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The Unreasonable Effectiveness of Number Theory

This book is based on the AMS Short Course, The Unreasonable Effectiveness of Number Theory, held in Orono, Maine, in August 1991. This Short Course provided some views into the great breadth of applications of number theory outside cryptology and highlighted the power and applicability of number-theoretic ideas. Because number theory is one of the most accessible areas of mathematics, this book will appeal to a general mathematical audience as well as to researchers in other areas of science and engineering who wish to learn how number theory is being applied outside of mathematics. All of the chapters are written by leading specialists in number theory and provides excellent introduction to various applications.

Effectiveness. of. Number. Theory. in. Statistical. Mechanics. GEORGE. E.
ANDREWS. 1. Introduction. This paper will be somewhat more restricted than the
title implies. We shall concern ourselves only with additive number theory and
how it relates to statistical mechanics. Also it must be noted at the onset that this
paper was presented in a session entitled The Unreasonable Effectiveness of
Number Theory. The course description suggests that thirty years ago all
applications of ...

Critical Population and Error Threshold on the Sharp Peak Landscape for a Moran Model

The goal of this work is to propose a finite population counterpart to Eigen's model, which incorporates stochastic effects. The author considers a Moran model describing the evolution of a population of size of chromosomes of length over an alphabet of cardinality . The mutation probability per locus is . He deals only with the sharp peak landscape: the replication rate is for the master sequence and for the other sequences. He studies the equilibrium distribution of the process in the regime where

For the discovery time, there is no miracle: before discovering the master
sequence, the process is likely to explore a significant portion of the genotype
space, hence the discovery time should be of order card{A,T,G,C }l = 4l. These
simple heuristics indicate that the persistence time depends on the selection drift,
while the discovery time depends on the spatial entropy. Suppose that we send
m, l to ∞ simultaneously. If the discovery time is much larger than the persistence
time, then the ...

Ordered Exponential Fields

Model theoretic algebra has witnessed remarkable progress in the last few years. It has found profound applications in other areas of mathematics, notably in algebraic geometry and in singularity theory. Since Wilkie's results on the o-minimality of the expansion of the reals by the exponential function, and most recently even by all Pfaffian functions, the study of o-minimal expansions of the reals has become a fascinating topic. The quest for analogies between the semi-algebraic case and the o-minimal case has set a direction to this research. Through the Artin-Schreier Theory of real closed fields, the structure of the non-archimedean models in the semi-algebraic case is well understood. For the o-minimal case, so far there has been no systematic study of the non-archimedean models. The goal of this monograph is to serve this purpose. The author presents a detailed description of the non-archimedean models of the elementary theory of certain o-minimal expansions of the reals in which the exponential function is definable. The example of exponential Hardy fields is worked out with particular emphasis. The basic tool is valuation theory, and a sufficient amount of background material on orderings and valuations is presented for the convenience of the reader.

The Fields Institute--which generates this impressive series of books --is well known and the topics are written by top mathematicians. This monograph is priced for bookseller sales.

Lectures on Number Theory

This volume is a translation of Dirichlet's Vorlesungen uber Zahlentheorie which includes nine supplements by Dedekind and an introduction by John Stillwell, who translated the volume. Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions. The book is suitable as a textbook, yet it also offers a fascinating historical perspective that links Gauss with modern number theory.

Because it follows from 4>(a)ip(a) = f(a) = 0 (mod p) that at least one of the two
numbers 4>(a), ip{a) is divisible by p. Now if one of ... 54. See also the work from
Gauss's Nachlass: Analysis Residuorum, Gauss Werke, vol. II, 1863. 21 D. A. art.

Interpolation and Approximation by Rational Functions in the Complex Domain

The present work is restricted to the representation of functions in the complex domain, particularly analytic functions, by sequences of polynomials or of more general rational functions whose poles are preassigned, the sequences being defined either by interpolation or by extremal properties (i.e. best approximation). Taylor's series plays a central role in this entire study, for it has properties of both interpolation and best approximation, and serves as a guide throughout the whole treatise. Indeed, almost every result given on the representation of functions is concerned with a generalization either of Taylor's series or of some property of Taylor's series--the title ``Generalizations of Taylor's Series'' would be appropriate.

Interpolation formulas Hitherto we have studied primarily interpolation and
approximation by polynomials in the complex variable 2. We shall now
commence the study of interpolation and approximation by more general rational
functions, ...

Abelian Coverings of the Complex Projective Plane Branched Along Configurations of Real Lines

This work studies abelian branched coverings of smooth complex projective surfaces from the topological viewpoint. Geometric information about the coverings (such as the first Betti numbers of a smooth model or intersections of embedded curves) is related to topological and combinatorial information about the base space and branch locus. Special attention is given to examples in which the base space is the complex projective plane and the branch locus is a configuration of lines.

Since, in particular, Y is smooth at p, q must also be a smooth point of X. We will
now assume p is a point in B. Let U be a small ball around p isomorphic to a
complex disk, so that, for any two distinct points q1 and q2 in p"(p), the connected
 ...

Fourier Transforms in the Complex Domain

With the aid of Fourier-Mellin transforms as a tool in analysis, the authors were able to attack such diverse analytic questions as those of quasi-analytic functions, Mercer's theorem on summability, Milne's integral equation of radiative equilibrium, the theorems of Munz and Szasz concerning the closure of sets of powers of an argument, Titchmarsh's theory of entire functions of semi-exponential type with real negative zeros, trigonometric interpolation and developments in polynomials of the form $\sum^N_1A_ne^{i\lambda_nx}$, lacunary series, generalized harmonic analysis in the complex domain, the zeros of random functions, and many others.

... that while their power has not been completely exhausted, some radically new
idea is needed if we are to bring this theory into its definitive shape. CHAPTER
VIII GENERALIZED HARMONIC ANALYSIS m THE COMPLEX DOMAIN 33.

Linear Differential Equations in the Complex Domain

Problems of Analytic Continuation

This book is a translation of a 1976 book originally written in Japanese. The main attention is paid to intrinsic aspects of problems related to linear ordinary differential equations in complex domains. Examples of the problems discussed in the book include the Riemann problem on the Riemann sphere, a characterization of regular singularities, and a classification of meromorphic differential equations. Since the original book was published, many new ideas have developed, such as applications of D-modules, Gevrey asymptotics, cohomological methods, $k$-summability, and studies of differential equations containing parameters. Five appendices, added in the present edition, briefly cover these new ideas. In addition, more than 100 references have been added. This book introduces the reader to the essential facts concerning the structure of solutions of linear differential equations in the complex domain and illuminates the intrinsic meaning of older results by means of more modern ideas. A useful reference for research mathematicians, this book would also be suitable as a textbook in a graduate course or seminar.

As the study of topology, algebraic geometry, and functions of several complex
variables has advanced, many methods which are useful in such fields have
been introduced into the territory of differential equations. In this book, focusing ...