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For more than thirty years the senior author has been trying to
learn algebraic geometry. In the process he discovered that many of
the classic textbooks in algebraic geometry require substantial
knowledge of cohomology, homological algebra, and sheaf theory. In
an attempt to demystify these abstract concepts and facilitate
understanding for a new generation of mathematicians, he along with
co-author wrote this book for an audience who is familiar with
basic concepts of linear and abstract algebra, but who never has
had any exposure to the algebraic geometry or homological algebra.
As such this book consists of two parts. The first part gives a
crash-course on the homological and cohomological aspects of
algebraic topology, with a bias in favor of cohomology. The second
part is devoted to presheaves, sheaves, Cech cohomology, derived
functors, sheaf cohomology, and spectral sequences. All important
concepts are intuitively motivated and the associated proofs of the
quintessential theorems are presented in detail rarely found in the
standard texts.
This book is a unique work which provides an in-depth exploration
into the mathematical expertise, philosophy, and knowledge of H W
Gould. It is written in a style that is accessible to the reader
with basic mathematical knowledge, and yet contains material that
will be of interest to the specialist in enumerative combinatorics.
This book begins with exposition on the combinatorial and algebraic
techniques that Professor Gould uses for proving binomial
identities. These techniques are then applied to develop formulas
which relate Stirling numbers of the second kind to Stirling
numbers of the first kind. Professor Gould's techniques also
provide connections between both types of Stirling numbers and
Bernoulli numbers. Professor Gould believes his research success
comes from his intuition on how to discover combinatorial
identities.This book will appeal to a wide audience and may be used
either as lecture notes for a beginning graduate level
combinatorics class, or as a research supplement for the specialist
in enumerative combinatorics.
This book provides the mathematical fundamentals of linear algebra
to practicers in computer vision, machine learning, robotics,
applied mathematics, and electrical engineering. By only assuming a
knowledge of calculus, the authors develop, in a rigorous yet down
to earth manner, the mathematical theory behind concepts such as:
vectors spaces, bases, linear maps, duality, Hermitian spaces, the
spectral theorems, SVD, and the primary decomposition theorem. At
all times, pertinent real-world applications are provided. This
book includes the mathematical explanations for the tools used
which we believe that is adequate for computer scientists,
engineers and mathematicians who really want to do serious research
and make significant contributions in their respective fields.
Volume 2 applies the linear algebra concepts presented in Volume 1
to optimization problems which frequently occur throughout machine
learning. This book blends theory with practice by not only
carefully discussing the mathematical under pinnings of each
optimization technique but by applying these techniques to linear
programming, support vector machines (SVM), principal component
analysis (PCA), and ridge regression. Volume 2 begins by discussing
preliminary concepts of optimization theory such as metric spaces,
derivatives, and the Lagrange multiplier technique for finding
extrema of real valued functions. The focus then shifts to the
special case of optimizing a linear function over a region
determined by affine constraints, namely linear programming.
Highlights include careful derivations and applications of the
simplex algorithm, the dual-simplex algorithm, and the primal-dual
algorithm. The theoretical heart of this book is the mathematically
rigorous presentation of various nonlinear optimization methods,
including but not limited to gradient decent, the
Karush-Kuhn-Tucker (KKT) conditions, Lagrangian duality,
alternating direction method of multipliers (ADMM), and the kernel
method. These methods are carefully applied to hard margin SVM,
soft margin SVM, kernel PCA, ridge regression, lasso regression,
and elastic-net regression. Matlab programs implementing these
methods are included.
This textbook explores advanced topics in differential geometry,
chosen for their particular relevance to modern geometry
processing. Analytic and algebraic perspectives augment core
topics, with the authors taking care to motivate each new concept.
Whether working toward theoretical or applied questions, readers
will appreciate this accessible exploration of the mathematical
concepts behind many modern applications. Beginning with an
in-depth study of tensors and differential forms, the authors go on
to explore a selection of topics that showcase these tools. An
analytic theme unites the early chapters, which cover
distributions, integration on manifolds and Lie groups, spherical
harmonics, and operators on Riemannian manifolds. An exploration of
bundles follows, from definitions to connections and curvature in
vector bundles, culminating in a glimpse of Pontrjagin and Chern
classes. The final chapter on Clifford algebras and Clifford groups
draws the book to an algebraic conclusion, which can be seen as a
generalized viewpoint of the quaternions. Differential Geometry and
Lie Groups: A Second Course captures the mathematical theory needed
for advanced study in differential geometry with a view to
furthering geometry processing capabilities. Suited to classroom
use or independent study, the text will appeal to students and
professionals alike. A first course in differential geometry is
assumed; the authors' companion volume Differential Geometry and
Lie Groups: A Computational Perspective provides the ideal
preparation.
This book provides the mathematical fundamentals of linear algebra
to practicers in computer vision, machine learning, robotics,
applied mathematics, and electrical engineering. By only assuming a
knowledge of calculus, the authors develop, in a rigorous yet down
to earth manner, the mathematical theory behind concepts such as:
vectors spaces, bases, linear maps, duality, Hermitian spaces, the
spectral theorems, SVD, and the primary decomposition theorem. At
all times, pertinent real-world applications are provided. This
book includes the mathematical explanations for the tools used
which we believe that is adequate for computer scientists,
engineers and mathematicians who really want to do serious research
and make significant contributions in their respective fields.
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