Series: Springer Undergraduate Mathematics Series
2007, Approx. 265 p., 25 illus., Softcover
ISBN: 978-1-84628-971-2
Due: November 2007
About this textbook
Developed from a course taught over many years, this book provides a rigorous introduction to the theory for undergraduate students in mathematics. It covers two main topics:
the Sturm-Liouville eigenvalue problem and its solutions as an important qualitative theory in ordinary differential equations; and
the generalised Fourier series based on the eigenfunctions of the problem.
The material is clearly presented in a straightforward mathematical style and many physical problems are used to illustrate the applications. Only the standard background of calculus, ordinary differential equations and real analysis is assumed. The plentiful worked examples and exercises with solutions make the book ideal for self-study and therefore also accessible to undergraduate students in physics, engineering and related fields.
Table of contents
Inner product space.- The Sturm-Liouville theory.- Fourier series.- Orthogonal polynomials.- Bessel functions.- The Fourier transformation.- The Laplace transformation.- Solutions to selected exercises.- References.- Notation.- Index
Series: Problem Books in Mathematics
2008, Approx. 135 p., 29 illus., Hardcover
ISBN: 978-0-387-73511-5
Due: November 2007
About this textbook
Contains far more problems of a statistical nature than the competition
Will become a handy resource for professors looking for problems to assign for a course
Of interest to undergraduate math students, as well as a more general audience of amateur scientists
The present book is based on the view that cognitive skills are best acquired by solving challenging, non-standard probability problems. The author's own experience, both in learning and in teaching, is that challenging problems often provide more, and longer lasting, inductive insights than plain-style deductions from general concepts. Problems help to develop, and to sharpen our intuition for important probabilistical concepts and tools such as conditionaing or first-step analyses.
Many puzzles and problems presented here are either new within a problem solving context (although as topics in fundamental research they are of course long known) or are variations of classical problems which follow directly from elementary concepts. A small number of particularly instructive problems is taken from previous sources which in this case are generally given.
Table of contents
Preface.- Notation and Terminology.- Problems.- Hints.- Solutions.- References.- Index.
Series: Grundlehren der mathematischen Wissenschaften , Vol. 323
2nd ed., 2008, Approx. 710 p., Hardcover
ISBN: 978-3-540-37888-4
Due: December 2007
About this book
This book is a vital tool in the field, and it has just got better. The second edition is a corrected and extended version of the first. It deals with cohomological topics in number theory.
It offers the reader a virtually complete treatment of a vast array of central topics in algebraic number theory.
This is crucial, as there is so much material known to the experts, but whose detailed proof did not exist in the literature. Most notable amongst these is the celebrated duality theorem of Poitou and Tate, included here.
The first part of the book provides algebraic background including cohomology of profinite groups, duality groups, free products, and homotopy theory of modules.
You’ll find new sections too on spectral sequences and on Tate cohomology of profinite groups.
The second part deals with the famed Poitou-Tate duality, Hasse principles, and the theorem of Grunwald-Wang, among others.
New material is introduced here on duality theorems for unramified and tamely ramified extensions, a careful analysis of 2-extensions of real number fields and a complete proof of Neukirch’s theorem on solvable Galois groups with given local conditions.
Table of contents
I Algebraic Theory: Cohomology of Profinite Groups.- Some Homological Algebra.- Duality Properties of Profinite Groups.- Free Products of Profinite Groups.- Iwasawa Modules
II Arithmetic Theory: Galois Cohomology.- Cohomology of Local Fields.- Cohomology of Global Fields.- The Absolute Galois Group of a Global Field.- Restricted Ramification.- Iwasawa Theory of Number Fields; Anabelian Geometry.- Literature.- Index.
Series: Undergraduate Topics in Computer Science
2008, Softcover
ISBN: 978-1-84628-844-9
Due: March 2008
About this textbook
This book equips the student with essential intellectual tools that are needed from the very beginning of university studies in computing.
These consist of abilities and skills - to pass from a concrete problem to an abstract representation, reason with the abstract structure coherently and usefully, and return with booty to the specific situation. The most basic and useful concepts needed come from the worlds of sets (with also their employment as relations & functions), structures (notably trees & graphs), & combinatorics (alias principles of counting, with their application in the world of probability). Recurring in all these are 2 kinds of instrument of proof ? logical (notably inference by suppositions, reductio ad absurdum, & proof by cases), & mathematical (notably induction on the positive integers & on well-founded structures).
From this book the student can assimilate the basics of these worlds & set out on the paths of computing with understanding & a platform for further study as needed.
Table of contents
Before the Beginning.- Do I really need to know about this?.- Proving and disproving: Logical techniques.- Proving and disproving: Mathematical techniques.- First Steps.- Flat set theory.- Families of sets.- Relations.- Functions.- Structures.- Order.- Trees.- Graphs.- Logic.- Truth-functional connectives.- Quantifiers.- Counting.- The pigeonhole principles.- The addition and multiplication rules.- Selection rules.- Probability.- The Kolmogorov axioms for finitely additive probability.- Combining combinatorics and probability
2008, Approx. 340 p., Hardcover
ISBN: 978-0-8176-4702-5
A Birkhauser book
Due: May 2008
About this book
This monograph presents the first comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables. Such approximation methods are useful in many problems arising in science and engineering requiring an optimal mathematical representation of physical reality.
The work is structured in four main chapters and an appendix:
* Chapter 1: shape-preserving approximation and interpolation of real functions of one real variable by real polynomials
* Chapter 2: shape-preserving approximation of real functions of several real variables by multivariate real polynomials
* Chapter 3: shape-preserving approximation of analytic functions of one complex variable by complex polynomials in the unit disk
* Chapter 4: shape-preserving approximation of analytic functions of several complex variables on the unit ball or the unit polydisk by polynomials of several complex variables
* Appendix: related results of non-polynomial and non-spline approximations preserving shape, including by complexified operators
Three different constructive methods are applicable to all four categories of shape-preserving approximation by polynomials and thus serve as ‘red lines’ for the book: Bernstein-type, Shisha-type, and convolution-type.
Containing many open problems at the end of each chapter to spur future research as well as a rich and updated bibliography surveying the vast literature, the text will be useful to graduate students and researchers interested in approximation theory, mathematical analysis, numerical analysis, Computer Aided Geometric Design, robotics, data fitting, chemistry, fluid mechanics, and engineering.
Table of contents
Preface.- Shape Preserving Approximation by Real Univariate Polynomials.- Shape Preserving Approximation by Real Multivariate Polynomials.- Shape Preserving Approximation by Complex Univariate Polynomials.- Shape Preserving Approximation by Real Multivariate Polynomials.- Appendix: Some Related Topics.- References.- Index.