Hardcover
ISBN 978-3-030-02184-9
First book to explore Shrinkage Estimation as a global phenomenon
Focuses on point and loss estimation in multivariate normal and spherically
symmetric distributions
Authors are at the forefront of Shrinkage research
This book provides a coherent framework for understanding shrinkage estimation in statistics.
The term refers to modifying a classical estimator by moving it closer to a target which could
be known a priori or arise from a model. The goal is to construct estimators with improved
statistical properties. The book focuses primarily on point and loss estimation of the mean
vector of multivariate normal and spherically symmetric distributions. Chapter 1 reviews the
statistical and decision theoretic terminology and results that will be used throughout the book.
Chapter 2 is concerned with estimating the mean vector of a multivariate normal distribution
under quadratic loss from a frequentist perspective. In Chapter 3 the authors take a Bayesian
view of shrinkage estimation in the normal setting. Chapter 4 introduces the general classes of
spherically and elliptically symmetric distributions. Point and loss estimation for these broad
classes are studied in subsequent chapters. In particular, Chapter 5 extends many of the
results from Chapters 2 and 3 to spherically and elliptically symmetric distributions. Chapter 6
considers the general linear model with spherically symmetric error distributions when a
residual vector is available. Chapter 7 then considers the problem of estimating a location
vector which is constrained to lie in a convex set. Much of the chapter is devoted to one of two
types of constraint sets, balls and polyhedral cones. In Chapter 8 the authors focus on loss
estimation and data-dependent evidence reports. Appendices cover a number of technical
topics including weakly differentiable functions; examples where Steinfs identity doesnft hold;
Steinfs lemma and Stokesf theorem for smooth boundaries; harmonic, superharmonic and
subharmonic functions; and modified Bessel functions.
Hardcover
ISBN 978-3-030-00830-7
Includes more than 300 exercises
Useful for graduate students and for researchers that apply combinatorial
methods in different areas and levels of difficulty
Provides a theoretical background for several topics in combinatorial
mathematics
This text provides a theoretical background for several topics in combinatorial mathematics,
such as enumerative combinatorics (including partitions and Burnside's lemma), magic and
Latin squares, graph theory, extremal combinatorics, mathematical games and elementary
probability. A number of examples are given with explanations while the book also provides
more than 300 exercises of different levels of difficulty that are arranged at the end of each
chapter, and more than 130 additional challenging problems, including problems from
mathematical olympiads. Solutions or hints to all exercises and problems are included. The
book can be used by secondary school students preparing for mathematical competitions, by
their instructors, and by undergraduate students. The book may also be useful for graduate
students and for researchers that apply combinatorial methods in different areas.
Softcover
ISBN 978-981-13-2900-5
Introduces a new method to construct and classify matrix-valued symmetry
breaking operators in representation theory
Includes hot topics of automorphic forms and conformal geometry as
applications of branching rules in representation theory
Provides the complete classification of all conformally equivariant operators
on differential forms on the model space ( S<i>n, Sn-1})
This work provides the first classification theory of matrix-valued symmetry breaking operators
from principal series representations of a reductive group to those of its subgroup. The study
ofsymmetry breaking operators(intertwining operators for restriction) is an important and very
active research area in modern representation theory, which also interacts with various fields in
mathematics and theoretical physics ranging from number theory to differential geometry and
quantum mechanics. The first author initiated a program of the general study of symmetry
breaking operators. The present book pursues the program by introducing new ideas and
techniques, giving a systematic and detailed treatment in the case of orthogonal groups of real
rank one, which will serve as models for further research in other settings. In connection to
automorphic forms, this work includes a proof for a multiplicity conjecture by Gross and Prasad
for tempered principal series representations in the case (SO(n+ 1, 1),SO(n, 1)). The authors
propose a further multiplicity conjecture for nontempered representations. Viewed from
differential geometry, this seminal work accomplishes the classification of all conformally
covariant operators transforming differential forms on a Riemanniann manifoldXto those on a
submanifold in the model space (X,Y) = (Sn,Sn-1).
Hardcover
ISBN 978-3-030-01946-4
Presents very complete and up-to-date review articles
Includes detailed information on the state-of-the-art in the field
Discusses a variety of mathematical tools
This volume covers selected topics addressed and discussed during the workshop gPDE models
for multi-agent phenomena,h which was held in Rome, Italy, from November 28th to December
2nd, 2016. The content mainly focuses on kinetic equations and mean field games, which
provide a solid framework for the description of multi-agent phenomena. The book includes
original contributions on the theoretical and numerical study of the MFG system: the
uniqueness issue and finite difference methods for the MFG system, MFG with state constraints,
and application of MFG to market competition. The book also presents new contributions on
the analysis and numerical approximation of the Fokker-Planck-Kolmogorov equations, the
isotropic Landau model, the dynamical approach to the quantization problem and the
asymptotic methods for fully nonlinear elliptic equations. Chiefly intended for researchers
interested in the mathematical modeling of collective phenomena, the book provides an
essential overview of recent advances in the field and outlines future research directions.
Hardcover
ISBN 978-3-030-03349-1
Systematically treats the global geometry of equisingular families of
algebraic curves on algebraic surfaces
Accumulates the material spread over numerous, recent and classical journal
publications, and elaborates it into a unified theory which allows one to
approach all main problems in the subject and to answer several classical
questions in this area
Provides a guide to a variety of methods, results and applications of singular
algebraic curves and their families
Offers a detailed presentation of the background stuff (including the global
deformation theory and the original Viro patchworking construction) which
leads the reader to the main ideas of the theory
Singular algebraic curves have been in the focus of study in algebraic geometry from the very
beginning, and till now remain a subject of an active research related to many modern
developments in algebraic geometry, symplectic geometry, and tropical geometry. The
monograph suggests a unified approach to the geometry of singular algebraic curves on
algebraic surfaces and their families, which applies to arbitrary singularities, allows one to treat
all main questions concerning the geometry of equisingular families of curves, and, finally,
leads to results which can be viewed as the best possible in a reasonable sense. Various
methods of the cohomology vanishing theory as well as the patchworking construction with its
modifications will be of a special interest for experts in algebraic geometry and singularity
theory. The introductory chapters on zero-dimensional schemes and global deformation theory
can well serve as a material for special courses and seminars for graduate and post-graduate
students.Geometry in general plays a leading role in modern mathematics, and algebraic
geometry is the most advanced area of research in geometry. In turn, algebraic curves for
more than one century have been the central subject of algebraic geometry both in
fundamental theoretic questions and in applications to other fields of mathematics and
mathematical physics.
Hardcover
ISBN 978-981-13-2729-2
Is the first book to explain how to apply Fourier analysis in business cycle
theory in a mathematically rigorous manner
Provides an explanation of Fourier analysis of generalized functions, which is
indispensable for business cycle theory
Presents in-depth proofs of Bochner's theorem, a difficult theorem but one
with great importance in Fourier analysis
This is the first monograph that discusses in detail the interactions between Fourier analysis
and dynamic economic theories, in particular, business cycles. Many economic theories have
analyzed cyclical behaviors of economic variables. In this book, the focus is on a couple of
trials: (1) the Kaldor theory and (2) the Slutsky effect. The Kaldor theory tries to explain
business fluctuations in terms of nonlinear, 2nd-order ordinary differential equations (ODEs). In
order to explain periodic behaviors of a solution, the Hopf-bifurcation theorem frequently plays
a key role. Slutsky's idea is to look at the periodic movement as an overlapping effect of
random shocks. The Slutsky process is a weakly stationary process, the periodic (or almost
periodic) behavior of which can be analyzed by the Bochner theorem. The goal of this book is
to give a comprehensive and rigorous justification of these ideas. Therefore, the aim is first to
give a complete theory that supports the Hopf theorem and to prove the existence of periodic
solutions of ODEs; and second to explain the mathematical structure of the Bochner theorem
and its relation to periodic (or almost periodic) behaviors of weakly stationary processes.
Although these two targets are the principal ones, a large number of results from Fourier
analysis must be prepared in order to reach these goals. The basic concepts and results from
classical as well as generalized Fourier analysis are provided in a systematic way. Prospective
readers are assumed to have sufficient knowledge of real, complex analysis. However,
necessary economic concepts are explained in the text, making this book accessible even to
readers without a background in economics.
Hardcover
ISBN 978-3-030-00894-9
Introduces Fractional Calculus in an accessible manner, based on standard
integer calculus;
Supports the use of higher-level mathematical packages, such as
Mathematica or Maple;
Facilitates understanding the generalization (towards Fractional Calculus) of
important models and systems, such as Lorenz, Chua, and many others;
Provides a simultaneous introduction to analytical and numerical methods in
Fractional Calculus.
This book introduces a series of problems and methods insufficiently discussed in the field of
Fractional Calculus ? a major, emerging tool relevant to all areas of scientific inquiry. The
authors present examples based on symbolic computation, written in Maple and Mathematica,
and address both mathematical and computational areas in the context of mathematical
modeling and the generalization of classical integer-order methods. Distinct from most books,
the present volume fills the gap between mathematics and computer fields, and the transition
from integer- to fractional-order methods.