Series: Applied and Numerical Harmonic Analysis
2015, XXX, 690 p. 15 illus., 5 illus. in
Hardcover
ISBN 978-3-319-25611-5
* Extensive and updated exercises included at the end of every chapter
* Selected research topics presented with recommendations for more
advanced topics and further reading
* Basic results presented in an accessible way for both pure and
applied mathematicians
* Full proofs included in introductory chapters; only basic knowledge
of functional analysis required
This revised and expanded monograph presents the general theory for frames and Riesz
bases in Hilbert spaces as well as its concrete realizations within Gabor analysis, wavelet
analysis, and generalized shift-invariant systems. Compared with the first edition, more
emphasis is put on explicit constructions with attractive properties. Based on the exiting
development of frame theory over the last decade, this second edition now includes new
sections on the rapidly growing fields of LCA groups, generalized shift-invariant systems,
duality theory for as well Gabor frames as wavelet frames, and open problems in the field.
2015, VIII, 328 p. 40 illus., 20 illus. in
Hardcover
ISBN 978-1-4939-3406-5
* New edition extensively revised and updated
* Includes many new figures and examples
* New topics include random matrix theory and quantum chaos
* Includes recent work on modular forms and their corresponding
L-functions in higher rank, the heat equation on Pn solution, the
central limit theorem for Pn, densest lattice packing of spheres in
Euclidean space, and much more
This text explores the geometry and analysis of higher rank analogues of the symmetric
spaces introduced in volume one. To illuminate both the parallels and differences of the
higher rank theory, the space of positive matrices is treated in a manner mirroring that of
the upper-half space in volume one. This concrete example furnishes motivation for the
general theory of noncompact symmetric spaces, which is outlined in the final chapter.
The book emphasizes motivation and comprehensibility, concrete examples and explicit
computations (by pen and paper, and by computer), history, and, above all, applications in
mathematics, statistics, physics, and engineering.
The second edition includes new sections on Donald St. P. Richards’s central limit theorem
for O( n)-invariant random variables on the symmetric space of GL( n, R), on random
matrix theory, and on advances in the theory of automorphic forms on arithmetic groups.
Series: Universitext
1st ed. 2015, XII, 538 p. 77 illus., 61 illus. in
Softcover
ISBN 978-3-319-27074-6
* Contains nearly 300 compelling problems and exercises
* Contains several fascinating threads that emerge as larger groups of
transformations
* Presents isothermic immersions which have enjoyed a recent rebirth
in the field of integrable systems
Designed for intermediate graduate studies, this text will broaden students' core
knowledge of differential geometry providing foundational material to relevant topics
in classical differential geometry. The method of moving frames, a natural means for
discovering and proving important results, provides the basis of treatment for topics
discussed. Its application in many areas helps to connect the various geometries and
to uncover many deep relationships, such as the Lawson correspondence. The nearly
300 problems and exercises range from simple applications to open problems. Exercises
are embedded in the text as essential parts of the exposition. Problems are collected
at the end of each chapter; solutions to select problems are given at the end of the
book. MathematicaR, Matlab*, and Xfig are used to illustrate selected concepts and
results. The careful selection of results serves to show the reader how to prove the most
important theorems in the subject, which may become the foundation of future progress.
The book pursues significant results beyond the standard topics of an introductory
differential geometry course. A sample of these results includes the Willmore functional,
the classification of cyclides of Dupin, the Bonnet problem, constant mean curvature
immersions, isothermic immersions, and the duality between minimal surfaces in
Euclidean space and constant mean curvature surfaces in hyperbolic space. The book
concludes with Lie sphere geometry and its spectacular result that all cyclides of Dupin
are Lie sphere equivalent. The exposition is restricted to curves and surfaces in order to
emphasize the geometric interpretation of invariants and other constructions. Working in
low dimensions helps students develop a strong geometric intuition. Aspiring geometers
will acquire a working knowledge of curves and surfaces in classical geometries. Students
will learn the invariants of conformal geometry and how these relate to the invariants of
Euclidean, spherical, and hyperbolic geometry. They will learn the fundamentals of Lie
sphere geometry, which require the notion of Legendre immersions of a contact structure.
Prerequisites include a completed one semester standard course on manifold theory.
Series: Universitext
2015, XXI, 629 p. 65 illus.
Hardcover
ISBN 978-3-662-48790-7
* Thoroughness of coverage, from elementary to very advanced
* Clarity of exposition
* Originality and variety of exercises and examples
* Complete logical rigor of discussion
* Various new appendices
* Useful not only to mathematicians, but also to physicists and
engineers
VLADIMIR A. ZORICH is professor of mathematics at Moscow State University. His
areas of specialization are analysis, conformal geometry, quasiconformal mappings,
and mathematical aspects of thermodynamics. He solved the problem of global
homeomorphism for space quasiconformal mappings. He holds a patent in the
technology of mechanical engineering, and he is also known by his book Mathematical
Analysis of Problems in the Natural Sciences .
This second English edition of a very popular two-volume work presents a thorough first
course in analysis, leading from real numbers to such advanced topics as differential forms
on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic
functions; and distributions. Especially notable in this course are the clearly expressed
orientation toward the natural sciences and the informal exploration of the essence and
the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched
by a wealth of instructive exercises, problems, and fresh applications to areas seldom
touched on in textbooks on real analysis.
The main difference between the second and first English editions is the addition of a
series of appendices to each volume. There are six of them in the first volume and five in
the second. The subjects of these appendices are diverse. They are meant to be useful
to both students (in mathematics and physics) and teachers, who may be motivated by
different goals.
Some of the appendices are surveys, both prospective and retrospective. The final survey
establishes important conceptual connections between analysis and other parts of
mathematics.
Series: Lecture Notes of the Unione Matematica Italian, a, Vol. 18
1st ed. 2015, X, 238 p.
Softcover
ISBN 978-3-319-26764-7
* Winner of the 2015 Book Prize of the Unione Matematica Italiana
Providing an introduction to both classical and modern techniques in projective algebraic
geometry, this monograph treats the geometrical properties of varieties embedded in
projective spaces, their secant and tangent lines, the behavior of tangent linear spaces,
the algebro-geometric and topological obstructions to their embedding into smaller
projective spaces, and the classification of extremal cases. It also provides a solution of
Hartshorne’s Conjecture on Complete Intersections for the class of quadratic manifolds
and new short proofs of previously known results, using the modern tools of Mori Theory
and of rationally connected manifolds.
The new approach to some of the problems considered can be resumed in the principle
that, instead of studying a special embedded manifold uniruled by lines, one passes to
analyze the original geometrical property on the manifold of lines passing through a
general point and contained in the manifold. Once this embedded manifold, usually of
lower codimension, is classified, one tries to reconstruct the original manifold, following
a principle appearing also in other areas of geometry such as projective differential
geometry or complex geometry.