We consider the problem of determining bounds on the price of an option based on observable prices of other options and the no-arbitrage assumption. This problem has been addressed and solved for the case of European call options of a single maturity and a single underlying asset. In a multiple maturities case the existence of an equivalent martingale measure, which is responsible for securities prices, introduces much more difficulty to the problem, since then additional constraints must be imposed to guarantee the condition of martingale existence.

       We find exact bounds on the price of a European call option, if the set of priced options of different maturities is given. We also find exact bounds if the payoff function on the option is a piecewise linear function of the stock. We solve a similar problem in a two dimensional case. The methods developed for two-dimensional case are applicable to a multiple dimensional case, however the number of variables grows exponentially with the dimension.

       We also investigate how the optimal bounds in the one-dimensional case change, if, in addition to the set of observable prices, there is a condition on the variance of the underlying asset. We derive an exact solution for the tight upper bound on the variance and propose a polynomial time algorithm for obtaining the lower bound.

       In a portfolio allocation problem we prove that for large time horizons, there exists a myopic policy which leads to a distribution of the terminal wealth with the property that the probability of underperforming any other policy tends to zero as the horizon tends to infinity. We address the problem of maximization of the mean-variance function of the terminal wealth in a multi-period case. For general price dynamics of securities we propose a monte-carlo based method for the solution which is polynomial in the number of securities.

       With respect to the first part, we develop a new paradigm for proving lower bounds in propositional calculus. Our method is based on the purely computational concept of pseudorandom generator. Namely, we call a pseudorandom generator G_n:{0,1}ⁿ -> {0,1}^m hard for a propositional proof system P if P cannot efficiently prove the (properly encoded) statement G_n (x₁,...,x_n) b for any string b \in {0,1}^m. We consider a variety of "combinatorial" pseudorandom generators inspired by the Nisan-Wigderson generator on the one hand, and by the construction of Tseitin tautologies on the other. We prove that under certain circumstances these generators are hard for such proof systems as Resolution, Polynomial Calculus and Polynomial Calculus with Resolution (PCR).

       As to the second part, we prove that the problem of approximating the size of the shortest proof within factor 2^log1-o(1)n is NP-hard. This result is very robust in that it holds for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution, Horn resolution, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problem of approximating the length of proofs.

       Finally, we show that neither Resolution nor tree-like Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is fixed-parameter tractable by randomized algorithms with one-sided error.

1. We study the equilibrium and stability of electron bubbles. Electron bubbles are cavities formed around electrons injected into liquid helium. They can be treated as simple mathematical systems that minimize the energy with three terms: the energy of the electron proportional to a Laplace eigenvalue, the surface energy proportional to the surface area of the cavity, and the hydrostatic pressure proportional to its volume. This system possesses a surprising result: an instability of the 2S electron bubbles.
2. We compute the simple eigenvalues on a regular polygon with N sides. The polygon is treated as a perturbation of the unit circle and its eigenvalues are approximated by a Taylor series. The accuracy of our approach is measured by comparison with finite element estimates. For the lowest eigenvalue, the first Taylor term provides an estimate within 10-5 of the true value. The second term reduces the error to 10-7. We discuss how to utilize the available symmetry to obtain better finite element estimates. Finally, we briefly discuss the expansion of simple eigenvalues on regular polygons in powers of 1/N.

       A primary motivation for considering shifted complexes is a procedure called shifting. Shifting associates a shifted complex to any simplicial complex in a way which preserves meaningful information, while simplifying the structure of the complex. For example, shifting preserves the f-vector of a complex but always reduces the topology to a wedge of spheres. Shifting has proved to be a successful tool for answering questions regarding f-vectors.

       While most of the previous results on shifted complexes are algebraic or topological in nature, we explore the combinatorics of shifted complexes. We give intrinsic characterization theorems for shifted complexes and shifted matroid complexes. In addition, we show results on the enumeration of shifted complexes and connections to various combinatorial structures.

       Supersolvable lattices were introduced by Richard Stanley in 1972 and were shown to admit S_n EL-labellings. Examples include finite distributive lattices, the lattice of partitions of [n] and the lattice of subgroups of a supersolvable group (hence the terminology). We show that a lattice is supersolvable if and only if it has an S_n EL-labelling. As one of our tools, we introduce a naturally defined local action on the maximal chains of posets with S_n EL-labellings. We see that this action gives a representation of the Hecke algebra of type A at q=0. As a further desirable property, the character of this representation is closely related to the flag f-vector. We ask what other posets have an action with these properties and, in particular, we show that a finite graded lattice has such an action if and only if it has an S_n EL-labelling.

       It is an interesting problem, both from a mathematical and a physical point of view, to classify VA's which are generated by a Virasoro element L, a space g of even primary fields of conformal weight 1 (currents) and a space U of odd primary fields of conformal weight 3/2.

       I will discuss a way to approach this problem and describe the solution in the case g is a simple Lie algebra and U an irreducible g-module. I will also show how, under certain assumption on the values of the Kac-Moody levels, one can prove transitivity of the group action on the sphere. This generalizes a similar result of Kac for the case of Lie conformal algebras.

       In this thesis we construct tridiagonal matrix models for the general ( > 0) -Hermite and ( > 0, a > (m-1) / 2) -Laguerre ensembles, and investigate applications of these new ensembles, particularly in the areas of eigenvalue statistics. Using these models, we obtain Strong Laws of Large Numbers and first and second order perturbation theory for large , for all -Hermite and -Laguerre ensembles. In addition to this, we study the -ensembles in connection with Jack and Multivariate Orthogonal Polynomials, and obtain a duality principle.

       The thesis also contains an analysis of our Maple Library (MOPs: Multivariate Orthogonal Polynomials symbolically) which implements some new and some known algorithms for computing the Jack, Hermite, Laguerre, and Jacobi multivariate polynomials for arbitrary . This library can be used as a tool for conjecture-formulation and testing, for graphics, for statistical computations, or simply for getting acquainted with the mathematical concepts.