Series: Springer Monographs in Mathematics
2010, X, 560 p., Hardcover
ISBN: 978-3-642-05013-8
Due: March 2010
The very classical orthogonal polynomials named after Hermite, Laguerre and Jacobi, satisfy many common properties. For instance, they satisfy a second-order differential equation with polynomial coefficients and they can be expressed in terms of a hypergeometric function.
Replacing the differential equation by a second-order difference equation results in (discrete) orthogonal polynomial solutions with similar properties. Generalizations of these difference equations, in terms of Hahn's q-difference operator, lead to both continuous and discrete orthogonal polynomials with similar properties. For instance, they can be expressed in terms of (basic) hypergeometric functions.
Based on Favard's theorem, the authors first classify all families of orthogonal polynomials satisfying a second-order differential or difference equation with polynomial coefficients. Together with the concept of duality this leads to the families of hypergeometric orthogonal polynomials belonging to the Askey scheme. For each family they list the most important properties and they indicate the (limit) relations.
Furthermore the authors classify all q-orthogonal polynomials satisfying a second-order q-difference equation based on Hahn's q-operator. Together with the concept of duality this leads to the families of basic hypergeometric orthogonal polynomials which can be arranged in a q-analogue of the Askey scheme. Again, for each family they list the most important properties, the (limit) relations between the various families and the limit relations (for q --> 1) to the classical hypergeometric orthogonal polynomials belonging to the Askey scheme.
These (basic) hypergeometric orthogonal polynomials have several applications in various areas of mathematics and (quantum) physics such as approximation theory, asymptotics, birth and death processes, probability and statistics, coding theory and combinatorics.
Foreword by Tom H. Koornwinder.- Preface.- 1.Definitions and miscellaneous formulas.- 2.Polynomial solutions of eigenvalue problems.- 3.Orthogonality of the polynomial solutions.- Part I: Classical orthogonal polynomials.- 4.Orthogonal polynomial solutions of differential equations, Continuous classical orthogonal polynomials.- 5.Orthogonal polynomial solutions of real difference equations, Discrete classical orthogonal polynomials I.- 6.Orthogonal polynomial solutions of complex difference equations, Discrete classical orthogonal polynomials II.- 7.Orthogonal polynomial solutions in x(x+u) of real difference equations, Discrete classical orthogonal polynomials III.- 8.Orthogonal polynomial solutions in z(z+u) of complex difference equations, Discrete classical orthogonal polynomials IV. Askey scheme of hypergeometric orthogonal polynomials.- 9.Hypergeometric orthogonal polynomials.- Part II: Classical q-orthogonal polynomials.- 10.Orthogonal polynomial solutions of q-difference equation.- Classical q-orthogonal polynomials I.- 11.Orthogonal polynomial solutions in q-x of q-difference equations,Classical q-orthogonal polynomials II.- 12.Orthogonal polynomial solutions in q-x +uqx of real q-difference equations, Classical q-orthogonal polynomials III.- 13.Orthogonal polynomial solutions in a/z + uz/a of complex q-difference equations, Classical q-orthogonal polynomials IV.- 14.Basic hypergeometric orthogonal polynomials.- Bibliography.- Index.
Series: Operator Theory: Advances and Applications , Vol. 207
2010, Approx. 310 p., Hardcover
ISBN: 978-3-0346-0346-1
Due: April 2010
The book is a commemorative volume honoring the mathematician Paul R. Halmos (1916-2006), who contributed passionately to mathematics in manifold ways, among them by basic research, by unparalleled mathematical exposition, by unselfish service to the mathematical community, and, not least, by the inspiration others found in his dedication to that community.Halmos made fundamental contributions in several areas of mathematics.
This volume emphasises Halmos's contributions to operator theory, his venue for most of his mathematical life. The core of the volume is a series of expository articles by prominent operator theorists providing an overview of how operator theory prospered during the Halmos era, in no small measure thanks to Halmos's leadership and penetrating insights.
Preface. 1. About Paul Halmos.- Paul Halmos ? Expositor par excellence, by V. S. Sunder.- Paul Halmos: In his own words, by John Ewing.- Obituary: Paul Halmos, 1916?2006, by Heydar Radjavi and Peter Rosenthal.- Mathematical Review of How to write mathematics, by George Piranian.- 2. Publications of Paul R. Halmos.- Photos.- 4. Articles.- Jim Agler and John E. McCarthy, What can Hilbert spaces tell us about bounded functions in the bidisk?.- William Arveson, Dilation theory yesterday and today.- Sheldon Axler, Toeplitz operators.- Hari Bercovici, Dual algebras and invariant subspaces.- John B. Conway and Nathan S. Feldman, The state of subnormal operators.- Raul Curto and Mihai Putinar, Polynomially hyponormal operators.- Kenneth R. Davidson, Essentially normal operators.- Michael A. Dritschel and James Rovnyak, The operator Fejer-Riesz Theorem.- Paul S. Muhly, A Halmos doctrine and shifts on Hilbert space.- V. V. Peller, The behavior of functions of operators under perturbations.- Gilles Pisier, The Halmos similarity problem.- Heydar Radjavi and Peter Rosenthal, Paul Halmos and invariant subspaces.
Series: Pure and Applied Mathematics
ISBN: 9781439819463
Publication Date: 31/08/2010
Pages: 500
For at least forty years, translation planes, spreads, nets, and parallelisms have played a central role in finite geometry. Finite geometry is an important, wide-ranging research area in mathematics with strong connections to algebra and combinatorics. It also has relevant practical applications in coding theory and cryptography. Written by an active researcher in translation planes, spreads, nets, and parallelisms, this book introduces parallelisms and spreads in a comprehensive manner. Topics covered include partial spreads, spreadsets, quasifields, collineations, automorphisms and autotopisms.
Series: Pure and Applied Mathematics
ISBN: 9781439807026
Publication Date: 31/12/2010
Pages: 250
First realized in the 1980s by Lenstra, Lenstra, and Lovasz, the LLL algorithm was originally intended to factor polynomials with rational coefficients. It improved upon the existing lattice reduction algorithm in order to solve integer linear programming problems and was later adapted for use in crypanalysis. This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm. With numerous examples and suggested exercises, the text discusses various applications of lattice basis reduction to polynomial factorization, cryptography, number theory, and matrix canonical forms.
Introduction to lattices. Lattice reduction in two dimensions. Gram-Schmidt orthogonalization. The original LLL algorithm. Polynomial factorization. Knapsack cryptosystems. Integer relations among real numbers. Lattice reduction with dependent vectors. g. Deep insertions. Semi-reduction. Floating point arithmetic I. Floating point arithmetic II. Algorithms of Kannan and Helfrich. The Schnorr hierarchy. The Hermite normal form.
Series on Number Theory & Its Applications - Vol. 6
268pp Pub. date: Nov 2009
ISBN: 978-981-4289-84-9
This volume aims at collecting survey papers which give broad and enlightening perspectives of various aspects of number theory.
Kitaoka's paper is a continuation of his earlier paper published in the last proceedings and pushes the research forward. Browning's paper introduces a new direction of research on analytic number theory ? quantitative theory of some surfaces and Bruedern et al's paper details state-of-the-art affairs of additive number theory. There are two papers on modular forms ? Kohnen's paper describes generalized modular forms (GMF) which has some applications in conformal field theory, while Liu's paper is very useful for readers who want to have a quick introduction to Maass forms and some analytic-number-theoretic problems related to them. Matsumoto et al's paper gives a very thorough survey on functional relations of root system zeta-functions, Hoshi?Miyake's paper is a continuation of Miyake's long and fruitful research on generic polynomials and gives rise to related Diophantine problems, and Jia's paper surveys some dynamical aspects of a special arithmetic function connected with the distribution of prime numbers. There are two papers of collections of problems by Shparlinski on exponential and character sums and Schinzel on polynomials which will serve as an aid for finding suitable research problems. Yamamura's paper is a complete bibliography on determinant expressions for a certain class number and will be useful to researchers.
Thus the book gives a good-balance of classical and modern aspects in number theory and will be useful to researchers including enthusiastic graduate students.
Recent Progress on the Quantitative Arithmetic of Del Pezzo Surfaces (T D Browning)
Additive Representation in Thin Sequences, VIII: Diophantine Inequalities in Review (J Brudern et al.)
Recent Progress on Dynamics of a Special Arithmetic Function (C-H Jia)
Some Diophantine Problems Arising from the Isomorphism Problem of Generic Polynomials (A Hoshi & K Miyake)
A Statistical Relation of Roots of a Polynomial in Different Local Fields II (Y Kitaoka)
Generalized Modular Functions and Their Fourier Coefficients (W Kohnen)
Functional Relations for Zeta-Functions of Root Systems (Y Komori et al.)
A Quick Introduction to Maass Forms (J-Y Liu)
The Number of Non-Zero Coefficients of a Polynomial-Solved and Unsolved Problems (A Schinzel)
Open Problems on Exponential and Character Sums (I E Shparlinski)
Errata to gA General Modular Relation in Analytic Number Theoryh (H Tsukada)
Bibliography on Determinantal Expressions of Relative Class Numbers of Imaginary Abelian Number Fields (K Yamamura)
Readership: Graduate students and researchers in mathematics.