The Real Positive Definite Complection Problem
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Author | : Wayne Walton Barrett |
Publisher | : American Mathematical Soc. |
Total Pages | : 82 |
Release | : 1996 |
Genre | : Mathematics |
ISBN | : 0821804731 |
Given a partial symmetric matrix, the positive definite completion problem asks if the unspecified entries in the matrix can be chosen so as to make the resulting matrix positive definite. Applications include probability and statistics, image enhancement, systems engineering, geophysics, and mathematical programming. The positive definite completion problem can also be viewed as a mechanism for addressing a fundamental problem in Euclidean geometry: which potential geometric configurations of vectors (i.e., configurations with angles between some vectors specified) are realizable in a Euclidean space. The positions of the specified entries in a partial matrix are naturally described by a graph. The question of existence of a positive definite completion was previously solved completely for the restrictive class of chordal graphs and this work solves the problem for the class of cycle completable graphs, a significant generalization of chordal graphs. These are graphs for which knowledge of completability for induced cycles (and cliques) implies completability of partial symmetric matrices with the given graph.
Author | : Wayne W. Barrett |
Publisher | : |
Total Pages | : 69 |
Release | : 1996 |
Genre | : |
ISBN | : |
Author | : Liviu I. Nicolaescu |
Publisher | : American Mathematical Soc. |
Total Pages | : 98 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 0821806211 |
In this book, an index theorem is proved for arbitrary families of elliptic boundary value problems for Dirac operators and a surgery formula for the index of a family of Dirac operators on a closed manifold. Also obtained is a very general result on the cobordism invariance of the index of a family. All results are established by first symplectically rephrasing the problems and then using a generalized symplectic reduction technique. This provides a unified approach to all possible parameter spaces and all possible symmetries of a Dirac operator (eigh symmetries in the real case and two in the complex case). This text will also be of interest to those working in geometry and topology.
Author | : H. Andréka |
Publisher | : American Mathematical Soc. |
Total Pages | : 146 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 0821805959 |
"We prove that any variety of relation algebras which contains an algebra with infinitely many elements below the identity, or which contains the full group relation algebra on some infinite group (or on arbitrarily large finite groups), must have an undecidable equational theory. Then we construct an embedding of the lattice of all subsets of the natural numbers into the lattice of varieties of relation algebras such that the variety correlated with a set [italic capital]X of natural numbers has a decidable equational theory if and only if [italic capital]X is a decidable (i.e., recursive) set. Finally, we construct an example of an infinite, finitely generated, simple, representable relation algebra that has a decidable equational theory.'' -- Abstract.
Author | : Abraham Berman |
Publisher | : World Scientific |
Total Pages | : 218 |
Release | : 2003-04-11 |
Genre | : Mathematics |
ISBN | : 9814486000 |
A real matrix is positive semidefinite if it can be decomposed as A=BB′. In some applications the matrix B has to be elementwise nonnegative. If such a matrix exists, A is called completely positive. The smallest number of columns of a nonnegative matrix B such that A=BB′ is known as the cp-rank of A.This invaluable book focuses on necessary conditions and sufficient conditions for complete positivity, as well as bounds for the cp-rank. The methods are combinatorial, geometric and algebraic. The required background on nonnegative matrices, cones, graphs and Schur complements is outlined.
Author | : Marloes Maathuis |
Publisher | : CRC Press |
Total Pages | : 612 |
Release | : 2018-11-12 |
Genre | : Mathematics |
ISBN | : 0429874235 |
A graphical model is a statistical model that is represented by a graph. The factorization properties underlying graphical models facilitate tractable computation with multivariate distributions, making the models a valuable tool with a plethora of applications. Furthermore, directed graphical models allow intuitive causal interpretations and have become a cornerstone for causal inference. While there exist a number of excellent books on graphical models, the field has grown so much that individual authors can hardly cover its entire scope. Moreover, the field is interdisciplinary by nature. Through chapters by leading researchers from different areas, this handbook provides a broad and accessible overview of the state of the art. Key features: * Contributions by leading researchers from a range of disciplines * Structured in five parts, covering foundations, computational aspects, statistical inference, causal inference, and applications * Balanced coverage of concepts, theory, methods, examples, and applications * Chapters can be read mostly independently, while cross-references highlight connections The handbook is targeted at a wide audience, including graduate students, applied researchers, and experts in graphical models.
Author | : Robin Hartshorne |
Publisher | : American Mathematical Soc. |
Total Pages | : 111 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 0821806483 |
Content Description #"November 1997, volume 130, number 617 (first of 4 numbers)."#On t.p. "P" is blackboard bold.#Includes bibliographical references.
Author | : Charles R. Johnson |
Publisher | : American Mathematical Soc. |
Total Pages | : 272 |
Release | : 1990 |
Genre | : Mathematics |
ISBN | : 0821801546 |
This volume contains the lecture notes prepared for the AMS Short Course on Matrix Theory and Applications, held in Phoenix in January, 1989. Matrix theory continues to enjoy a renaissance that has accelerated in the past decade, in part because of stimulation from a variety of applications and considerable interplay with other parts of mathematics. In addition, the great increase in the number and vitality of specialists in the field has dispelled the popular misconception that the subject has been fully researched.
Author | : Leslie Hogben |
Publisher | : CRC Press |
Total Pages | : 1838 |
Release | : 2013-11-26 |
Genre | : Mathematics |
ISBN | : 1466507292 |
With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and
Author | : Panos M. Pardalos |
Publisher | : American Mathematical Soc. |
Total Pages | : 272 |
Release | : 1998 |
Genre | : Mathematics |
ISBN | : 0821808257 |
This volume presents refereed papers presented at the workshop Semidefinite Programming and Interior-Point Approaches for Combinatorial Problems: held at The Fields Institute in May 1996. Semidefinite programming (SDP) is a generalization of linear programming (LP) in that the non-negativity constraints on the variables is replaced by a positive semidefinite constraint on matrix variables. Many of the elegant theoretical properties and powerful solution techniques follow through from LP to SDP. In particular, the primal-dual interior-point methods, which are currently so successful for LP, can be used to efficiently solve SDP problems. In addition to the theoretical and algorithmic questions, SDP has found many important applications in combinatorial optimization, control theory and other areas of mathematical programming. The papers in this volume cover a wide spectrum of recent developments in SDP. The volume would be suitable as a textbook for advanced courses in optimization. It is intended for graduate students and researchers in mathematics, computer science, engineering and operations.