Series: Algebra and Applications , Vol. 8
2008, XI, 517 p., Hardcover
ISBN: 978-1-4020-6946-8
Due: March 2008
About this book
Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields in much the same way as the classical algebraic geometry arose from the study of polynomial equations with numerical coefficients. The first stage of the development of the theory is associated with its founder J. F. Ritt (1893 - 1951) and R. Cohn whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrew the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings.
This book reflects the contemporary level of difference algebra; it contains a systematic study of partial difference algebraic structures and their applications, as well as the coverage of the classical theory of ordinary difference rings and field extensions. The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. It will be also of interest to researchers in computer algebra, theory of difference equations and equations of mathematical physics. The book is self-contained; it requires no prerequisites other than knowledge of basic algebraic concepts and mathematical maturity of an advanced undergraduate.
Table of contents
Preface. 1. Preliminaries. 1.1 Basic terminology and background material. 1.2 Elements of the theory of commutative rings. 1.3 Graded and filtered rings and modules. 1.4 Numerical polynomials. 1.5 Dimension polynomials of sets of m-tuples. 1.6 Basic facts of the field theory. 1.7 Derivations and modules of differentials. 1.8 Grobner Bases. 2. Basic concepts of difference algebra. 2.1 Difference and inversive difference rings. 2.2 Rings of difference and inversive difference polynomials. 2.3 Difference ideals. 2.4 Autoreduced sets of difference and inversive difference polynomails. Characteristic sets. 2.5 Ritt difference rings. 2.6 Varieties of difference polynomials. 3. Difference modules. 3.1 Rings of difference operators. Difference modules. 3.2 Dimension polynomials of difference modules. 3.3 Grobner Bases with respects to several orderings and multivariable dimension polynomials of difference modules. 3.4 Inversive difference modules. 3.5 s*-Dimension polynomials and their invariants. 3.6 Dimension of general difference modules. 4. Difference field extensions. 4.1 Transformal dependence. Difference transcendental bases and difference transcendental degree. 4.2 Dimension polynomials of difference and inversive difference field extensions. 4.3 Limit degree. 4.4 The fundamental theorem on finitely generated difference field extensions. 4.5 Some results on ordinary difference field extensions. 4.6 Difference algebras. 5. Compatibility, Replicability, and Monadicity. Difference specializations. 5.1 Compatible and incompatible difference field extensions. 5.2 Difference kernels over ordinary difference fields. 5.3 Difference specializations. 5.4 Babbitt's decomposition. Criterion of compatibility. 5.5 Replicability. 5.6 Monadicity. 6. Difference kernels over partial difference fields. Difference valuation rings. 6.1 Difference kernels over partial difference fields and their prolongations. 6.2 Realizations of difference kernels over partial difference fields. 6.3 Difference valuation rings and extensions of difference specializations. 7. Systems of algebraic difference equations. 7.1 Solutions of ordinary difference polynomials. 7.2 Existence theorem for ordinary algebraic difference equations. 7.3 Existence of solutions of difference polynomials in the case of two translations. 7.4 Singular and Multiple Realizations. 7.5 Review of further results on varieties of ordinary difference polynomials. 7.6 Ritt's number. Greenspan's and Jacobi's Bounds. 7.7 Dimension polynomials and the strength of a system of algebraic difference equations. 8. Elements of the difference galois theory. 8.1 Galois correspondence for difference field extensions. 8.2 Picard-Vessiot theory of linear homogeneous difference equations. 8.3 Picard-Vessiot rings and galois theory of difference equations. Bibliography. Index.
Series: Operator Theory: Advances and Applications, Vol. 182
Subseries: Linear Operators and Linear Systems
2008, Approx. 250 p., Hardcover
ISBN: 978-3-7643-8731-0
Due: March 2008
About this book
In this monograph the natural evolution operators of autonomous first-order differential equations with exponential dichotomy on an arbitrary Banach space are studied in detail. Characterizations of these so-called exponentially dichotomous operators in terms of their resolvents and additive and multiplicative perturbation results are given. The general theory of the first three chapters is then followed by applications to Wiener-Hopf factorization and Riccati equations, transport equations, diffusion equations of indefinite Sturm-Liouville type, noncausal infinite-dimensional linear continuous-time systems, and functional differential equations of mixed type.
Written for:
Table of contents
Preface.- 1. Exponentially Dichotomous Operators and Bisemigroups.- 2.- Perturbing Exponentially Dichotomous Operators.- 3. Abstract Cauchy Problems.- 4. Riccati Equations and Wiener-Hopf Factorization.- 5. Transport Equations.- 6. Indefinite Sturm-Liouville Problems.- 7. Noncausal Continuous Time Systems.- 8. Mixed-type Functional Differential Equations.- Bibliography.- Index.
Collection: Progress in Mathematics , Vol. 267
2008, Approx. 208 p., Relie
ISBN: 978-3-7643-8709-9
a paraitre: mars 2008
A propos de ce livre
The classical theory of Sturm sequences provides an algorithm for determining the number of roots of a polynomial with real coefficients contained in an open interval. The main purpose of this monograph is to show that a suitable generalization of the theory of Sturm sequences provides, among others: a notion of Maslov index for an algebraic loop of lagrangians defined over a commutative ring; a proof of the fundamental theorem of (algebraic) hermitian K-theory (theorem due to M. Karoubi); a proof of the theorems of (topological) Bott periodicity (in the spirit of the work of F. Latour); the computation of the relative K2-group, symplectic-linear, for all commutative ring (in the spirit of the work of R. Sharpe).
Sommaire
1. Introduction.- 2. Algebre lineaire symplectique.- 3. Sur la "composante connexe" du point base dans la lagrangienne infinie.- 4. Enonce et demonstration du theoreme fondamental de la K-theorie hermitienne.- 5. Suites de Sturm et H2 de l'homomorphisme hyperbolique.- 6. Generalisations.- References.
2008, Approx. 360 p., Softcover
ISBN: 978-3-7643-8721-1
Due: April 2008
This book is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. It consists of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance.
The two parts have some connections, due to the fact that the space of probability measures provides an important model to which the "metric" theory applies, but the book is conceived in such a way that the two parts can be read independently, the first one by the reader more interested in non-smooth analysis and analysis in metric spaces, and the second one by the reader more orientated towards the applications in partial differential equations, measure theory and probability.
Table of contents
1. Introduction.- Part I. Gradient flow in metric spaces - 2. Curves and gradients in metric spaces - 3. Existence of curves of maximal slope - 4. Proofs of the convergence theorems - 5. Generation of contraction semigroups.- Part II. Gradient flow in the Wasserstein spaces of probability measures - 6. Preliminary results on measure theory - 7. The optimal transportation problem - 8. The Wasserstein distance and its behaviour along geodesics - 9. A.c. curves and the continuity equation - 10. Convex functionals - 11. Metric slope and subdifferential calculus - 12. Gradient flows and curves of maximal slope - 13. Appendix.- Bibliograph
Series: Springer Texts in Statistics
2008, Approx. 310 p., Hardcover
ISBN: 978-0-387-75964-7
Due: April 2008
About this textbook
Basic theoretical underpinnings are covered
Describes the principles that drive good designs and good statistics
Although statistical design is one of the oldest branches of statistics, its importance is ever increasing, especially in the face of the data flood that often faces statisticians. It is important to recognize the appropriate design, and to understand how to effectively implement it, being aware that the default settings from a computer package can easily provide an incorrect analysis. The goal of this book is to describe the principles that drive good design, paying attention to both the theoretical background and the problems arising from real experimental situations. Designs are motivated through actual experiments, ranging from the timeless agricultural randomized complete block, to microarray experiments, which naturally lead to split plot designs and balanced incomplete blocks.
George Casella is Distinguished Professor in the Department of Statistics at the University of Florida. He is active in many aspects of statistics, having contributed to theoretical statistics in the areas of decision theory and statistical confidence, to environmental statistics, and has more recently concentrated efforts in statistical genomics. He also maintains active research interests in the theory and application of Monte Carlo and other computationally intensive methods. He is listed as an ISI "Highly Cited Researcher."
In other capacities, Professor Casella has served as Theory and Methods Editor of the Journal of the American Statistical Association, 1996-1999, Executive Editor of Statistical Science, 2001-2004, and Co-Editor of the Journal of the Royal Statistical Society, Series B, 2009-2012. He has served on the Board of Mathematical Sciences of the National Research Council, 1999-2003, and many committees of both the American Statistical Association and the Institute of Mathematical Statistics. Professor Casella has co-authored five textbooks: Variance Components, 1992; Theory of Point Estimation, Second Edition, 1998; Monte Carlo Statistical Methods, Second Edition, 2004; Statistical Inference, Second Edition, 2001, and Statistical Genomics of Complex Traits, 2007.
Table of contents
Basics.- Completely randomized designs.- Complete block designs.- Interlude: assessing the effects of blocking.- Split plot designs.- Confounding in blocks.