Series: Lecture Notes in Mathematics, Vol. 2174
* Offers a fast introduction to the theory of pseudodifferential
equations over non-Archimedean fields and their connections with
mathematical physics, probability and number theory Provides
a very general theory of parabolic-type equations and their Markov
processes motivated by the models of hierarchic complex systems
introduced by Avetisov et al. in around 2000 Combines methods of
PDEs, probability and number theory
Focusing on p-adic and adelic analogues of pseudodifferential equations, this monograph
presents a very general theory of parabolic-type equations and their Markov processes
motivated by their connection with models of complex hierarchic systems. The Gelfand-
Shilov method for constructing fundamental solutions using local zeta functions is
developed in a p-adic setting and several particular equations are studied, such as the padic
analogues of the Klein-Gordon equation. Pseudodifferential equations for complexvalued
functions on non-Archimedean local fields are central to contemporary harmonic
analysis and mathematical physics and their theory reveals a deep connection with
probability and number theory. The results of this book extend and complement the
material presented by Vladimirov, Volovich and Zelenov (1994) and Kochubei (2001),
which emphasize spectral theory and evolution equations in a single variable, and
Albeverio, Khrennikov and Shelkovich (2010), which deals mainly with the theory and
applications of p-adic wavelets.
X, 160 p. 1 illus.
Softcover
ISBN 978-3-319-46737-5
Series: Operator Theory: Advances and Applications, Vol. 258
* Presents papers written by internationally leading researchers
* Shares a broad picture of the current research in operator theory and
partial differential equations
* Motivates further research in a wide variety of fields in Mathematical
Analysis and its applications
This volume is dedicated to the eminent Georgian mathematician Roland Duduchava on
the occasion of his 70th birthday. It presents recent results on Toeplitz, Wiener-Hopf, and
pseudodifferential operators, boundary value problems, operator theory, approximation
theory, and reflects the broad spectrum of Duduchava's research. The book is addressed to
a wide audience of pure and applied mathematicians.
1st ed. 2017, Approx. 305 p.
Hardcover
ISBN 978-3-319-47077-1
Series: Progress in Mathematics, Vol. 320
* Offers a unique synthesis of techniques: tools from complex algebraic
geometry are applied to fundamental questions in number theory
and Diophantine geometry
* Investigates the connection between derived equivalences and
existence of rational points on varieties, especially on K3 surfaces
* Includes a founding paper in the emerging theory of universal
triviality of the Chow group of 0-cycles and its relationship to stable
rationality problems
The contributions in this book explore various contexts in which the derived category of
coherent sheaves on a variety determines some of its arithmetic. This setting provides
new geometric tools for interpreting elements of the Brauer group. With a view towards
future arithmetic applications, the book extends a number of powerful tools for analyzing
rational points on elliptic curves, e.g. isogenies among curves, torsion points, modular
curves, and the resulting descent techniques, as well as higher-dimensional varieties like
K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces,
the book is intended to foster cross-pollination between the fields of complex algebraic
geometry and number theory.
1st ed. 2017, Approx. 300 p.
Hardcover
ISBN 978-3-319-46851-8
190pp Nov 2016
ISBN: 978-981-3108-50-9 (hardcover)
ISBN: 978-981-3108-51-6 (softcover)
The aim of the book is to give a smooth analytic continuation from calculus to complex analysis by way of plenty of practical examples and worked-out exercises. The scope ranges from applications in calculus to complex analysis in two different levels.
If the reader is in a hurry, he can browse the quickest introduction to complex analysis at the beginning of Chapter 1, which explains the very basics of the theory in an extremely user-friendly way. Those who want to do self-study on complex analysis can concentrate on Chapter 1 in which the two mainstreams of the theory * the power series method due to Weierstrass and the integration method due to Cauchy * are presented in a very concrete way with rich examples. Readers who want to learn more about applied calculus can refer to Chapter 2, where numerous practical applications are provided. They will master the art of problem solving by following the step by step guidance given in the worked-out examples.
This book helps the reader to acquire fundamental skills of understanding complex analysis and its applications. It also gives a smooth introduction to Fourier analysis as well as a quick prelude to thermodynamics and fluid mechanics, information theory, and control theory. One of the main features of the book is that it presents different approaches to the same topic that aids the reader to gain a deeper understanding of the subject.
A Quick Introduction to Complex Analysis with Applications:
The Quickest Introduction to Complex Analysis
Complex Number System
Power Series and Euler's Identity
Residue Calculus
Review on Vector-Valued Functions
Cauchy*Riemann Equation
Inverse Functions
Around Jensen's Formula
Residue Calculus Again
Partial Fraction Expansion
Second-Order Systems and the Laplace Transform
Robust Controller for Servo Systems
Paley*Wiener Theorem
Bernstein Polynomials
Some Far-Reaching Principles in Mathematics
Applicable Real and Complex Functions:
Preliminaries
Algebra of Complex Numbers
Power Series Again
Improper Integrals
Differentiation
Differential Calculus of One and Several Variables
Computation of Definite Integrals
Cauchy Integral Theorem
Cauchy Integral Formula
Taylor Expansions and Extremal Values
Complex Power Series
Laurent Expansions
Differential Equations
Inverse Functions
Rudiments of the Fourier Transform
Paley*Wiener Theorem and Signal Transmission
Appendices:
Integration
Answers and Hints
Advanced undergraduate mathematics, physics and engineering students; researchers in the field of complex analysis; also suitable for self-study.
196pp Nov 2016
ISBN: 978-981-3146-21-1 (hardcover)
ISBN: 978-981-3143-64-7 (softcover)
This solutions manual thoroughly goes through the exercises found in Undergraduate Convexity: From Fourier and Motzkin to Kuhn and Tucker. Several solutions are accompanied by detailed illustrations and intuitive explanations. This book will pave the way for students to easily grasp the multitude of solution methods and aspects of Undergraduate Convexity.
Fourier*Motzkin Elimination
Affine Subspaces
Convex Subsets
Polyhedra
Computations with Polyhedra
Closed Convex Subsets and Separating Hyperplanes
Convex Functions
Differentiable Functions of Several Variables
Convex Functions of Several Variables
Convex Optimization
Appendices: Analysis; Linear (In)dependence and the Rank of a Matrix
Undergraduates focusing on convexity and optimization.
372pp Dec 2016
ISBN: 978-981-3146-93-8 (hardcover)
ISBN: 978-981-3148-02-4 (softcover)
The book offers a good introduction to topology through solved exercises. It is mainly intended for undergraduate students. Most exercises are given with detailed solutions.
In the second edition, some significant changes have been made, other than
the additional exercises. There are also additional proofs (as exercises)
of many results in the old section "What You Need To Know", which
has been improved and renamed in the new edition as "Essential Background".
Indeed, it has been considerably beefed up as it now includes more remarks
and results for readers', convenience. The interesting sections "True
or False" and "Tests" have remained as they were, apart
from a very few changes
Sets, Functions et al.
Metric Spaces
Topological Spaces
Continuity and Convergence
Compact Spaces
Connected Spaces
Complete Metric Spaces
Function Spaces
Undergraduate students and lecturers in topology
200pp Mar 2017
ISBN: 978-981-4733-43-4 (hardcover)
ISBN: 978-981-4733-44-1 (softcover)
This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.
Grauduate students and researchers interested in understanding from a viewpoint of combinatorics on crystal base theory.