Basic General Topology for Graduate Studies
Pages: 824
ISBN: 978-981-12-7733-7 (hardcover)
Point-Set Topology with Topics: Basic General Topology for Graduate Studies can be used as an introduction to a general topology course for both undergraduate and graduate level courses. Presenting topological concepts which apply directly to functional analysis, this book will also be of interest to scholars working in those fields.
Part I contains a summary of normed vector spaces and metric spaces, which will ease the reader into the main subject matter of general topology. Parts II to VI form the core of the text, presenting in diverse ways and at a range of levels the material usually covered in basic general topology courses. After having worked through the most fundamental concepts of topology, the reader will be exposed, in part VII, to more specialized or advanced topics of point-set topology. These appear in the form of a sequence of chapters, many of which can be studied independently.
Each chapter is followed by a collection of 'concept review' questions whose answers can usually be found in the main body of the text. These questions highlight the main concepts presented in that chapter. They serve to attract the attention of the reader to ideas that are often overlooked when first encountered. The efforts required in answering such questions correctly will provide the student with the ability to solve the more complex problems collected at the end of each section.
Norms and Metrics:
Norms on Vector Spaces
Metrics on Sets
Topological Spaces: Fundamental Concepts:
A Topology on a Set
Set Closures, Interiors and Boundaries
Bases of Topological Spaces
Continuity on Topological Spaces
Product Spaces
The Quotient Topology
Topological Spaces: Separation Axioms:
Separation with Open Sets
Separation with Continuous Functions
Limit Points in Topological Spaces:
Limit Points in First Countable Spaces
Limit Points of Nets
Limit Points of Filters
Compact Spaces and Relatives:
Compactness: Definition and Basic Properties
Countably Compact Spaces
Lindelof Spaces
Sequentially and Feebly Compact Spaces
Locally Compact Spaces
Paracompact Topological Spaces
The Connected Property:
Connected Spaces and Properties
Topics:
Compactifications of Completely Regular Spaces
Singular Sets and Singular Compactifications
On C-Embeddings and Pseudocompactness
Realcompact Spaces
Perfect Functions
Perfect and Freudenthal Compactifications
Spaces Whose Elements are Sequences
Completing Incomplete Metric Spaces
The Uniform Space and the Uniform Topology
The Stone-Weierstrass Theorem
Metrizability
The Stone Space
Baire Spaces
The Class of F-Spaces
Appendix
Bibliography
Index
Pages: 592
ISBN: 978-981-12-6838-0 (hardcover)
ISBN: 978-981-12-6944-8 (softcover)
This is a thoroughly revised and expanded third edition of a successful university textbook that provides a broad introduction to key areas of stochastic modelling. The previous edition was developed from lecture notes for two one-semester courses for third-year science and actuarial students at the University of Melbourne.
This book reviews the basics of probability theory and presents topics on Markov chains, Markov decision processes, jump Markov processes, elements of queueing theory, basic renewal theory, elements of time series and simulation. It also features elements of stochastic calculus and introductory mathematical finance. This makes the book suitable for a larger variety of university courses presenting the fundamentals of modern stochastic modelling.
To make the text covering a lot of material more appealing and accessible to the reader, instead of rigorous proofs we often give only sketches of the arguments, with indications as to why a particular result holds and also how it is related to other results, and illustrate them by examples. It is in this aspect that the present, third edition differs from the second one: the included background material and argument sketches have been extended, made more graphical and informative. The whole text was reviewed and streamlined wherever possible to make the book more attractive and useful for readers. Wherever possible, the book includes references to more specialised texts on respective topics that contain both proofs and more advanced material.
Introduction
Basics of Probability Theory
Markov Chains
Markov Decision Processes
The Exponential Distribution and Poisson Process
Jump Markov Processes
Elements of Queueing Theory
Elements of Renewal Theory
Elements of Time Series
Elements of Simulation
Martingales and Stochastic Calculus
Diffusion Processes
Elements of Mathematical Finance
Pages: 300
ISBN: 978-981-12-8540-0 (hardcover)
The theory of algebraic hyperstructures, in particular the theory of Krasner hyperrings, has seen a spectacular development in the last 20 years, which is why a book dedicated to the study of these is so vital. Krasner hyperrings are a generalization of hyperfields, introduced by Krasner in order to study complete valued fields. A Krasner hyperring (R, +, .) is an algebraic structure, where (R, +) is a canonical hypergroup, (R, .) is a semigroup having zero as a bilaterally absorbing element and the multiplication is distributive with respect to the hyperoperation +.
Krasner Hyperring Theory presents an elaborate study on hyperstructures, particularly Krasner hyperrings, across 10 chapters with extensive examples. It contains the results of the authors, but also of other researchers in the field, focusing especially on recent research. This book is especially addressed to doctoral students or researchers in the field, as well as to all those interested in this interesting part of algebra, with applications in other fields.
Canonical Hypergroups
Introduction to Krasner Hyperrings
Homomorphisms and Isomorphisms
Generalizations of Hyperideals
Lower and Upper Approximations in Krasner Hyperrings
Derived Hyperstructures from Hyperconics
Fundamental Relations on Krasner Hyperrings
Some Special Hyperrings
Differential Krasner Hyperrings
Ordered Krasner Hyperrings
Pages: 700
ISBN: 978-981-12-8578-3 (hardcover)
Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory.
An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills.
All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers.
As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book.
Propositional Logic: The Foundation of Mathematical Reasoning
Set Theory and Introduction to Boolean Algebra: A Naive Approach
Proof Techniques, an Active Approach to do Mathematics
An Introduction to Predicate Logic: One Step Further
Functions: Back to Basics
Elementary Number Theory: The Basics, Primes, Congruences and a Bit More
Relations: Equivalence and Order Relations and Everything in Between
Basic Combinatorics: The Art of Counting Without Counting
Graph Theory: The Basics, First Results and Interesting Applications
More on Graph Theory: Planar Graphs, Coloring, Euler and Hamiltonian Special Paths
Trees: An Elite Family of Graphs
Pages: 180
ISBN: 978-981-12-8527-1 (hardcover)
ISBN: 978-981-128-560-8 (softcover)
The goal of the book is to provide insight into many enjoyable and fascinating aspects of geometry, and to reveal interesting geometrical properties. The emphasis is on the practical applications of theory in the problem-solving process. The chapters cover a myriad of topics among which are the classic theorems and formulas such as Archimedes' Law of the Lever, the Pythagorean Theorem, Heron's formula, Brahmagupta's formula, Appollonius's Theorem, Euler's line properties, the Nine-Point Circle, Fagnano's Problem, the Steiner-Lehmus Theorem, Napoleon's Theorem, Ceva's Theorem, Menelaus's Theorem, Pompeiu's Theorem, and Morley's Miracle. The book focuses on geometric thinking ? what it means, how to develop it, and how to recognize it. "Geometrical Kaleidoscope" consists of a kaleidoscope of topics that seem to not be related at first glance. However, that perception disappears as you go from chapter to chapter and explore the multitude of surprising relationships, unexpected connections, and links. Readers solving a chain of problems will learn from them general techniques, rather than isolated instances of the application of a technique. In spite of the many problems' challenging character, their solutions require no more than a basic knowledge covered in a high school geometry curriculum. There are plenty of problems for readers to work out for themselves (solutions are provided at the end of the book).
In the 2nd edition of the book there are many new ideas and additional explanations that help the reader better understand the solutions of problems and connect the chapters to one another. A new chapter "Alternative proofs of the Pythagorean Theorem" is added. It covers seven different proofs of the famous theorem and discusses its generalizations and applications. There is also Appendix and Index added, which were missing in the first edition of the book.
Preface
Medians, Centroid, and Center of Gravity of a System of Points
Altitudes and the Orthocenter of a Triangle and Some of Its Properties
The Orthic Triangle and Its Properties
The Angle Bisector of a Triangle and Its Properties
The Area of a Quadrilateral
The Theorem of Ratios of the Areas of Similar Polygons
A Pivotal Approach. Applying Rotation in Problem Solving
Auxiliary Elements in Problem Solving
Constructions-Siblings
Session of One Interesting Construction Problem
Alternative Proofs of the Pythagorean Theorem
Morley's Theorem
Solutions and Answers to the Problems
Appendix
Index
References