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This book is an updated and extended version of Completely Positive
Matrices (Abraham Berman and Naomi Shaked-Monderer, World
Scientific 2003). It contains new sections on the cone of
copositive matrices, which is the dual of the cone of completely
positive matrices, and new results on both copositive matrices and
completely positive matrices.The book is an up to date
comprehensive resource for researchers in Matrix Theory and
Optimization. It can also serve as a textbook for an advanced
undergraduate or graduate course.
This monograph is a revised set of notes on recent applications of
the theory of cones, arising from lectures I gave during my stay at
the Centre de recherches mathematiques in Montreal. It consists of
three chapters. The first describes the basic theory. The second is
devoted to applications to mathematical programming and the third
to matrix theory. The second and third chapters are independent.
Natural links between them, such as mathematical programming over
matrix cones, are only mentioned in passing. The choice of
applications described in this paper is a reflection of my p"r9onal
interests, for examples, the complementarity problem and iterative
methods for singular systems. The paper definitely does not contain
all the applications which fit its title. The same remark holds for
the list of references. Proofs are omitted or sketched briefly
unless they are very simple. However, I have tried to include
proofs of results which are not widely available, e.g. results in
preprints or reports, and proofs, based on the theory of cones, of
classical theorems. This monograph benefited from helpful
discussions with professors Abrams, Barker, Cottle, Fan, Plemmons,
Schneider, Taussky and Varga.
The book is based on lecture notes of a course 'from elementary
number theory to an introduction to matrix theory' given at the
Technion to gifted high school students. It is problem based, and
covers topics in undergraduate mathematics that can be introduced
in high school through solving challenging problems. These topics
include Number theory, Set Theory, Group Theory, Matrix Theory, and
applications to cryptography and search engines.
Here is a valuable text and research tool for scientists and
engineers who use or work with theory and computation associated
with practical problems relating to Markov chains and queuing
networks, economic analysis, or mathematical programming.
Originally published in 1979, this new edition adds material that
updates the subject relative to developments from 1979 to 1993.
Theory and applications of nonnegative matrices are blended here,
and extensive references are included in each area. You will be led
from the theory of positive operators via the Perron-Frobenius
theory of nonnegative matrices and the theory of inverse
positivity, to the widely used topic of M-matrices. On the way,
semigroups of nonnegative matrices and symmetric nonnegative
matrices are discussed. Later, applications of nonnegativity and
M-matrices are given; for numerical analysis the example is
convergence theory of iterative methods, for probability and
statistics the examples are finite Markov chains and queuing
network models, for mathematical economics the example is
input-output models, and for mathematical programming the example
is the linear complementarity problem. Nonnegativity constraints
arise very naturally throughout the physical world. Engineers,
applied mathematicians, and scientists who encounter nonnegativity
or generalizations of nonnegativity in their work will benefit from
topics covered here, connecting them to relevant theory.
Researchers in one area, such as queuing theory, may find useful
the techniques involving nonnegative matrices used by researchers
in another area, say, mathematical programming. Exercises and
biographical notes are included with each chapter.
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