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Author: Wayne Walton Barrett
Publisher: American Mathematical Soc.
ISBN: 0821804731
Category : Mathematics
Languages : en
Pages : 82
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Book Description
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 Walton Barrett
Publisher: American Mathematical Soc.
ISBN: 0821804731
Category : Mathematics
Languages : en
Pages : 82
Get Book Here
Book Description
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: Panos M. Pardalos
Publisher: American Mathematical Soc.
ISBN: 0821808257
Category : Mathematics
Languages : en
Pages : 272
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Book Description
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.
Author: Dragana S. Cvetković Ilić
Publisher: American Mathematical Society
ISBN: 1470469871
Category : Mathematics
Languages : en
Pages : 170
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Book Description
Completion problems for operator matrices are concerned with the question of whether a partially specified operator matrix can be completed to form an operator of a desired type. The research devoted to this topic provides an excellent means to investigate the structure of operators. This book provides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory. It will be useful for graduate students and researchers interested in operator theory and the problem of matrix completions.
Author: Marloes Maathuis
Publisher: CRC Press
ISBN: 0429874235
Category : Mathematics
Languages : en
Pages : 612
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Book Description
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: Károly Bezdek
Publisher: Springer Science & Business Media
ISBN: 3319002007
Category : Mathematics
Languages : en
Pages : 341
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Book Description
Optimization has long been a source of both inspiration and applications for geometers, and conversely, discrete and convex geometry have provided the foundations for many optimization techniques, leading to a rich interplay between these subjects. The purpose of the Workshop on Discrete Geometry, the Conference on Discrete Geometry and Optimization, and the Workshop on Optimization, held in September 2011 at the Fields Institute, Toronto, was to further stimulate the interaction between geometers and optimizers. This volume reflects the interplay between these areas. The inspiring Fejes Tóth Lecture Series, delivered by Thomas Hales of the University of Pittsburgh, exemplified this approach. While these fields have recently witnessed a lot of activity and successes, many questions remain open. For example, Fields medalist Stephen Smale stated that the question of the existence of a strongly polynomial time algorithm for linear optimization is one of the most important unsolved problems at the beginning of the 21st century. The broad range of topics covered in this volume demonstrates the many recent and fruitful connections between different approaches, and features novel results and state-of-the-art surveys as well as open problems.
Author: Abraham Berman
Publisher: World Scientific
ISBN: 9814486000
Category : Mathematics
Languages : en
Pages : 218
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Book Description
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: Seth Sullivant
Publisher: American Mathematical Society
ISBN: 1470475103
Category : Mathematics
Languages : en
Pages : 506
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Book Description
Algebraic statistics uses tools from algebraic geometry, commutative algebra, combinatorics, and their computational sides to address problems in statistics and its applications. The starting point for this connection is the observation that many statistical models are semialgebraic sets. The algebra/statistics connection is now over twenty years old, and this book presents the first broad introductory treatment of the subject. Along with background material in probability, algebra, and statistics, this book covers a range of topics in algebraic statistics including algebraic exponential families, likelihood inference, Fisher's exact test, bounds on entries of contingency tables, design of experiments, identifiability of hidden variable models, phylogenetic models, and model selection. With numerous examples, references, and over 150 exercises, this book is suitable for both classroom use and independent study.
Author: James Damon
Publisher: American Mathematical Soc.
ISBN: 0821804812
Category : Mathematics
Languages : en
Pages : 130
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Book Description
Almost free divisors and complete intersections form a general class of nonisolated hypersurface and completer intersection singularities. They also include discriminants of mappings, bifurcation sets, and certain types of arrangements of hyperplanes such as Coxeter arrangements and generic arrangements. Associated to the singularities of this class is a "singular Milnor fibration" which has the same homotopy properties as the Milnor fibration for isolated singularities. This memoir deduces topological properties of singularities in a number of situations including: complements of hyperplane arrangements, various nonisolated complete intersections, nonlinear arrangements of hypersurfaces, functions on discriminants, singularities defined by compositions of functions, and bifurcation sets.
Author: Moshé Flato
Publisher: American Mathematical Soc.
ISBN: 0821806831
Category : Mathematics
Languages : en
Pages : 328
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Book Description
The purpose of this work is to present and give full proofs of new original research results concerning integration of and scattering for the classical Maxwell-Dirac equations.
Author: Christopher Keil McCord
Publisher: American Mathematical Soc.
ISBN: 0821806920
Category : Mathematics
Languages : en
Pages : 106
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Book Description
The phase space of the spatial three-body problem is an open subset in R18. Holding the ten classical integrals of energu, center of mass, linear and angular momentum fixed defines an eight dimensional manifold. For fixed nonzero angular momentum, the topology of this manifold depends only on the energy. This volume computes the homology of this manifold for all energy values. This table of homology shows that for negative energy, the integral manifolds undergo seven bifurcations. Four of these are the well-known bifurcations due to central configurations, and three are due to "critical points at infinity". This disproves Birkhoffs conjecture that the bifurcations occur only at central configurations.
