https://doi.org/10.1142/12053 | December 2020
Pages: 156
ISBN: 978-981-122-865-0 (hardcover)
Interdisciplinary Mathematical Sciences: Volume 21
The nature of time in a nonautonomous dynamical system is very different from that in autonomous systems, which depend only on the time that has elapsed since starting rather than on the actual time itself. Consequently, limiting objects may not exist in actual time as in autonomous systems. New concepts of attractors in nonautonomous dynamical system are thus required.
In addition, the definition of a dynamical system itself needs to be generalised to the nonautonomous context. Here two possibilities are considered: two-parameter semigroups or processes and the skew product flows. Their attractors are defined in terms of families of sets that are mapped onto each other under the dynamics rather than a single set as in autonomous systems. Two types of attraction are now possible: pullback attraction, which depends on the behaviour from the system in the distant past, and forward attraction, which depends on the behaviour of the system in the distant future. These are generally independent of each other.
The component subsets of pullback and forward attractors exist in actual time. The asymptotic behaviour in the future limit is characterised by omega-limit sets, in terms of which form what are called forward attracting sets. They are generally not invariant in the conventional sense, but are asymptotically invariant in general and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant.
Much of this book is based on lectures given by the authors in Frankfurt and Wuhan. It was written mainly when the first author held a "Thousand Expert" Professorship at the Huazhong University of Science and Technology in Wuhan.
Dynamical Systems:
Autonomous Dynamical Systems
Nonautonomous Dynamical Systems: Processes
Skew Product Flows
Entire Solutions and Invariant Sets
Pullback Attractors:
Attractors
Nonautonomous Equilibrium Solutions
Attractors for Processes
Examples of Pullback Attractors for Processes
Attractors of Skew Product Flows
Forward Attractors and Attracting Sets:
Limitations of Pullback Attractors of Processes
Forward Attractors
Omega-Limit Sets and Forward Attracting Sets
Random aAttractors:
Random Dynamical Systems
Mean-Square Random Dynamical Systems
Upper level undergraduate, graduate students, researchers in Mathematics and in areas of applications such as biomathematics, ecology, meteorology, medicine, etc.
https://doi.org/10.1142/12040 | December 2020
Pages: 324
ISBN: 978-981-122-823-0 (hardcover)
ISBN: 978-981-123-007-3 (softcover)
Problem Solving in Mathematics and Beyond: Volume 22
Mathematical Labyrinths. Pathfinding provides an overview of various non-standard problems and the approaches to their solutions. The essential idea is a framework laid upon the reader on how to solve nonconventional problems ? particularly in the realm of mathematics and logic. It goes over the key steps in approaching a difficult problem, contemplating a plan for its solution, and discusses set of mental models to solve math problems.
The book is not a routine set of problems. It is rather an entertaining and educational journey into the fascinating world of mathematical reasoning and logic. It is about finding the best path to a solution depending on the information given, asking and answering the right questions, analyzing and comparing alternative approaches to problem solving, searching for generalizations and inventing new problems. It also considers as an important pedagogical tool playing mathematical and logical games, deciphering mathematical sophisms, and interpreting mathematical paradoxes.
It is suitable for mathematically talented and curious students in the age range 10?20. There are many 'Eureka'- type, out of the ordinary, fun problems that require bright idea and insight. These intriguing and thought-provoking brainteasers and logic puzzles should be enjoyable by the audience of almost any age group, from 6-year-old children to 80-year-old and older adults.
Entering the Labyrinth
What? How? Why?
The Clue is in the Question
Thinking Outside the Box. Sophisms and Paradoxes
"Precise Steps" Problems. Playing Preferans
Euclidean Plane Transformations/li>
Geometrical Constructions with Restricted Elements
Inventing a Problem
Related Problems
Alternative Solutions Search
Eureka!
General public, secondary school and college undergraduate students and teachers, educationalists.
https://doi.org/10.1142/11990 | February 2021
Pages: 350
ISBN: 978-981-122-621-2 (hardcover)
P-adic Analytic Functions describes the definition and properties of p-adic
analytic and meromorphic functions in a complete algebraically closed ultrametric
field.
Various properties of p-adic exponential-polynomials are examined, such as the Hermite Lindemann theorem in a p-adic field, with a new proof. The order and type of growth for analytic functions are studied, in the whole field and inside an open disk. P-adic meromorphic functions are studied, not only on the whole field but also in an open disk and on the complemental of a closed disk, using Motzkin meromorphic products. Finally, the p-adic Nevanlinna theory is widely explained, with various applications. Small functions are introduced with results of uniqueness for meromorphic functions. The question of whether the ring of analytic functions ? in the whole field or inside an open disk ? is a Bezout ring is also examined.
Ultrametric Fields:
Basic Definitions and Properties of Ultrametric fields
Monotonous and Circular filters
Ultrametric Absolute Values for Rational Functions
Hensel Lemma
Extensions of Ultrametric fields: The field ??
Normal Extensions of ?? Inside ??
Spherically Complete Extensions
Transcendence Order and Transcendence Type
Analytic Elements and Analytic Functions:
Algebras R(D)
Analytic Elements
Composition of Analytic Elements
Multiplicative Spectrum of H(D)
Power and Laurent Series
Krasner?Mittag?Leffer Theorem
Factorization of Analytic Elements
Algebras H(D)
Derivative of Analytic Elements
Properties of the Function ƒÕ for Analytic Elements
Vanishing Along a Monotonous filter
Quasi-Minorated Elements
Zeros of Power Series
Image of a Disk
Quasi-invertible Analytic Elements
Logarithm and Exponential in a p-adic Field
Problems on p-adic Exponentials
Divisors of Analytic Functions
Michel Lazard's Problem
Motzkin Factorization and Roots of Analytic Functions
Order of Growth for Entire Functions
Type of Growth for Entire Functions
Growth of the Derivative of an Entire Function
Growth of an Analytic Functions in an Open Disk
Meromorphic Functions and Nevanlinna Theory:
Meromorphic Functions in ??
Residues of Meromorphic Functions
Meromorphic Functions Out of a Hole
Nevanlinna Theory in ?? and in an Open Disk
Nevanlinna Theory Out of a Hole
Immediate Applications of the Nevanlinna Theory
Branched Values
Exceptional Values of Functions and Derivatives
The p-adic Hayman Conjecture
Small Functions
Bibliography
Definitions
Notations
The target audience consists of researchers and post-graduate students in ultrametric analysis and number theory. Also appropriate for researchers in Levi-Civita fields in p-adic physics.
https://doi.org/10.1142/11943 | February 2021
Pages: 328
ISBN: 978-981-122-448-5 (hardcover)
This volume aims to bridge between elementary textbooks on calculus and established books on advanced analysis. It provides elucidation of the reversible process of differentiation and integration through two featured principles: the chain rule and its inverse ? the change of variable ? as well as the Leibniz rule and its inverse ? the integration by parts. The chain rule or differentiation of composite functions is ubiquitous since almost all (a.a.) functions are composite functions of (elementary) functions and with the change of variable method as its reverse process. The Leibniz rule or differentiation of the product of two functions is essential since it makes differentiation nonlinear and with the method of integration by parts as its reverse process.
Readers will find numerous worked-out examples and exercises in this volume. Detailed solutions are provided for most of the common exercises so that readers remain enthusiastically motivated in solving and understanding the concepts better.
The intention of this volume is to lead the reader into the rich fields of advanced analysis and to obtain a much better view of useful mathematics.
Preface
Preliminaries
Continuity of Real Numbers
Limits and Continuity
Properties of Continuous Mappings
Differentiation
Properties of Differentiable Functions
Extremal Values, Mean Value Theorems and Taylor Expansions in One Variable
Extremal Values, Mean Value Theorems and Taylor Expansion in Several Variables
Algorithms
Theory of Riemann?Stieltjes Integration
Theory of Riemann Integration
Limit Values and Summability
ODE and FDE
Vector Analysis
Theory of Distributions
Topological Spaces
Appendices:
Basics from Linear Algebra
Diagonalization of Matrices
Circulants
Formulas for Vector Operations
Alternating Algebras
Bibliography
Index
University students who want to know what analysis is, and researchers in various disciplines who want to have a glimpse of analysis for possible applications to their own fields. Also useful for self-study.
https://doi.org/10.1142/11983 | March 2021
Pages: 390
ISBN: 978-981-122-585-7 (hardcover)
ISBN: 978-981-122-769-1 (softcover)
This is a self-contained book that covers the standard topics in introductory analysis and that in addition, constructs the natural, rational, real and complex numbers, also handles complex-valued functions, sequences, and series.
The book teaches how to write proofs. Fundamental proof-writing logic is covered in Chapter 1 and is repeated and enhanced in two appendices. Many examples of proofs appear with words in a different font for what should be going on in the proof writer's head.
The book contains many examples and exercises to solidify the understanding. The material is presented rigorously with proofs and with many worked-out examples. Exercises are varied, many involve proofs, and some provide additional learning materials.
How We will Do Mathematics
Concepts with Which We will Do Mathematics
Construction of The Basic Number Systems
Limits of Functions
Continuity
Differentiation
Integration
Sequences
Infinite Series and Power Series
Exponential and Trigonometric Functions
Advice on Writing Mathematics
What You Should Never Forget
Appendix A: Advice on Writing Mathematics
Appendix B: What One Should Never Forget
Introduction to proofs and analysis on real and complex functions for undergraduates.