Due 2019-10-27
1st ed. 2019, X, 110 p. 33
illus., 5 illus. in color.
Printed book
Softcover
ISBN 978-3-030-27967-7
Entirely self-contained and aimed to fully accompany a single-semester
graduate course
Many classical proofs have been simplified and streamlined
Contains numerous useful exercises
This open access book focuses on the interplay between random walks on planar maps and
Koebe’s circle packing theorem. Further topics covered include electric networks, the He?
Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local
limits of finite planar maps and the almost sure recurrence of simple random walks on these
limits. One of its main goals is to present a self-contained proof that the uniform infinite planar
triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A
planar map is a graph that can be drawn in the plane without crossing edges, together with a
specification of the cyclic ordering of the edges incident to each vertex.One widely applicable
method of drawing planar graphs is given by Koebe’s circle packing theorem (1936). Various
geometric properties of these drawings, such as existence of accumulation points and bounds
on the radii, encode important probabilistic information, such as the recurrence/transience of
simple random walks and connectivity of the uniform spanning forest. This deep connection is
especially fruitful to the study of random planar maps. The book is aimed at researchers and
graduate students in mathematics and is suitable for a single-semester course; only a basic
knowledge of graduate level probability theory is assumed.
Due 2019-10-22
1st ed. 2019, X, 190 p. 126
illus., 2 illus. in color.
Printed book
Softcover
ISBN 978-3-030-27122-0
Initiates a theory of new categorical structures that generalize the simplicial
Segal property to higher dimensions
Explores many important constructions that exhibit the 2-Segal property
Offers an entry point for readers new to the field by including an elementary
formulation of 2-Segal spaces and numerous examples
This monograph initiates a theory of new categorical structures that generalize the simplicial
Segal property to higher dimensions. The authors introduce the notion of a d-Segal space,
which is a simplicial space satisfying locality conditions related to triangulations of ddimensional
cyclic polytopes. Focus here is on the 2-dimensional case. Many important
constructions are shown to exhibit the 2-Segal property, including Waldhausen’s S-construction,
Hecke-Waldhausen constructions, and configuration spaces of flags. The relevance of 2-Segal
spaces in the study of Hall and Hecke algebras is discussed. Higher Segal Spaces marks the
beginning of a program to systematically study d-Segal spaces in all dimensions d. The
elementary formulation of 2-Segal spaces in the opening chapters is accessible to readers with
a basic background in homotopy theory. A chapter on Bousfield localizations provides a
transition to the general theory, formulated in terms of combinatorial model categories, that
features in the main part of the book. Numerous examples throughout assist readers entering
this exciting field to move toward active research; established researchers in the area will
appreciate this work as a reference.
Due 2019-10-21
1st ed. 2019, X, 181 p. 17
illus.
Printed book
Softcover
ISBN 978-3-030-28296-7
An up-to-date and user-friendly introduction to the rapidly developing field of
L2-invariants
Proceeds quickly to the research level after thoroughly covering all the basics
Contains many motivating examples, illustrations, and exercises
This book introduces the reader to the most important concepts and problems in the field of 2-
invariants. After some foundational material on group von Neumann algebras, L2-Betti numbers
are defined and their use is illustrated by several examples. The text continues with Atiyah's
question on possible values of L2-Betti numbers and the relation to Kaplansky's zero divisor
conjecture. The general definition of L2-Betti numbers allows for applications in group theory. A
whole chapter is dedicated to Luck's approximation theorem and its generalizations. The final
chapter deals with L2-torsion, twisted variants and the conjectures relating them to torsion
growth in homology. The text provides a self-contained treatment that constructs the required
specialized concepts from scratch. It comes with numerous exercises and examples, so that
both graduate students and researchers will find it useful for self-study or as a basis for an
advanced lecture course.graduate students and researchers will find it useful for self-studying
or as basis for an advanced lecture course.
Due 2019-11-22
1st ed. 2019, XII, 608 p. 7
illus., 1 illus. in color.
Printed book
Hardcover
ISBN 978-3-030-28949-2
Features self-contained chapters that can be read independently
Presents cutting-edge research from the frontiers of functional equations and
analytic inequalities active fields
Contains an extensive list of references
This volume presents cutting edge research from the frontiers of functional equations and
analytic inequalities active fields. It covers the subject of functional equations in a broad sense,
including but not limited to the following topics: Hyperstability of a linear functional equation
on restricted domains Hyers?Ulam’s stability results to athree point boundary value problem of
nonlinear fractional order differential equations Topologicaldegree theory and Ulam’s stability
analysis of a boundary value problem of fractional differentialequations General Solution and
Hyers-Ulam Stability of Duo TriginticFunctional Equation in Multi-Banach Spaces Stabilities of
Functional Equations via Fixed PointTechnique Measure zero stabilityproblem for the Drygas
functional equation with complex involution Fourier Transforms and UlamStabilities of Linear
Differential Equations Hyers?Ulam stability of a discrete diamond?alpha derivative equation
Approximatesolutions of an interesting new mixed type additive-quadratic-quartic functional
equation. The diverse selection of inequalities covered includes Opial, Hilbert-Pachpatte,
Ostrowski, comparison of means, Poincare, Sobolev, Landau, Polya-Ostrowski, Hardy,HermiteHadamard,
Levinson, and complex Korovkin type. The inequalities are also in the
environmentsof Fractional Calculus and Conformable Fractional Calculus. Applications from this
book's results can be found in many areas of pure and applied mathematics, especially in
ordinary and partialdifferential equations and fractional differential equations. As such, this
volume is suitable forresearchers, graduate students and related seminars, and all science and
engineering libraries.
Due 2019-11-13
1st ed. 2019, Approx. 125 p.
Printed book
Softcover
ISBN 978-3-030-29529-5
Self-contained introduction to Carleman estimates for three typical second
order partial differential equations
All Carleman estimates are derived from a fundamental identity for a second
order partial differential operator
Of wide interest to researchers in the field
This book provides a brief, self-contained introduction to Carleman estimates for three typical
second order partial differential equations, namely elliptic, parabolic, and hyperbolic equations,
and their typical applications in control, unique continuation, and inverse problems. There are
three particularly important and novel features of the book. First, only some basic calculus is
needed in order to obtain the main results presented, though some elementary knowledge of
functional analysis and partial differential equations will be helpful in understanding them.
Second, all Carleman estimates in the book are derived from a fundamental identity for a
second order partial differential operator; the only difference is the choice of weight functions.
Third, only rather weak smoothness and/or integrability conditions are needed for the
coefficients appearing in the equations. Carleman Estimates for Second Order Partial
Differential Operators and Applications will be of interest to all researchers in the field.
Due 2019-11-07
1st ed. 2019, X, 425 p. 1
illus.
Printed book
Softcover
ISBN 978-3-030-27152-7
Focuses on different problems in physics and other applied sciences
Some of the exercises supplement theoretical material, while others relate to the real world
Uniquely geared toward graduate students and researchers in applied
mathematics, physics, and neighboring fields of science
Presents main results and techniques in Functional Analysis and uses them to
explore other areas of mathematics and applications
This advanced graduate textbook presents main results and techniques in Functional Analysis
and uses them to explore other areas of mathematics and applications. Special attention is
paid to creating appropriate frameworks towards solving significant problems involving
dierential and integral equations. Exercises at the end of each chapter help the reader to
understand the richness of ideas and methods offered by Functional Analysis. Some of the
exercises supplement theoretical material, while others relate to the real world. This textbook,
with its friendly exposition, focuses on different problems in physics and other applied sciences
and uniquely provides solutions to most of the exercises. The text is aimed toward graduate
students and researchers in applied mathematics, physics, and neighboring fields of science.
Due 2019-11-13
1st ed. 2019, XII, 188 p.
Printed book
Hardcover
ISBN 978-3-662-59902-0
Provides new notions and results of the theory of nonlinear expectations and
related stochastic analysis
Summarizes the latest studies on G-Martingale representation theorem and
Ito’s integrals
Includes exercises that help reader master and learn in each chapter
This book is focused on the recent developments on problems of probability model uncertainty
by using the notion of nonlinear expectations and, in particular, sublinear expectations. It
provides a gentle coverage of the theory of nonlinear expectations and related stochastic
analysis. Many notions and results, for example, G-normal distribution, G-Brownian motion, GMartingale
representation theorem, and related stochastic calculus are first introduced or
obtained by the author. This book is based on Shige Peng’s lecture notes for a series of
lectures given at summer schools and universities worldwide. It starts with basic definitions of
nonlinear expectations and their relation to coherent measures of risk, law of large numbers
and central limit theorems under nonlinear expectations, and develops into stochastic integral
and stochastic calculus underG-expectations. It ends with recent research topic onG-Martingale
representation theorem andG-stochastic integral for locally integrable processes. With exercises
topracticeat the end of each chapter, thisbook can be used as a graduate textbook for students
in probability theory and mathematical finance.Each chapter also concludes with a sectionNotes
and Comments,which gives history and further references on the material covered in that
chapter. Researchers and graduate students interested in probability theory and mathematical
finance will find this book very useful.