The approach of layer-damping coordinate transformations to treat singularly perturbed equations is a relatively new, and fast growing area in the field of applied mathematics. This monograph aims to present a clear, concise, and easily understandable description of the qualitative properties of solutions to singularly perturbed problems as well as of the essential elements, methods and codes of the technology adjusted to numerical solutions of equations with singularities by applying layer-damping coordinate transformations and corresponding layer-resolving grids. The first part of the book deals with an analytical study of estimates of the solutions and their derivatives in layers of singularities as well as suitable techniques for obtaining results. In the second part, a technique for building the coordinate transformations eliminating boundary and interior layers, is presented. Numerical algorithms based on the technique which is developed for generating layer-damping coordinate transformations and their corresponding layer-resolving meshes are presented in the final part of this volume. This book will be of value and interest to researchers in computational and applied mathematics.

This book presents an operator theoretic approach to robust control analysis for linear time-varying systems. It emphasizes the conceptual similarity with the H control theory for time-invariant systems and at the same time clarifies the major difficulties confronted in the time varying case. The necessary operator theory is developed from first principles and the book is as self-contained as possible. After presenting the necessary results from the theories of Toeplitz operators and nest algebras, linear systems are defined as input- output operators and the relationship between stabilization and the existance of co-prime factorizations is described. Uniform optimal control problems are formulated as model-matching problems and are reduced to four block problems. Robustness is considered both from the point of view of fractional representations and the "time varying gap" metric, and the relationship between these types of uncertainties is clarified. The book closes with the solution of the orthogonal embedding problem for time varying contractive systems. This book will be useful to both mathematicians interested in the potential applications of operator theory in control and control engineers who wish to deal with some of the more mathematically sophisticated extension of their work.

The linear theory of oscillations traditionally operates with frequency representa- tions based on the concepts of a transfer function and a frequency response. The universality of the critria of Nyquist and Mikhailov and the simplicity and obvi- ousness of the application of frequency and amplitude - frequency characteristics in analysing forced linear oscillations greatly encouraged the development of practi- cally important nonlinear theories based on various forms of the harmonic balance hypothesis [303]. Therefore mathematically rigorous frequency methods of investi- gating nonlinear systems, which appeared in the 60s, also began to influence many areas of nonlinear theory of oscillations. First in this sphere of influence was a wide range of problems connected with multidimensional analogues of the famous van der Pol equation describing auto- oscillations of generators of various radiotechnical devices. Such analogues have as a rule a unique unstable stationary point in the phase space and are Levinson dis- sipative. One of the pioneering works in this field, which started the investigation of a three-dimensional analogue of the van der Pol equation, was K. O. Friedrichs's paper [123]. The author suggested a scheme for constructing a positively invariant set homeomorphic to a torus, by means of which the existence of non-trivial periodic solutions was established. That scheme was then developed and improved for dif- ferent classes of multidimensional dynamical systems [131, 132, 297, 317, 334, 357, 358]. The method of Poincare mapping [12, 13, 17] in piecewise linear systems was another intensively developed direction.

With a long history of innovation in the market, Larson/Edwards' Calculus, International Metric Edition has been widely praised by a generation of students and professors for solid and effective pedagogy that addresses the needs of a broad range of teaching and learning styles and environments. This edition clearly presents and effectively demonstrates the concepts and rules of calculus with a student-focused approach. It offers a wealth of learning support and digital resources - all thoroughly updated and refined using proven learning design principles that remove typical barriers to learning to create a carefully planned, inclusive experience for all students.

It is hard to imagine that another elementary analysis book would contain ma terial that in some vision could qualify as being new and needed for a discipline already abundantly endowed with literature. However, to understand analysis, be ginning with the undergraduate calculus student through the sophisticated math ematically maturing graduate student, the need for examples and exercises seems to be a constant ingredient to foster deeper mathematical understanding. To a talented mathematical student, many elementary concepts seem clear on their first encounter. However, it is the belief of the authors, this understanding can be deepened with a guided set of exercises leading from the so called "elementary" to the somewhat more "advanced" form. Insight is instilled into the material which can be drawn upon and implemented in later development. The first year graduate student attempting to enter into a research environment begins to search for some original unsolved area within the mathematical literature. It is hard for the student to imagine that in many circumstances the advanced mathematical formulations of sophisticated problems require attacks that draw upon, what might be termed elementary techniques. However, if a student has been guided through a serious repertoire of examples and exercises, he/she should certainly see connections whenever they are encountered."

This and the next volume of the OT series contain the proceedings of the Work shop on Operator Theory and its Applications, IWOTA 95, which was held at the University of Regensburg, Germany, July 31 to August 4, 1995. It was the eigth workshop of this kind. Following is a list of the seven previous workshops with reference to their proceedings: 1981 Operator Theory (Santa Monica, California, USA) 1983 Applications of Linear Operator Theory to Systems and Networks (Rehovot, Israel), OT 12 1985 Operator Theory and its Applications (Amsterdam, The Netherlands), OT 19 1987 Operator Theory and Functional Analysis (Mesa, Arizona, USA), OT 35 1989 Matrix and Operator Theory (Rotterdam, The Netherlands), OT 50 1991 Operator Theory and Complex Analysis (Sapporo, Japan), OT 59 1993 Operator Theory and Boundary Eigenvalue Problems (Vienna, Austria), OT 80 IWOTA 95 offered a rich programme on a wide range of latest developments in operator theory and its applications. The programme consisted of 6 invited plenary lectures, 54 invited special topic lectures and more than 100 invited session talks. About 180 participants from 25 countries attended the workshop, more than a third came from Eastern Europe. The conference covered different aspects of linear and nonlinear spectral prob lems, starting with problems for abstract operators up to spectral theory of ordi nary and partial differential operators, pseudodifferential operators, and integral operators. The workshop was also focussed on operator theory in spaces with indefinite metric, operator functions, interpolation and extension problems.

Recent years have witnessed an increasingly close relationship growing between potential theory, probability and degenerate partial differential operators. The theory of Dirichlet (Markovian) forms on an abstract finite or infinite-dimensional space is common to all three disciplines. This is a fascinating and important subject, central to many of the contributions to the conference on Potential Theory and Degenerate Partial Differential Operators', held in Parma, Italy, February 1994.

Calculus and linear algebra are two dominant themes in contemporary mathematics and its applications. The aim of this book is to introduce linear algebra in an intuitive geometric setting as the study of linear maps and to use these simpler linear functions to study more complicated nonlinear functions. In this way, many of the ideas, techniques, and formulas in the calculus of several variables are clarified and understood in a more conceptual way. After using this text a student should be well prepared for subsequent advanced courses in both algebra and linear differential equations as well as the many applications where linearity and its interplay with nonlinearity are significant. This second edition has been revised to clarify the concepts. Many exercises and illustrations have been included to make the text more usable for students.

2 Radiant sets 236 3 Co-radiant sets 239 4 Radiative and co-radiative sets 241 5 Radiant sets with Lipschitz continuous Minkowski gauges 245 6 Star-shaped sets and their kernels 249 7 Separation 251 8 Abstract convex star-shaped sets 255 References 260 11 DIFFERENCES OF CONVEX COMPACTA AND METRIC SPACES OF CON- 263 VEX COMPACTA WITH APPLICATIONS: A SURVEY A. M. Rubinov, A. A. Vladimirov 1 Introduction 264 2 Preliminaries 264 3 Differences of convex compact sets: general approach 266 4 Metric projections and corresponding differences (one-dimensional case) 267 5 The *-difference 269 6 The Demyanov difference 271 7 Geometric and inductive definitions of the D-difference 273 8 Applications to DC and quasidifferentiable functions 276 9 Differences of pairs of set-valued mappings with applications to quasidiff- entiability 278 10 Applications to approximate subdifferentials 280 11 Applications to the approximation of linear set-valued mappings 281 12 The Demyanov metric 282 13 The Bartels-Pallaschke metric 284 14 Hierarchy of the three norms on Qn 285 15 Derivatives 287 16 Distances from convex polyhedra and convergence of convex polyhedra 289 17 Normality of convex sets 290 18 D-regular sets 291 19 Variable D-regular sets 292 20 Optimization 293 References 294 12 CONVEX APPROXIMATORS.

This monograph develops a generalised energy flow theory to investigate non-linear dynamical systems governed by ordinary differential equations in phase space and often met in various science and engineering fields. Important nonlinear phenomena such as, stabilities, periodical orbits, bifurcations and chaos are tack-led and the corresponding energy flow behaviors are revealed using the proposed energy flow approach. As examples, the common interested nonlinear dynamical systems, such as, Duffing's oscillator, Van der Pol's equation, Lorenz attractor, Roessler one and SD oscillator, etc, are discussed. This monograph lights a new energy flow research direction for nonlinear dynamics. A generalised Matlab code with User Manuel is provided for readers to conduct the energy flow analysis of their nonlinear dynamical systems. Throughout the monograph the author continuously returns to some examples in each chapter to illustrate the applications of the discussed theory and approaches. The book can be used as an undergraduate or graduate textbook or a comprehensive source for scientists, researchers and engineers, providing the statement of the art on energy flow or power flow theory and methods.

This book was undertaken to provide a text and reference on the theory and practice of the FFT and its common usage. This book is organized in only four chapters, and is intended as a tutorial on the use of the FFf and its trade space. The trade space of the FFT is the parameters in its usage and the relationships between them - the sampie rate, the total number of points or the interval over which processing occurs in a single FFf, the selectivity of tuning to a given frequency over signals out-of-band, and the bandwidth over which a signal appears. The examples given in this text are in FORTRAN 9512003. FORTRAN 2003 was frozen as a standard while this work was in progress. The listings given here are intended as an aid in understanding the FFT and associated algorithms such as spectral window weightings, with the goal of making the best of them more accessible to the reader. The code I use here provides a simple bridge between the material in the text and implementation in FORTRAN 2003, C++, Java, MATLAB (c), and other modem languages. The examples are sufficiently simple to be translated into older languages such as C and FORTRAN 77 if desired.

This is a collection of original and review articles on recent advances and new directions in a multifaceted and interconnected area of mathematics and its applications. It encompasses many topics in theoretical developments in operator theory and its diverse applications in applied mathematics, physics, engineering, and other disciplines. The purpose is to bring in one volume many important original results of cutting edge research as well as authoritative review of recent achievements, challenges, and future directions in the area of operator theory and its applications. The intended audience are mathematicians, physicists, electrical engineers in academia and industry, researchers and graduate students, that use methods of operator theory and related fields of mathematics, such as matrix theory, functional analysis, differential and difference equations, in their work.

This book publishes a collection of original scientific research articles that address the state-of-art in using partial differential equations for image and signal processing. Coverage includes: level set methods for image segmentation and construction, denoising techniques, digital image inpainting, image dejittering, image registration, and fast numerical algorithms for solving these problems.

The subject of the book is Diophantine approximation and Nevanlinna theory. Not only does the text provide new results and directions, it also challenges open problems and collects latest research activities on these subjects made by the authors over the past eight years. Some of the significant findings are the proof of the Green-Griffiths conjecture by using meromorphic connections and Jacobian sections, and a generalized abc-conjecture. The book also presents the state of the art in the studies of the analogues between Diophantine approximation (in number theory) and value distribution theory (in complex analysis), with a method based on Vojta's dictionary for the terms of these two fields. The approaches are relatively natural and more effective than existing methods.

The book is self-contained and appended with a comprehensive and up-to-date list of references. It is of interest to a broad audience of graduate students and researchers specialized in pure mathematics.

This two-volume work mainly addresses undergraduate and graduate students in the engineering sciences and applied mathematics. Hence it focuses on partial differential equations with a strong emphasis on illustrating important applications in mechanics. The presentation considers the general derivation of partial differential equations and the formulation of consistent boundary and initial conditions required to develop well-posed mathematical statements of problems in mechanics. The worked examples within the text and problem sets at the end of each chapter highlight engineering applications. The mathematical developments include a complete discussion of uniqueness theorems and, where relevant, a discussion of maximum and miniumum principles. The primary aim of these volumes is to guide the student to pose and model engineering problems, in a mathematically correct manner, within the context of the theory of partial differential equations in mechanics.

This is the first systematic presentation of the capacitory approach and symmetrization in the context of complex analysis. The content of the book is original - the main part has not been covered by existing textbooks and monographs. After an introduction to the theory of condenser capacities in the plane, the monotonicity of the capacity under various special transformations (polarization, Gonchar transformation, averaging transformations and others) is established, followed by various types of symmetrization which are one of the main objects of the book. By using symmetrization principles, some metric properties of compact sets are obtained and some extremal decomposition problems are solved. Moreover, the classical and present facts for univalent and multivalent meromorphic functions are proven. This book will be a valuable source for current and future researchers in various branches of complex analysis and potential theory.

This collection of original articles and surveys, emerging from a 2011 conference in Bertinoro, Italy, addresses recent advances in linear and nonlinear aspects of the theory of partial differential equations (PDEs). Phase space analysis methods, also known as microlocal analysis, have continued to yield striking results over the past years and are now one of the main tools of investigation of PDEs. Their role in many applications to physics, including quantum and spectral theory, is equally important. Key topics addressed in this volume include: *general theory of pseudodifferential operators *Hardy-type inequalities *linear and non-linear hyperbolic equations and systems *Schroedinger equations *water-wave equations *Euler-Poisson systems *Navier-Stokes equations *heat and parabolic equations Various levels of graduate students, along with researchers in PDEs and related fields, will find this book to be an excellent resource. Contributors T. Alazard P.I. Naumkin J.-M. Bony F. Nicola N. Burq T. Nishitani C. Cazacu T. Okaji J.-Y. Chemin M. Paicu E. Cordero A. Parmeggiani R. Danchin V. Petkov I. Gallagher M. Reissig T. Gramchev L. Robbiano N. Hayashi L. Rodino J. Huang M. Ruzhanky D. Lannes J.-C. Saut F. Linares N. Visciglia P.B. Mucha P. Zhang C. Mullaert E. Zuazua T. Narazaki C. Zuily

This Festschrift contains five research surveys and thirty-four shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10-14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Roeckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker-Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art.

Spectral theoryis an important part of functional analysis.It has numerousapp- cations in many parts of mathematics and physics including matrix theory, fu- tion theory, complex analysis, di?erential and integral equations, control theory and quantum physics. In recent years, spectral theory has witnessed an explosive development. There are many types of spectra, both for one or several commuting operators, with important applications, for example the approximate point spectrum, Taylor spectrum, local spectrum, essential spectrum, etc. The present monograph is an attempt to organize the available material most of which exists only in the form of research papers scattered throughout the literature. The aim is to present a survey of results concerning various types of spectra in a uni?ed, axiomatic way. The central unifying notion is that of a regularity, which in a Banach algebra isasubsetofelementsthatareconsideredtobe nice .AregularityRinaBanach algebraA de?nes the corresponding spectrum ? (a)={ C: a / ? R} in R the same wayas the ordinaryspectrum is de?ned by means of invertible elements, ?(a)={ C: a / ? Inv(A)}. Axioms of a regularity are chosen in such a way that there are many natural interesting classes satisfying them. At the same time they are strong enough for non-trivial consequences, for example the spectral mapping theorem. Spectra ofn-tuples ofcommuting elements ofa Banachalgebraaredescribed similarly by means of a notion of joint regularity. This notion is closely related to ? the axiomatic spectral theory of Zelazko and S lodkowski."

In this text, integral geometry deals with Radon's problem of representing a function on a manifold in terms of its integrals over certain submanifolds-hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial di erential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation. From the reviews: "Integral Geometry is a fascinating area, where numerous branches of mathematics meet together. the contents of the book is concentrated around the duality and double vibration, which is realized through the masterful treatment of a variety of examples. the book is written by an expert, who has made fundamental contributions to the area." -Boris Rubin, Louisiana State University

On February 15-17, 1993, a conference on Large Scale Optimization, hosted by the Center for Applied Optimization, was held at the University of Florida. The con ference was supported by the National Science Foundation, the U. S. Army Research Office, and the University of Florida, with endorsements from SIAM, MPS, ORSA and IMACS. Forty one invited speakers presented papers on mathematical program ming and optimal control topics with an emphasis on algorithm development, real world applications and numerical results. Participants from Canada, Japan, Sweden, The Netherlands, Germany, Belgium, Greece, and Denmark gave the meeting an important international component. At tendees also included representatives from IBM, American Airlines, US Air, United Parcel Serice, AT & T Bell Labs, Thinking Machines, Army High Performance Com puting Research Center, and Argonne National Laboratory. In addition, the NSF sponsored attendance of thirteen graduate students from universities in the United States and abroad. Accurate modeling of scientific problems often leads to the formulation of large scale optimization problems involving thousands of continuous and/or discrete vari ables. Large scale optimization has seen a dramatic increase in activities in the past decade. This has been a natural consequence of new algorithmic developments and of the increased power of computers. For example, decomposition ideas proposed by G. Dantzig and P. Wolfe in the 1960's, are now implement able in distributed process ing systems, and today many optimization codes have been implemented on parallel machines."

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Jacobs University, Bremen, GermanyKatrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Covering some of the key areas of optimal control theory (OCT), a rapidly expanding field, the authors use new methods to set out a version of OCT's more refined'maximum principle.' The results obtainedhave applicationsin production planning, reinsurance-dividend management, multi-model sliding mode control, and multi-model differential games.

This book" "explores material that will be of great interest to post-graduate students, researchers, and practitioners in applied mathematics and engineering, particularly in the area of systems and control."

Survey on Classical Inequalities provides a study of some of the well known inequalities in classical mathematical analysis. Subjects dealt with include: Hardy-Littlewood-type inequalities, Hardy's and Carleman's inequalities, Lyapunov inequalities, Shannon's and related inequalities, generalized Shannon functional inequality, operator inequalities associated with Jensen's inequality, weighted Lp -norm inequalities in convolutions, inequalities for polynomial zeros as well as applications in a number of problems of pure and applied mathematics. It is my pleasure to express my appreciation to the distinguished mathematicians who contributed to this volume. Finally, we wish to acknowledge the superb assistance provided by the staff of Kluwer Academic Publishers. June 2000 Themistocles M. Rassias Vll LYAPUNOV INEQUALITIES AND THEIR APPLICATIONS RICHARD C. BROWN Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA. email address: [email protected] DON B. HINTON Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA. email address: [email protected] Abstract. For nearly 50 years Lyapunov inequalities have been an important tool in the study of differential equations. In this survey, building on an excellent 1991 historical survey by Cheng, we sketch some new developments in the theory of Lyapunov inequalities and present some recent disconjugacy results relating to second and higher order differential equations as well as Hamiltonian systems. 1. Introduction Lyapunov's inequality has proved useful in the study of spectral properties of ordinary differential equations. Typical applications include bounds for eigenvalues, stability criteria for periodic differential equations, and estimates for intervals of disconjugacy.

This textbook features applications including a proof of the Fundamental Theorem of Algebra, space filling curves, and the theory of irrational numbers. In addition to the standard results of advanced calculus, the book contains several interesting applications of these results. The text is intended to form a bridge between calculus and analysis. It is based on the authors lecture notes used and revised nearly every year over the last decade. The book contains numerous illustrations and cross references throughout, as well as exercises with solutions at the end of each section.