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Presenting a range of substantive applied problems within Bayesian Statistics along with their Bayesian solutions, this book arises from a research program at CIRM in France in the second semester of 2018, which supported Kerrie Mengersen as a visiting Jean-Morlet Chair and Pierre Pudlo as the local Research Professor.

This book describes a novel approach to the study of Siegel modular forms of degree two with paramodular level. It introduces the family of stable Klingen congruence subgroups of GSp(4) and uses this family to obtain new relations between the Hecke eigenvalues and Fourier coefficients of paramodular newforms, revealing a fundamental dichotomy for paramodular representations. Among other important results, it includes a complete description of the vectors fixed by these congruence subgroups in all irreducible representations of GSp(4) over a nonarchimedean local field.Siegel paramodular forms have connections with the theory of automorphic representations and the Langlands program, Galois representations, the arithmetic of abelian surfaces, and algorithmic number theory. Providing a useful standard source on the subject, the book will be of interest to graduate students and researchers working in the above fields.

This volume contains lecture notes on key topics in geometric analysis, a growing mathematical subject which uses analytical techniques, mostly of partial differential equations, to treat problems in differential geometry and mathematical physics.

Since the early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. course held in Summer 2006, when some of the most world recognized experts in the field reviewed the rigorous setting of mixed finite elements and revisited it after more than 30 years of practice.

Each year young mathematicians congregate in Saint Flour, France, and listen to extended lecture courses on new topics in Probability Theory. This title features notes, representing a course given by Terry Lyons in 2004, that provide an account of the key results forming the foundation of the theory of rough paths.

Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type.

This open access book focuses on the interplay between random walks on planar maps and Koebe's circle packing theorem.

Collating different aspects of Vector-valued Partial Differential Equations and Applications, this volume is based on the 2013 CIME Course with the same name which took place at Cetraro, Italy, under the scientific direction of John Ball and Paolo Marcellini.

Studying the fine properties of solutions to Stochastic (Partial) Differential Equations with reflection at a boundary, this book begins with a discussion of classical one-dimensional diffusions as the reflecting Brownian motion, devoting a chapter to Bessel processes, and moves on to function-valued solutions to SPDEs.

This volumebrings together four lecture courses on modern aspects of water waves. Thelectures provide a useful source for those who want to begin to investigate howmathematics can be used to improve our understanding of water wave phenomena.

Written by leading experts in an emerging field, this book offers a unique view of the theory of stochastic partial differential equations, with lectures on the stationary KPZ equation, fully nonlinear SPDEs, and random data wave equations.

Pluripotential theory is a very powerful tool in geometry, complex analysis and dynamics. This CIME course focused on complex Monge-Ampere equations, applications of pluripotential theory to Kahler geometry and algebraic geometry and to holomorphic dynamics.

Providing an introduction to isogeometric methods with a focus on their mathematical foundations, this book is composed of four chapters, each devoted to a topic of special interests for isogeometric methods and their theoretical understanding.

This book is devoted to the study of certain integral representations for Neumann, Kapteyn, Schloemilch, Dini and Fourier series of Bessel and other special functions, such as Struve and von Lommel functions.

The central theme of this reference book is the metric geometry of complex analysis in several variables.

Presents an introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This book addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, and moment maps.

Collecting together the lecture notes of the CIME Summer School held in Cetraro in July 2018, the aim of the book is to introduce a vast range of techniques which are useful in the investigation of complex manifolds.

This book sets out an account of the tools which Frobenius used to discover representation theory for nonabelian groups and describes its modern applications.

The main goal of this book is to provide an overview of the state of the art in the mathematical modeling of complex fluids, with particular emphasis on its thermodynamical aspects.

This three-chapter volume concerns the distributions of certain functionals of Levy processes. The first chapter, by Makoto Maejima, surveys representations of the main sub-classes of infinitesimal distributions in terms of mappings of certain Levy processes via stochastic integration.

These lecture notes provide an introduction to the applications of Brownian motion to analysis and more generally, connections between Brownian motion and analysis.

This is meant to be Quillen on Quillen as it happened forty years ago, an informal text for a second-semester graduate student on topology, category theory and K-theory, a potential preface to studying Quillen's own landmark papers and an informal glimpse of his great mind.

This book collects together a unique set of articles dedicated to several fundamental aspects of the Navier¿Stokes equations. As is well known, understanding the mathematical properties of these equations, along with their physical interpretation, constitutes one of the most challenging questions of applied mathematics. Indeed, the Navier-Stokes equations feature among the Clay Mathematics Institute's seven Millennium Prize Problems (existence of global in time, regular solutions corresponding to initial data of unrestricted magnitude). The text comprises three extensive contributions covering the following topics: (1) Operator-Valued H¿-calculus, R-boundedness, Fourier multipliers and maximal Lp-regularity theory for a large, abstract class of quasi-linear evolution problems with applications to Navier¿Stokes equations and other fluid model equations; (2) Classical existence, uniqueness and regularity theorems of solutions to the Navier¿Stokes initial-value problem, along with space-time partial regularity and investigation of the smoothness of the Lagrangean flow map; and (3) A complete mathematical theory of R-boundedness and maximal regularity with applications to free boundary problems for the Navier¿Stokes equations with and without surface tension.Offering a general mathematical framework that could be used to study fluid problems and, more generally, a wide class of abstract evolution equations, this volume is aimed at graduate students and researchers who want to become acquainted with fundamental problems related to the Navier¿Stokes equations.

Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems.

Presenting some recent results on the construction and the moments of Levy-type processes, the focus of this volume is on a new existence theorem, which is proved using a parametrix construction. Levy-type processes behave locally like Levy processes but, in contrast to Levy processes, they are not homogeneous in space.

Focusing on special matrices and matrices which are in some sense `near' to structured matrices, this volume covers a broad range of topics of current interest in numerical linear algebra.