
historical reasons.   The mean width of a solid in space
is completely characterized by axioms 1, 2, 3, and 4".  In
particular, it is invariant, that is,  it does not depend
on position. For example, the formula for the mean width of
a sphere of radius $r$ is computed to be

$$\mu_1(S_r) = 4r. $$

Thus we see that in three dimensions each of the three
elementary symmetric functions of three variables  leads to
an invariant measure that enjoys equal rights with volume.
The first two of these measures are well known, namely,
volume and area.  The third, the mean width,  is at present
almost totally unknown. I know of no person who has an
intuitive feeling for the mean width, similar to the
intuitive feeling we have for volume and area.

\bigskip

Let us conjecture  a possible application of the mean
width.  A potato grower knows that a potato's volume is
important, because it determines the nutritional content of
the potato. The potato grower also knows that the surface
area of a potato is important, because it is rumored that
the vitamins in a potato are concentrated in the skin.  We
may conjecture that as soon as the potato grower will
become aware of the mean width, he or she will find a
nutritional interpretation of the mean width of a potato.
I am indebted to Steve Schanuel for this example.

\bigskip

A similar kind of  reasoning works in $n$ dimensions. We
discover  $n$ different invariant measures, each of them
well defined on all polyhedra and on all finite unions of compact
convex sets. Each of the $n$ elementary symmetric functions
of $n$ variables leads to the definition of a new invariant
measure which is a different generalization of the notion
of volume. These $n$ measures  are called the intrinsic
volumes. The intrinsic volumes are first defined on an
orthogonal  polytope $P$ whose sides equal $x_1,x_2,\dots ,
x_n$ by setting

$$\mu_k(P) = e_k(x_1,x_2,\dots ,x_n),$$

where $ e_k(x_1,x_2,\dots ,x_n) $ is the $k$-th elementary
symmetric function. Here, the subscript $k$ ranges from $1$
to $n$.

\bigskip

One then proceeds to extend the definition of the intrinsic
volumes to more general sets, by a technique which we will
shortly see.

\bigskip

The intrinsic volumes are independent of each other, except
for certain inequalities they satisfy. Mathematicians are
presently working on determining these as yet unknown
inequalities among the intrinsic volumes. These
inequalities generalize the classical isoperimetric
inequality that relates volume to area.   At present, we
know very little about the intrinsic volumes;  they have
not been around  for long and very little research has been
done on them.  We do not even know the formula for the
intrinsic volumes of an $n$-simplex.

\bigskip

Now you are thinking: this is all fine and dandy, but how
is the extension of the intrinsic volumes from
parallelotopes to more general sets carried out? And
besides, isn't there any intuitive  interpretation we can
give the intrinsic volumes?

\bigskip

We will answer both these questions simultaneously. Let us
go back to three-dimensional space.  You all know that the
set of all straight lines in space - not necessarily
through the origin - forms a nice algebraic variety, called
the Grassmannian. The group of all Euclidean rigid motions
acts on the Grassmannian, and there is an invariant measure
on the Grassmannian under the action of the group of
Euclidean motions.  This invariant measure is unique except
for a constant factor. A similar statement may be made
about the set of all planes, and more generally for the set
of all linear varieties of dimension $k$ in Euclidean space
of dimension $n$. Remember that these linear varieties need
not pass through the origin.

\bigskip

In the practice of mathematics, computation with invariant
measures on Grassmanians  is  rare; most
mathematicians would be hard put even to recall an explicit
formula for the invariant measures on Grassmanians.  Let us
take a few minutes to get a feeling for the invariant
measure on the set of all straight lines in three-space.
As is customary, we begin by giving this measure a name:
let us call it $\lambda^3_1$; the upper index $3$ stands
for three-dimensional space, and the lower index stands for
the dimension of a line, namely, one. To repeat, we use the
notation $\lambda^3_1$ to denote the invariant measure on
the set of all straight lines in three-space.

\bigskip

Consider  a rectangle $R$ placed anywhere in space, and
consider the set of all straight lines that meet the
rectangle $R$. Can we compute the measure of this set of
lines without knowing the formula for the invariant measure
on the Grassmannian of all lines in three-space? Of course
we can. A straight line meets the rectangle
$R$ either at a point or not at all; therefore, the value
of the measure of the set of all lines meeting $R$ depends
only on the area $\mu_2(R)$ of the rectangle
$R$.  If  we take another rectangle
$R'$ whose area is double the area of $R$, then the measure
of the set of all lines meeting $R'$ is double the measure
of the set of all lines meeting $R$. Proceeding along these
lines, we get to Cauchy's functional equation, and we infer
that  the measure of the set of all straight lines meeting
a rectangle $R$ equals a constant times the area
$\mu_2(R)$. Since we are at liberty to choose a
normalization of the measure, let us agree to set this
constant equal to one.

\bigskip

But instead of working with a rectangle we could have
worked with any planar figure
$C$ whatsoever, placed in an arbitrary position in space.
The measure of the set of lines meeting $C$ equals the area
$\mu_2(C)$, by the same reasoning. We stress the assumption
that $C$ must lie in a plane. To conclude: even without
knowing the formula for the invariant measure
$\lambda^3_1$, we can nevertheless compute the value of
such a measure on certain sets of lines.

\bigskip

Let us now take a more sophisticated set of straight lines. We take
a set $D$ in three-space that is the union of disjoint
sets $C_1, C_2,\cdots, C_n$, where each of the $C_i$ is
contained in a different plane, and we ask for the measure
of the set of all straight lines meeting $D$.  Such a
computation can be carried out, but it is a combinatorial
nightmare; so much so, that we are forced to do what
mathematicians do when confronted with combinatorial
nightmares: they change the problem ever so slightly. In this
case we take a hint from the way probabilists work.  Let
$X_D(\omega)$ equal the number of times the straight line
$\omega$ meets the set
$D$.  Instead of computing a measure, let us compute the
integral

$$ \int X_D(\omega) d\lambda^3_1(\omega) ,$$

where $\omega$  ranges over the Grassmannian, that is, over
the set of all straight lines in space. We will see that we
can compute this integral without knowing the measure
$\lambda^3_1$ on the Grassmannian.  Since

$$ D = \bigcup_{i=1}^n C_i ,$$

and since the $C_i$ are disjoint, we have

$$  \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n
\int X_{C_i}(\omega) d\lambda^3_1(\omega). $$

But we have chosen each of the sets $C_i$ to lie in a
plane, so that a straight line meets $C_i$ either once or
not at all. It follows that

$$  \int X_{C_i}(\omega) d\lambda^3_1(\omega) = \mu_2(C_i)
$$

and therefore

$$  \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n
\mu_2(C_i) . $$

What is this identity telling us? The right hand side
equals the area of the surface $D$. Nothing stops us from
passing to the limit, and making the following assertion.
Let $E$ be "any" surface in space, and let $X_E(\omega) $
be the number of times the straight line $\omega$ meets the
surface $E$. Then the integral

 $$ \int X_E(\omega) d\lambda^3_1(\omega) $$

ranging over all straight lines $\omega$, equals the
surface area of $E$.

In probabilistic language:
 the average number of times a randomly chosen straight
line meets the surface $E$ equals the surface area of
$E$.

\bigskip

Let us now retrace our steps, and repeat the same reasoning
taking the set of all planes in space, instead of the set of
all straight lines. The invariant measure on this
Grassmannian is denoted by $\lambda_2^3$, where again the
upper index stands for three dimensional space, and the
lower index for the dimension of a plane. Since a plane
meets a straight line segment either at a point or not at
all,  the same argument shows that the measure of the set
of all planes that meet a line segment $L$ equals
$\mu_1(L)$, namely, the length of the segment $L$; more
generally, if $F$ is any curve "whatsoever" in space, and
if $X_F(\omega)$ equals the number of times the plane
$\omega$ meets the curve $F$, then repeating the argument
we used for straight lines we infer that the integral

$$ \int X_F(\omega) d\lambda^3_2(\omega)  $$

equals the length of the curve $F$. The  variable of
integration $\omega$ now ranges over planes, not over
straight lines. Here again we compute  an integral without
knowing the measure.

\bigskip

We are now very close to getting an intuitive
interpretation of the mean width. Recall the parallelotope
$P$ with sides equal to $x_1,x_2,x_3$.
To measure the planes meeting the parallelotope $P$, we first consider a
family of {\em parallel} planes, all sharing the same fixed unit normal
$u$.  In other words, consider the set of all planes parallel to the
plane $u^{\perp}$.  Without loss of generality, place the parallelotope
in space so that one of the vertices of $P$ is at the origin, and
such that the vector $u$ lie in the octant of space opposite to the
parallelotope $P$.  (We can do this generically.)  Denote the edges of
$P$ that meet the origin by $x_1,x_2,x_3$.
Given the fixed unit vector $u$, and its family of normal planes, let us
take the curve $F$ to be a path along the edges (line segments)

$$[0,x_1], \; [x_1,(x_1+x_2)], \; \hbox{ and } \;
[(x_1+x_2),(x_1+x_2+x_3)],$$

 in that order.  A plane parallel to
$u^{\perp}$ meets the parallelotope $P$ if and only if it meets the
curve $F$ on the parallelotope at exactly one point.  Therefore, the
measure of the set of all planes parallel to $u^{\perp}$ that meet the
parallelotope $P$ is proportional to the length of the curve $F$.
Averaging over all unit vectors $u$ (and hence, over all families of
parallel planes), we conclude that the measure of the set of all planes
meeting a parallelotope equals the mean width of the parallelotope,
except for a constant factor which we will again set to be one.

\bigskip

In view of this realization, we can immediately see how to
define the mean width of any closed convex set: it equals
the measure of the set of all planes that meet the convex
set. Thus, we have shown that the mean width may be
extended to all closed convex sets in space.

\bigskip

We are now in a position to give  a probabilistic
interpretation of the mean width of a convex set.

\bigskip

Take two compact convex sets $A$ and $B$ in three dimensional
Euclidean space, and suppose that $A$ is contained in $B$.
Let  us begin by belaboring the obvious.  Suppose that we
take a point at random belonging to the larger set $B$.
What is the probability that the point shall belong to the
smaller set $A$?  The answer is clear: such a probability
equals the ratio of the volume of $A$ by the volume of
$B$.

\bigskip

 Instead of choosing a point at random, let us
choose a straight line at random in space.  Assuming that
such a straight line meets the larger set $B$, what is the
probability that such a straight line will also meet the
smaller set $A$?

We have already computed the answer to this question,
albeit implicitly.  Such a probability equals the surface
area of the set $A$, divided by the surface area of the set
$B$.

\bigskip

You can tell what is coming next. We now  take a random
plane in space. Assuming that the plane meets the larger
set $B$, what is the probability that it will also meet the
smaller set $A$?  The answer is the following: such a
probability equals  the mean width of $A$, divided by the
mean width of $B$.

\bigskip

In Euclidean $n$-space, we obtain by much the same
reasoning interpretations of the intrinsic volume
$\mu_k(C)$ of a compact convex set $C$ as the Grassmannian
measure of the set of all linear varieties of dimension
$n-k$ that meet the convex set $C$, and a similar
probabilistic interpretation holds.

\bigskip

What comes next? There are at least two questions still
open. First, are there any other invariant measures besides
the intrinsic volumes, and second, how can the definition
of the intrinsic volumes be extended to more general
subsets of
$n$-space than convex sets. The answers to both
these questions are closely related.

\bigskip

The answer to the first question is negative. We are
missing one measure, and to discover it, we will engage for
a minute in the kind of mathematical reasoning that
physicists find unbearably pedantic, just to show
physicists  that such reasoning does pay off.

\bigskip

Let us ask ourselves the question: what is the value of the
symmetric function of order zero of a set of $n$ variables
$x_1,x_2,\dots , x_n$, say $e_0(x_1,x_2,\dots , x_n)$?  I
will give you the answer, and will leave it to you to
justify this answer after the lecture is over.  The answer
is the following:  $e_0= 1$ if $n >0$, that is, if the set
of variables  $x_1,x_2,\dots , x_n$ is non empty,  and
$e_0 = 0 $ if the set of variables is empty.

\bigskip

We are led to believe that  there
may exist an invariant measure in $n$-space associated
with the symmetric function of order zero. We set

$$\mu_0(C) = 1 $$

if $C$ is any non empty  compact convex set, and of course
$\mu_0(\emptyset ) = 0$. Does such a measure exist? It does indeed exist,
and the fact that it exists is, in my opinion, one of the
most remarkable discoveries ever made in mathematics.

\bigskip

We will prove that such a measure is well defined on any
set which is a finite union of compact convex sets. We do
this by employing a classical device borrowed from
functional analysis: instead of defining a measure, we
define a linear functional on all simple functions, that
is, on all real  functions $f(\omega)$ defined for $\omega
\in R^n $ which are linear  combinations of indicator
functions of compact convex sets. Let us first begin with
the case $n=1$, that is, let
$\omega$ range over points on the line. Define a linear
functional $\chi_1$ on simple functions as follows:

$$ \chi_1(f) = \sum (f(\omega) - f(\omega+)) ,$$

where the sum ranges over all real numbers $\omega$. The
meaning of the plus sign is best gleaned from an example.

Let $f$ be the indicator function of the closed segment
$[a,b]$.  Then $f(\omega) - f(\omega+) = 0$ for all $\omega
$ except $\omega = b $, because we have $f(b) = 1$ but
$f(b+) = 0$. Thus, we see that $\chi_1(f) = 1 $ if $f$ is
the indicator function of an interval $[a,b]$.

\bigskip

Now let us go over to $n$ dimensions, proceeding by
induction. Do not worry, this won't take long. Take a straight line
$L$ and for every point
$\omega$ in
$L$ let
$H_{\omega} $ be the hyperplane through the point $\omega $
perpendicular to the line $L $. If $f$ is a simple function
defined in $n$ space, and if
$\omega$ is a point on the straight line $L$, let
$f_{\omega}$ be the restriction of
$f$ to the hyperplane
$H_{\omega}
$.   Define a linear functional $\chi_n $ as follows:

$$ \chi_n(f) = \sum \chi_{n-1}(f_{\omega}) -
\chi_{n-1}(f_{\omega +}) ,$$

where the sum ranges over all points $\omega$ on the line
$L$. There is
only a finite set of $\omega's $ for which the summand is
nonzero.  When $f$ is the indicator function of a non empty
compact convex set, then an argument similar to the
preceding shows that $\chi_n(f) = 1 $.  Thus, we may define
a measure $\mu_0(G) = \chi_n(f) $, where $G$ is any finite
union of compact convex sets, and $f$ is the indicator
function of the set $G$.  We have thus proved the existence
of a
 measure $\mu_0$ which is defined on all finite unions of
compact convex sets, and which takes the value one on all
non empty compact convex sets. This measure has a long
history: it is   the Euler characteristic.

\bigskip

Now you are thinking: if this is the Euler characteristic,
then it is up to you to show that it coincides with what we
ordinarily believe to be the Euler characteristic.  Let us
conclude this lecture by deriving the formula of
Euler-Schl\"afli-Poincar\'e  for polyhedra. As a
matter of fact, this formula can be encapsulated into a
simpler formula, one that is easy to remember.

\bigskip

Let $C$ be a non empty compact convex polytope of dimension
$n$, and let $int(C)$ be the interior of $C$. Then we have
the following fundamental formula for the Euler
characteristic of $int(C)$:

$$ \mu_0(int(C)) = (-1)^n .$$

Indeed, if  $f$ is the indicator function of the set
$int(C)$, we have:

$$  \mu_0(int(C))  = \sum \chi_{n-1}(f_{\omega}) -
\chi_{n-1}(f_{\omega + }) ,$$

where the sum ranges over all points $\omega$ on the line
$L$ as above. But by induction, we see that every term on the right
hand side equals zero, except when $\omega$ is the first
point on the line $L$ for which  the intersection $C \cap
H_{\omega} $ is not empty.  If $\omega_{\ell} $ is such a
first point, then we have

$$  \chi_{n-1}(f_{\omega_{\ell}}) = 0 $$

because the point $\omega_{\ell} $ is on the boundary of
$C$, and

$$  \chi_{n-1}(f_{\omega_{\ell} +}) = (-1)^{n-1} $$

by the induction hypothesis, because $ f_{\omega_{\ell} +}$ is the
indicator function of the set
 $int(C)
\cap H_{\omega_{\ell}+}
$, which is the interior of a convex polyhedron one
dimension lower.

Putting all this together, we obtain

$$  \mu_0(int(C))  = \sum \chi_{n-1}(f_{\omega_{\ell}}) -
\chi_{n-1}(f_{\omega_{\ell}+}) =  - (-1)^{n-1} = (-1)^n ,$$

as desired.

We are now in a position to state the famous Euler formula
for polyhedra. What is a polyhedron? A polyhedron is a
finite union of convex polyhedra. Given a polyhedron, we
must define a system of faces (of all dimensions, ranging from dimension
$0$ (a point) to dimension $n$). We will say that a set
$\bf{F}$ of convex polyhedra is a system of faces for an
arbitrary polyhedron $K$ when the elements of
$\bf{F}$, called faces,  are non empty compact convex sets
$F$ with disjoint interiors such that

$$ K =  \bigcup_{F \in \bf{F} }int(F). $$

Caution: the interior of a face of dimension $k$ is to
be taken relative to the linear space of dimension $k$ that
contains the face, and the interior of a point is a point.

\bigskip

Under these conditions we may take the Euler characteristic
of both sides, and using the fact that any two interiors of
faces are disjoint we obtain (using the fact that the measure of the
disjoint union of a family of sets equals the sum of the measures of the
individual sets):

$$ \mu_0(K) = \sum_{F \in \bf{F} } \mu_0(int(F)) = f_0 - f_1
+ f_2 - \dots +
\dots , $$

where $f_i$ equals the number of faces of dimension $i$.
This is Euler's formula.

\bigskip

We can now answer the second of the questions we had left
open: how to  extend the definition of the intrinsic volumes
from compact convex sets  to all finite unions of compact
convex sets.  If $G$ is such a finite union of compact
convex sets,  then  we set

$$ \mu_k (G) = \int \mu_0(G \cap \omega )
d\lambda_{n-k}^n(\omega), $$

where $\omega$ ranges over all linear varieties of dimension $n-k$ in
$n$-space.
The left hand side defines   a measure, and when $G$ is a compact convex
set it
agrees with the definition we have already given. It is
therefore the desired extension. The Euler
characteristic does all the work for us.

\bigskip

We are now in a position to state the main theorem of
geometric probability.

We will say that an invariant measure $\mu$ on Euclidean
$n$-space, defined on all finite unions of compact convex
sets, is continuous, when

$$ \lim_{C_n \rightarrow C} \mu(C_n) = \mu(C) $$

for all sequences $C_n$ of compact convex sets converging
to the compact convex set
$C$.

We have the

\bigskip

{\bf Main Theorem of Geometric Probability}

\bigskip

 The
$n+1$ intrinsic volumes  $\mu_0, \mu_1,\dots , \mu_n $ are
a basis of the space of all continuous  invariant measures
defined on all finite unions of compact convex sets.

\bigskip

The first proof of this theorem is due to Hadwiger; the
first  elementary proof was published last year by Dan
Klain of Georgia Tech.

\bigskip

In closing, let me try to answer the question you are about
to ask: what has this got to do with geometric probability,
anyway?

\bigskip

I will attempt a sketchy answer. Consider two compact convex sets $A$
and $B$. We imagine $B$ to be fixed in
$n$-space, and that we "drop" the rigid set $A$ at random.  What is
the probability that $A$ meets $B$? We answer this question
in three steps. First, we realize that by keeping $B$ fixed
and varying $A$ by the group of Euclidean Motions, we define an invariant
measure on
convex sets
$B$. Second, we apply Hadwiger's theorem, and infer that
such an invariant measure equals a linear combination of
the $n+1$ intrinsic volumes, with coefficients depending on $A$ and not
on   $B$. Third, we determine these coefficients by taking
suitable
$B$'s.  The end result is an identity which is known as the
kinematic formula, which has been the object of much
research in this century, still going on today.

\bigskip

Thank you for your attention.

\bigskip

{\bf Bibliography}

\bigskip

Daniel A. Klain and Gian-Carlo Rota, Introduction to Geometric
Probability (Lezioni Lincee), Cambridge University Press, 1997

\newpage

\begin{center} {\bf INVARIANT THEORY, OLD AND NEW}\\

being\\

The second Colloquium Lecture\\

\bigskip

delivered at the Annual Meeting of the American Mathematical Society\\

Baltimore, January 8, 1997.\\

\bigskip

Gian-Carlo Rota\\ Department of Mathematics\\  MIT, room 2-351\\ 77
Massachusetts
Avenue\\
Cambridge MA 02139-4307\\

\end{center}

\bigskip

\bigskip

Invariant theory is the great Romantic story of mathematics. For one
hundred and fifty
years,
from its beginnings with Boole to the time, around the middle of this
century,  when it
branched
off into several independent disciplines, mathematicians of all countries
were brought
together
by their common faith in invariants: in England, Cayley, MacMahon,
Sylvester and
Salmon, and
later, Alfred Young,  Aitken, Littlewood and Turnbull. In Germany, Clebsch,
Gordan,
Grassmann, Sophus Lie, Study; in France, Hermite, Jordan and Laguerre; in
Italy, Capelli,
Brioschi, Trudi and Corrado Segre, in America, Glenn, Dickson, Carus (of
the Carus
Monographs), Eric Temple Bell and later Hermann Weyl. Seldom in  history
has an
international  community of  scholars  felt so  united  by a common
scientific ideal for so
long a
stretch of time. In our century,    Lie theory and algebraic geometry,
differential algebra and
algebraic combinatorics are offsprings of invariant theory. No other
mathematical theory,
with
the exception of the theory of functions of a complex variable, has had as
deep and lasting
an
influence on the development of mathematics.

\bigskip

Eventually, invariant theory was to become a victim of its own success: the
very term
"invariant
theory" is nowadays understood in such a  wide variety of senses that it
has become all but
meaningless.     It is no wonder that you are baffled by the title of this
lecture, and curious
to hear
what will be said about invariant theory in the next forty-eight minutes.

\bigskip

Like the Arabian phoenix arising from its ashes, classical  invariant
theory, once
pronounced
dead, is once again at the forefront of mathematics. The old treatises are
being dusted off
the
shelves of library basements and reread, reinterpreted and presented in a
language that
meets the
standard of rigor of our day. The program of classical invariant theory,
that had for some
time
been given up as hopeless, is again being pursued, and success may at last
be within reach.

\bigskip

We will review  two turning points in the history of invariant theory.
The first, the
"new"
one,  happened around the turn of the century, and its effects are still
being felt all over
mathematics. The second,  the "old" one, happened very early in the game,
and led to a
serious
misunderstanding that lasts to this day.

\bigskip

A pedestrian definition of invariant theory might go as follows: invariant
theory is the study
of
orbits of group actions. Such a definition is correct, but it must be
supplemented by a
programmatic statement.  Hermann Weyl, in the introduction to his book "The
Classical
Groups", was  the first in this century to  give a sweeping overview of the
program of
invariant
theory. He summarized this program in two basic assertions. The first
states that "All
geometric
facts are expressed by the vanishing of invariants", and the second states
that "all
invariants are
invariants of tensors".

\bigskip

Let us briefly comment on  these lofty statements. What is a geometric
fact? A geometric
fact is
a fact about space  that is independent of the choice of a coordinate
system. Geometric
facts are
described by means of equations which require a choice of   coordinates. In
a vector space
$V$
of dimension $n$ one chooses a coordinate system
$ x_1, x_2,\dots ,x_n
$ . Since Descartes, we have learned to express geometric facts by
equations  in the
coordinates
$ x_1, x_2,\dots ,x_n $.  However, about  one hundred years ago,
mathematicians and
physicists  made the shocking discovery  that the usual type of equations,
that is, equations
in
the commutative ring generated by the variables
$ x_1, x_2,\dots ,x_n $, are inadequate for the description of  a lot of
geometric and physical facts.
Motivated by this discovery, they introduced  a  more general ring. This is
the ring of non
commutative polynomials in the coordinates  $ x_1, x_2,\dots ,x_n $.
Homogeneous
elements
of this ring, that is, homogeneous  non commutative polynomials in the
variables  $ x_1,
x_2,\dots ,x_n $, are called tensors. If we believe Hermann Weyl's
philosophy, then we
will be
satisfied that  equations in the tensor algebra  suffice for the
description of any geometric
fact we
will ever meet.  Furthermore, if these equations are to  express geometric
properties, then
they
must hold no matter what coordinate system is chosen; in other words,
equations that
describe
geometric facts must be invariant under changes of coordinates.    The
program of
invariant
theory, from Boole to our day, is precisely the translation of  geometric
facts into invariant
algebraic equations expressed in terms of  tensors.

\bigskip

This program of translation of geometry into  algebra  was to be carried
out in two steps.
The
first step consisted in decomposing tensor algebra into irreducible
components under
changes of
coordinates. The second step consisted in devising an efficient notation
for the expression
of
invariants for each irreducible component.  The first step was successfully
carried out in
this
century; the second was abandoned sometime in the twenties and only
recently has
it resurfaced.

\bigskip

The decomposition of tensor algebra into irreducible components was
discovered around
the turn
of the century   almost simultaneously by Issai Schur and Alfred Young.
The gist of this
decomposition is one of the great advances in mathematics of all times, and
it may be
worthwhile
to  present it in a form that can be made available  to undergraduates.

\bigskip

Let us consider
 functions of three variables, such as
$f(x_1, x_2, x_3)$.
Two well known classes of functions of three variables are symmetric
functions, defined to
satisfy
 the equations

$$f_s(x_1,x_2,x_3) = f_s(x_{i_1},x_{i_2},x_{i_3}) $$

for every permutation sending the indices $(1,2,3)$ to $(i_1,i_2,i_3)$,
and  skew symmetric functions, defined by  the equations

$$f_a(x_1,x_2,x_3) = \pm f_a(x_{i_1},x_{i_2},x_{i_3}), $$

where the sign is $+1$ or $-1$ according as the permutation sending  the
indices
$(1,2,3)$ to $(i_1,i_2,i_3)$ is even or odd.

\bigskip

It is not true that a function of three variables is the sum of  a symmetric
  function and  a skew-symmetric  function.  A third type of function is
required,  which is
called  a cyclic function,  which is defined by  the equation

$$f_c(x_1,x_2,x_3) + f_c(x_3,x_1,x_2) + f_c(x_2,x_3,x_1)  = 0.$$

Every function of three variables can be uniquely written as the sum
 of a symmetric function, a skew symmetric function, and a cyclic function,
in symbols:

$$f(x_1,x_2,x_3) =  f_s(x_1,x_2,x_3) +f_a(x_1,x_2,x_3) +f_c(x_1,x_2,x_3).$$

Each of the three symmetry classes is invariant under
permutations; this fact is obvious for symmetric and skew symmetric
functions but not
quite so
obvious for cyclic functions.  These three invariant subspaces  play for
the group of
permutations
of a set of three elements  a role analogous to the  role of the
eigenvectors of a
symmetric
matrix.

\bigskip

For  functions $f(x_1,x_2,x_3,x_4) $ of four variables there are five symmetry
classes, which are defined as follows:

\bigskip

1. Symmetric functions.

\bigskip

2. Skew symmetric functions.

\bigskip

3. Cyclic symmetric functions, satisfying the four equations

$$f(x_1,x_2,x_3,x_4)+ f(x_1,x_4,x_2,x_3)+f(x_1,x_3,x_4,x_2) = 0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_4,x_2,x_1,x_3)+f(x_3,x_2,x_4,x_1) = 0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_4,x_1,x_3,x_2)+f(x_2,x_4,x_3,x_1) = 0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_3,x_1,x_2,x_4)+f(x_2,x_3,x_1,x_4)=0.$$

\bigskip

4. Functions satisfying the four equations

$$f(x_1,x_2,x_3,x_4)+f(x_2,x_1,x_3,x_4)+f(x_1,x_2,x_4,x_3)+
f(x_2,x_1,x_4,x_3) =
0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_3,x_2,x_1,x_4)+f(x_1,x_4,x_3,x_2)+
f(x_3,x_4,x_1,x_2)=
0,$$

$$f(x_1,x_2,x_3,x_4)+f(x_1,x_3,x_2,x_4)+f(x_4,x_2,x_3,x_1)+
f(x_4,x_3,x_2,x_1)=
0,$$

$$\sum sign(\sigma ) f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma
4})= 0.$$

\bigskip

5. Functions satisfying the equations

$$f(x_1,x_2,x_3,x_4)-f(x_2,x_1,x_3,x_4)-f(x_1,x_2,x_4,x_3)+
f(x_2,x_1,x_4,x_3) =
0,$$

$$f(x_1,x_2,x_3,x_4)-f(x_3,x_2,x_1,x_4)-f(x_1,x_4,x_3,x_2)+
f(x_3,x_4,x_1,x_2)=
0,$$

$$f(x_1,x_2,x_3,x_4)-f(x_1,x_3,x_2,x_4)-f(x_4,x_2,x_3,x_1)+
f(x_4,x_3,x_2,x_1)=
0,$$

$$\sum  f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma 4})= 0.$$

\bigskip

Every function of four variables is uniquely expressible as the sum of five
functions, each
one
belonging to one of these symmetry classes.  Each symmetry class is
invariant under
permutations.

\bigskip

More generally, every function of $n$ variables $f(x_1,x_2,\dots ,x_n)$ can
be uniquely
written as
the sum of $p_n$ functions, each one belonging to a different symmetry
class. Here,
$p_n$ equals
the number of partitions of the integer $n$. Each symmetry class is defined
by equations
which are
not difficult to find.

\bigskip

This decomposition holds for tensors as well, after some cosmetic changes
of notation.
To this day, only
two symmetry classes of tensors have been studied in any detail.
 Symmetric tensors are ordinary commutative polynomials such as we
learned to use in analytic geometry. Skew symmetric tensors are polynomials
in the
coordinates  $
x_1, x_2,\dots ,x_n $ provided that  the variables are assumed to  satisfy  the
equations
$x_ix_j = - x_jx_i $.
Tensors belonging to symmetry classes other than the classes of symmetric and
 skew symmetric tensors  also occur in geometry and physics. However, these
symmetry
classes
have been studied very little, and they are a long way from being understood.

\bigskip

So much for  the word   "new" in the  title of this lecture; let us next do
some  justice
to the word "old".  We will describe  the most peculiar  feature of classical
invariant theory, namely, the symbolic or umbral notation, to which Eric
Temple Bell
dedicated
his Colloquium Lectures in 1927.  We will consider the simplest group,
namely, the group
of translations of the line. The unusual  features of the symbolic method
will already be
 apparent in this  special case.

Let  $p(x)$ and $ q(x) $ be monic polynomials in the variable $x$.
 We write them in the following quaint notation:

$$ p(x) = x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} + \dots
+ {n
\choose
n-1}a_{n-1}x + a_n , $$

and

$$ q(x) = x^k + {k \choose 1}b_1x^{k-1} + {k \choose 2}b_2 x^{k-2} +
\dots + {k \choose k-1}b_{k-1}x + b_k . $$

We  assume that the polynomial $q(x)$ is of lower degree than
 the polynomial $p(x)$, that is, that $k \leq n $.

\bigskip

Define the translation operator $T^c$ on a polynomial $p(x)$ as follows:

$$ T^c p(x) = p(x + c) .$$

Let us write

$$ p(x+c) = x^n +{n \choose 1}p_1(c) x^{n-1} + {n \choose 2}p_2(c) x^{n-2}
+ \dots +
{n
\choose n-1}p_{n-1}(c) x + p_n(c) , $$

The $j$-th coefficient $p_j(c)$ of the polynomial $p(x+c) $ is
computed to be

$$ p_j(c) =  a_j + {j \choose 1} a_{j-1} c + {j \choose 2 }a_{j-2}c^2 +
\dots + c^j . $$

A polynomial I($a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) $ in the
 variables
$a_1$, $a_2$, \dots ,$ a_n$, $b_1$,$ b_2$, \dots ,$ b_k $  is said to be an
invariant
of the two
polynomials $p(x)$, $q(x)$ when

$$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) = I(p_1(c),p_2(c),
\dots , p_n(c),
q_1(c), q_2(c),
\dots , q_k(c) ) $$

for all complex numbers $c$. By abuse of notation, we write $I(p(x), q(x)) $
and we speak
of $I$ as
being an invariant of the polynomials $p(x)$ and $q(x)$. In this abusive
notation, a
polynomial
$I$ is said to be an  invariant of the polynomials $p(x)$ and
$q(x)$ whenever

$$ E(T^cp(x), T^cq(x) ) = E(p(x), q(x)) $$

for all constants $c$.

\bigskip

Invariant theory is concerned with the problem of finding all  invariants
of a given set of
polynomials, as well as their significance.

\bigskip

What is meant by the "significance" of an invariant? We will  appeal to
Hermann
Weyl. "Every" property of polynomials which is invariant under the group of
translations is
expressed by the vanishing of a set of invariants. In other words, "any"
set of polynomials
which
is invariant under translations is the same set as a set of polynomials
obtained by setting to
zero a
set of invariants of such polynomials.

\bigskip

It is impossible to understand the above  statement without examples.

Let us consider the simplest and oldest example.  The property of  a
quadratic polynomial

$$ q(x) =  x^2 + 2b_1 x + b_2 .$$

of having a double root is invariant under translations; in other words, if
the polynomial
$q(x)$
has a double root, so does the polynomial $q(x+c)$  for any constant
$c$ . Following Hermann Weyl, we look for an invariant whose vanishing
expresses this
property. Sure enough, it  is easy to check that the discriminant

$$ D( b_1, b_2) = b_1^2 - b_2 $$

is the desired  invariant.

This example, due to Boole, was the spark that led to the birth of
invariant theory.

\bigskip

One often hears the sentence "Hilbert killed invariant theory",  repeated
as an excuse to
ignore
all that went on in invariant theory after Hilbert. I don't know who made
up this infamous
sentence.  It is not true.  Hilbert loved invariant theory, and he went on
publishing striking
papers in  invariant theory  well after he proved the theorem that is
nowadays called the
Hilbert
basis theorem, the theorem that is supposed to have killed invariant
theory. Some of the
most
fascinating results in invariant theory were discovered in the first twenty
years of this
century, a
long time after Hilbert proved his basis theorem.

\bigskip

What then is the reason for the temporary demise of invariant theory in
this century? One
reason is
 the endemic  use of a notation that lacked rigor and that
amounted to little more than handwaving in print. This is the symbolic or
umbral notation.

\bigskip

Dieudonn\'e  wrote  that half the success of a piece of mathematics depends
on  a  proper
choice of notation. It would be interesting to make a list of unfortunate
notations that killed
various chapters of mathematics, as well as a list of felicitous notations
that promoted the
development of other branches of mathematics. The symbolic or umbral
notation that was
used
by invariant theorists through the nineteen twenties was a catastrophe.  A
number of
mathematicians tried to make sense of the symbolic method without success,
the three most
notable ones being Hermann Weyl,  Eric Temple Bell, and  Edward Hegeler
Carus. Eric
Temple Bell failed to properly define  umbral notation, and his book
"Algebraic Arithmetic"
remains to this day the book of seven seals.
 If Hermann Weyl and Eric Temple Bell  had lived fifty years longer,
so as to benefit of the development of what was in their time called
"modern" algebra,  they
would
undoubtedly have succeeded in properly defining umbral notation.

\bigskip

In our day, it does not take much work to accomplish this task.
Do not be alarmed: it  will only take a
few minutes.  Before  I start spouting out definitions, let me say what I
am not going to
say.
Umbral notation can be shown to be equivalent, or "cryptomorphic", to use a
term invented
by my
late friend Garrett Birkhoff, to another notation that has gained great
notoriety in our day: I
mean the
notation of Hopf algebras. I will not justify this Sybilline pronouncement,
not because it is
difficult
to do so, but because it would be too boring to do so.

\bigskip

Let  us go on to the definition of umbral notation.

\bigskip

Side by side with the polynomials $p(x)$ and $q(x)$, we consider another
polynomial
algebra
${\bf C}[ x, \alpha , \beta  ] $ in three variables $x$ , $ \alpha $ and
$\beta $,  together
with a
linear functional $E$ defined on the underlying vector space
${\bf C}[x, \alpha, \beta ]$.  The  definition of the  linear functional $E$ is
the key point.  It is carried out
in the following steps:

\bigskip

Step 1. Set

$$ E(x^j ) = x^j $$

for all non negative integers $j$   , in particular $E(1) = 1$. Thus, the
range of the linear functional $E$ is $C[x]$.

\bigskip

Step 2. Set

$$ E(\alpha^j) = a_j ,$$

in particular, we have $ E(\alpha^j) = 0 $ if $j > n $.

\bigskip

Step 3. Set

$$ E(\beta^j) = b_j ,$$

in particular, we have  $ E(\beta^j) = 0 $ if $j > k $.

\bigskip

Step 4. This is the main step. Set

$$ E(\alpha^i \beta^j x^{\ell}) = E(\alpha^i ) E( \beta^j )x^{\ell}. $$

Following Sylvester, the variables $\alpha $ and $\beta $ are called umbrae.
In other words, the linear functional $E$ is multiplicative on distinct umbrae.

\bigskip

Step 5.

Extend by linearity.

\bigskip

This completes the definition of the linear functional $E$.

We next come to the most disquieting feature of umbral notation.  Let
$f(\alpha , \beta , x)
$ and
$g(\alpha , \beta , x) $ be two polynomials in the variables $
\alpha , \beta , x $.  We write

$$ f(\alpha , \beta , x) \cong g(\alpha , \beta , x) $$

to mean

$$ E(f(\alpha , \beta , x)) = E( g(\alpha , \beta , x) ) . $$

Read $\cong $ as "equivalent to".

The "classics" went a bit  too far, they wrote

  $$ f(\alpha , \beta , x) = g(\alpha , \beta , x) $$

that is, they replaced the symbol $\cong$ by ordinary equality. This was an
excessive abuse of notation.   The "classics" were
 aware of the error, and while they avoided computational errors
 by clever artistry, they were unable to  settle on a correct notation.

\bigskip

The umbral or symbolic method consists of replacing all occurrences of the
coefficients of
the
polynomials $p(x)$ and $q(x)$ by umbrae and equivalences.  For example,

$$ p(x) \cong (x + \alpha)^n $$

and

$$ q(x) \cong (x+ \beta)^k .$$

Let us carefully check  the first equivalence.

By definition, the equivalence means the same as

$$ E(p(x)) = E( (x + \alpha)^n).$$

Since $E(x^j) = x^j $  for all non negative integers $j$, this identity can
be rewritten as

$$ p(x) = E( (x + \alpha)^n).$$

Expanding the right hand side by the binomial theorem, we obtain

$$ E( (x + \alpha)^n) = $$
$$ E(  x^n + {n \choose 1}\alpha  x^{n-1} +  {n \choose 2}\alpha^2
x^{n-2} + \dots + {n \choose n-1}\alpha^{n-1}x +\alpha^n)  $$

By linearity this equals

$$ x^n + {n \choose 1}E(\alpha) x^{n-1} + {n \choose 2}E(\alpha^2) x^{n-2}
+ \dots +
{n
\choose n-1}E(\alpha^{n-1})x + E(\alpha^n) . $$

Evaluating the linear functional $E$, we see that this in turn equals

$$ x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} +
\dots + {n \choose n-1}a_{n-1}x + a_n , $$

as desired.

\bigskip

The expression

$$ (x + \alpha)^n $$

is called  an umbral representation of the polynomial $p(x)$.

 In umbral notation, a     complex number $r$ is a root of the polynomial
 equation $p(x) = 0$ if and only if

$$ (r + \alpha)^n \cong 0. $$

Similarly, in umbral notation  the   polynomial $T^cp(x)= p(x+c)$ may be
represented as
follows:

$$ p(x+c ) \cong (x+\alpha + c)^n, $$

and this yields the umbral expression  for the coefficients
$p_j(c)$ of the polynomial $p(x+c)$, namely

$$ p_j(c) \cong (\alpha + c)^j . $$

Let us next see how umbral notation is related to    invariants.
 Let us assume  that the two polynomials $p(x)$ and $q(x)$ have the same degree
$n$. Then an  invariant $A$ of the polynomials $p(x), q(x)$ may be defined as
follows:

$$ A(q(x), p(x)) \cong (\beta - \alpha)^n .$$

The evaluation  of the invariant $A$ in terms of the   coefficients of
$p(x)$ and
$q(x)$ proceeds as follows:

$$ A(q(x), p(x)) = E((\beta - \alpha)^n) = $$

$$E(\beta^n -{n \choose 1} \beta^{n-1}\alpha +   \dots
+ (-1)^{n-1}{n \choose n-1}\beta
\alpha^{n-1} + (-1)^n \alpha^n) = $$

$$ E(\beta^n)  - E({n \choose 1} \beta^{n-1}\alpha) +
 \dots +  (-1)^{n-1}E({n \choose n-1}\beta
\alpha^{n-1}) + (-1)^n E(\alpha^n)= $$

$$ E(\beta^n)  - {n \choose 1} E(\beta^{n-1})E(\alpha) +   \dots +
(-1)^{n-1}{n \choose
n-1}E(\beta) E(\alpha^{n-1}) + (-1)^n E(\alpha^n) = $$

$$ b_n - {n
\choose 1} b_{n-1}a_1 + {n \choose 2}b_{n-2}a_2 - \dots + \dots +
(-1)^{n-1}{n \choose
n-1}b_1a_{n-1} + (-1)^n a_n. $$

Why is $A$ an invariant? This is best seen in umbral notation:

$$ A(T^cq(x), T^cp(x)) \cong (\beta + c - \alpha - c )^n = (\beta -
\alpha)^n. $$

The invariant $A$ is called the apolar invariant; two polynomials $p(x)$
and $q(x)$ having
the
property that $A(q(x), p(x)) = 0 $ are said to be apolar. In umbral
notation,  two
polynomials
are apolar whenever

$$ (\beta - \alpha)^n \cong 0. $$

 The concept of apolarity has a distinguished pedigree going all the way
back to
Apollonius.

\bigskip

What is the "significance" of the apolar invariant? What does it mean for two
polynomials to be apolar? This question is answered by

\bigskip

{\bf Theorem 1}. Suppose that $r$ is a root of the polynomial $q(x)$, that is,
that $q(r)= 0$.
Then the
polynomials $q(x)$ and $p(x) = (x - r)^n $ are apolar.

\bigskip

Proof. For $p(x) = (x-r)^n $ we  have $\alpha^j \cong (-r)^j $, and hence

$$ A(q(x),p(x)) \cong (\beta - (-r))^n = (\beta + r)^n \cong 0, $$

as desired.

\bigskip

{\bf Corollary}. If the polynomial $q(x)$ has $n$ distinct roots $r_1,r_2,\dots
, r_n $, and if
the
polynomial $p(x)$ is apolar to $q(x)$, then there exist constants
$c_1,c_2,\dots ,c_n $ for
which

$$ p(x) = c_1 (x-r_1)^n + c_2 (x-r_2)^n + \dots + c_n (x-r_n)^n . $$

\bigskip

Proof. The  dimension of the affine subspace of all
monic polynomials $p(x)$ which are apolar to $q(x)$ equals $n$. But if  the
polynomial
$q(x)$
has simple roots, then by the above theorem the polynomials $(x-r_1)^n,
(x-r_2)^n,\dots , (x-
r_n)^n $ are
linearly  independent and apolar to $q(x)$. Hence the polynomial $p(x)$ is
a linear
combination  of these polynomials. This completes the proof.

\bigskip

Thus, we see that apolarity  gives a trivial  answer to the following
question: when can a
polynomial $p(x)$ be written as a linear combination of polynomials  of the
form $(x-
r_1)^n,
(x-r_2)^n,\dots , (x-r_n)^n $?

\bigskip

A beautiful theorem on apolarity was proved by the British mathematician
John Hilton
Grace.
We state it without proof:

\bigskip

{\bf Grace's Theorem}. If two polynomials $p(x)$ and $q(x)$ of degree $n$ are
apolar, then
every
disk in the complex plane containing every zero of $p(x)$ also contains at
least one zero of
$q(x)$.

\bigskip

Grace's Theorem is an instance of what might be called a sturdy theorem.
For almost one
hundred years it has resisted all attempts at generalization.  Almost all
known  results
about the distribution of zeros of polynomials in the complex plane are
corollaries of
Grace's
theorem.

\bigskip

We will next generalize the apolar invariant  to the case of two
polynomials $p(x)$ and
$q(x)$ of
different degrees $n$ and $k$, with $ k \leq n$. To this end, we  slightly
generalize the
definition of invariant, as follows.

\bigskip

A polynomial $ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x ) $ in the
variables

$a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k , x $  is said to be an
invariant of the
polynomials
$p(x)$, $q(x)$ when

$$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x) = $$
$$I(p_1(c),p_2(c), \dots , p_n(c), q_1(c),
q_2(c), \dots , q_k(c), x+ c ) $$

for all complex numbers $c$.

\bigskip

Sometimes these more general invariants are called covariants.

\bigskip

We   define a more general apolar invariant as follows:

$$ A(q(x), p(x)) \cong (\beta - \alpha)^k (x- \alpha )^{n-k}. $$

Again, we say that two polynomials $p(x)$ and $q(x)$ are apolar when
$A(q(x), p(x))$ is
identically zero, that is, zero for all $x$. Theorem 1 remains valid as
stated.  That is, if
$q(r) = 0$ then the polynomial $p(x) = (x-r)^n $ is apolar to $q(x)$.

\bigskip

Let us consider a special case.   Suppose that $q(x)$ is a quadratic polynomial
and $p(x)$ is a cubic polynomial:

$$ q(x) = x^2 + 2b_1 x + b_2 $$

and

$$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 .$$

Then we have, in umbral notation

$$ A(q(x), p(x)) \cong (\beta - \alpha)^2 (x - \alpha) =
\\ (\beta^2 - 2 \alpha \beta + \alpha^2)x - \alpha \beta^2 + 2 \alpha^2
\beta -
\alpha^2. $$

Evaluating the linear functional $E$, we obtain the following explicit
expression for the
apolar
invariant:

$$ A(q(x), p(x)) = E((\beta^2 - 2 \alpha \beta + \alpha^2)x  - \alpha
\beta^2 + 2
\alpha^2 \beta - \alpha^2) =$$
$$ E (\beta^2 ) - 2E( \alpha \beta) +E( \alpha^2)x +E ( -
\alpha \beta^2) + 2 E(\alpha^2 \beta)  -E( \alpha^2) =$$
 $$E (\beta^2 ) - 2E( \alpha)E( \beta) +E( \alpha^2)x +E ( - \alpha)E(
\beta^2) + 2
E(\alpha^2)E(
\beta)  -E( \alpha^2) = $$
$$ (b_2 - 2 a_1b_1 + a_2)x  - a_1 b_2 + 2 a_2 b_1 - a_3. $$

Thus,  a quadratic polynomial $q(x)$ and a cubic polynomial $p(x)$ are
apolar if and only
if
their coefficients satisfy the two equations

$$ b_2 - 2 a_1b_1 + a_2 = 0 $$

$$ - a_1 b_2 + 2 a_2 b_1 - a_3 = 0 . $$

Using these equations, we can prove two important theorems:

\bigskip

{\bf Theorem 2}. There is in general one quadratic polynomial which is
apolar to
a given cubic
polynomial.

\bigskip

Proof. Indeed, the above equations may be
rewritten as

$$ b_2 - 2 a_1b_1 = -  a_2  $$

$$ - a_1 b_2 + 2 a_2 b_1 = a_3. $$

The solutions $b_1, b_2$ for given   $a_1, a_2, a_3 $ are in general unique.

\bigskip

{\bf Theorem 3}.  There  is always a two-dimensional space of cubic polynomials
which are
apolar to
a given quadratic polynomial.

\bigskip

Proof. Indeed, given  $b_1, b_2$ we may solve for  $a_1, a_2, a_3 $  from
the equations

$$- 2 a_1b_1 + a_2 = - b_2 $$

$$  - a_1 b_2 + 2 a_2 b_1 = a_3 .$$

These equations always have a double infinity of solutions, as they used to
say in the old
days.

\bigskip

Theorems 2 and 3 provide a simple and  explicit  method for  solving a
cubic equation. It
goes as
follows.

\bigskip

Given the cubic polynomial

$$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 ,$$

first, by Theorem 2 we  find a unique quadratic polynomial $q(x)$ which is
apolar to
$p(x)$. In
general, such a  quadratic polynomial $q(x)$ has two distinct roots $r_1$
and $r_2$.  By
Theorem 1, the cubic polynomials $ (x-r_1)^3$ and $(x-r_2)^3$ are apolar to
$q(x)$.
Second, by
Theorem 3, the affine linear space of cubic polynomials apolar to $q(x)$
has dimension
two. Since
$p(x)$ is apolar to $q(x)$, we conclude  that $p(x)$ is a linear
combination of  $ (x-
r_1)^3$ and
$(x-r_2)^3$.  In symbols:

$$ p(x) = c (x-r_1)^3 + (1-c) (x-r_2)^3 $$

for some constant $c$. Observe that $c$, $r_1$ and $r_2$ are  computed by
solving
linear and quadratic equations.

\bigskip

In this way, the solution of the cubic equation $p(x) = 0$ is reduced to
the solution of the
equation

$$ c (x-r_1)^3 = -  (1-c) (x-r_2)^3 ,$$

and this equation is easily solved by taking a cube root.

\bigskip

This method of solving a cubic equation is the only one I can remember.

\bigskip

Let me digress with a personal anecdote. A few years ago, I was lecturing
on this material
at a symposium in combinatorics that took place at the University at
Minnesota. Persi
Diaconis
was sitting in the front row, and I could tell as I started to lecture that
he was falling asleep;
he
eventually began to doze off. But the moment  I mentioned the magic words
"solving a
cubic
equation" he woke up with a start and said: "Really! How?"

\bigskip

The  preceding two theorems are easily generalized.

\bigskip

{\bf Theorem 4}.  The dimension of the space of all (monic) polynomials of
degree
$k$ that
are
apolar to a polynomial of degree $n$ equals $2k-n$, in general, when $ k
\leq n $.

\bigskip

{\bf Theorem 5}.  The dimension of the space of all (monic) polynomials of
degree
$n$ that
are
apolar to a polynomial of degree $k$  equals $k$, if $k \leq n$.

\bigskip

Let us  try to solve an equation of degree 5 in much the same way as we
solved a cubic equation. Given
 the quintic polynomial

$$ p(x) = x^5 + 5a_1x^4 + 10 a_2 x^3 + 10 a_3 x^2 + 5 a_4 x + a_5 = 0 ,$$

Theorem 4 assures us that there is in general a unique cubic polynomial
$q(x)$ which is
apolar to
$p(x)$. In general, this cubic polynomial has three distinct roots $r_1,
r_2, r_3$. By
Theorem 1,
the polynomials $(x-r_1)^5, (x-r_2)^5, (x-r_3)^5 $ are linearly independent
and  apolar to
$q(x)$ . By Theorem 5, the dimension of the space of all polynomials apolar
to $q(x)$
equals
$3$. But the polynomial $p(x)$ is apolar to $q(x)$. Hence, $p(x)$ can be
written in the
form

$$ p(x) = c_1(x-r_1)^5 + c_2(x-r_2)^5 + c_3(x - r_3)^5 $$

for suitable constants $c_i$. Thus, we see that a generic polynomial of
degree 5 can be
written as a
linear combination of three fifth powers of linear polynomials. These are
computed by
solving
linear, quadratic and cubic equations.  This reduction to canonical form of
the quintic is as
close
as we can come to solving a quintic equation by radicals.

\bigskip

At this point, someone in the audience will raise his or her hand and say:
"Excuse me, but
the
umbral method you have introduced is not even good enough to express the
discriminant of
a
quadratic equation! "

\bigskip

Quite right.

\bigskip

The  definitions of umbrae and of the linear functional $E$ have an obvious
generalization
to
any array of polynomials, say $p_1(x), p_2(x),\dots , p_{\ell}(x) $.  One
simply
considers  the
space of polynomials

$$ {\bf C}[x, \alpha_1 ,\alpha_2, \dots , \alpha_{\ell}  ] $$

and one sets

$$E(\alpha_{t}^j )$$

to equal the $j$-th coefficient of the polynomial $p_{t}(x)$. What is
crucial, the linear
functional $E$ is again  multiplicative on distinct umbrae:

$$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell} ) =  E(\alpha_1^i
)E(\alpha_2^j) E(
\alpha_3^k )\dots x^{\ell}.$$

Now comes the catch that in the old days was to turn into a notational
nightmare: the
polynomials say $p_1(x), p_2(x),\dots , p_{\ell}(x) $ need not be distinct.
In fact, the
most
important case occurs when each of the polynomials $p_1(x), p_2(x),\dots ,
p_{\ell}(x) $
is
equal to one and  the same polynomial $p(x)$. In this case, the definition
of the linear
functional
$E$ may be simplified as follows:

1. $$E(\alpha_i^j) = a_j $$  for every $i$ , and

2. $$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell}) =  a_i a_j a_k
\dots x^{\ell} $$

for all non negative integers $i,j, k, \dots \ell $.

\bigskip

Umbrae  $ \alpha_1 ,\alpha_2, \dots , \alpha_{\ell}  $ satisfying 1 and 2
are said to be
exchangeable. Thus, for exchangeable umbrae we have

$$ (x + \alpha_1)^n \cong (x+\alpha_2)^n. $$

Eric Temple Bell, who wrote $=$ in place of $\cong$, was baffled by the
fact that
two umbrae could be   exchangeable without being equal.

\bigskip

We can now state the main theorem of invariant  theory. We will consider  a
single
polynomial.

\bigskip

{\bf Theorem 6}.  Every invariant of a polynomial $p(x)$ is obtained by
evaluating some
polynomial
in the differences $\alpha_i - \alpha_j $ and $\alpha_i - x,$ where
$\alpha_i $ and $\alpha_j
$ are
exchangeable umbrae.  Conversely, every polynomial in such differences is
equivalent to an
invariant  of the polynomial $p(x)$.

\bigskip

The proof  is extremely simple, but will be omitted.

\bigskip

Let us review some classical examples.

\bigskip

The discriminant of a quadratic polynomial $p(x) = x^2 +2a_1x+a_2 $  may be
umbrally
represented as follows:

$$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2 / 2, $$

where $\alpha_1 $ and $\alpha_2 $ are exchangeable umbrae.  Indeed:

$$ E( (\alpha_1 - \alpha_2)^2 ) =E( \alpha_1^2) - 2 E(\alpha_1\alpha_2) +
E(\alpha_2
^2)  =a_2 - 2 a_1^2 + a_2 = 2 (a_2 - a_1^2), $$

as desired.

\bigskip

Let us next consider a cubic polynomial $p(x) = x^3 + 3a_1 x^2 + 3a_2 x +
a_3 $.  The
discriminant of this polynomial, let us call it $D(p(x)) $, equals, as you
know,  the
expression

$$ D(p(x)) = \frac{4(a_2 - a_1)^2(a_1a_3 - a_2^2) - (a_3 - a_1a_2)^2}{2}
.$$

The umbral expression of the discriminant is  easier to remember.

$$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_3 - \alpha_4)^2(\alpha_1 -
\alpha_4)(\alpha_2 - \alpha_3). $$

As you know, the discriminant vanishes if and only if the cubic equation
$p(x) = 0 $ has a
double root.

\bigskip

The Hessian of a cubic polynomial can be elegantly written in umbral
notation as follows:

$$ H(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_1 - x)(\alpha_2 - x).  $$

The Hessian vanishes if and only if the
cubic polynomial is the third power of a polynomial of degree one.

\bigskip

Allow me another digression. On hearing about the vanishing of the Hessian
as the
condition
that a cubic polynomial be a perfect cube, it comes naturally to ask the
general question:
which
invariant  of  a polynomial of degree $n$ vanishes if and only if the
polynomial is the $k$-
th
power of some polynomial of degree $n/k$? Here $k$ is a divisor of $n$.
For a long time I thought the answer to this question to be beyond reach,
until one day,
while
leafing despondently through the second volume of Hilbert's collected
papers, I
accidentally
discovered  that Hilbert had completely solved it. The solution can be
elegantly expressed
in
umbral notation.  This is only one of several striking results of Hilbert's
on invariant theory
that
have been forgotten.

\bigskip

Let us consider next an invariant of the quintic. Theorem 3 tells us that a
quintic
$p(x) = x^5 + 5a_1x^4 + 10 a_2x^3 + 5 a_3x^4 + a_5 $ has a unique apolar cubic
polynomial
$q(x)$. The polynomial $q(x)$ is an invariant of $p(x)$. Does it have a simple
expression in umbral notation? Indeed it does. The expression  is the
following:

$$ q(x) \cong (\alpha_2 - \alpha_3)^2(\alpha_3 - \alpha_1)^2(\alpha_1 -
\alpha_2)^2(\alpha_1 - x)(\alpha_2 - x)(\alpha_3 - x). $$

In the classical literature this invariant is denoted by the letter $j$.

What  property  will the quintic polynomial $p(x)$ have when the invariant
$j$ vanishes?

The answer to this question is pleasing. The invariant $j$ of a quintic
polynomial is
identically
equal to zero if and only if the quintic is apolar to some non trivial
polynomial
 of degree two. But then Theorem 5 tells us  that the quintic may be
written in the form

$$ p(x) = c(x-r_1)^5 + (1-c)(x-r_2)^5, $$

where $r_1, r_2 $ are the roots of a quadratic equation. Thus, the vanishing of
the invariant $j$ is a necessary and sufficient condition that the quintic
polynomial $p(x)$
may
be written as the sum of two rather than three fifth powers of linear
polynomials. When this
is
the  case, the fifth degree equation $p(x) = 0$ can be solved by radicals.

By  similar arguments, one  can compute all invariants whose vanishing
implies that the
equation of degree five is algorithmically solvable by radicals.
Twenty-three invariants
play a
relevant role, as Cayley was first to show.

\bigskip

Hilbert's theorem on finite generation of the ring of invariants can be
recast in the language
of
umbrae, and can be given a simple combinatorial proof that dispenses with
the Hilbert basis
theorem.

\bigskip

In closing, let us touch upon another  reason for the demise of the
symbolic  method in
invariant theory.

\bigskip

In mathematics, it is extremely difficult to tell the truth. The formal
exposition of a
mathematical
theory does not tell the whole truth. The truth of a
mathematical theory is more likely to be grasped while we listen to a
casual remark made by some
expert that gives away  some  hidden motivation,  when we finally pin down the
typical examples,
or
when we  discover what the real problems are that were stored behind
the showcase
problems.

\bigskip

Philosophers and psychiatrists should explain why it is that we
mathematicians are in the
habit of
systematically erasing our footsteps.  Scientists have always looked
askance at this strange
habit
of mathematicians, which has changed little from Pythagoras to our day.

\bigskip

The hidden purpose of the symbolic method in invariant theory was not
simply that of
finding
easy  expressions for invariants. A deeper faith was guiding this method. It
was the expectation that the expression of invariants by the symbolic
method would
eventually
guide us to single out the "relevant" or "important" invariants among an
infinite variety.  It
was the
hope  that the  significance of the vanishing of an invariant could be
gleaned from its
umbral expression. The vanishing of this faith was the real reason for the
demise of
classical
invariant theory, and the revival of this faith is the reason for its
present rebirth.

\bigskip

Whether or not we will succeed this second time where the classics failed
is a cliffhanger
that  will
probably be resolved in the next few years.  I would not be speaking to you
now if I did
not believe
in  success.

\bigskip

Thank you for your attention.

\bigskip

{\bf Bibliography}

\bigskip

J.P.S. Kung and Gian-Carlo Rota, The invariant theory of binary forms,
Bulletin of the American Mathematical Society, vol.10 (1984), pp. 27-85.

\bigskip

Frank D. Grosshans, Gian-Carlo Rota and Joel A. Stein, Invariant Theory
and Superalgebras, CBMS Regional Conferences in Mathematics, vol. 69,
American Mathematical Society,1987.

\bigskip

N. Metropolis and Gian-Carlo Rota, Symmetry classes: functions of three
variables, American Mathematical Monthly, vol. 98 (1991), pp. 328-332.

\bigskip

N. Metropolis, G.-C. Rota and Joel A. Stein, Theory of symmetry classes,
Proceedings of the National Academy of Sciences, vol. 88 (1991), pp.
8415-8419.

\bigskip

Richard Ehrenborg and Gian-Carlo Rota, Apolarity and canonical forms for
homogeneous polynomials, European Journal of Combinatorics, vo. 14
(1993), pp. 157-181.

\bigskip

Gian-Carlo Rota and B. D. Taylor, The classical umbral calculus, SIAM
Journal of Mathematical Analysis, vol. 25 (1994), pp.694-711.

\bigskip

Antonio Di Crescenzo and Gian-Carlo Rota, Sul calcolo umbrale, Ricerche
di Matematica, vol XLIII (1994), pp. 129-162.

\bigskip

N. Metropolis, Gian-Carlo Rota and Joe. A. Stein, Symmetry classes of
functions, Journal of Algebra, vo. 171 (1995), pp. 845-866.

\newpage

\begin{center} {\bf COMBINATORIAL SNAPSHOTS\\

being\\

The third of three Colloquium Lectures\\

delivered at the Annual Meeting of the American Mathematical Society\\

Baltimore, January 9, 1998\\

Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\ Cambridge MA 02139-4307\\}

\end{center}

\bigskip

\bigskip

When I was in high school, my English teacher gave me to read   an essay by
James
Thurber, called "The secret life of Walter Mitty".  After rereading this essay
every few years,  I  decided that everyone  has a Walter Mitty complex.
One  way
to understand a person  might be to discover that person's Walter Mitty
fantasies.

Most  of our mathematical thoughts in high school or in college were Walter
Mitty
fantasies. When we learned a new piece of math, we would find ourselves
fantasizing on its possible generalizations. As soon as we understood binomial
coefficients, we fantasized about their generalization to the case when the
denominator is negative; the moment we learned about derivatives, we launched
into derivatives of fractional order.  If we were ever exposed to the Riemann
zeta function, we would  romanticize some new interpretation of this function
that would give away its secret.

\bigskip

This lecture should have been given   another title.  It should be called "The
later life of Walter Mitty". It  will  consist of a sequence of displays of
"chutzpah" by a Walter Mitty who has lost his shyness. Each snapshot will deal
with  some    youthful  fantasy that has
partially worked out.

\newpage

FIRST SNAPSHOT: AN EXAMPLE OF PROFINITE COMBINATORICS.

\bigskip

Let  us begin with a piece of history-fiction, and fantasize how Riemann might
have discovered the Riemann zeta function.

\bigskip

Professor Riemann was aware that arithmetic density is of fundamental
importance
in number theory. If $A$ is a subset of the set of  positive integers
$N$,
then the arithmetic density of the set $A$ is defined to be

$$ dens(A)=\lim_{n\rightarrow \infty} {1\over n} \|{A\cap \{1,2, \dots ,
n\}}\|.$$

\bigskip

whenever the limit exists. For example $dens(N) = 1$. If $A_p$ is the set of
multiples of the prime $p$, then $dens(A_p) = \frac{1}{p}$; what is more
appealing, one easily computes that $dens (A_p \cap A_q)  = \frac{1}{pq} $
for any
two primes $p$ and $q$. If density were a (countably additive) probability
measure, we would infer that  the events that a randomly chosen number is
divisible by either of two primes are independent. Unfortunately, arithmetic
density shares some but not all
properties of a probability measure. It is most emphatically not countably
additive.

\bigskip

After a period of soul searching, Professor Riemann was able to find a
remedy to
some deficiencies of arithmetic density by a brilliant leap of imagination. He
chose a real number $s > 1$, and defined the measure of a positive integer
$n$ to
equal $\frac{1}{n^s}$; in this way, the measure of the set $N$ turned out
to equal

$$ \zeta (s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$

\bigskip

Therefore, he could  define a (countably additive) probability measure
$P_s$ on
the set
$N$ of positive integers by setting

$$ P_s(A) = \frac{1}{\zeta (s)} \sum_{n \in A }\frac{1}{n^s}. $$

Riemann then proceeded to verify what he had sensed all along, namely, the
fundamental property

$$P_s(A_p \cap A_q) = P_s(A_p)P_s(A_q) = \frac{1}{pq}. $$

In other words, the  events $A_p$ and $A_q$ that a randomly chosen integer
$n$  be divisible by one of the two primes $p$ or $q$ are independent
relative to the probability
$P_s $.

The Riemann zeta function was good for something, after all.

\bigskip

I will now use a rhetorical device that was effectively employed by one
of my undergraduate teachers, Professor Bochner.   In the classroom,
Professor Bochner
would prefix the statement of a theorem by the words: "Subject to technical
assumptions, the following is true"; without, of course, ever disclosing
what his
technical assumptions were.

\bigskip

Professor Riemann then proceeded to show that, subject to technical assumptions
on the set $A$,

$$ \lim_{s\rightarrow 1}P_s(A) = dens (A). $$

\bigskip

Thus, even  though arithmetic density is not a probability, it is under
suitable
conditions the limit of probabilities.

\bigskip

Long after Riemann was gone, it was shown, again subject to technical
assumptions, that the probabilities $P_s$ are the only probabilities defined on
the set $N$ of natural integers for which the events of divisibility by
different
primes are independent. This fact seems to lend support to the program of
proving
results of number theory by probabilistic methods based upon the Riemann zeta
function.

\bigskip

Why didn't Professor Riemann ever publish this wonderful idea of his? The
answer
is not hard to find. True, some theorems of number theory can be proved
probabilistically by this limiting process, for example Dirichlet's theorem on
primes in arithmetic progression.  However, deeper number theoretic results
have
to this day eluded this approach, for example, no one has succeeded in proving
the prime number theorem by this method. Professor Riemann, aware of this
deficiency, threw his notes into the wastebasket and proceeded to link the
Riemann zeta function to the distribution of primes in an altogether different
way, by stating the hypothesis that bears his name and that remains unproved to
this day.

\bigskip

Why am I telling you this bit of history-fiction? Because I want to propose
another probabilistic interpretation of the Riemann zeta function that is quite
different
from the interpretation just outlined.

\bigskip

Let us consider a problem in combinatorial enumeration. Let us take  a
cyclic group of order $r$, say $C_r $.   Every character
$\chi $ of the group $C_r$ has a kernel which is a subgroup of $C_r$. More
generally, every  sequence $\chi_1, \chi_2, \dots , \chi_s $ of characters of
$C_r$ has a joint kernel which is also a subgroup of $C_r$; the joint
kernel of a
sequence of characters is simply the intersection of the kernels of each of the
characters in the sequence  $\chi_1, \chi_2, \dots , \chi_s $ . If a sequence
$\chi_1, \chi_2, \dots , \chi_s $ of $s$ characters is chosen independently and
at random, what is the probability that the joint kernel of the sequence
equals a
given subgroup $C_n $ of $C_r $?

\bigskip

The probability of  the event that  the  kernel of a randomly chosen
character
will contain the subgroup $C_n$ equals $\frac{1}{n}$, since there are $r$
characters of the group $C_r$ and $\frac{r}{n}$ such characters  will vanish on
$C_n $. Therefore, the probability that the joint kernel of a randomly chosen
sequence  $\chi_1, \chi_2, \dots , \chi_s $  of $s $ characters shall
contain the
subgroup $C_n $ equals $(\frac{1}{n})^s $. Let us denote by $P_{C_n}$ the
probability that the joint kernel of the characters   $\chi_1, \chi_2, \dots ,
\chi_s $ shall equal  the subgroup $C_n $. Then we have the identity

$$ { 1 \over n^s} = \sum_{n|d|r} P_{C_d}.$$

Here, we use the fact that the partially ordered set of subgroups of a cyclic
group $C_r$ is isomorphic to the partially ordered set of divisors of the
integer
$r$.

\bigskip

We now use the M\"obius inversion formula of number theory, thereby obtaining

$$ P_{C_n} = \sum_{n|d|r} \mu (d/n){1 \over d^s}.$$

Here, $\mu(j) $ is the M\"obius function of number theory.

\bigskip

After the change of variable  $d = nj $  we can recast the right hand side as
follows:

$$ P_{C_n} = { 1 \over n^s}  \sum_{j}\mu (j){1 \over j^s}.$$

The variable $j$ on the right ranges over some subset of  divisors of the
integer
$r$, which we need not worry about.

\bigskip

Now if the sum on the right ranged
 over all positive integers $j$, then the right hand side would equal

$${ 1 \over n^s} \frac{1}{\zeta(s)},$$

\bigskip

 that is, it could be expressed in terms of
 the inverse of the Riemann zeta function.  If we could change our
combinatorial problem to get an unrestricted sum on the right hand side,
then we
would have a probabilistic interpretation of the Riemann zeta function.

\bigskip

This is done by replacing the finite cyclic group $C_n$ by a profinite cyclic
group.

\bigskip

Consider the  group $C_{\infty}$ of rational numbers modulo one. For every
positive integer $n$, the group $C_{\infty}$ has a unique finite subgroup $C_n$
with $n$ elements.  The character group
$C_{\infty}^*$ of $C_{\infty}$ is a compact group; it has a Haar measure
which is
a probability measure $P$. The group $C_{\infty}^* $ is the desired profinite
group on which we can generalize the preceding computation.

\bigskip

The set of all characters of the group
$C_{\infty}$ (that is, the set of all elements of the group  $C_{\infty}^* $)
which vanish on a subgroup
$C_n$ of $C_{\infty}$ has Haar measure equal to
$\frac{1}{n}$.  Thus, if we choose a sequence  $\chi_1, \chi_2, \dots ,
\chi_s $
of $s $ characters of
$C_{\infty}$  independently and at random, the probability that their joint
kernel will contain the group $C_n$ equals $(\frac{1}{n})^s$ .  If we again
denote by  $ P_{C_n} $ the probability that the joint kernel of a sequence
$\chi_1, \chi_2, \dots , \chi_s $  of $s $ characters equals the group $C_n$,
then we have the identity

$${ 1 \over n^s} = \sum_{n|d} P_{C_d},$$

where the sum on the right is now infinite. Again by the M\"obius inversion
formula we obtain

$$ P_{C_n} = \sum_{n|d} \mu (d/n){ 1 \over d^s} =
 \frac{1}{n^s}\frac{1}{\zeta(s)}.$$

This is the promised   probabilistic interpretation of the  Riemann zeta
function. Some properties of the Riemann zeta function can be proved
probabilistically using this
interpretation, for example, the product formula. It remains to be seen which
other  properties of the Riemann zeta function can be proved in this way.

\bigskip

The preceding argument is  an instance of a generalization of an
enumeration problem  on a finite set to an enumeration on a profinite
set.  Such a replacement of a finite set by a profinite "set" works in
other
 combinatorial problems. Will we ever have a profinite combinatorics on
profinite
sets  side by side with combinatorics on finite sets?

\newpage

SECOND SNAPSHOT: THE CYCLIC DERIVATIVE.

\bigskip

The ordinary derivative of a polynomial in one variable has been generalized by
Hausdorff to polynomials and formal power series in non commutative
variables as
follows.  Consider the associative algebra ${\bf{C}} \langle a,b,
\dots , c , x \rangle $ generated by a set of letters $\{a, b, \dots, c, x\}$.
The letter $x$ is called a variable, all other letters are called constants. A
monomial in this associative algebra is what you think it should be: it is a
word  like

$$ m = axbax^3bcxd.$$

A polynomial is a linear combination of monomials, and a formal power series is
defined as an infinite sum of monomials, with suitable restrictions on the
growth
of degrees of the summands. Formal power series in non commutative
variables form
an algebra ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $. We will
denote by $f(x)$ such a formal power series.

\bigskip

The Hausdorff derivative of the monomial $m$ is computed as follows:

$$ H (m) = H(axbax^3bcxd) = abax^3bcxd +3 axbax^2bcxd + axbax^3bcd. $$

This definition is extended by linearity to polynomials and to formal power
series.

\bigskip

If $m'$ is another monomial, we have the expected  rule for finding the
Hausdorff
derivative of a product:  $H(m m') = H(m)m' + m H(m'). $

\bigskip

The Hausdorff derivative suffers from a major weakness.  There seems to be no
analog of the chain rule for the differentiation of a function of a
function. For
example, the Hausdorff derivative of the polynomial $(ax)^n$, when the letters
$a$ and $x $ do not commute, is not equal to $n (ax)^{n-1}a $. It is a mess.

\bigskip

There is another notion of derivative that does satisfy a simple chain rule
under
functional composition. It is  the cyclic derivative, denoted  by the
letter $D$.

\bigskip

The cyclic derivative is defined as follows. First define  the truncation
operator $T$ as follows:

\bigskip

a. if the first letter of a monomial $m$ is not the variable $x$, set $T(m)
= 0;$

\bigskip

b. if the first letter of a monomial $m$ is the variable $x$, so that $m = x m'
$, set $T(m) = m'. $

\bigskip

c. Extend by linearity to  ${\bf C}\langle \langle a,b,\dots , c, x \rangle
\rangle $.

\bigskip

The cyclic derivative of a monomial $m$ is defined in terms of the truncation
operator as  follows:

\bigskip

a. Let $p$ be the polynomial obtained by adding all cyclic permutations of the
monomial $m$.

\bigskip

b. Set $D(m) = T(p).$

\bigskip

c. Extend by linearity to all formal power series.

\bigskip

 For example, the  cyclic derivative of the above monomial $m$  is computed in
the following steps:

\bigskip

Step 1. Write down all cyclic permutations on the monomial $ axbax^3bcxd$.
These
are

$$ xbax^3bcxda , bax^3bcxdax , ax^3bcxdaxb, x^3bcxdaxba ,
 x^2bcxdaxbax , xbcxdaxbax^2 ,$$
$$bcxdaxbax^3 , cxdaxbax^3b , xdaxbax^3bc , daxbax^3bcx . $$

\bigskip

Step 2. In the above list, perform one of the following operations:

\bigskip

a. if the first letter of a monomial is not $x$, remove the monomial from the
list;

\bigskip

b. if the first letter of a monomial is $x$, remove the first letter.

\bigskip

When we perform operations a. and b. on the each of the monomials in the
above list, we
obtain a shorter list, namely:

$$ bax^3bcxda ,  x^2bcxdaxba , xbcxdaxbax,  bcxdaxbax^2 , daxbax^3bc. $$

Step 3. Add the monomials thus obtained to get the cyclic derivative:

\begin{eqnarray*}D(m) & = & D(axbax^3bcxd) \\
 & = & bax^3bcxda +  x^2bcxdaxba + xbcxdaxbax+  bcxdaxbax^2 +  daxbax^3bc.
\end{eqnarray*}

Another example: the cyclic derivative of the monomial $axbxcxdx$ equals

$$D(axbxcxdx) = bxcxdxa + cxdxaxb + dxaxbxcx + axbxcxd. $$

The cyclic derivative of the monomial  $(ax)^n$ is the following:

\begin{eqnarray*} D((ax)^n) & = & D(axax \dots ax)\\ & = & axax\dots axa +
axax\dots axa + \dots axax\dots axa \\ & = & n
(ax)^{n-1} a.\end{eqnarray*}

Similarly, one computes

$$ D(x+a)^n = n(x+a)^{n-1}$$

and, for formal power series,

$$ D(e^{x+a}) = e^{x+a} $$

and

$$ D(e^{ax}) = e^{ax}a. $$

Remember,  the letters $a$ and $x $ do not commute! In  these
examples, the corresponding
Hausdorff derivative is a mess.

The  cyclic derivative enjoys all properties expected of the ordinary
derivative;
in particular, it satisfies the  chain rule for the composition of two formal
power series.

\bigskip

To state the rules for taking cyclic derivatives, we need one more operator,
called  the wrapping operator.

\bigskip

The  wrapping operator is defined as follows.
Let
$c_1,c_2,\dots ,c_n $ be any letters. If $g(x)$ is any formal power series, set

$$ \langle C c_1c_2\dots c_n | g(x) \rangle\\ = $$
$$c_1c_2\dots
c_n  g(x) + c_2\dots c_n  g(x)c_1 + c_3 \dots c_n  g(x)c_1c_2 + \dots + c_n
g(x)c_1c_2\dots c_{n-1}.$$

\bigskip

If $f(x)$ is any formal power series, the wrapping operator

$$ \langle C f(x) | g(x) \rangle $$

is defined by linearity.

\bigskip

Define

$$ \langle D(f(x)) | g(x) \rangle = T \langle C f(x) | g(x) \rangle. $$

For example:

$$ \langle D(f(x)) | 1 \rangle = D(f(x)). $$

\bigskip

The cyclic derivative of the product of two "functions" is given by the
following
identity:

$$ D(f(x)g(x)) = \langle D(f(x)) | g(x) \rangle + \langle D(g(x)) | f(x)
\rangle. $$

For example, one obtains

\bigskip

$$ D((1-ax)^{-1}(1-bx)^{-1})  =  (1-ax)^{-1}(1-bx)^{-1}(1-ax)^{-1}a $$

$$ +(1-bx)^{-1}(1-ax)^{-1}(1-bx)^{-1}b .$$

\bigskip

No such identity holds for the Hausdorff derivative.

\bigskip

The cyclic derivative of the  product of any sequence  of formal power
series  is
similarly computed by the wrapping operator:

$$D(f_1(x)f_2(x)\dots f_n(x)) = $$
 $$\langle D(f_1(x)) | f_2(x) \dots f_n(x)
\rangle + \langle D(f_2(x)) | f_3(x) \dots f_n(x)f_1(x) \rangle + $$
$$ \dots\\
 +
\langle D(f_n(x)) | f_1(x)f_2(x) \dots f_{n-1}(x) \rangle . $$

\bigskip

We come now to the main property of the cyclic derivative: the chain rule.
Given
two   formal power series $f(x) $ and $g(x) $ in ${\bf C}\langle \langle
a,b,\dots , c, x \rangle \rangle $, assume that the formal power series
$g(x) $ does not have a constant term.  Under these circumstances, the
composition $f(g(x))$ is well defined by replacing  $g(x) $ for every
occurrence
of the variable $x$ in the formal power series $f(x)$.

Let us write $ D_g(f(x)) $ to denote the formal power series  obtained by
substituting
$g(x)$ in place of every occurrence of $x$ in the cyclic derivative
$D(f(x))$ of
the formal power series  $f(x)$.  Then the chain rule for the cyclic derivative
goes as follows:

$$ D(f(g(x))) = \langle Dg(x) | D_g(f(x)) \rangle. $$

For example, we have

$$ D(e^{axbx}) = bx e^{axbx} a + e^{axbx}axb. $$

A more elegant example is the following:

$$ D(e^{(1-ax)^{-1}}) = (1-ax)^{-1} e^{(1-ax)^{-1}}(1-ax)^{-1} a. $$

One can prove that the cyclic derivative of a rational formal power series  in
non commutative letters is again a rational non commutative power series,
and that the cyclic derivative of an algebraic formal power series in non
commutative
letters is
again an algebraic formal power series.

\bigskip

Despite the  evidence that the cyclic derivative is the natural notion of
derivative for non commutative algebras, the theory as it is at present is
not satisfying. The cyclic derivative is an empirical discovery. It needs
to be ensconced in some broader algebraic theory, much like the Hausdorff
derivative has been ensconced in the theory of Hopf algebras.

\newpage

THIRD SNAPSHOT: LOGARITHMS AND THE BINOMIAL THEOREM.

\bigskip

The  Euler-MacLaurin summation formula is one of the most remarkable
formulas of
 mathematics.  For a suitable function $f(x)$ of a real or complex variable,
it is stated as follows:

$$ f(x) + f(x+1) + f(x+2) + \dots + f(x+n) =$$
$$ B_0 \int_x^{x+n+1}f(y ) dy + B_1
(f(x+n+1) - f(x)) $$
$$ + {B_2 \over 2!} D (f(x+n+1) - f(x))  + {B_3
\over 3!}  D^2 (f(x+n+1)- f(x))  + \dots .$$

The $B_n$ are the Bernoulli numbers and  $D$ is the ordinary derivative
operator.

\bigskip

The Euler-MacLaurin formula has proved very useful for over two hundred years.
Nonetheless, the Euler-MacLaurin formula suffers from a serious deficiency.
The
series on the   right hand side is almost never convergent, unless it
reduces to
a finite sum.

\bigskip

Our question is the following:  is there a vector space of functions which
contains as many of the elementary functions as possible, and a topology on
such
a vector space, relative to which the right hand side of the Euler-MacLaurin
formula is a convergent series?

\bigskip

The answer to this question is unexpectedly related to the answer to another
 question. What is the "right" generalization of the
binomial
coefficients ${n \choose k}$ when $k$ is allowed to be a negative integer? This
question leads in turn to a third question: how shall we know whether a
generalization of the binomial coefficients is  "right"? The answer to
this third question is easy: a generalization of the binomial coefficients is
"right" if it leads to a sensible generalization of the binomial theorem:

\bigskip

$$ (a+ x)^n = \sum_{k=0}^n {n \choose k}a^k x^{n-k}. $$

\bigskip

When I was young, I used to think of the binomial theorem as trivial. I think I
have learned my lesson. A well-known philosopher, I can't remember his
name, wrote
that the whole universe can be inferred from  a grain of sand. He should have
added that a great deal  of mathematics can be derived by meditating upon the
binomial theorem.

\bigskip

Let us take the bull by the horns, and state the "right" generalization of the
binomial coefficients.  We proceed in the most pedestrian way, by first
generalizing the definition of the factorial.  Thus, let $n$ be any integer,
positive or negative. We define the Roman factorial $[n]!$ as follows:

$$ [n]! = n! $$

if $ n \geq 0  $, and

$$ [n]! = {(-1)^{n+1} \over (-n-1)!} $$

 if $ n < 0 $.

\bigskip

Where does this definition come from ? I could simply say that it works,
but that
would not be the whole truth. The value of   the Roman factorial $ [n]! $
for $n$
negative equals the residue of the gamma function at the integer $n-1$.

Using the Roman factorial, we define the Roman coefficients as follows:

$$ { n \brack k} = { [n]! \over [k]![n-k]!}. $$

When  $ n \geq k \geq 0 $, the Roman coefficients coincide
with the binomial coefficients.  For all integers $n$ and $k$, the Roman
coefficients share all elementary properties of binomial coefficients, such as
Pascal's triangle, etc.   However, there are some surprises in store, for
example, for $k$ positive we find

$$ {0 \brack -k}= {0 \brack k} = { (-1)^{k+1} \over k }. $$

\bigskip

Does this make any sense? Well, yes, because we can find a generalization
of the
binomial theorem that goes with this.  It is the following. Recall the power
series expansion of the logarithm:

$$ log(x+a) = log x + \sum_{k=1}^{\infty}{ (-1)^{k+1} \over k} {a^k \over
x^k}.$$

We can recast this power series expansion in terms of the Roman coefficients as
follows:

$$ log(x+a) = log x + \sum_{k=1}^{\infty} {0 \brack k} {a^k \over x^k}. $$

This is beginning to look like a generalization of the binomial theorem,
with the
logarithm playing the roles of zero-th power. Another  power series expansion
where the Roman coefficients make their appearance is the following:

$$ (x+a)(log(x+a) - 1) = x(log x - 1) + a log x + \sum_{k=2}^{\infty}{1
\brack   k} a^k x^{1 -k}. $$

Do we see a pattern? Well, let us try yet another power series expansion:

$$ (x +a)^2(log(x+a) - 1 - {1 \over 2}) =$$
$$ x(log x - 1 - {1 \over 2}) + {2
\brack 1}a x(log x - 1) + {2 \brack 2}a^2 log x + \sum_{k=3}^{\infty} {2
\brack k} a^k x^{2-k}. $$

Now we can leap to a generalization. For suitable functions $f(x)$, set

$$ D^{-1}f(x) $$

to be the unique indefinite integral of the function $f(x)$ which has constant
term equal to zero. Do not worry, this will make sense in a moment.

\bigskip

Define

$$ \lambda_n^{(1)} (x) = [n]! D^{-n}log x. $$

Here, $n$ is any integer, positive or negative. The functions
$\lambda_n^{(1)}(x)$
are called the harmonic logarithms of order one.  For $n$ positive we have

$$   \lambda_n^{(1)} (x) = x^n(log x - 1 - {1\over 2} - {1 \over 3} - \dots -{1
\over n}) $$

and

$$  \lambda_{-n}^{(1)} (x) = {1 \over x^n}.$$

Of course we also have

$$  \lambda_0^{(1)} (x) = log x.$$

We are now in a position to state the generalization of the binomial
theorem that
is associated with the harmonic logarithms. It goes as follows:

$$  \lambda_n^{(1)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k
\lambda_{n-k}^{(1)} (x).
$$

The  three identities above are special cases of this identity, for $n =
0,1,2$.

\bigskip

The generalization of the binomial theorem to harmonic logarithms gives nothing
new for negative exponents, where it reduces to the identity

$$(x+a)^{-n} = \sum_{k=0}^{\infty} { -n \choose k} a^k x^{-n-k}.$$

However, for positive exponents we obtain a genuine and baffling generalization
of the binomial theorem. It states that the functions $  \lambda_n^{(1)} (x) $,
for $n $
positive, satisfy the ordinary binomial theorem, modulo negative powers of
$x$.  In other words, we have the following identity:

$$ (x+a)^n(log(x+a) - 1 - {1\over 2} - {1 \over 3} - \dots - {1 \over n})
\cong $$

$$\sum_{k=0}^n {n \choose k}a^k x^{n-k}(log(x) - 1 - {1\over 2} - {1
\over 3} -
\dots - {1 \over {n-k}}). $$

The identity is valid modulo negative powers of $x$. Miracles of
cancellation are
occurring in this identity. I wish I knew a combinatorial or probabilistic
interpretation of this logarithmic generalization of the binomial theorem.

\bigskip

So far, we have assumed that all series converge in the topology of the complex
numbers. We will now change the topology, while retaining the convergence.

\bigskip

The motivation for the logarithmic topology we are about to define is the
algebra of
formal Laurent series.  This topological algebra may be defined by defining a
topology on the algebra of rational functions in the variable $x$, and then
completing this algebra relative to the topology. The topology is so chosen
as to
have $lim_{n \rightarrow \infty}x^{-n} = 0 $. Every element of the completed
algebra turns out to be a formal Laurent series, that is, a series of the form

$$ \sum_{n < d} a_n x^n. $$

We want to perform an analogous completion process on another algebra: the
algebra generated by all functions of the form $x^n(logx)^t$, where $n$ is any
integer, positive or negative, and where $t$ is a non negative integer.

In
order
to specify which elements of this algebra are to converge to zero, we need a
better behaved basis of this algebra. This basis is provided by the harmonic
logarithms of arbitrary order
$t$. They are defined as follows:

$$\lambda_n^{(t)}(x) =[n]! D^n (log x)^t $$

for every nonnegative integer $t$ and for every integer $n$ .  For
example, we have

$$\lambda_n^{(0)}(x) = x^n $$

for every nonnegative integer $n$, and

$$ \lambda_n^{(0)}(x) = 0 $$

for negative $n$.

\bigskip

Explicit expressions are known for the harmonic logarithms. For the harmonic
logarithms of order $2$ we have $\lambda_0^{(2)}(x) = (logx)^2 $ and for $n$
positive

$$\lambda_n^{(2)}(x) = x^n \big(  (logx)^2 - ( 2 + {2 \over 2} + \dots + {2
\over n})log x + 2 + {2 \over 2} (1+ {1 \over 2}) + \dots + {2 \over n}(1+ {1
\over 2} + \dots + {1 \over n}) \big), $$

and

$$ \lambda_{-n}^{(2)}(x) = 2x^{-n} \big( log x - 1 - {1\over 2} - \dots - {1
\over n-1} \big). $$

For  every non negative integer $t$ the harmonic logarithms of order $t$
satisfy
the same generalization of the binomial theorem that we have already seen
for the
harmonic logarithms of order $1$:

$$  \lambda_n^{(t)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k
\lambda_{n-k}^{(t)} (x).
$$

The harmonic logarithms are a basis of the algebra generated by all functions
$x^n(logx)^t$.  We define a topology on this algebra by requiring that

$$ lim_{n \rightarrow - \infty}\lambda_n^{(t)}(x) = 0 $$

 for every non negative
integer $t$.  This topology  is called the logarithmic topology.  The
completion
of this algebra relative to the logarithmic topology is the algebra of formal
power series of logarithmic type, or logarithmic algebra.

\bigskip

Every element of the logarithmic algebra is a linear combination of convergent
power series of the form

$$ f(x) = \sum_{t, n \leq d} b_{n,t} \lambda_n^{(t)}(x) $$

ranging over a finite set of values of $t$.

\bigskip

We can now return to the Euler-MacLaurin summation formula:

\bigskip

{\bf Theorem}. For every element $f(x)$ of the logarithmic algebra  the right
hand side of the Euler-MacLaurin series converges in the logarithmic topology.

\bigskip

For example, the following infinite series is convergent in the logarithmic
topology:

$$ log x + log(x+1) + log(x+2) + \dots + log(x+n) = $$
 $$B_0 ((x+n+1)log(x+n+1) - xlogx - n - 1 ) + B_1 ( log(x+n+1) - logx ) +$$
$${B_2 \over 2!}  ({1 \over x+n+1 } - {1 \over x}) + \dots ) . $$

\bigskip

Another example is the following. As you know, the sum

$$ x^k + (x+1)^k + (x+2)^k + \dots + (x+n)^k $$

can be expressed in closed form by the Euler-Maclaurin formula. The preceding
theorem leads to analogous closed form expressions for sums of the form

$$ x^klogx + (x+1)^klog(x+1) + (x+2)^klog(x+2)+ \dots + (x+n)^klog(x+n) .$$

The harmonic logarithms have other applications, let us mention one in
closing.
\bigskip

Recall the definition of the shift operator of the calculus of finite
differences:

$$E^af(x) = f(x+a).$$

For $n$ a non negative integer, define the operator $E_1$ as follows:

$$ E_1 \lambda_n^{(0)}(x) = \lambda_n^{(1)}(x). $$

In ordinary notation, this is the same as saying

$$ E_1 x^n = x^n(log x - 1 - 1/2 - 1/3 - \dots - 1/n). $$

One can prove the following two propositions:

\bigskip

{\bf Proposition}. The operators $E^a$ and $E_1$ commute.

\bigskip

{\bf Proposition}. The restriction of the derivative operator $D$ to the
subalgebra of the logarithmic algebra generated by the harmonic logarithms
$\lambda_n^{(t)}(x)$ for positive
$t$ (that is, excluding the non negative powers of $x$) is invertible.

\bigskip

 These two
propositions   can be used to obtain "logarithmic extensions" of special
functions. Let us conclude with the simplest example: let us compute  the
logarithmic extension of the sequence of lower factorials, namely, the
polynomials
$ (x)_n = x(x-1)(x-2)\dots (x-n+1)$. This sequence
of polynomials satisfies the difference equation

$$ \Delta (x)_n = n(x)_{n-1},$$

where $\Delta$ is the difference operator: $\Delta f(x) = f(x+1) - f(x).$

This sequence can be extended to negative $n$ by setting

$$ (x)_{-n} = { 1 \over {(x+1)(x+2)...(x+n)} }, $$

and we have

$$ \Delta (x)_{-n} = -n(x)_{-n-1}.$$

For positive $n$, we may define the logarithmic extension of this sequence by
setting

$$  (x)_{-n}^{(1)} = (x)_{-n} ={ 1 \over {(x+1)(x+2)...(x+n)} }. $$

For example, $(x)_{-1}^{(1)} = { 1\over x+1}$.

\bigskip

The elements $ (x)_{-n}^{(1)}$ belong to the submodule of the logarithmic
algebra spanned by $\lambda_n^{(1)}(x)$, as $n $ ranges over all integers.  On
this submodule, the operator
$\Delta$ is invertible, and we can therefore set

$$ (x)_n^{(1)} = \Delta^{-n-1}{1 \over x+1}$$

for all non negative integers $n$.

\bigskip

It  turns out that the  element    $(x)_0^{(1)}$ is given by the following
series, convergent in the
logarithmic topology:

$$ (x)_0^{(1)} = log(x+1) + {B_1 \over 1+x} - {B_2 \over 2(1+x)^2} + {B_3
\over { 3(1+x)^3 }} - \dots . $$

But this is a familiar object: it is the $\Psi$- function,  heuristically
introduced by Gauss.  Gauss motivated the
$\Psi$-function as the "right" solution of the difference equation

$$ \Delta \Psi(x+1) = {1 \over { x+1}}. $$

We have now rigorously verified  Gauss's guess. Further computations show that
the elements $ (x)_1^{(1)}$ and $ (x)_2^{(1)} $ also coincide with special
functions introduced by Gauss, namely, the digamma and trigamma functions,
which
are at last rigorously defined by infinite series convergent in the logarithmic
topology.

\bigskip

In a similar vein, one defines logarithmic extensions of the Bernoulli
polynomials, the Hermite polynomials, etc., and one finds that the
asymptotic expansions of these polynomials reappear naturally as members of
the
logarithmic
extensions of these functions. As a matter of fact, the logarithmic topology
allows us to replace asymptotic expansions by  series which are convergent in
the logarithmic topology.

\bigskip

In closing, two open problems may be mentioned.

\bigskip

First, no closed form expression is known for  the coefficients of the
expansion
of a product

$$\lambda_n^{(t)}(x) \lambda_k^{(s)}(x) $$

into a logarithmic power series.  Second, we do not know a combinatorial or
probabilistic  interpretation of the Roman coefficients ${n \brack k}$ in
general.

\bigskip

Thank you for listening.

\bigskip

{\bf Bibliography}

\bigskip

J.P. S. Kung, M. Ram Murthy and Gian-Carlo Rota, The Redei zeta function,
Journal of Number Theory, vol. 12 (1980), pp. 421-436.

\bigskip

Gian-Carlo Rota, Bruce Sagan and Paul R. Stein, A cyclic derivative in
noncommutative algebra, Journal of Algebra, vol. 64 (1980), pp.54-75/

\bigskip

Daniel E. Loeb and Gian-Carlo Rota, Formal power series of logarithmic
type, Advances in Mathematics, vol.75 (1989), pp. 1-118.

\bigskip

Kenneth S. Alexander, Kenneth Baclawski and Gian-Carlo Rota, A stochastic
interpretation of the Riemann zeta function, Proceedings of the National
Academy of Sciences, vo. 90 (1993), pp. 697-699.

\bigskip

 J.P.S. Kung, ed., Gian-Carlo Rota on Combinatorics, Birkh\"auser
Boston, 1996, ISBN 0-8176-3713-3 or ISBN 3-7643-3713-3

\bigskip

\end{document}

--============_-1322120415==_============--

Date:           Wed, 18 Mar 1998 14:45:10 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

Note, the seminar is basically booked through the end of the semester
If anyone didn't get a chance to speak who wanted to, please let me
know so I can schedule a talk in the fall.

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Alex Suciu
Title:   Characteristic varieties of real and complex arrangements 
Abstract: 
The k-th Fitting ideal of the Alexander invariant of
an arrangement A of n complex hyperplanes defines a
characteristic subvariety, V_k(A), of the complex algebraic
torus (C^*)^n.  The characteristic varieties of an arrangement
provide subtle and effectively computable homotopy-type invariants
of its complement.  In joint work with Daniel Cohen, we show that
the tangent cone at the identity of the top characteristic variety
V_1(A) coincides with R_1(A), the first-cohomology support locus
of the Orlik-Solomon algebra.  We conclude that the variety V_1(A)
is combinatorially determined, and that Falk's variety R_1(A) is the
union of a subspace arrangement in C^n.  We illustrate these techniques
by computing the top characteristic varieties of braid arrangements
and monomial arrangements.

If A is a real 2-arrangement (in the sense of Goresky and McPherson),
the characteristic varieties are no longer subtori through the origin.
The nature of these varieties vividly illustrates the difference
between real and complex arrangements.  In joint work with Daniel Matei,
we study the homotopy types of complements of arrangements of n
transverse planes in R^4, obtaining a complete classification for n<=6,
and lower bounds for the number of homotopy types in general.  Furthermore,
we show that the homotopy type of the complement of a 2-arrangement in R^4
is not determined by its cohomology ring, thereby answering a question
of Ziegler.

The papers on which the talk will be based can be found at
<a href="http://www.math.neu.edu/~suciu/publications.html">
http://www.math.neu.edu/~suciu/publications.html</a>.

April 1: Jim Haglund
      Further investigations involving polynomials with only real roots 
April 8: Irena Peeva
      Monomial ideals, real Boolean subspace arrangements, and 
      order dimension of lattices 
April 10: Karen Collins
      Breaking symmetries of S5 and S6 
April 17: Anders Buch
      Chern class formulas for degeneracy loci 
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 
April 29: Patricia Hersh
      Shuffle posets of multisets 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 25 Mar 1998 10:18:30 -0500
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        second announcement

             Come to the Twenty-ninth one day conference on

                     Combinatorics and Graph Theory

                        Saturday, April 4, 1998

                         10 a.m. to 4:30 p.m.
                                 at
                            Smith College
                         Northampton MA 01063

                               Schedule

10:00  Van Vu (Yale University)
        Witness on the Upper Bound of Chromatic (Choice) Number
         of Random Graphs

11:10  Richard Stanley (MIT)
        Spanning Trees and a Conjecture of Kontsevich

12:10  Lunch

 2:00  Eckhard Steffen (Princeton University)
        Snarks

 3:10  Therese Biedl (McGill University)
        Efficient Algorithms For Petersen's Theorem

The next meeting is scheduled for May 2nd.

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Fri, 27 Mar 1998 15:06:47 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminars in April

Hope you are all enjoying spring break.  No talk today.  Below is the
schedule for April.  I just added a second talk this coming week.  All
abstracts are on the web as usual:
	http://www-math.mit.edu/~combin

April 1: Jim Haglund
      Further investigations involving polynomials with only real roots 
April 3: Manjul Bhargava
      The factorial function... and generalizations 
April 8: Irena Peeva
      Monomial ideals, real Boolean subspace arrangements, and order 
      dimension of lattices 
April 10: Karen Collins
      Breaking symmetries of S5 and S6 
April 15: Patricia Hersh
      Shuffle posets of multisets 
April 17: Anders Buch
      Chern class formulas for degeneracy loci 
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 
April 29: To be announced.

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Mon, 30 Mar 1998 12:12:05 -0500 (EST)
From:           Jim Propp <propp(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        test message --- please ignore

I'm just checking to make sure that auto-archiving is once again working
properly.

Jim Propp

Date:           Tue, 31 Mar 1998 13:07:13 -0500
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        third announcement

             Come to the Twenty-ninth one day conference on

                     Combinatorics and Graph Theory

                        Saturday, April 4, 1998

                         10 a.m. to 4:30 p.m.
                                 at
                            Smith College
                         Northampton MA 01063

                               Schedule

10:00  Van Vu (Yale University)
        Witness on the Upper Bound of Chromatic (Choice) Number
         of Random Graphs

11:10  Richard Stanley (MIT)
        Spanning Trees and a Conjecture of Kontsevich

12:10  Lunch

 2:00  Eckhard Steffen (Princeton University)
        Snarks

 3:10  Therese Biedl (McGill University)
        Efficient Algorithms For Petersen's Theorem

The next meeting is scheduled for May 2nd.

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Tue, 31 Mar 1998 14:14:23 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar tomorrow

*********Combinatorics Seminar TOMORROW at 4:15 in 2-338**********
                 refreshments served at 3:45 

                   ** Pretalk at 3:30-4:00**

Speaker: Jim Haglund (MIT)
Title:   Further investigations involving polynomials with only real
	roots 
Preprint: http://www-math.mit.edu/~combin/preprints
Abstract:
We present a number of conjectures involving rook polynomials
having only real zeros.  Many of these generalize a previous
conjecture of the author, K. Ono, and D. G. Wagner, namely that if
$A$ is a real $n \times n$ matrix which is weakly increasing 
down columns, then the permanent of $zA+J_n$ has only real zeros.  
In some cases these conjectures are motivated by factorization theorems 
for Ferrers boards.  Connections between results of Laguerre and Szeg\"o
on transformations which send polynomials with only real roots to other 
such polynomials are discussed.  We also present a weighted version
of the Poset Conjecture of enumerative combinatorics.

Upcoming Events:

April 3: Manjul Bhargava
      The factorial function... and generalizations 
April 8: Irena Peeva
      Monomial ideals, real Boolean subspace arrangements, and order 
      dimension of lattices 
April 10: Karen Collins
      Breaking symmetries of S5 and S6 
April 15: Patricia Hersh
      Shuffle posets of multisets 
April 17: Anders Buch
      Chern class formulas for degeneracy loci 
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 1 Apr 1998 16:39:32 -0400
To:             combopt(at-sign)math.uwaterloo.ca (Combinatorics and Optimization Waterloo),
                combinatorics(at-sign)math.mit.edu, combinatorics(at-sign)euclid.UCSD.EDU,
                combinatorics(at-sign)cs.toronto.edu, blaw(at-sign)fields.toronto.edu
From:           bergeron(at-sign)pascal.math.yorku.ca (Nantel Bergeron)
Subject:        FPSAC/SFCA 98 

-------------------------------------------------------------------------
10-th Conference on Formal Power Series and Algebraic Combinatorics
10ieme conference sur les Series Formelles et la Combinatoire Algebriques

June 14 - 19 juin, 1998

Fields Institute, Toronto

-------------------------------------------------------------------------
OnLine Registration: reduced registration fees... before April 15
Inscription en ligne: frais reduit... avant le 15 avril

            "http://www.math.yorku.ca/bergeron"

-------------------------------------------------------------------------
Full Schudule and other informations:
Horraire complet et autres informations:

            "http://www.math.yorku.ca/bergeron"

-------------------------------------------------------------------------
INVITED TALKS/CONFERENCIER INVITES:

       G. Benkart (USA)
       P. Cameron (England)
       P. Dehornoy (France)
       B. Derrida (France)
       P. Diaconis (USA)
       C. Godsil (Canada)
       K. Ono (USA)
       J. Y. Thibon (France)
       B. Sturmfels (USA)

-------------------------------------------------------------------------
For more information contact:
Pour plus de renseignements contactez:

              FPSAC/SFCA 98
              Fields Institute
              222 College Street Toronto,
              ON M5T 3J1 Canada

              fpsac98(at-sign)fields.utoronto.ca
              bergeron(at-sign)mathstat.yorku.ca.

-------------------------------------------------------------------------
ORGANIZATION/COMITE D'ORGANIZATION:
N. Bergeron, Chairman (York U.),
M. Delest (U. de Bordeaux), F. Sottile (U. Toronto), W. W hiteley (York U.).

-------------------------------------------------------------------------

Nantel Bergeron                                   bergeron(at-sign)mathstat.yorku.ca
Associate Prof. Mathematics                          nantel(at-sign)math.harvard.edu
York University                                         nantel(at-sign)lacim.uqam.ca
http://www.math.yorku.ca/bergeron

Date:           Fri, 3 Apr 1998 12:01:56 -0500 (EST)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Manjul Bhargava
Title: The factorial function... and generalizations 

Abstract:  Though ubiquitous in combinatorics, the factorial function also
occurs surprisingly often in number theory.  A close examination of these
occurrences leads to a series of beautiful generalizations of the factorial
function, which also turn out to be quite useful in a variety of
combinatorial, number-theoretic, and ring-theoretic problems.  In particular,
a fundamental problem about integer-valued polynomials, put forth by P\'olya
in 1919, is now resolved.

Upcoming events:

April 8: Irena Peeva
      Monomial ideals, real Boolean subspace arrangements, and order 
      dimension of lattices 
April 10: Karen Collins
      Breaking symmetries of S5 and S6 
April 15: Patricia Hersh
      Shuffle posets of multisets 
April 17: Anders Buch
      Chern class formulas for degeneracy loci 
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 8 Apr 1998 13:25:20 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Irena Peeva (MIT)
Title:  Monomial ideals, real Boolean subspace arrangements, and order 
        dimension of lattices 
Abstract:
The talk is on a joint work with V. Gasharov, V. Reiner, and V. Welker.
We give a formula for the graded Betti numbers of a monomial ideal
in terms of the homology of the lower intervals in the lattice of least
common multiples of the minimal monomial ideal generators; the formula 
has the same flavour as the Goresky-MacPherson formula for the dimensions 
of the cohomology of the complement of a subspace arrangement.  This leads 
to a relation of the cohomological properties of real Boolean arrangements 
and square-free monomial ideals.  Another application of the formula is a 
relation between the homology and the order dimension of an arbitrary 
finite lattice.

Upcoming Events:

April 10: Karen Collins
      Breaking symmetries of S5 and S6 
April 15: Patricia Hersh
      Shuffle posets of multisets 
April 17: Anders Buch
      Chern class formulas for degeneracy loci 
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Fri, 10 Apr 1998 12:50:01 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Karen Collins
Title: Breaking symmetries of S5 and S6 
Abstract: A labeling of the vertices of a graph G is said to be
r-distinguishing, provided no automorphism of the graph
preserves all of the vertex labels. The distinguishing
number of a graph G, D(G), is the minimum r such that G
has an r-distinguishing labeling.  For any group J, we
define D(J) := { D(G) | Aut(G) is isomorphic to J }.  It
has been shown that D(S3) = {2,3}, and D(S4) = {2,4}.
In this talk, we will sketch the proof that D(S5) = {2,3,5}
and present other distinguishing results.

Upcoming Events:

April 15: Patricia Hersh
      Shuffle posets of multisets 
April 17: Anders Buch
      Chern class formulas for degeneracy loci 
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Sun, 12 Apr 1998 10:14:18 -0400
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        May 2nd 

               Come to the Thirtieth one day conference on

                     Combinatorics and Graph Theory

                        Saturday, May 2, 1998

                         10 a.m. to 4:30 p.m.
                                  at
                            Smith College
                         Northampton MA 01063

                               Schedule

10:00  Dan Rockmore (Dartmouth College)
        FFTs for $SL_2(F_q)$ - Theory and Applications

11:10  Randall McCutcheon (Wesleyan Univ.)
        TBA

12:10  Lunch

 2:00  Joel Spencer (New York Univ. -- Courant Institute)
            Sixty Years of Ramsey R(3,k)

 3:10  Lakshmibai (Northeastern Univ.)
        Flag varieties -- their Geometry & Combinatorics

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Wed, 15 Apr 1998 13:57:18 -0400 (EDT)
From:           Jim Propp <propp(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        Discrete Dinner

After a two-year hiatus, the BOSTON AREA DISCRETE DINNER is back.
The proposed dates for this term are May 6, May 8, and May 13.

For each of the proposed dates, please let me know what your probability
of attendance would be, if the dinner were held on that day.

If you have any particular recommendations for local restaurants, feel
free to submit those as well.  Also feel free to pass this message on
to combinatorialists and discrete mathematicians who aren't on the 
recipient list.

Jim Propp (propp(at-sign)math.mit.edu)

Date:           Wed, 15 Apr 1998 15:00:51 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Patricia Hersh
Title: Shuffle posets of multisets 

Abstract:
We study posets defined by Stanley as a multiset generalization
of Greene's shuffle posets.  We compute the flag f-vector 
in terms of the quasi-symmetric function encoding $F_P$ defined 
by Ehrenborg.  The result is a symmetric function which is 
expressible as a sum of Schur functions with nonnegative integer 
coefficients.  We find a topological chain decomposition which
yields $F_P$ and similarly gives the characteristic 
polynomial, zeta polynomial and rank generating function for
shuffle posets of multisets.  This decomposition also leads to a 
symmetric group action on maximal chains with Frobenius 
characteristic related to $F_P$. 

Upcoming Events:

April 17: Anders Buch
      Chern class formulas for degeneracy loci 
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 
**April 29: Henrik Eriksson
      Sorting bridge hands and DNA.

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Fri, 17 Apr 1998 13:57:36 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Anders Buch
Title: Chern class formulas for degeneracy loci 
Preprint:"http://www-math.mit.edu/~combin/abstracts/apr98/quiver.ps"

Abstract:
We study a general class of degeneracy loci associated to a sequence of
vector bundles with maps between them and arbitrary rank conditions on 
the maps and their compositions.  The cohomology classes of such loci are
described by polynomials in the Chern classes of the vector bundles.  
These polynomials generalize all known types of Schubert polynomials.  We
give explicit formulas for the polynomials as linear combinations with
integer coefficients of products of Schur determinants.  We furthermore
conjecture that all coefficients are positive and given by counting
tableaux. 

Upcoming Events:
April 22: Andrzej Rucinski
      On graphs with linear Ramsey numbers 
April 24: Lou Shapiro
      Path pairs and Catalan probabilities 
April 29: Henrik Eriksson.
      Sorting bridge hands and DNA 
May 6:David Ingerman (NYU) 
      Inverse boundary problems, arrangements of pseudo-lines and 
      total positivity 
May 8: Christian Lenart (MIT) 
      Combinatorial Aspects of the K-theory of flag varieties 
May 13: Jim Propp (MIT)
      Diabolo tilings of fortresses 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Mon, 20 Apr 1998 10:16:15 -0400
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        second announcement

               Come to the Thirtieth one day conference on

                     Combinatorics and Graph Theory

                        Saturday, May 2, 1998

                         10 a.m. to 4:30 p.m.
                                  at
                            Smith College
                         Northampton MA 01063

                               Schedule

10:00  Dan Rockmore (Dartmouth College)
        FFTs for $SL_2(F_q)$ - Theory and Applications

11:10  Randall McCutcheon (Wesleyan Univ.)
        Sets of recurrence in semigroups and Ramsey theory

12:10  Lunch

 2:00  Joel Spencer (New York Univ. -- Courant Institute)
            Sixty Years of Ramsey R(3,k)

 3:10  Lakshmibai (Northeastern Univ.)
        Flag varieties -- their Geometry & Combinatorics

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Wed, 22 Apr 1998 15:58:56 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Andrzej Rucinski
Title:  On graphs with linear Ramsey numbers 
Abstract: For a fixed graph $H$, the \emph{Ramsey number} $r(H)$ is defined
to be the least integer $N$ such that in any 2-coloring of the
edges of the complete graph $K_{N}$, some monochromatic copy
of $H$ is always formed. Let ${\mathcal H}(n,\Delta)$ denote the
class of graphs $H$ having $n$ vertices and maximum degree $\Delta$.
It was shown by Chvat\'al, R\"odl, Szemer\'edi and Trotter that for each
$\Delta$ there exists $c(\Delta)$ such that $r(H) < c(\Delta)n$
for all $H \in {\mathcal H}(n,\Delta)$. That is, the Ramsey numbers
grow \emph{linearly} with the size of $H$. However, their proof
relied on the well-known regularity lemma of Szemeredi and only
gave an upper bound for $c(\Delta)$ which grew like an exponential
tower of $2's$ of height $\Delta$.

In this talk we avoid the use of the regularity lemma altogether,
and show that one can in fact take, for some fixed $c$, 
$c(\Delta) < 2^{c\Delta(\log \Delta)^{2}}$ in the general case, and even 
$c(\Delta) < 2^{c\Delta\log \Delta}$ if $H$ is bipartite. 
In particular, we improve an old upper bound on the Ramsey number of the 
$n$-cube due to Beck \cite{B}.
We also show that for all $n$,
there are $H' \in {\mathcal H}(n,\Delta)$ with $r(H') > 2^{c'\Delta}n$
for a fixed $c' > 0$, which is not so far from our upper bound.

This is a joint work with R.L.Graham and V.R\"odl.

Upcoming Events:

April 24: Lou Shapiro
      Path pairs and Catalan probabilities 
April 29: Henrik Eriksson.
      Sorting bridge hands and DNA 
May 6:David Ingerman (NYU) 
      Inverse boundary problems, arrangements of pseudo-lines and 
      total positivity 
May 8: Christian Lenart (MIT) 
      Combinatorial Aspects of the K-theory of flag varieties 
May 13: Jim Propp (MIT)
      Diabolo tilings of fortresses 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Fri, 24 Apr 1998 13:08:49 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Lou Shapiro
Title:  Path pairs and Catalan probabilities 
Abstract. Path pairs are pairs of paths starting at the origin, each
consisting of unit east and north steps, with the upper path never
going below the lower path, and with the two rapths having a common
endpoint. They are in many ways the mother of all Catalan settings.
Togrther with generating functions and k-motzkin paths we can derive
probabalistic statements about many Catalan phenomena. For instance
as n gets large the number of elements in the visible blocks of a
noncrossing partition approaches 8. A variety of such results will be
discussed including walking through an Eulerian polygonal maze.

Upcoming Events:
April 29: Henrik Eriksson.
      Sorting bridge hands and DNA 
May 6:David Ingerman (NYU) 
      Inverse boundary problems, arrangements of pseudo-lines and 
      total positivity 
May 8: Christian Lenart (MIT)  ***Pretalk at 3:30 in 2-338***
      Combinatorial Aspects of the K-theory of flag varieties 
May 13: Jim Propp (MIT)
      Diabolo tilings of fortresses 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Fri, 24 Apr 1998 15:55:13 -0400
To:             combinatorics(at-sign)math.mit.edu
From:           Donna Beers <beers(at-sign)artemis.simmons.edu>
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Lou Shapiro
Title:  Path pairs and Catalan probabilities 
Abstract. Path pairs are pairs of paths starting at the origin, each
consisting of unit east and north steps, with the upper path never
going below the lower path, and with the two rapths having a common
endpoint. They are in many ways the mother of all Catalan settings.
Togrther with generating functions and k-motzkin paths we can derive
probabalistic statements about many Catalan phenomena. For instance
as n gets large the number of elements in the visible blocks of a
noncrossing partition approaches 8. A variety of such results will be
discussed including walking through an Eulerian polygonal maze.

Upcoming Events:
April 29: Henrik Eriksson.
      Sorting bridge hands and DNA 
May 6:David Ingerman (NYU) 
      Inverse boundary problems, arrangements of pseudo-lines and 
      total positivity 
May 8: Christian Lenart (MIT)  ***Pretalk at 3:30 in 2-338***
      Combinatorial Aspects of the K-theory of flag varieties 
May 13: Jim Propp (MIT)
      Diabolo tilings of fortresses 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Mon, 27 Apr 1998 13:33:57 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        [daisymae(at-sign)math.mit.edu: Applied Mathematics Colloquium -- Monday, April 27, 1998]

Just in case you are not on the Applied Math Colloquium list, here a
talk you may be interested in.

Date:           Mon, 27 Apr 1998 11:15:03 -0400 (EDT)
From:           Shirley Entzminger-Merritt <daisymae(at-sign)math.mit.edu>
To:             amc(at-sign)math.mit.edu, gradstu(at-sign)math.mit.edu
Subject:        Applied Mathematics Colloquium -- Monday, April 27, 1998  

				T   O   D   A   Y  

			APPLIED MATHEMATICS COLLOQUIUM

TOPIC:		COMBINATORICS AND TOPOLOGY OF GRAPH PROPERTIES

SPEAKER:	PROFESSOR ANDERS BJORNER
		Department of Mathematics
		Royal Institute of Technology
		Sweden

DATE:		MONDAY, APRIL 27, 1998

TIME:		4:15 P.M.

LOCATION:	Building 2, Room 105

Refreshments will be served at 3:45 p.m. in Building 2, Room 349

Applied Math Colloquium:  http://www-math.mit.edu/amc/spring98
Math Department:  http://www-math.mit.edu

					Massachusetts Institute of Technology
					Department of Mathematics
					Cambridge, MA  02139

Date:           Wed, 29 Apr 1998 10:17:52 -0400
To:             kcollins(at-sign)mail.wesleyan.edu
From:           kcollins(at-sign)wesleyan.edu (Karen L. Collins)
Subject:        third announcement

               Come to the Thirtieth one day conference on

                     Combinatorics and Graph Theory

                        Saturday, May 2, 1998

                         10 a.m. to 4:30 p.m.
                                  at
                            Smith College
                         Northampton MA 01063

                               Schedule

10:00  Dan Rockmore (Dartmouth College)
        FFTs for $SL_2(F_q)$ - Theory and Applications

11:10  Randall McCutcheon (Wesleyan Univ.)
        Sets of recurrence in semigroups and Ramsey theory

12:10  Lunch

 2:00  Joel Spencer (New York Univ. -- Courant Institute)
            Sixty Years of Ramsey R(3,k)

 3:10  Lakshmibai (Northeastern Univ.)
        Flag varieties -- their Geometry & Combinatorics

The conferences are supported by an NSF grant which allows us
to provide a modest transportation allowance to those attendees
who are not local.  We also gratefully acknowledge support from
Smith College and Wesleyan University.

Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is:  http://math.smith.edu/~rhaas/coneweb.html

Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)math.smith.edu

Karen Collins (Wesleyan University), (860) 685-2169,
kcollins(at-sign)wesleyan.edu

Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)math.smith.edu

Date:           Wed, 29 Apr 1998 13:17:22 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: Henrik Eriksson.
Title:  Sorting bridge hands and DNA 
Abstract: Most bridge players start by sorting their hand, repeatedly
moving blocks of cards. Open problem: Can this be done in at most
seven moves? Solved problem: Sort the reversed permutation in seven
moves!

Block moves are fundamental mutations of DNA-molecules, so the block
move distance can also be used for phylogenic estimates.  The talk
will review what is known about the problem and what we believe to be
true. (Joint work with Kimmo Eriksson, KTH.)

Upcoming Events:
May 6:David Ingerman (NYU) 
      Inverse boundary problems, arrangements of pseudo-lines and 
      total positivity 
May 8: Christian Lenart (MIT)  ***Pretalk at 3:30 in 2-338***
      Combinatorial Aspects of the K-theory of flag varieties 
May 13: Jim Propp (MIT)
      Diabolo tilings of fortresses 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Fri, 01 May 1998 11:25:15 +0100
From:           "a. Iarrobino" <iarrobin(at-sign)neu.edu>
Subject:        of possible interest
To:             combinatorics(at-sign)math.mit.edu

Combinatorics related talk at Northeastern GAS seminar:

Monday May 4 at 1:30 PM    Geometry-Algebra-Singularities Seminar:

                        "Total positivity and double Bruhat cells"
                        Andrei Zelevinsky (Northeastern)
                        Northeastern University, 509 Lake Hall

Abstract:

  An invertible matrix is totally nonnegative if all its minors
(including all matrix entries) are nonnegative real numbers.
We discuss the generalization (due to G. Lusztig) of this classical notion
to any semisimple group.
The natural geometric framework for this study is provided by the decomposition
of the group into a disjoint union of double Bruhat cells
(intersections of cells in two Bruhat decompositions with respect to
opposite Borel subgroups.
  This is joint work with S. Fomin

Date:           Mon, 4 May 1998 12:30:12 -0400 (EDT)
From:           Jim Propp <propp(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        Discrete Dinner

The Boston area

                D I S C R E T E   D I N N E R

will be held at

                    P H O   P A S T E U R

(a Vietnamese restaurant at 35 Dunster Street in Harvard Square) at

                        6 : 0 0   P M
on

              W E D N E S D A Y ,   M A Y   1 3

R.S.V.P. to Jim Propp (propp(at-sign)math.mit.edu) by Friday, so that we can
make a reservation for a table of suitable size.  (If the number of 
people in your party is a random variable, please give its expected 
value.)

See you then,

Jim Propp

Date:           Wed, 6 May 1998 15:07:06 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

Speaker: David Ingerman (NYU, ingerman(at-sign)boheme.cims.nyu.edu)
Title:  Inverse boundary problems, arrangements of pseudo-lines and 
        total positivity 
Abstract: Abstract: 

Let $\Gamma$ be a planar graph embedded in a closed disk.  Each
positive weight function $\gamma$ on the edges of $\Gamma$ induces a
response map of the graph (the linear map from Dirichlet to Neumann
boundary data of $\gamma$-harmonic functions.)  We will consider
inverse boundary problems of obtaining information about the weighted
graph from the response map.  Our motivation for the study of these
problems comes from their continuous PDE analogs (and originally from
oil production industry). Surprisingly the answers turn out to be
combinatorial in nature. For example: Every weighted graph is
determined by its response matrix up to simple Reidemeister type
transformations. The quotient of the weighted graphs by the
transformations is essentially the set of circular permutations.
There is a natural one-to-one correspondence between the graphs for
which the inverse problem has unique solution and the arrangements of
pseudo-lines on the plane.
   The response matrices are completely characterized by their
total positivity property (sign conditions on determinants of minors.)

Upcoming Events:
May 8: Christian Lenart (MIT, lenart(at-sign)math.mit.edu)  
      Combinatorial Aspects of the K-theory of flag varieties		
***Pretalk at 3:30 in 2-338***
      
May 13: Jim Propp (MIT, propp(at-sign)math.mit.edu)
      Diabolo tilings of fortresses 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Fri, 8 May 1998 13:42:21 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

*********Combinatorics Seminar Today at 4:15 in 2-338**********
                 refreshments served at 3:45 

         	***Pretalk at 3:30 in 2-338***

Speaker: Christian Lenart (MIT, lenart(at-sign)math.mit.edu)  
Title:   Combinatorial Aspects of the K-theory of flag varieties
Abstract:
In this talk we present some results concerning Grothendieck
polynomials, which are representatives for the classes corresponding to
Schubert varieties in the K-theory of the flag variety. Grothendieck
polynomials are nonhomogeneous polynomials whose lowest homogeneous
component is the Schubert polynomial indexed by the same permutation;
hence they offer more information about the geometry of the flag variety
than Schubert polynomials. The first part of the talk is devoted to
Grothendieck polynomials corresponding to Grassmannian permutations; we
discuss their expansion in the basis of Schur polynomials and a Pieri
rule. In the second part of the talk, we sketch the proof of a
conjecture of A. Lascoux concerning the expansion of an arbitrary
Grothendieck polynomial in the basis of Schubert polynomials, and
present a combinatorial interpretation for the coefficients of this
expansion. The proof is based on certain noncommutative analogs of
Schubert polynomials, for which we prove a Pieri rule and a Cauchy
identity, thus extending the work of S. Fomin and C. Greene on
noncommutative Schur functions.

Upcoming Events:
      
May 13: Jim Propp (MIT, propp(at-sign)math.mit.edu)
      Diabolo tilings of fortresses 

Happy Summer Vacation, we will continue the combinatorics seminar on
Wednesday, September 9th. 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Wed, 13 May 1998 12:50:00 -0400 (EDT)
From:           Sara Billey <sara(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        seminar today

  *********Combinatorics Seminar Today at 4:15 **********
                 refreshments served at 3:45 

          ***NOTE: Seminar will be held in 2-131***

Speaker: Jim Propp (MIT, propp(at-sign)math.mit.edu)
Title:   Diabolo tilings of fortresses 
Abstract: 
Diabolo tilings of fortresses are combinatorial objects that share
some features with domino tilings of Aztec diamonds, rhombus tilings
of hexagons, and alternating sign matrices, but that also exhibit
novel phenomena that are not yet entirely understood.  After defining
these tilings, I will present Mihai Ciucu's simple proof of Bo-Yin
Yang's formula for the number of diabolo tilings of the fortress of
order n, using the graph-theoretic method of "urban renewal".  Then I
will show how extensions of urban renewal allow one to efficiently
sample uniformly from the set of such tilings, and to encode
statistical properties of random tilings via generating functions.

These techniques suggest, and should eventually allow us to prove, that
in addition to (homogeneous) "frozen regions" and a (non-homogeneous)
"liquid region", diabolo tilings of fortresses also exhibit "gaseous"
behavior in the center of the tiling, as seen in
"http://www-math.mit.edu/~propp/hidden/fort200.ps.gz"
a random diabolo tiling of a fortress of order 200 and
"http://www-math.mit.edu/~propp/hidden/fort300.5.ps.gz"
a color-plot of the local statistics 
for the ensemble of random diabolo tilings of a fortress of order 300.
Other displays used in this talk can be found at
"http://www-math.mit.edu/~propp/diabolo.ps.gz"

This is joint work with Mihai Ciucu, Henry Cohn, Rick Kenyon, David Wilson,
and the undergraduate members of the Tilings Research Group.
The talk presupposes no knowledge of tiling theory or statistical
mechanics.

Note that the talk will take place in room 2-131 (not the usual
meeting place for the seminar).

Upcoming Events:  Discrete Dinner tonight.
      
Happy Summer Vacation, we will continue the combinatorics seminar on
Wednesday, September 9th. 

*******************************************************
* Sara Billey                                         *
* Applied Mathematics Instructor                      *
* Massachusetts Institute of Technology               *
* Web page: http://www-math.mit.edu/~sara/            *
* Room: 2-334 Phone: (617)-253-7775                   *
*******************************************************

Date:           Tue, 9 Jun 1998 13:42:29 -0400 (EDT)
From:           Shirley Entzminger-Merritt <daisymae(at-sign)math.mit.edu>
Reply-To:       Shirley Entzminger-Merritt <daisymae(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        Special Lecture -- Friday, June 12, 1998

			COMBINATORICS SEMINAR
                                                                            
SPEAKER: 	Ezra Miller
		University of California, Berkeley                                         
                                                                            
TITLE: 		Alexander duality for arbitrary monomial ideals                      
                                                                            
DATE:		Friday, June 12, 1998                                                       
                                                                            
TIME: 		2:00 p.m.                                                             
                                                                            
LOCATION: 	MIT 
		Building 2, Room 338                                                           
                                                                            
Abstract:                                                                   
                                                                            
Given a simplicial complex $X$ on a finite vertex set, the {\it Alexander
dual} $X^\vee$ of $X$ is another simplicial complex (on the same vertex
set) whose homology is isomorphic to the cohomology of $X$.  This
construction carries over to the corresponding face ideals $I$ and
$I^\vee$, which are squarefree monomial ideals in a polynomial ring.  The
talk will generalize the construction of $I^\vee$ from $I$ to the
nonsquarefree case.  Several equivalent constructions of the {\it
Alexander dual ideal} will be outlined, relating duality in various
lattices to duality in regular cell complexes and the switching of minimal
generators for irreducible components of an ideal. 

Date:           Wed, 10 Jun 1998 12:58:01 -0400 (EDT)
From:           Shirley Entzminger-Merritt <daisymae(at-sign)math.mit.edu>
To:             combinatorics(at-sign)math.mit.edu
Subject:        Special Lecture -- Friday, June 12, 1998 -- CANCELLED

  NOTE:  THIS SEMINAR HAS BEEN CANCELLED.

 			COMBINATORICS SEMINAR
                                                                             
SPEAKER: 	Ezra Miller
 		University of California, Berkeley                                         
                                                                             
TITLE: 		Alexander duality for arbitrary monomial ideals                      
                                                                             
DATE:		Friday, June 12, 1998 -- CANCELLED                                                       
                                                                             
TIME: 		2:00 p.m.                                                             
                                                                             
LOCATION: 	MIT 
 		Building 2, Room 338