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This book presents advances in matrix and tensor data processing in
the domain of signal, image and information processing. The
theoretical mathematical approaches are discusses in the context of
potential applications in sensor and cognitive systems engineering.
The topics and application include Information Geometry,
Differential Geometry of structured Matrix, Positive Definite
Matrix, Covariance Matrix, Sensors (Electromagnetic Fields,
Acoustic sensors) and Applications in Cognitive systems, in
particular Data Mining.
This book presents a substantial part of matrix analysis that is
functional analytic in spirit. Topics covered include the theory of
majorization, variational principles for eigenvalues, operator
monotone and convex functions, and perturbation of matrix functions
and matrix inequalities. The book offers several powerful methods
and techniques of wide applicability, and it discusses connections
with other areas of mathematics.
This book presents advances in matrix and tensor data processing in
the domain of signal, image and information processing. The
theoretical mathematical approaches are discusses in the context of
potential applications in sensor and cognitive systems engineering.
The topics and application include Information Geometry,
Differential Geometry of structured Matrix, Positive Definite
Matrix, Covariance Matrix, Sensors (Electromagnetic Fields,
Acoustic sensors) and Applications in Cognitive systems, in
particular Data Mining."
The aim of this book is to present a substantial part of matrix analysis that is functional analytic in spirit. Much of this will be of interest to graduate students and research workers in operator theory, operator algebras, mathematical physics and numerical analysis. The book can be used as a basic text for graduate courses on advanced linear algebra and matrix analysis. It can also be used as supplementary text for courses in operator theory and numerical analysis. Among topics covered are the theory of majorization, variational principles for eigenvalues, operator monotone and convex functions, perturbation of matrix functions and matrix inequalities. Much of this is presented for the first time in a unified way in a textbook. The reader will learn several powerful methods and techniques of wide applicability, and see connections with other areas of mathematics. A large selection of matrix inequalities will make this book a valuable reference for students and researchers who are working in numerical analysis, mathematical physics and operator theory.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume I
includes the introductory material, the papers on limit theorems
and review articles.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume II
includes the papers on PDE, SDE, diffusions, and random media.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume
III includes the papers on large deviations.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume IV
includes the papers on particle systems.
This book represents the first synthesis of the considerable body
of new research into positive definite matrices. These matrices
play the same role in noncommutative analysis as positive real
numbers do in classical analysis. They have theoretical and
computational uses across a broad spectrum of disciplines,
including calculus, electrical engineering, statistics, physics,
numerical analysis, quantum information theory, and geometry.
Through detailed explanations and an authoritative and inspiring
writing style, Rajendra Bhatia carefully develops general
techniques that have wide applications in the study of such
matrices. Bhatia introduces several key topics in functional
analysis, operator theory, harmonic analysis, and differential
geometry--all built around the central theme of positive definite
matrices. He discusses positive and completely positive linear
maps, and presents major theorems with simple and direct proofs. He
examines matrix means and their applications, and shows how to use
positive definite functions to derive operator inequalities that he
and others proved in recent years. He guides the reader through the
differential geometry of the manifold of positive definite
matrices, and explains recent work on the geometric mean of several
matrices. Positive Definite Matrices is an informative and useful
reference book for mathematicians and other researchers and
practitioners. The numerous exercises and notes at the end of each
chapter also make it the ideal textbook for graduate-level courses.
Like the first volume, this is a special collection of articles
describing the work of some of the best-known mathematicians from
India. It contains eight articles written by experts, each of whom
has chosen one major research contribution by an Indian
mathematician and explained its context, significance, and impact.
This is done in a way that makes the main ideas accessible to
someone whose own research interests might be in a different area.
Included here are commentaries on important works by: R. C. Bose,
S. S. Shrikhande and E. T. Parker H. Cramer and C. R. Rao V. B.
Mehta and A. Ramanathan R. Narasimha K. R. Parthasarathy, R. Ranga
Rao, and V. S. Varadarajan R. Parthasarathy D. N. Verma N. Wiener
and P. R. Masani
Perturbation Bounds for Matrix Eigenvalues contains a unified
exposition of spectral variation inequalities for matrices. The
text provides a complete and self-contained collection of bounds
for the distance between the eigenvalues of two matrices, which
could be arbitrary or restricted to special classes. The book's
emphasis on sharp estimates, general principles, elegant methods,
and powerful techniques, makes it a good reference for researchers
and students. For the SIAM Classics edition, the author has added
over 60 pages of material covering recent results and discussing
the important advances made in the theory, results, and proof
techniques of spectral variation problems in the two decades since
the book's original publication. This updated edition is
appropriate for use as a research reference for physicists,
engineers, computer scientists, and mathematicians interested in
operator theory, linear algebra, and numerical analysis. It is also
suitable for a graduate course in linear algebra or functional
analysis.
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