Carl D. Meyer
Matrix Analysis and Applied Linear Algebra
Table of Contents
Chapter 1. Linear Equations
Chapter 2. Rectangular Systems And Echelon Forms
Chapter 3. Matrix Algebra
Chapter 4. Vector Spaces
Chapter 5. Norms, Inner Products, and Orthogonality
Chapter 6. Determinants
Chapter 7. Eigenvalues and Eigenvectors
Chapter 8. Perron-Frobenius Theory Of Nonnegative Matrices
Index
Calude, C.S., University of Auckland, New Zealand
Paun, G., Romanian Academy of Sciences, Bucharest, Romania
Finite Versus Infinite
Contributions to an Eternal Dilemma
2000. X, 371 pp.
1-85233-251-4
The finite-infinite interplay is central in the human thinking, from ancient philosophers and mathematicians
(Zenon, Pythagoras), to modern mathematics (Cantor, Hilbert) and computer science (Turing, Goedel). Recent
developments in mathematics and computer science suggest radically new answers to classical questions such as:
Does infinity exist?
Where does infinity come from?
How can we reconcile the finiteness of the human brain with the infinity of ideas it produces?
Well-known authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. While mathematical in spirit, contributions have many connections with computer science, cognitive science, linguistics, philosophy, physics, biology and semiotics.
Contents: Smoothing Data: When Finite and Infinite Dimensional Help Each Other (Umberto Amato, Dan Tudor
Vuza).- Rearrangements of Real Functions derived from the Combinatorics of Young Tableaux (Hari Bercovici).- A Possible Distance for Sequences of Equal and Unequal Length (Massimo Borelli, Andrea Sgarro).- Solving Finitely Refutable Mathematical Problems (Cristian Calude, Helmut Juergensen, Shane Legg).- State Complexity of Regular Languages: Finite versus Infinite (Cezar Campeanu, Kai Salmoaa, Sheng Yu).- A Century of Controversy over the Foundations of Mathematics (Gregory Chaitin).- Continuity, Discontinuity, and Negotiation of Meaning in Distributed Virtual Environments (Patrick Coppock).- Finite versus Infinite in Singularity Dynamics (Florin Diacu).- An Interesting Serendipitous Real Number (John Ewing, Ciprian Foias).- Algebraic Representations of Regular Array Languages (Rudolf Freund, Alexandru Mateescu, Arto Salomaa).- Rough Set Processing of Vague Information using Fuzzy Similarity Relations (Salvatore Greco, Benedetto Matarazzo, Roman Slowinski).- Relativised Code Concepts and Multi-Tube DNA Dictionaries (Tom Head).- Uniform Scattered Factors (Lucian Llie, Ion Petre, Grzegorz Rozenberg).- Pregroups: A New Algebraic Approach to Sentence Structure (Jim Lambek).- Splicing Normalization and Regularity (Vincenzo Manca).- DNA Pattern Multigrammars (Brian Mayoh).- Finite and Infinite in Geography (Georges Nicolas).- Ultrafilters, Dictators and Gods (Piergiorgio Odifreddi).- Mathematical Structures for the Chomsky Grammar of English (Luis Peris-Vine).- On the Behaviour of Semiautomata (Sergiu Rudeanu).- Infinity: An Outline of Conceptions in Mythology, Cosmology and Natural Philosophy (Jouko Seppanen).- Finite versus Infinite Neural Computation (Hava Sigeleman).- On Information-Theoretical Aspects of Relational Databases (Dan Simovici, Szymon Jaroszewicz).- A Version of Omega for which ZFC cannot Predict a Single Bit (Robert Solovay).- On the Power of Reading the Whole Infinite Input Tape (Ludwig Staiger).- Finite and Infinite in Logic (Victor Vianu).
Series: Discrete Mathematics and Theoretical Computer Science.
Hurley, D.J., University College Cork, Ireland
Vandyck, M.A., University College Cork, IrelandGeometry, Spinors and Applications
2000. XVII, 369 pp.
1-85233-223-9
This text is a self-contained, comprehensive treatment of the tensor and spinor calculus of space-time manifolds with as few technicalities as correct treatment allows. Both the physical and geometrical motivation of all concepts are discussed, helping the reader to go through the technical details in a confident manner. Several physical theories are discussed and developed beyond standard treatment using results in the book. Both the traditional "index" and modern "coordinate-free" notations are used side-by-side in the book, making it accessible to beginner graduate students in mathematics and physics. The methods developed offer new insights into standard areas of physics, such as classical mechanics or electromagnetism, and takes readers to the frontiers of knowledge of spinor calculus.
Contents: Preface.- Part I Preliminaries and Algebraic Aspects of Spinors: General Vector Spaces.
Vector Spaces with a Metric.- Part II Preliminaries and Geometrical Aspects of Spinors: Manifolds in
General. Lie Groups as Special Manifolds. Fibre Bundles as Special Manifolds.- Part III General Spinorial
Differentiation: Geometrical Definition of C31 (R) Spinors. Differentiation of Spinor Fields. Interplay between
Differentiations. The Invariant Formalism.- Part IV Illustrations and Applications: Newtonian Mechanics and
C30 (R). Electro-Magnetism. Cartan Formalism. Geometrical Gravitational Theories.- A: Infeld-van der Waerden Symbols.- B: Maxwells's Equations: Complements.
Series: Springer Praxis Books.
Puu, T., University of Umea, Finland
Attractors, Bifurcations, and Chaos
Nonlinear Phenomena in Economics
2000. XII, 507 pp. 186 figs., 2 tabs.
3-540-66862-4
The first part of this book is an introduction to mathematical methods in modern nonlinear dynamics and deals with
differential equations, both ordinary and partial, further with discrete maps, and with catastrophe theory.
Phenomena such as bifurcations and deterministic chaos are given considerable emphasis, both in the
methodological part, and in the second part, containing various applications in economics and in regional science.
Coexistence of attractors and the multiplicity of development paths in nonlinear systems are central topics. The
applications focus issues such as business cycles, oligopoly, interregional trade dynamics, and economic
development theory, focusing increasing diversity and complexity. Some of the applications are cast as processes
in discrete time, others are modelled in continuous time. Some have a layout involving even space and hence
address issues of economic pattern formation in geographical space.
Keywords: Nonlinearity, Chaos, Bi ! furcation, Dynamics, Economic Dynamics
Kolokoltsov, V.N., The Nottingham Trent University, Nottingham, UK
Semiclassical Analysis for Diffusions and Stochastic Processes
2000. VIII, 347 pp.
3-540-66972-8
The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic
evolution equations describing the transition probabilities of various Markov processes. These include (i)
diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Levy processes, (iii) complex stochastic Schrodinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.
Keywords: semiclassical approximation, diffusion, Markov processes, Stochastic Hamilton-Jacobi and
Schr?dinger equations, path integral
Contents: I Gaussian diffusions.- II Boundary value problem for Hamiltonian systems.- III Semiclassical
approximation for regular diffusion.- IV Invariant degenerate diffusion on cotangent bundles.- V Transition
probability densities for stable jump-diffusion.- VI Semiclassical asymptotics for the localised Feller-Courr?ge
processes.- VII Complex stochastic diffusions or stochastic Schr?dinger equations.- VIII Some topics in
semiclassical spectral analysis.- IX Path integration for the Schr?dinger, heat and complex stochastic diffusion
equations.
Series: Lecture Notes in Mathematics.VOL. 1724
Maric, V., Novi Sad, Yugoslavia
Regular Variation and Differential Equations
2000. X, 127 pp.
3-540-67160-9
DM 44,-
Recommended List Price
This is the first book offering an application of regular variation to the qualitative theory of differential equations. The notion of regular variation, introduced by Karamata (1930), extended by several scientists, most significantly de Haan (1970), is a powerful tool in studying asymptotics in various branches of analysis and in probability theory. Here, some asymptotic properties (including non-oscillation) of solutions of second order linear and of some non-linear equations are proved by means of a new method that the well-developed theory of regular variation has yielded. A good graduate course both in real analysis and in differential equations suffices for understanding the book.
Keywords: Second order linear differential equations, Differential equations of Thomas-Fermi type, Boundary
layer equation
Contents: I Linear equations: Existence of regular solutions: Preliminaries.- The case f(x)<0.- II- and
I-varying solutions.- The case of f(x) of arbitrary sign.- Regular boundedness of solutions.- Generalizations.-
Examples.- Comments.- Asymptotic behaviour of regular solutions: Slowly varying solutions.- The case of f(x) of arbitrary sign.- The case of f(x)<0.- Regularly varying solutions.- On zeros of oscillating solutions. Examples.- Comments II Nonlinear equations: Equations of Thomas Fermi type: Introduction and preliminaries.- The case of regularly varying f and o/.- The case of rapidly varying f or o/.- An equations arising in boundary-layer theory: Introduction.- Existence and uniqueness.- Estimates and asymptotic behaviour of solutions.- Comments.- Appendix: Properties of regularly varying and related Functions.- References.- Index.
Series: Lecture Notes in Mathematics.VOL. 1726
Kravanja, P., Katholieke Universiteit Leuven, Heverlee, Belgium
Barel, M.V., Katholieke Universiteit Leuven, Heverlee, BelgiumComputing the Zeros of Analytic Functions
2000. VII, 111 pp.
3-540-67162-5
Computing all the zeros of an analytic function and their respective multiplicities, locating clusters of zeros and
analytic fuctions, computing zeros and poles of meromorphic functions, and solving systems of analytic equations are problems in computational complex analysis that lead to a rich blend of mathematics and numerical analysis.
This book treats these four problems in a unified way. It contains not only theoretical results (based on formal
orthogonal polynomials or rational interpolation) but also numerical analysis and algorithmic aspects,
implementation heuristics, and polished software (the package ZEAL) that is available via the CPC Program
Library. Graduate studets and researchers in numerical mathematics will find this book very readable.
Keywords: computational complex analysis, zeros of analytic functions, formal orthogonal polynomials, rational
interpolation
Contents: Zeros of analytic functions: Introduction.- Formal orthogonal polynomials.- An accurate algorithm to
compute zeros of FOPs.- Numerical examples.- The software package ZEAL.- A derivative-free approach.-
Clusters of zeros of analytic functions: How to abtain the centre of a cluster and its weight.- A numerical
example.- Rational interpolation at roots of unity.- More numerical examples.-
Zeros and ples of meromorphic functions: Introduction.- Theoretical considerations and numerical algorithm.- A
numerical example.-
Systems of analytic equations: Introduction.- A multidimensional logarithmic residue formula.- The algorithm.-
Numerical examples.
Series: Lecture Notes in Mathematics.VOL. 1727
Gatermann, K., Konrad-Zuse-Zentrum fur Informationstechnik, Berlin, Germany
Computer Algebra Methods for Equivariant Dynamical Systems
2000. XV, 153 pp.
3-540-67161-7
This book starts with an overview of the research of Grobner bases which have many applications in various areas of mathematics since they are a general tool for the investigation of polynomial systems.
The next chapter describes algorithms in invariant theory including many examples and time tables. These
techniques are applied in the chapters on symmetric bifurcation theory and equivariant dynamics.
This combination of different areas of mathematics will be interesting to researchers in computational algebra
and/or dynamics.
Keywords: Grobner bases, invariant theory, symmetry, dynamical systems
Contents: Grobner bases: Buchberger's algorithm.- The consequence of grading.- Definitions and the
relation to Grobner bases.- Computation of a Hilbert series.- The Hilbert series driven Buchberger algorithm.-
The computation with algebraic extensions.- Detection of Gr?bner bases.- Dynamic Buchberger algorithm.-
Elimination.- Algorithms of the computation of invariants and equivariants: Using the Hilbert series.-
Invariants.- Equivariants.- Using the nullcone.- Using a homogeneous system of parameters.- Computing
uniqueness.- Symmetric bifurcation theory.- Local bifurcation analysis.- An example of secondary Hopf
bifurcation.- Orbit space reduction.- Exact computation of steady states.- Differential equations on the orbit
space.- Using Noether normalization.- Further reading.- References.- Index.
Series: Lecture Notes in Mathematics.VOL. 1728