Due 2018-09-16
1st ed. 2018, XI, 1034 p.
Hardcover
ISBN 978-981-13-0937-3
Product category
Undergraduate textbook
Discusses major topics in real and complex analysis
Includes the essential analysis that is needed for the study of functional analysis
Presents applications of complex analysis to analytic number theory
Features over 800 step-by-step, fully solved examples
Is useful to undergraduate students of mathematics and engineering
This book offers a clear presentation of major topics in real and complex analysis. Intended for
undergraduate mathematics and engineering students, it covers the essential analysis that is
needed for the study of functional analysis, developing the concepts rigorously with sufficient
detail and with minimum prior knowledge of the fundamentals of advanced calculus required.
Divided into seven chapters, it discusses the proof of the prime number theorem, Picard’s little
theorem, Riemann’s zeta function, Euler’s gamma function and Riemann hypothesis. It also
addresses the applications of complex analysis to analytic number theory in a manner that
does not require any prior exposure to number theory. Further, it includes extensive exercises
and their solutions with each concept. The book examines several useful theorems in the
realm of real and complex analysis, most of which are the work great mathematicians of the
19th and 20th centuries, such as Baire, Banach, Cauchy, Dirichlet, Fatou, Fourier, Fubini,
Hadamard, Jordan, Lebesgue, Liouville, Minkowski, Morera, Picard, Poisson, Radon, Riemann,
Riesz, Schwarz, Taylor, Weierstrass, and Young, to name just a few.
Due 2018-09-30
1st ed. 2018, X, 562 p. 13
illus.
Hardcover
ISBN 978-3-319-95176-8
Graduate/advanced undergraduate textbook
Series
Birkhauser Advanced Texts Basler
Combines the standard material on algebraic field extensions with some
related but advanced topics
Each chapter starts with a motivating introduction and ends with a set of
specially adapted exercises, some of them with solutions in the appendix
Can be used as an accompanying textbook for a course, but also as a source
for more specialized seminars
The entire text is self-contained, up to a few rudimentary facts from Linear Algebra
The material presented here can be divided into two parts. The first, sometimes referred to as
abstract algebra, is concerned with the general theory of algebraic objects such as groups,
rings, and fields, hence, with topics that are also basic for a number of other domains in
mathematics. The second centers around Galois theory and its applications. Historically, this
theory originated from the problem of studying algebraic equations, a problem that, after
various unsuccessful attempts to determine solution formulas in higher degrees, found its
complete clarification through the brilliant ideas of E. Galois. The study of algebraic equations
has served as a motivating terrain for a large part of abstract algebra, and according to this,
algebraic equations are visible as a guiding thread throughout the book. To underline this
point, an introduction to the history of algebraic equations is included. The entire book is selfcontained,
up to a few prerequisites from linear algebra. It covers most topics of current
algebra courses and is enriched by several optional sections that complement the standard
program or, in some cases, provide a first view on nearby areas that are more advanced. Every
chapter begins with an introductory section on "Background and Overview," motivating the
material that follows and discussing its highlights on an informal level. Furthermore, each
section ends with a list of specially adapted exercises, some of them with solution proposals in
the appendix. The present English edition is a translation and critical revision of the eighth
German edition of the Algebra book by the author.
Due 2018-08-31
1st ed. 2018, VIII, 372 p.
Softcover
ISBN 978-3-319-96573-4
Product category
Graduate/advanced undergraduate textbook
Series
Modern Birkhauser Classics
Multivariable Linear Systems and Projective Algebraic Geometry
Introduces seven representations of a multivariable linear system and
establishes the major results of the underlying theory
Makes the basic ideas of algebraic geometry accessible to engineers and
applied scientists
Emphasizes constructive methods and clarity in examples, rather than
abstraction
"An introduction to the ideas of algebraic geometry in the motivated context of system theory."
This describes this two volume work which has been specifically written to serve the needs of
researchers and students of systems, control, and applied mathematics. Without sacrificing
mathematical rigor, the author makes the basic ideas of algebraic geometry accessible to
engineers and applied scientists. The emphasis is on constructive methods and clarity rather
than on abstraction. While familiarity with Part I is helpful, it is not essential, since a
considerable amount of relevant material is included here. Part I,Scalar Linear Systems and
Affine Algebraic Geometry,contains a clear presentation, with an applied flavor , of the core
ideas in the algebra-geometric treatment of scalar linear system theory. Part II extends the
theory to multivariable systems. After delineating limitations of the scalar theory through
carefully chosen examples, the author introduces seven representations of a multivariable
linear system and establishes the major results of the underlying theory. Of key importance is
a clear, detailed analysis of the structure of the space of linear systems including the full set
of equations defining the space. Key topics also covered are the Geometric Quotient Theorem
and a highly geometric analysis of both state and output feedback. Prerequisites are the basics
of linear algebra, some simple topological notions, the elementary properties of groups, rings,
and fields, and a basic course in linear systems.
Due 2018-09-29
1st ed. 2018, X, 170 p. 35 illus.
Softcover
ISBN 978-3-319-95263-5
Series
Frontiers in Mathematics
New Results in Modern Theory of Inverse Problems and an Application in
Laser Optics
Presents recent research results in a consistent notation
Contributes to two very active fields of research
Shows that variational source condition always yields convergence rate results
The book collects and contributes new results on the theory and practice of ill-posed
inverseproblems. Different notions of ill-posedness in Banach spaces for linear and nonlinear
inverse problems arediscussed not only in standard settings but also in situations up to now
not covered by the literature.Especially, ill-posedness of linear operators with uncomplemented
null spaces is examined. Tools for convergence rate analysis of regularization methods are
extended to a wider field ofapplicability. It is shown that the tool known as variational source
condition always yieldsconvergence rate results. A theory for nonlinear inverse problems with
quadratic structure is developed as well ascorresponding regularization methods. The new
methods are applied to a difficult inverse problemfrom laser optics. Sparsity promoting
regularization is examined in detail from a Banach space point of view. Extensiveconvergence
analysis reveals new insights into the behavior of Tikhonov-type regularization withsparsity
enforcing penalty.
Due 2018-09-18
1st ed. 2018, Approx. 150 p.
Hardcover
ISBN 978-3-319-96261-0
Series
Springer Monographs in Mathematics
Offers a comprehensive introduction to coding theory: the reader does not
need a lot of background
Illustrates links between coding theory and statistical inference
Presents applications to order identification in Hidden Markov chain models
The purpose of these notes is to highlight the far-reaching connections between Information
Theory and Statistics. Universal coding and adaptive compression are indeed closely related to
statistical inference concerning processes and using maximum likelihood or Bayesian methods.
The book is divided into four chapters, the first of which introduces readers to lossless coding,
provides an intrinsic lower bound on the codeword length in terms of Shannon’s entropy, and
presents some coding methods that can achieve this lower bound, provided the source
distribution is known. In turn, Chapter 2 addresses universal coding on finite alphabets, and
seeks to find coding procedures that can achieve the optimal compression rate, regardless of
the source distribution. It also quantifies the speed of convergence of the compression rate to
the source entropy rate. These powerful results do not extend to infinite alphabets. In Chapter
3, it is shown that there are no universal codes over the class of stationary ergodic sources
over a countable alphabet. This negative result prompts at least two different approaches: the
introduction of smaller sub-classes of sources known as envelope classes, over which adaptive
coding may be feasible, and the redefinition of the performance criterion by focusing on
compressing the message pattern. Finally, Chapter 4 deals with the question of order
identification in statistics.