Bøger i Student Mathematical Library serien

Offers a reader-friendly introduction to the theory of symmetric functions. It includes fundamental topics such as the monomial, elementary, homogeneous, and Schur function bases. The prerequisites for the book are minimal-familiarity with linear algebra, partitions, and generating functions.

Presents problems in abstract algebra for strong undergraduates or beginning graduate students. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations); rings (including basic ideal theory and factorization in integral domains); linear algebra (emphasizing linear transformations); and fields (including Galois theory).

Introduces functional analysis to undergraduate mathematics students who possess a basic background in analysis and linear algebra. By studying how the Volterra operator acts on vector spaces of continuous functions, its readers will sharpen their skills, reinterpret what they already know, and learn fundamental Banach-space techniques.

Provides a bridge between an undergraduate course in Real Analysis and a first graduate-level course in Measure Theory and Integration. The main goal of this book is to prepare students for what they may encounter in graduate school, but will be useful for many beginning graduate students as well.

Introducing the heat equation and the closely related notion of harmonic functions from a probabilistic perspective, this book includes chapters on: the discrete case, random walk and the heat equation on the integer lattice; the continuous case, Brownian motion and the usual heat equation; and martingales and fractal dimension.

Number theory is the equal of Euclidean geometry - some would say it is superior to Euclidean geometry - as a model of pure, logical, deductive thinking. This title explains number theory in a way that gives deductive reasoning, including algorithms and computations, the central role.

Examines the theory and practice of integer factorisation from an historic perspective. It describes about twenty algorithms for factoring and a dozen other number theory algorithms that support the factoring algorithms. Most algorithms are described both in words and in pseudocode to satisfy both number theorists and computer scientists.

This is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein's Erlangen Program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal - although some remain more equal than others.

Gives an introduction to basic concepts in ergodic theory such as recurrence, ergodicity, the ergodic theorem, mixing, and weak mixing. In particular, this book includes a detailed construction of the Lebesgue measure on the real line and an introduction to measure spaces up to the Caratheodory extension theorem.

Presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. This book starts with the basics of polytope theory. It introduces Schlegel and Gale diagrams as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes.

Helps in the understanding of continuous and differentiable functions. This book emphasises on real functions of a single variable. It contains topics that include: continuous functions, the intermediate value property, uniform continuity, mean value theorems, Taylors formula, convex functions, and sequences and series of functions.

Ramanujan is recognized as one of the great number theorists of the twentieth century. This book provides an introduction to his work in number theory. It also examines subjects that have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research.

Offers an introduction to the theory of finite fields and to some of their many practical applications. After covering their construction and elementary properties, the authors discuss the trace and norm functions, bases for finite fields, and properties of polynomials over finite fields. Each of the remaining chapters details applications.

Based on classes in probability for advanced undergraduates held at the IAS/Park City Mathematics Institute (Utah), this title is derived from both lectures (Chapters 1-10) and computer simulations (Chapters 11-13) that were held during the program. It concludes with a number of problems ranging from routine to very difficult.

It is rarely taught in undergraduate or graduate curricula that the only conformal maps in Euclidean space of dimension greater than 2 are those generated by similarities and inversions in spheres. This is in stark contrast to the conformal maps in the plane. This book gives a treatment of this paucity of conformal maps in higher dimensions.

Introduces $p$-adic numbers from the point of view of number theory, topology, and analysis. Covering several topics from real analysis and elementary topology, this book includes totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, and surjectivity of isometries of compact metric spaces.

Presents the characteristic zero invariant theory of finite groups acting linearly on polynomial algebras. The author assumes basic knowledge of groups and rings, and introduces more advanced methods from commutative algebra along the way. The theory is illustrated by numerous examples.

Introduces algebraic geometric coding theory. This book covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors.

Suitable for undergraduate students as well as professional mathematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn's Lemma, this text introduces the main subjects of 'naive' (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, and operations on ordinals.