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In [2] Armond showed that the heads and tails of the colored Jones polynomial exist for adequate links. This was also shown independently by Garoufalidis and Le for alternating links in [8]. Here we study coefficients of the "difference quotient" of the colored Jones polynomial. We begin with the fundamentals of knot theory. A brief introduction to skein theory is also included to illustrate those necessary tools. In Chapter 3 we give an explicit expression for the first coefficient of the relative difference. In Chapter 4 we develop a formula of t_2, the number of regions with exactly 2 crossings in the diagram of a link, for a specific class of alternating links, and then improve with this result the upper bound of the volume for a hyperbolic alternating link which Dasbach and Tsvietkova gave in the coefficients of the colored Jones polynomial in [7].

It is known that the geometric Satake equivalence is intimately related to the Springer correspondence when restricting to small representations of the Langlands dual group (see a paper by Achar and Henderson and one by Achar, Henderson, and Riche). This dissertation relates the derived geometric Satake equivalence of Bezrukavnikov and Finkelberg and the derived Springer correspondence of Rider when we restrict to small representations of the Langlands dual group under consideration. The main theorem of the before-mentioned paper of Achar, Henderson, and Riche sits inside this derived relationship as its degree zero piece.

We focus on the two-dimensional Falicov-Kimball model in the neutral case with U >> 0. We determine the ground state ion configurations for all rational densities in (1/3, 2/5), as well as densities 1/6 and 2/11. We also determine the ground states for a sequence of densities starting at 1/5 and converging to 0. On the interval (1/3, 2/5) we show that the ground states are periodic, having the same structure as the one-dimensional ground states. For densities between 1/6 and 2/11 we show that the ground states exhibit a phase separation.

Two new general results about dynamical systems are obtained using the characteristics of their complex time series solutions. These series are obtained locally around movable singularities in the complex time domain via methods which are an extension of the Painleve-Kovalevskaya test for integrability and which therefore have the advantage of being algorithmic in nature. The first of these results applies to autonomous polynomial vector fields and provides necessary and sufficient conditions for the existence of an open set of initial conditions for which the solutions will diverge to infinity as time (i.e. the independent variable) approaches some finite real value. The conditions for blow-up involve only the asymptotic leading order coefficient of the local series representation for the general solution around the complex time singularities. Additional analyses lead to the second result, which involves exponentially small separatrix splitting. When an autonomous system of ODE's possessing a homoclinic or heteroclinic orbit is perturbed by a rapidly oscillating non-autonomous term, the resulting splitting distance of the separatrix becomes exponentially small. Therefore, any first order approximation technique for measuring this splitting, e.g. the Melnikov vector, apparently loses its validity. An accurate expression for the splitting distance is valuable because it can be used to detect the presence of chaos in the system. Using only the local asymptotic forms of the solutions to the linearized variational equations and of the perturbation term, sufficient conditions on the perturbation amplitude such that the Melnikov vector gives the proper leading order splitting distance are found. This result applies to autonomous polynomial vector fields with periodic perturbations for which the amplitude of the perturbation is inversely proportional to some algebraic order of the frequency, and it depends only on the asymptotic form of the solutions near the complex time singularities.

<p> Ohta has given a detailed study of the ordinary part of <i>p</i>-adic Eichler-Shimura cohomology groups (resp., generalized <i> p</i>-adic Eichler-Shimura cohomology groups) from the perspective of <i> p</i>-adic Hodge theory. Assuming various hypotheses, he is able to use the structure of these groups to give a simple proof of the Iwasawa main conjecture over <b>Q.</b> The goal of this thesis is to extend Ohta&rsquo;s arguments with a view towards removing these hypotheses.</p>

<p> Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials. </p><p> In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides an alternate proof of Kosters' degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result given by Zan and Cao.</p>

<p>We present a new approach to perform calculations with the certain standard classes in cohomology of the moduli spaces of curves. It is based on an important lemma of Ionel relating the intersection theoriy of the moduli space of curves and that of the space of admissible coverings. As particular results, we obtain expressions of Hurwitz numbers in terms of the intersections in the tautological ring, expressions of the simplest intersection numbers in terms of Hurwitz numbers, an algorithm of calculation of certain correlators which are the subject of the Witten conjecture, an improved algorithm for intersections related to the Boussinesq hierarchy, expressions for the Hodge integrals over two-pointed ramification cycles, cut-and-join type equations for a large class of intersection numbers, etc.</p>

A mathematical model describing transient processes in isoelectric focusing (IEF) of M biprotic ampholytes is proposed. The problem consists of nonlinear partial differential equations and algebraic equations under nonlinear boundary conditions. Different models of IEF have been studied. For each model problem, we investigated the qualitative properties such as the local existence, global boundedness, stabilizations, and steady-state structures of its solutions. We have shown that, for all models the solutions of the evolution problem stabilize to the steady-state solutions, which have separate peaks at a certain point (the so-called isoelectric point). This means that for transient IEF processes, the concentrations of ampholytes will focus on their isoelectric points as time goes on. All these analytic results showed good agreement with the laboratory experiments and computer simulations.

In order to obtain the equations of motion for a particle in a classical gauge field, a variational principle is considered. The theory is general in that the structural group is an arbitrary r-dimensional Lie group and the base space is an arbitrary n-dimensional psuedo-Riemannian manifold. An n + r dimensional principal fiber bundle is constructed in order to introduce the usual gauge potentials and field strengths. In addition, a set of r quantities (called "coupling parameters") which transform as the components of an adjoint type (0,1) object and also depend upon the parameter of the particle's trajectory are constructed. The gauge potentials and coupling parameters are evaluated on the identity section of the principle bundle, and the Lagrangian is assumed to be a C³ scalar function of these and of the components of the metric tensor and tangent vector on the base space. The Lagrangian is not gauge-invariant, but it is stipulated that when the arguments of the Euler-Lagrange vector (evaluated on the identity section) are replaced by their counterparts (which may be evaluated on an arbitrary section) the resulting vector must be gauge-invariant. A novel application of methods from the theory of functional equations is applied together with standard techniques inherent in the theory of differential equations to show that the arguments of the Lagrangian must occur together in certain prescribed combinations. The invariance postulates uniquely determine the Lagrangian in terms of its arguments other than the coupling parameters and r functions of the coupling parameters. The Lagrangian is shown to separate into a free-field term and an interaction term, and the functions of the coupling parameters are found to be the components of an adjoint type (0,1) quantity whose adjoint absolute derivative vanishes. This agrees with the equations of certain approaches to the Yang-Mills theory for isotopic spin particles.¹ Standard initial conditions are shown to determine a unique (local) solution to the derived equations of motion. ftn¹ The equations have the same formal structure as systems obtained in the classical limit of quantum mechanical results found by Wong (1), pp. 691-693.