The Algebraic Representation of Partial Functions

The paper presents a generalization of the theorem which states that any (everywhere defined) function from a finite field GF(p^n) into itself may be represented at a polynomial over GF(p^n). The generalization is to partial functions over GF(p^n) and exhibits representations of a partial function f by the sum of a polynomial and a sum of terms of the form a/(x-b)i, where b is one of the points at which f is undefined. Three such representation theorems are given. The second is the analog of the Mittag-Leffler Theorem of the theory of functions of a single complex variable. The main result of the paper is that the sum of the degree of the polynomial part of the representation and the degrees of the principal parts of the representation need be no more than max(|A|, |B|) where A is the set upon which the function is defined and B is the set upon which the function is undefined

Using cylindrical algebraic decomposition and local Fourier analysis to study numerical methods: two examples

Local Fourier analysis is a strong and well-established tool for analyzing the convergence of numerical methods for partial differential equations. The key idea of local Fourier analysis is to represent the occurring functions in terms of a Fourier series and to use this representation to study certain properties of the particular numerical method, like the convergence rate or an error estimate. In the process of applying a local Fourier analysis, it is typically necessary to determine the supremum of a more or less complicated term with respect to all frequencies and, potentially, other variables. The problem of computing such a supremum can be rewritten as a quantifier elimination problem, which can be solved with cylindrical algebraic decomposition, a well-known tool from symbolic computation. The combination of local Fourier analysis and cylindrical algebraic decomposition is a machinery that can be applied to a wide class of problems. In the present paper, we will discuss two examples. The first example is to compute the convergence rate of a multigrid method. As second example we will see that the machinery can also be used to do something rather different: We will compare approximation error estimates for different kinds of discretizations.Comment: The research was funded by the Austrian Science Fund (FWF): J3362-N2

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl

L-functions for Classical Groups: The Integral Representations, Algebraicity and the p-adic Interpolations

The main theme of this thesis is the study of special values of L-functions through integral representations. We present an integral representation of the standard L-functions for classical groups via the doubling method. Our computations, comparing with the well-known result for partial L-functions by Piatetski-Shapiro and Rallis, include all ramified local integrals with the explicit choice of local sections for Eisenstein series. When the classical group admits a Shimura variety, we have a well-defined notion of algebraic modular forms. In this case, we calculate the Fourier expansion of Eisenstein series from which the properties of their special values can be easily read off. Utilizing our integral representations, we then prove the algebraicity of certain special L-values for modular forms on some classical groups. Furthermore, by our proper choice of the local sections for Eisenstein series, we construct the p-adic L-functions interpolating these special L-values. Generalizing the classical doubling method, Cai, Friedberg, Ginzburg and Kaplan presents an integral representation for Sp(2n)×GL(k) by the twisted doubling method. In the final chapter of the thesis, we present another integral representation for the L-functions of Sp(2n)×GL(k) via a non-unique model and obtain some analytic results

Exact Results for the Distribution of the Partial Busy Period for a Multi-Server Queue

Exact explicit results are derived for the distribution of the partial busy period of the M/M/c multi-server queue for a general number of servers. A rudimentary spectral method leads to a representation that is amenable to efficient numerical computation across the entire ergodic region. An alternative algebraic approach yields a representation as a finite sum of Marcum Q-functions depending on the roots of certain polynomials that are explicitly determined for an arbitrary number of servers. Asymptotic forms are derived in the limit of a large number of servers under two scaling regimes, and also for the large-time limit. Connections are made with previous work. The present work is the first to offer tangible exact results for the distribution when the number of servers is greater than two