Introductory Mathematics: Algebra and Analysis


Author: Geoffrey C. Smith
Publisher: Springer Science & Business Media
ISBN: 1447106199
Category: Mathematics
Page: 215
View: 2004
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This text provides a lively introduction to pure mathematics. It begins with sets, functions and relations, proof by induction and contradiction, complex numbers, vectors and matrices, and provides a brief introduction to group theory. It moves onto analysis, providing a gentle introduction to epsilon-delta technology and finishes with continuity and functions. The book features numerous exercises of varying difficulty throughout the text.

Calculus of One Variable


Author: K.E. Hirst
Publisher: Springer Science & Business Media
ISBN: 1846282225
Category: Mathematics
Page: 268
View: 9177
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Adopts a user-friendly approach, with an emphasis on worked examples and exercises, rather than abstract theory The computer algebra and graphical package MAPLE is used to illustrate many of the ideas and provides an additional aid to teaching and learning Supplementary material, including detailed solutions to exercises and MAPLE worksheets, is available via the web

Real Analysis


Author: John M. Howie
Publisher: Springer Science & Business Media
ISBN: 1447103416
Category: Mathematics
Page: 276
View: 9977
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Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions. All the concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. This book offers a fresh approach to a core subject and manages to provide a gentle and clear introduction without sacrificing rigour or accuracy.

Elements of Logic via Numbers and Sets

Visualization of Mathematical Objects with Mathematica. CD-ROM
Author: D.L. Johnson
Publisher: Springer Science & Business Media
ISBN: 9783540761235
Category: Mathematics
Page: 174
View: 3054
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This is an elementary text, aimed at first-year undergraduates, which has been designed to bridge the gap between school and university mathematics and to emphasise the importance of proofs - both how to follow a proof and how to construct a proof. The book lays the foundation for most of the key subjects studied in an undergraduate degree program, and provides numerous exercises and a bibliography with suggestions for further and background reading.

Groups, Rings and Fields


Author: David A.R. Wallace
Publisher: Springer Science & Business Media
ISBN: 1447104250
Category: Mathematics
Page: 248
View: 5666
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This is a basic introduction to modern algebra, providing a solid understanding of the axiomatic treatment of groups and then rings, aiming to promote a feeling for the evolutionary and historical development of the subject. It includes problems and fully worked solutions, enabling readers to master the subject rather than simply observing it.

Basic Linear Algebra


Author: Thomas S. Blyth,Edmund F. Robertson
Publisher: Springer Science & Business Media
ISBN: 1447134966
Category: Mathematics
Page: 201
View: 6726
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Basic Linear Algebra is a text for first year students, working from concrete examples towards abstract theorems, via tutorial-type exercises. The book explains the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations, and complex numbers. Linear equations are treated via Hermite normal forms, which provides a successful and concrete explanation of the notion of linear independence. Another highlight is the connection between linear mappings and matrices, leading to the change of basis theorem which opens the door to the notion of similarity. The authors are well known algebraists with considerable experience of teaching introductory courses on linear algebra to students at St Andrews. This book is based on one previously published by Chapman and Hall, but it has been extensively updated to include further explanatory text and fully worked solutions to the exercises that all 1st year students should be able to answer.

Linear Functional Analysis


Author: N.A
Publisher: 清华大学出版社有限公司
ISBN: 9787302121398
Category: Functional analysis
Page: 283
View: 7893
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Codes: An Introduction to Information Communication and Cryptography


Author: Norman L. Biggs
Publisher: Springer Science & Business Media
ISBN: 9781848002739
Category: Computers
Page: 274
View: 5886
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Many people do not realise that mathematics provides the foundation for the devices we use to handle information in the modern world. Most of those who do know probably think that the parts of mathematics involvedare quite ‘cl- sical’, such as Fourier analysis and di?erential equations. In fact, a great deal of the mathematical background is part of what used to be called ‘pure’ ma- ematics, indicating that it was created in order to deal with problems that originated within mathematics itself. It has taken many years for mathema- cians to come to terms with this situation, and some of them are still not entirely happy about it. Thisbookisanintegratedintroductionto Coding.Bythis Imeanreplacing symbolic information, such as a sequence of bits or a message written in a naturallanguage,byanother messageusing (possibly) di?erentsymbols.There are three main reasons for doing this: Economy (data compression), Reliability (correction of errors), and Security (cryptography). I have tried to cover each of these three areas in su?cient depth so that the reader can grasp the basic problems and go on to more advanced study. The mathematical theory is introduced in a way that enables the basic problems to bestatedcarefully,butwithoutunnecessaryabstraction.Theprerequisites(sets andfunctions,matrices,?niteprobability)shouldbefamiliartoanyonewhohas taken a standard course in mathematical methods or discrete mathematics. A course in elementary abstract algebra and/or number theory would be helpful, but the book contains the essential facts, and readers without this background should be able to understand what is going on. vi Thereareafewplaceswherereferenceismadetocomputeralgebrasystems.

Essential Topology


Author: Martin D. Crossley
Publisher: Springer Science & Business Media
ISBN: 9781852337827
Category: Mathematics
Page: 224
View: 5537
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This thoroughly modern introduction to undergraduate topology brings the most exciting and useful aspects of modern topology to the reader. Containing all the key results of basic topology, this book concentrates on uniting the most interesting aspects of the subject with aspects that are most useful to research. It is suitable for self-study, and will leave the reader both motivated and well prepared for further study.

Elementary Number Theory


Author: Gareth A. Jones,Josephine M. Jones
Publisher: Springer Science & Business Media
ISBN: 144710613X
Category: Mathematics
Page: 302
View: 7181
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An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.

Numbers and Functions

Steps into Analysis
Author: R. P. Burn
Publisher: Cambridge University Press
ISBN: 1316033783
Category: Mathematics
Page: N.A
View: 6377
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The transition from studying calculus in schools to studying mathematical analysis at university is notoriously difficult. In this third edition of Numbers and Functions, Professor Burn invites the student reader to tackle each of the key concepts in turn, progressing from experience through a structured sequence of more than 800 problems to concepts, definitions and proofs of classical real analysis. The sequence of problems, of which most are supplied with brief answers, draws students into constructing definitions and theorems for themselves. This natural development is informed and complemented by historical insight. Carefully corrected and updated throughout, this new edition also includes extra questions on integration and an introduction to convergence. The novel approach to rigorous analysis offered here is designed to enable students to grow in confidence and skill and thus overcome the traditional difficulties.

Complex Analysis and Differential Equations


Author: Luis Barreira,Claudia Valls
Publisher: Springer Science & Business Media
ISBN: 1447140087
Category: Mathematics
Page: 415
View: 5464
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This text provides an accessible, self-contained and rigorous introduction to complex analysis and differential equations. Topics covered include holomorphic functions, Fourier series, ordinary and partial differential equations. The text is divided into two parts: part one focuses on complex analysis and part two on differential equations. Each part can be read independently, so in essence this text offers two books in one. In the second part of the book, some emphasis is given to the application of complex analysis to differential equations. Half of the book consists of approximately 200 worked out problems, carefully prepared for each part of theory, plus 200 exercises of variable levels of difficulty. Tailored to any course giving the first introduction to complex analysis or differential equations, this text assumes only a basic knowledge of linear algebra and differential and integral calculus. Moreover, the large number of examples, worked out problems and exercises makes this the ideal book for independent study.

Numerical Linear Algebra

A Concise Introduction with MATLAB and Julia
Author: Folkmar Bornemann
Publisher: Springer
ISBN: 3319742221
Category: Mathematics
Page: 153
View: 1236
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This book offers an introduction to the algorithmic-numerical thinking using basic problems of linear algebra. By focusing on linear algebra, it ensures a stronger thematic coherence than is otherwise found in introductory lectures on numerics. The book highlights the usefulness of matrix partitioning compared to a component view, leading not only to a clearer notation and shorter algorithms, but also to significant runtime gains in modern computer architectures. The algorithms and accompanying numerical examples are given in the programming environment MATLAB, and additionally – in an appendix – in the future-oriented, freely accessible programming language Julia. This book is suitable for a two-hour lecture on numerical linear algebra from the second semester of a bachelor's degree in mathematics.

Groups


Author: Camilla R. Jordan,David A. Jordan
Publisher: Butterworth-Heinemann
ISBN: 034061045X
Category: Mathematics
Page: 207
View: 9943
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Introduction to mathematical groups

Introduction to Lie Algebras


Author: K. Erdmann,Mark J. Wildon
Publisher: Springer Science & Business Media
ISBN: 1846284902
Category: Mathematics
Page: 251
View: 4094
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Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.

A First Course in Discrete Mathematics


Author: Ian Anderson
Publisher: Springer Science & Business Media
ISBN: 085729315X
Category: Mathematics
Page: 200
View: 1160
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Drawing on many years'experience of teaching discrete mathem atics to students of all levels, Anderson introduces such as pects as enumeration, graph theory and configurations or arr angements. Starting with an introduction to counting and rel ated problems, he moves on to the basic ideas of graph theor y with particular emphasis on trees and planar graphs. He de scribes the inclusion-exclusion principle followed by partit ions of sets which in turn leads to a study of Stirling and Bell numbers. Then follows a treatment of Hamiltonian cycles, Eulerian circuits in graphs, and Latin squares as well as proof of Hall's theorem. He concludes with the constructions of schedules and a brief introduction to block designs. Each chapter is backed by a number of examples, with straightforw ard applications of ideas and more challenging problems.

Sets, Logic and Categories


Author: Peter J. Cameron
Publisher: Springer Science & Business Media
ISBN: 1447105893
Category: Mathematics
Page: 182
View: 5027
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Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

Differential Equations


Author: Viorel Barbu
Publisher: Springer
ISBN: 3319452614
Category: Mathematics
Page: 224
View: 1110
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This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.

Introductory Mathematics: Applications and Methods


Author: Gordon S. Marshall
Publisher: Springer Science & Business Media
ISBN: 1447134125
Category: Mathematics
Page: 226
View: 3463
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This book is aimed at undergraduate students embarking on the first year of a modular mathematics degree course. It is a self-contained textbook making it ideally suited to distance learning and a useful reference source for courses with the traditional lecture/tutorial structure. The theoretical content is firmly based but the principal focus is on techniques and applications. The important aims and objectives are presented clearly and then reinforced using complete worked solutions within the text. There is a natural increase in difficulty and understanding as each chapter progresses, always building upon the basic elements. It is assumed that the reader has studied elementary calculus at Advanced level and is at least familiar with the concept of function and has been exposed to basic differentiation and integration techniques. Although these are covered in the book they are presented as a refresher course to jog the student's memory rather than to introduce the topic for the first time. The early chapters cover the topics of matrix algebra, vector algebra and com plex numbers in sufficient depth for the student to feel comfortable -when they reappear later in the book. Subsequent chapters then build upon the student's 'A' level knowledge in the area of real variable calculus, including partial differentiation and mUltiple inte grals. The concluding chapter on differential equations motivates the student's learning by consideration of applications taken from both physical and eco nomic contexts.

Mathematical Analysis

An Introduction
Author: Andrew Browder
Publisher: Springer Science & Business Media
ISBN: 1461207150
Category: Mathematics
Page: 335
View: 1763
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Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. To this extent, more than one proof is included for a theorem - where this is considered beneficial - so as to stimulate the students' reasoning for alternate approaches and ideas. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. Also included are some beautiful applications of this theory, such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions. Throughout, reference is made to earlier sections, so as to reinforce the main ideas by repetition. Unique in its applications to some topics not usually covered at this level.