MATRIX is Australia's international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the eight programs held at MATRIX in 2018: - Non-Equilibrium Systems and Special Functions - Algebraic Geometry, Approximation and Optimisation - On the Frontiers of High Dimensional Computation - Month of Mathematical Biology - Dynamics, Foliations, and Geometry In Dimension 3 - Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type - Functional Data Analysis and Beyond - Geometric and Categorical Representation Theory The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on a topic related to the MATRIX program; the remaining contributions are predominantly lecture notes or short articles based on talks or activities at MATRIX.

This volume offers a collection of carefully selected, peer-reviewed papers presented at the BIOMAT 2018 International Symposium, which was held at the University Hassan II, Morocco, from October 29th to November 2nd, 2018. The topics covered include applications of mathematical modeling in hepatitis B, HIV and Chikungunya infections; tumor cell dynamics; inflammatory processes; chemotherapeutic drug effects; and population dynamics. Also discussing the application of techniques like the generalized stochastic Milevsky-Promislov model, numerical simulations and convergence of discrete and continuous models, it is an invaluable resource on interdisciplinary research in mathematical biology for students, researchers, and professionals. Held every year since 2001, the BIOMAT International Symposium gathers together, in a single conference, researchers from Mathematics, Physics, Biology, and affine fields to promote the interdisciplinary exchange of results, ideas and techniques, promoting truly international cooperation for problem discussion. The 2018 edition of BIOMAT International Symposium received contributions by authors from seventeen countries: Algeria, Brazil, Cameroon, Canada, Chad, Colombia, France, Germany, Hungary, Italy, Mali, Morocco, Nigeria, Poland, Portugal, Russia, and Senegal. Selected papers presented at the 2017 edition of this Symposium were also published by Springer, in the volume "Trends in Biomathematics: Modeling, Optimization and Computational Problems" (978-3-319-91091-8).

The material of the present book has been used for graduate-level courses at the University of Ia i during the past ten years. It is a revised version of a book which appeared in Romanian in 1993 with the Publishing House of the Romanian Academy. The book focuses on classical boundary value problems for the principal equations of mathematical physics: second order elliptic equations (the Poisson equations), heat equations and wave equations. The existence theory of second order elliptic boundary value problems was a great challenge for nineteenth century mathematics and its development was marked by two decisive steps. Undoubtedly, the first one was the Fredholm proof in 1900 of the existence of solutions to Dirichlet and Neumann problems, which represented a triumph of the classical theory of partial differential equations. The second step is due to S. 1. Sobolev (1937) who introduced the concept of weak solution in partial differential equations and inaugurated the modern theory of boundary value problems. The classical theory which is a product ofthe nineteenth century, is concerned with smooth (continuously differentiable) sollutions and its methods rely on classical analysis and in particular on potential theory. The modern theory concerns distributional (weak) solutions and relies on analysis of Sob ole v spaces and functional methods. The same distinction is valid for the boundary value problems associated with heat and wave equations. Both aspects of the theory are present in this book though it is not exhaustive in any sense.

In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. Subsequent chapters then develop such topics as Hermitian exterior algebra and the Hodge - operator, harmonic theory on compact manifolds, differential operators on a Kahler manifold, the Hodge decomposition theorem on compact Kahler manifolds, the Hodge-Riemann bilinear relations on Kahler manifolds Griffiths' period mapping, quadratic transformations, and Kodaira's vanishing and embedding theorems. The third edition of this standard reference contains a new appendix by Oscar Garcia-Prada which gives an overview of the developments in the field during the decades since the book appeared.

In April 2007, the Deutsche Forschungsgemeinschaft (DFG) approved the Priority Program 1324 "Mathematical Methods for Extracting Quantifiable Information from Complex Systems." This volume presents a comprehensive overview of the most important results obtained over the course of the program. Mathematical models of complex systems provide the foundation for further technological developments in science, engineering and computational finance. Motivated by the trend toward steadily increasing computer power, ever more realistic models have been developed in recent years. These models have also become increasingly complex, and their numerical treatment poses serious challenges. Recent developments in mathematics suggest that, in the long run, much more powerful numerical solution strategies could be derived if the interconnections between the different fields of research were systematically exploited at a conceptual level. Accordingly, a deeper understanding of the mathematical foundations as well as the development of new and efficient numerical algorithms were among the main goals of this Priority Program. The treatment of high-dimensional systems is clearly one of the most challenging tasks in applied mathematics today. Since the problem of high-dimensionality appears in many fields of application, the above-mentioned synergy and cross-fertilization effects were expected to make a great impact. To be truly successful, the following issues had to be kept in mind: theoretical research and practical applications had to be developed hand in hand; moreover, it has proven necessary to combine different fields of mathematics, such as numerical analysis and computational stochastics. To keep the whole program sufficiently focused, we concentrated on specific but related fields of application that share common characteristics and as such, they allowed us to use closely related approaches.

The purpose of this book is to present modern developments and applications of the techniques of modulus or extremal length of path families in the study of m- n pings in R, n? 2, and in metric spaces. The modulus method was initiated by Lars Ahlfors and Arne Beurling to study conformal mappings. Later this method was extended and enhanced by several other authors. The techniques are geom- ric and have turned out to be an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on rather recent research papers and extends the modulus method beyond the classical applications of the modulus techniques presented in many monographs. Helsinki O. Martio Donetsk V. Ryazanov Haifa U. Srebro Holon E. Yakubov 2007 Contents 1 Introduction and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Moduli and Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. 2 Moduli in Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. 3 Conformal Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2. 4 Geometric De nition for Quasiconformality . . . . . . . . . . . . . . . . . . . . 13 2. 5 Modulus Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2. 6 Upper Gradients and ACC Functions . . . . . . . . . . . . . . . . . . . . . . . . . 17 p n 2. 7 ACC Functions in R and Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 p 2. 8 Linear Dilatation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2. 9 Analytic De nition for Quasiconformality. . . . . . . . . . . . . . . . . . . . . . 31 n 2. 10 R as a Loewner Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2. 11 Quasisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Moduli and Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3. 2 QED Exceptional Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3. 3 QED Domains and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3. 4 UniformandQuasicircleDomains . . . . . . . . . . . . . . . . . . . . . . . . . . . .

In this book we suggest a unified method of constructing near-minimizers for certain important functionals arising in approximation, harmonic analysis and ill-posed problems and most widely used in interpolation theory. The constructions are based on far-reaching refinements of the classical Calderon Zygmund decomposition. These new Calderon Zygmund decompositions in turn are produced with the help of new covering theorems that combine many remarkable features of classical results established by Besicovitch, Whitney and Wiener. In many cases the minimizers constructed in the book are stable (i.e., remain near-minimizers) under the action of Calderon Zygmund singular integral operators.

The book is divided into two parts. While the new method is presented in great detail in the second part, the first is mainly devoted to the prerequisites needed for a self-contained presentation of the main topic. There we discuss the classical covering results mentioned above, various spectacular applications of the classical Calderon Zygmund decompositions, and the relationship of all this to real interpolation. It also serves as a quick introduction to such important topics as spaces of smooth functions or singular integrals."

This book lays the foundations for a theory on almost periodic stochastic processes and their applications to various stochastic differential equations, functional differential equations with delay, partial differential equations, and difference equations. It is in part a sequel of authors recent work on almost periodic stochastic difference and differential equations and has the particularity to be the first book that is entirely devoted to almost periodic random processes and their applications. The topics treated in it range from existence, uniqueness, and stability of solutions for abstract stochastic difference and differential equations.

This book presents a unique fusion of two different research topics. One is related to the traditional mathematical problem of chases and escapes. The problem mainly deals with a situation where a chaser pursues an evader to analyze their trajectories and capture time. It dates back more than 300 years and has developed in various directions such as differential games. The other topic is the recently developing field of collective behavior, which investigates origins and properties of emergent behavior in groups of self-driving units. Applications include schools of fish, flocks of birds, and traffic jams. This book first reviews representative topics, both old and new, from these two areas. Then it presents the combined research topic of "group chase and escape", recently proposed by the authors. Although the combination is simple and straightforward, the book describes the emergence of rather intricate behavior, provoking the interest of readers for further developments and applications of related topics.

The book illustrates the theoretical results of fractional derivatives via applications in signals and systems, covering continuous and discrete derivatives, and the corresponding linear systems. Both time and frequency analysis are presented. Some advanced topics are included like derivatives of stochastic processes. It is an essential reference for researchers in mathematics, physics, and engineering.

This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending problems. The modern mimetic discretization technology developed in part by the Authors allows one to solve these equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The book provides a practical guide for those scientists and engineers that are interested in the computational properties of the mimetic finite difference method such as the accuracy, stability, robustness, and efficiency. Many examples are provided to help the reader to understand and implement this method. This monograph also provides the essential background material and describes basic mathematical tools required to develop further the mimetic discretization technology and to extend it to various applications.

This book offers a brief, practically complete, and relatively simple introduction to functional analysis. It also illustrates the application of functional analytic methods to the science of continuum mechanics. Abstract but powerful mathematical notions are tightly interwoven with physical ideas in the treatment of nontrivial boundary value problems for mechanical objects.

This second edition includes more extended coverage of the classical andabstract portions of functional analysis. Taken together, the first three chapters now constitute a regular text on applied functional analysis. This potential use of the book is supported by a significantly extended set of exercises with hints and solutions. A new appendix, providing a convenient listing of essential inequalities and imbedding results, has been added.

The book should appeal to graduate students and researchers in physics, engineering, and applied mathematics.

Reviews of first edition:

"This book covers functional analysis and its applications to continuum mechanics. The presentation is concise but complete, and is intended for readers in continuum mechanics who wish to understand the mathematical underpinnings of the discipline. Detailed solutions of the exercises are provided in an appendix." (L Enseignment Mathematique, Vol. 49 (1-2), 2003)

"The reader comes away with a profound appreciation both of the physics and its importance, and of the beauty of the functional analytic method, which, in skillful hands, has the power to dissolve and clarify these difficult problems as peroxide does clotted blood. Numerous exercises test the reader s comprehension at every stage. Summing Up: Recommended." (F. E. J. Linton, Choice, September, 2003)

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This book applies a step-by-step treatment of the current state-of-the-art of ordinary differential equations used in modeling of engineering systems/processes and beyond. It covers systematically ordered problems, beginning with first and second order ODEs, linear and higher-order ODEs of polynomial form, theory and criteria of similarity, modeling approaches, phase plane and phase space concepts, stability optimization and ending on chaos and synchronization.

Presenting both an overview of the theory of the introductory differential equations in the context of applicability and a systematic treatment of modeling of numerous engineering and physical problems through linear and non-linear ODEs, the volume is self-contained, yet serves both scientific and engineering interests. The presentation relies on a general treatment, analytical and numerical methods, concrete examples and engineering intuition.

The scientific background used is well balanced between elementary and advanced level, making it as a unique self-contained source for both theoretically and application oriented graduate and doctoral students, university teachers, researchers and engineers of mechanical, civil and mechatronic engineering.

One of the main aims of this book is to exhibit some fruitful links between renewal theory and regular variation of functions. Applications of renewal processes play a key role in actuarial and financial mathematics as well as in engineering, operations research and other fields of applied mathematics. On the other hand, regular variation of functions is a property that features prominently in many fields of mathematics. The structure of the book reflects the historical development of the authors' research work and approach - first some applications are discussed, after which a basic theory is created, and finally further applications are provided. The authors present a generalized and unified approach to the asymptotic behavior of renewal processes, involving cases of dependent inter-arrival times. This method works for other important functionals as well, such as first and last exit times or sojourn times (also under dependencies), and it can be used to solve several other problems. For example, various applications in function analysis concerning Abelian and Tauberian theorems can be studied as well as those in studies of the asymptotic behavior of solutions of stochastic differential equations. The classes of functions that are investigated and used in a probabilistic context extend the well-known Karamata theory of regularly varying functions and thus are also of interest in the theory of functions. The book provides a rigorous treatment of the subject and may serve as an introduction to the field. It is aimed at researchers and students working in probability, the theory of stochastic processes, operations research, mathematical statistics, the theory of functions, analytic number theory and complex analysis, as well as economists with a mathematical background. Readers should have completed introductory courses in analysis and probability theory.

This book collects up-to-date papers from world experts in a broad variety of relevant applications of approximation theory, including dynamical systems, multiscale modelling of fluid flow, metrology, and geometric modelling to mention a few. The 14 papers in this volume document modern trends in approximation through recent theoretical developments, important computational aspects and multidisciplinary applications. The book is arranged in seven invited surveys, followed by seven contributed research papers. The surveys of the first seven chapters are addressing the following relevant topics: emergent behaviour in large electrical networks, algorithms for multivariate piecewise constant approximation, anisotropic triangulation methods in adaptive image approximation, form assessment in coordinate metrology, discontinuous Galerkin methods for linear problems, a numerical analyst's view of the lattice Boltzmann method, approximation of probability measures on manifolds. Moreover, the diverse contributed papers of the remaining seven chapters reflect recent developments in approximation theory, approximation practice and their applications. Graduate students who wish to discover the state of the art in a number of important directions of approximation algorithms will find this a valuable volume. Established researchers from statisticians through to fluid modellers will find interesting new approaches to solving familiar but challenging problems. This book grew out of the sixth in the conference series on "Algorithms for Approximation", which took place from 31st August to September 4th 2009 in Ambleside in the Lake District of the United Kingdom.

This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: "This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews

This text contains a collection of 28 contributions on the topics of bifurcation theory and dynamical systems, mostly from the point of view of symmetry breaking, which has been revealed to be a powerful tool in the understanding of pattern formation and in the scientific application of these theories. Computational aspects of these theories are also considered. It is designed for graduate and postgraduate students of nonlinear applied mathematics, as well as any scientist or engineer interested in pattern formation and nonlinear instabilities. Dynamical systems and bifurcation theory are mathematical tools which are suitable for the study of time evolution and changes in the physical world. The introduction of the concept of symmetry and symmetry breaking in the theories enlarges their spectrum of application and their predictive capability in the evolution of physical systems.

The Stroh formalism is a powerful and elegant mathematical method developed for the analysis of the equations of anisotropic elasticity. The purpose of this exposition is to introduce the essence of this formalism and demonstrate its effectiveness in both static and dynamic elasticity. The equations of elasticity are complicated, because they constitute a system and, particularly for the anisotropic cases, inherit many parameters from the elasticity tensor. The Stroh formalism reveals simple structures hidden in the equations of anisotropic elasticity and provides a systematic approach to these equations. This work will appeal to students and researchers in applied mathematics, mechanics, and engineering science.

These proceedings were prepared in connection with the international conference Approximation Theory XIII, which was held March 7-10, 2010 in San Antonio, Texas. The conference was the thirteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 144 participants. Previous conferences in the series were held in Austin, Texas (1973, 1976, 1980, 1992), College Station, Texas (1983, 1986, 1989, 1995), Nashville, Tennessee (1998), St. Louis, Missouri (2001), Gatlinburg, Tennessee (2004), and San Antonio, Texas (2007). Along with the many plenary speakers, the contributors to this proceedings provided inspiring talks and set a high standard of exposition in their descriptions of new directions for research. Many relevant topics in approximation theory are included in this book, such as abstract approximation, approximation with constraints, interpolation and smoothing, wavelets and frames, shearlets, orthogonal polynomials, univariate and multivariate splines, and complex approximation.

A collection of self contained, state-of-the-art surveys. The authors have made an effort to achieve readability for mathematicians and scientists from other fields, for this series of handbooks to be a new reference for research, learning and teaching.

Partial differential equations represent one of the most rapidly developing topics in mathematics. This is due to their numerous applications in science and engineering on the one hand and to the challenge and beauty of associated mathematical problems on the other.

Key features:

- Self-contained volume in series covering one of the most rapid developing topics in mathematics.
- 7 Chapters, enriched with numerous figures originating from numerical simulations.
- Written by well known experts in the field.
- Self-contained volume in series covering one of the most rapid developing topics in mathematics.
- 7 Chapters, enriched with numerous figures originating from numerical simulations.
- Written by well known experts in the field.

This book provides an introduction to the topological classification of smooth structurally stable diffeomorphisms on closed orientable 2- and 3-manifolds.The topological classification is one of the main problems of the theory of dynamical systems and the results presented in this book are mostly for dynamical systems satisfying Smale's Axiom A. The main results on the topological classification of discrete dynamical systems are widely scattered among many papers and surveys. This book presents these results fluidly, systematically, and for the first time in one publication. Additionally, this book discusses the recent results on the topological classification of Axiom A diffeomorphisms focusing on the nontrivial effects of the dynamical systems on 2- and 3-manifolds. The classical methods and approaches which are considered to be promising for the further research are also discussed.< The reader needs to be familiar with the basic concepts of the qualitative theory of dynamical systems which are presented in Part 1 for convenience. The book is accessible to ambitious undergraduates, graduates, and researchers in dynamical systems and low dimensional topology. This volume consists of 10 chapters; each chapter contains its own set of references and a section on further reading. Proofs are presented with the exact statements of the results. In Chapter 10 the authors briefly state the necessary definitions and results from algebra, geometry and topology. When stating ancillary results at the beginning of each part, the authors refer to other sources which are readily available.

This book is devoted to the mathematical foundation of boundary integral equations. The combination of ?nite element analysis on the boundary with these equations has led to very e?cient computational tools, the boundary element methods (see e.g., the authors [139] and Schanz and Steinbach (eds.) [267]). Although we do not deal with the boundary element discretizations in this book, the material presented here gives the mathematical foundation of these methods. In order to avoid over generalization we have con?ned ourselves to the treatment of elliptic boundary value problems. The central idea of eliminating the ?eld equations in the domain and - ducing boundary value problems to equivalent equations only on the bou- ary requires the knowledge of corresponding fundamental solutions, and this idea has a long history dating back to the work of Green [107] and Gauss [95, 96]. Today the resulting boundary integral equations still serve as a major tool for the analysis and construction of solutions to boundary value problems.

This book shows how to derive, test and analyze numerical methods for solving differential equations, including both ordinary and partial differential equations. The objective is that students learn to solve differential equations numerically and understand the mathematical and computational issues that arise when this is done. Includes an extensive collection of exercises, which develop both the analytical and computational aspects of the material. In addition to more than 100 illustrations, the book includes a large collection of supplemental material: exercise sets, MATLAB computer codes for both student and instructor, lecture slides and movies.

This book features a collection of recent findings in Applied Real and Complex Analysis that were presented at the 3rd International Conference "Boundary Value Problems, Functional Equations and Applications" (BAF-3), held in Rzeszow, Poland on 20-23 April 2016. The contributions presented here develop a technique related to the scope of the workshop and touching on the fields of differential and functional equations, complex and real analysis, with a special emphasis on topics related to boundary value problems. Further, the papers discuss various applications of the technique, mainly in solid mechanics (crack propagation, conductivity of composite materials), biomechanics (viscoelastic behavior of the periodontal ligament, modeling of swarms) and fluid dynamics (Stokes and Brinkman type flows, Hele-Shaw type flows). The book is addressed to all readers who are interested in the development and application of innovative research results that can help solve theoretical and real-world problems.

The book presents the modern state of the art in the mathematical theory of compressible Navier-Stokes equations, with particular emphasis on the applications to aerodynamics. The topics covered include: modeling of compressible viscous flows; modern mathematical theory of nonhomogeneous boundary value problems for viscous gas dynamics equations; applications to optimal shape design in aerodynamics; kinetic theory for equations with oscillating data; new approach to the boundary value problems for transport equations. The monograph offers a comprehensive and self-contained introduction to recent mathematical tools designed to handle the problems arising in the theory.