Advances in techniques that reduce or eliminate the type of meshes associated with finite elements or finite differences are reported in the papers that form this volume. As design, analysis and manufacture become more integrated, the chances are that software users will be less aware of the capabilities of the analytical techniques that are at the core of the process. This reinforces the need to retain expertise in certain specialised areas of numerical methods, such as BEM/MRM, to ensure that all new tools perform satisfactorily within the aforementioned integrated process. The maturity of BEM since 1978 has resulted in a substantial number of industrial applications of the method; this demonstrates its accuracy, robustness and ease of use. The range of applications still needs to be widened, taking into account the potentialities of the Mesh Reduction techniques in general. The included papers originate from the 45th conference on Boundary Elements and other Mesh Reduction Methods (BEM/MRM) and describe theoretical developments and new formulations, helping to expand the range of applications as well as the type of modelled materials in response to the requirements of contemporary industrial and professional environments.

Quite a number of phenomena in science and technology, industrial and/or agricultural production and transport, medical and/or biological flows and movements, social and/or economical developments, etc., depend on many variables, and are very much complicated. Although the detailed knowledge is accumulated in respective fields, it is meaningful to model and analyze the essential part of the phenomena in terms of smaller number of variables, which falls into partial differential equations. This book aims at providing students and researchers the basic ideas and the methods to solve problems in various fields. Particular attention is paid to bridge the gap between mathematics and the real world. To do this, we start from a simple system with intuitively understandable physical background, extract the essential part, formulate into mathematical tools, and then generalize for further application. Here logical thinking in depth and wide linking to various fields are sought to construct intellectual network.

Containing the proceedings from the 41st conference on Boundary Elements and other Mesh Reduction Methods (BEM/MRM), this book is a collection of high quality papers that report on advances in techniques that reduce or eliminate the type of meshes associated with such methods as finite elements or finite differences. As design, analysis and manufacture become more integrated the chances are that the users will be less aware of the capabilities of the analytical techniques that are at the core of the process. This reinforces the need to retain expertise in certain specialised areas of numerical methods, such as BEM/MRM, to ensure that all new tools perform satisfactorily in the integrated process. The maturity of BEM since 1978 has resulted in a substantial number of industrial applications that demonstrate the accuracy, robustness and easy use of the technique. Their range still needs to be widened, taking into account the potentialities of the Mesh Reduction techniques in general. The papers in this volume help to expand the range of applications as well as the type of materials in response to industrial and professional requirements.

Quite a number of phenomena in science and technology, industrial and/or agricultural production and transport, medical and/or biological flows and movements, social and/or economical developments, etc., depend on many variables, and are very much complicated. Although the detailed knowledge is accumulated in respective fields, it is meaningful to model and analyze the essential part of the phenomena in terms of smaller number of variables, which falls into partial differential equations. This book aims at providing students and researchers the basic ideas and the methods to solve problems in various fields. Particular attention is paid to bridge the gap between mathematics and the real world. To do this, we start from a simple system with intuitively understandable physical background, extract the essential part, formulate into mathematical tools, and then generalize for further application. Here logical thinking in depth and wide linking to various fields are sought to construct intellectual network.

This volume considers resistance networks: large graphs which are connected, undirected, and weighted. Such networks provide a discrete model for physical processes in inhomogeneous media, including heat flow through perforated or porous media. These graphs also arise in data science, e.g., considering geometrizations of datasets, statistical inference, or the propagation of memes through social networks. Indeed, network analysis plays a crucial role in many other areas of data science and engineering. In these models, the weights on the edges may be understood as conductances, or as a measure of similarity. Resistance networks also arise in probability, as they correspond to a broad class of Markov chains.The present volume takes the nonstandard approach of analyzing resistance networks from the point of view of Hilbert space theory, where the inner product is defined in terms of Dirichlet energy. The resulting viewpoint emphasizes orthogonality over convexity and provides new insights into the connections between harmonic functions, operators, and boundary theory. Novel applications to mathematical physics are given, especially in regard to the question of self-adjointness of unbounded operators.New topics are covered in a host of areas accessible to multiple audiences, at both beginning and more advanced levels. This is accomplished by directly linking diverse applied questions to such key areas of mathematics as functional analysis, operator theory, harmonic analysis, optimization, approximation theory, and probability theory.

There is an extensive literature in the form of papers (but no books) on lattice dynamical systems. The book focuses on dissipative lattice dynamical systems and their attractors of various forms such as autonomous, nonautonomous and random. The existence of such attractors is established by showing that the corresponding dynamical system has an appropriate kind of absorbing set and is asymptotically compact in some way.There is now a very large literature on lattice dynamical systems, especially on attractors of all kinds in such systems. We cannot hope to do justice to all of them here. Instead, we have focused on key areas of representative types of lattice systems and various types of attractors. Our selection is biased by our own interests, in particular to those dealing with biological applications. One of the important results is the approximation of Heaviside switching functions in LDS by sigmoidal functions.Nevertheless, we believe that this book will provide the reader with a solid introduction to the field, its main results and the methods that are used to obtain them.

Hilbert-type inequalities, including Hilbert's inequalities proved in 1908, Hardy-Hilbert-type inequalities proved in 1934, and Yang-Hilbert-type inequalities first proved around 1998, play an important role in analysis and its applications. These inequalities are mainly divided in three classes: integral, discrete and half-discrete. During the last twenty years, there have been many research advances on Hilbert-type inequalities, and especially on Yang-Hilbert-type inequalities.In the present monograph, applying weight functions, the idea of parametrization as well as techniques of real analysis and functional analysis, we prove some new Hilbert-type integral inequalities as well as their reverses with parameters. These inequalities constitute extensions of the well-known Hardy-Hilbert integral inequality. The equivalent forms and some equivalent statements of the best possible constant factors associated with several parameters are considered. Furthermore, we also obtain the operator expressions with the norm and some particular inequalities involving the Riemann-zeta function and the Hurwitz-zeta function. In the form of applications, by means of the beta function and the gamma function, we use the extended Hardy-Hilbert integral inequalities to consider several Hilbert-type integral inequalities involving derivative functions and upper limit functions. In the last chapter, we consider the case of Hardy-type integral inequalities. The lemmas and theorems within provide an extensive account of these kinds of integral inequalities and operators.Efforts have been made for this monograph hopefully to be useful, especially to graduate students of mathematics, physics and engineering, as well as researchers in these domains.

This book is a general introduction to the statistical analysis of networks, and can serve both as a research monograph and as a textbook. Numerous fundamental tools and concepts needed for the analysis of networks are presented, such as network modeling, community detection, graph-based semi-supervised learning and sampling in networks. The description of these concepts is self-contained, with both theoretical justifications and applications provided for the presented algorithms.Researchers, including postgraduate students, working in the area of network science, complex network analysis, or social network analysis, will find up-to-date statistical methods relevant to their research tasks. This book can also serve as textbook material for courses related to thestatistical approach to the analysis of complex networks.In general, the chapters are fairly independent and self-supporting, and the book could be used for course composition "a la carte". Nevertheless, Chapter 2 is needed to a certain degree for all parts of the book. It is also recommended to read Chapter 4 before reading Chapters 5 and 6, but this is not absolutely necessary. Reading Chapter 3 can also be helpful before reading Chapters 5 and 7. As prerequisites for reading this book, a basic knowledge in probability, linear algebra and elementary notions of graph theory is advised. Appendices describing required notions from the above mentioned disciplines have been added to help readers gain further understanding.

A generalized Radon transform (GRT) maps a function to its weighted integrals along a family of curves or surfaces. Such operators appear in mathematical models of various imaging modalities. The GRTs integrating along smooth curves and surfaces (lines, planes, circles, spheres, amongst others) have been studied at great lengths for decades, but relatively little attention has been paid to transforms integrating along non-smooth trajectories. Recently, an interesting new class of GRTs emerged at the forefront of research in integral geometry. The two common features of these transforms are the presence of a 'vertex' in their paths of integration (broken rays, cones, and stars) and their relation to imaging techniques based on physics of scattered particles (Compton camera imaging, single scattering tomography, etc).This book covers the relevant imaging modalities, their mathematical models, and the related GRTs. The discussion of the latter comprises a thorough exploration of their known mathematical properties, including injectivity, inversion, range description and microlocal analysis. The mathematical background required for reading most of the book is at the level of an advanced undergraduate student, which should make its content attractive for a large audience of specialists interested in imaging. Mathematicians may appreciate certain parts of the theory that are particularly elegant with connections to functional analysis, PDEs and algebraic geometry.

First Semester Calculus for Students of Mathematics and Related Disciplines equips students with a strong working knowledge of the fundamental principles of calculus, providing an engaging and accessible entry point into this critical field of study. It prepares students for more advanced courses in calculus and also helps them understand how to apply basic principles of calculus to solve problems within a wide range of disciplines, including business, biology, engineering, science, liberal arts and, of course, mathematics. The text employs rigorous treatment of early calculus topics and detailed explanations to facilitate deeper understanding of later material. Over the course of five chapters, students learn about symbolic logic, continuity and limits, derivatives, antiderivatives, and applications of each. Throughout, students are provided with rich guidance and copious opportunities to deepen their personal understanding of the subject matter. In the second edition, a more efficient layout better highlights major theorems and definitions. Additionally, over 300 new exercises have been added to further aid student learning. Highly readable and innovative, yet pedagogically solid and very applicable, First Semester Calculus for Students of Mathematics and Related Disciplines is an ideal resource for a variety of courses that apply concepts of calculus to solve mathematical and real-world problems.

In many branches of mathematical analysis and mathematical physics, the Hardy operator and Hardy inequality are fundamentally important and have been intensively studied ever since the pioneer researches. This volume presents new properties of higher-dimensional Hardy operators obtained by the authors and their collaborators over the last decade. Its prime focus is on higher-dimensional Hardy operators that are based on the spherical average form.The key motivation for this monograph is based on the fact that the Hardy operator is generally smaller than the Hardy-Littlewood maximal operator, which leads to, on the one hand, the operator norm of the Hardy operator itself being smaller than the latter. On the other hand, the former characterizing the weight function class or function spaces is greater than the latter.

Uncertainties in GPS Positioning: A Mathematical Discourse describes the calculations performed by a GPS receiver and the problems associated with ensuring that the derived location is a close match to the actual location. Inaccuracies in calculating a location can have serious repercussions, so this book is a timely source for information on this rapidly evolving technology.

Hoermander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hoermander and is intimately connected to the geometry of vector fields.Motivations for the study of Hoermander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hoermander operators and vector fields, together with the necessary background and prerequisites.The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it.

This book aims to provide an overview of the special functions of fractional calculus and their applications in diffusion and random search processes. The book contains detailed calculations for various examples of anomalous diffusion, random search and stochastic resetting processes, which can be easily followed by the reader, who will be able to reproduce the obtained results. The book will be intended for advanced undergraduate and graduate students and researchers in physics, mathematics and other natural sciences due to the various examples which will be provided in the book.

The Qualitative Theory of Ordinary Differential Equations (ODEs) occupies a rather special position both in Applied and Theoretical Mathematics. On the one hand, it is a continuation of the standard course on ODEs. On the other hand, it is an introduction to Dynamical Systems, one of the main mathematical disciplines in recent decades. Moreover, it turns out to be very useful for graduates when they encounter differential equations in their work; usually those equations are very complicated and cannot be solved by standard methods.The main idea of the qualitative analysis of differential equations is to be able to say something about the behavior of solutions of the equations, without solving them explicitly. Therefore, in the first place such properties like the stability of solutions stand out. It is the stability with respect to changes in the initial conditions of the problem. Note that, even with the numerical approach to differential equations, all calculations are subject to a certain inevitable error. Therefore, it is desirable that the asymptotic behavior of the solutions is insensitive to perturbations of the initial state.Each chapter contains a series of problems (with varying degrees of difficulty) and a self-respecting student should solve them. This book is based on Raul Murillo's translation of Henryk Zoladek's lecture notes, which were in Polish and edited in the portal Matematyka Stosowana (Applied Mathematics) in the University of Warsaw.

The author's goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all mathematics (pure or applied) and physics programs include a compulsory course in mathematical analysis. This book will serve as can serve a main textbook of such (one semester) courses. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many math-centric options.

An Invitation to Applied Mathematics: Differential Equations, Modeling, and Computation introduces the reader to the methodology of modern applied mathematics in modeling, analysis, and scientific computing with emphasis on the use of ordinary and partial differential equations. Each topic is introduced with an attractive physical problem, where a mathematical model is constructed using physical and constitutive laws arising from the conservation of mass, conservation of momentum, or Maxwell's electrodynamics. Relevant mathematical analysis (which might employ vector calculus, Fourier series, nonlinear ODEs, bifurcation theory, perturbation theory, potential theory, control theory, or probability theory) or scientific computing (which might include Newton's method, the method of lines, finite differences, finite elements, finite volumes, boundary elements, projection methods, smoothed particle hydrodynamics, or Lagrangian methods) is developed in context and used to make physically significant predictions. The target audience is advanced undergraduates (who have at least a working knowledge of vector calculus and linear ordinary differential equations) or beginning graduate students. Readers will gain a solid and exciting introduction to modeling, mathematical analysis, and computation that provides the key ideas and skills needed to enter the wider world of modern applied mathematics.