Students and researchers from all fields of mathematics are invited to read and treasure this special Proceedings. A conference was held 25 -29 September 2017 at Noah's On the Beach, Newcastle, Australia, to commemorate the life and work of Jonathan M. Borwein, a mathematician extraordinaire whose untimely passing in August 2016 was a sorry loss to mathematics and to so many members of its community, a loss that continues to be keenly felt. A polymath, Jonathan Borwein ranks among the most wide ranging and influential mathematicians of the last 50 years, making significant contributions to an exceptional diversity of areas and substantially expanding the use of the computer as a tool of the research mathematician. The contributions in this commemorative volume probe Dr. Borwein's ongoing legacy in areas where he did some of his most outstanding work: Applied Analysis, Optimization and Convex Functions; Mathematics Education; Financial Mathematics; plus Number Theory, Special Functions and Pi, all tinged by the double prisms of Experimental Mathematics and Visualization, methodologies he championed.

Mathematical Physics with Partial Differential Equations, Second Edition, is designed for upper division undergraduate and beginning graduate students taking mathematical physics taught out by math departments. The new edition is based on the success of the first, with a continuing focus on clear presentation, detailed examples, mathematical rigor and a careful selection of topics. It presents the familiar classical topics and methods of mathematical physics with more extensive coverage of the three most important partial differential equations in the field of mathematical physics-the heat equation, the wave equation and Laplace's equation. The book presents the most common techniques of solving these equations, and their derivations are developed in detail for a deeper understanding of mathematical applications. Unlike many physics-leaning mathematical physics books on the market, this work is heavily rooted in math, making the book more appealing for students wanting to progress in mathematical physics, with particularly deep coverage of Green's functions, the Fourier transform, and the Laplace transform. A salient characteristic is the focus on fewer topics but at a far more rigorous level of detail than comparable undergraduate-facing textbooks. The depth of some of these topics, such as the Dirac-delta distribution, is not matched elsewhere. New features in this edition include: novel and illustrative examples from physics including the 1-dimensional quantum mechanical oscillator, the hydrogen atom and the rigid rotor model; chapter-length discussion of relevant functions, including the Hermite polynomials, Legendre polynomials, Laguerre polynomials and Bessel functions; and all-new focus on complex examples only solvable by multiple methods.

This book features a selection of articles based on the XXXV Bialowieza Workshop on Geometric Methods in Physics, 2016. The series of Bialowieza workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, are previously unpublished, at the cutting edge of current research, typically grounded in geometry and analysis, and with applications to classical and quantum physics. In 2016 the special session "Integrability and Geometry" in particular attracted pioneers and leading specialists in the field. Traditionally, the Bialowieza Workshop is followed by a School on Geometry and Physics, for advanced graduate students and early-career researchers, and the book also includes extended abstracts of the lecture series.

The book employs oscillatory dynamical systems to represent the Universe mathematically via constructing classical and quantum theory of damped oscillators. It further discusses isotropic and homogeneous metrics in the Friedman-Robertson-Walker Universe and shows their equivalence to non-stationary oscillators. The wide class of exactly solvable damped oscillator models with variable parameters is associated with classical special functions of mathematical physics. Combining principles with observations in an easy to follow way, it inspires further thinking for mathematicians and physicists. Contents Part I: Dissipative geometry and general relativity theory Pseudo-Riemannian geometry and general relativity Dynamics of universe models Anisotropic and homogeneous universe models Metric waves in a nonstationary universe and dissipative oscillator Bosonic and fermionic models of a Friedman-Robertson-Walker universe Time dependent constants in an oscillatory universe Part II: Variational principle for time dependent oscillations and dissipations Lagrangian and Hamilton descriptions Damped oscillator: classical and quantum theory Sturm-Liouville problem as a damped oscillator with time dependent damping and frequency Riccati representation of time dependent damped oscillators Quantization of the harmonic oscillator with time dependent parameters

This volume contains the proceedings of the Kovalevsky symposium held in Stockholm 2000. The first part is devoted to the life of S. Kovalevsky, the first female professor of mathematics, who influenced the development of European science during the last century. Historical notes by G. Mittag-Leffler and copies of official documents related to her life as well as several articles on her life and mathematics are presented. The main articles by J.-E. BjArk describe her life and professorship at Stockholm University. Part two of the volume contains 23 contributions in pure and applied mathematics, and in mathematical physics resulting from the lectures delivered within the program of the symposium.

This book derives new Hardy inequalities with double singular weights - at an interior point and on the boundary of the domain. We focus on the optimality of Hardy constant and on its attainability. Applications include: results about existence\nonexistence and controllability for parabolic equations with double singular potentials; estimates from below of the fi rst eigenvalue of p-Laplacian with Dirichlet boundary conditions.

This volume is to be regarded as the fifth in the series of Harish-Chandra's collected papers, continuing the four volumes already published by Springer-Verlag. Because of manifold illnesses in the last ten years of his life, a large part of Harish-Chandra's work remained unpublished. The present volume deals with those unpublished manuscripts involving real groups, and includes only those pertaining to the theorems which Harish-Chandra had announced without proofs. An attempt has been made by the volume editors to bring out this material in a more coherent form than in the handwritten manuscripts, although nothing essentially new has been added and editorial comments are kept to a minimum. The papers deal with several topics: characters on non-connected real groups, Fourier transforms of orbital integrals, Whittaker theory, and supertempered characters. The generality of Harish-Chandra's results in these papers far exceeds anything in print. The volume will be of great interest to all mathematicians interested in Lie groups, and all who have an interest in the opus of a twentieth century giant. Harish-Chandra was a great mathematician, perhaps one of the greatest of the second half of the twentieth century.

"Difference Equations, Second Edition," presents a practical introduction to this important field of solutions for engineering and the physical sciences. Topic coverage includes numerical analysis, numerical methods, differential equations, combinatorics and discrete modeling. A hallmark of this revision is the diverse application to many subfields of mathematics.

* Phase plane analysis for systems of two linear equations
* Use of equations of variation to approximate solutions
* Fundamental matrices and Floquet theory for periodic systems
* LaSalle invariance theorem
* Additional applications: secant line method, Bison problem, juvenile-adult population model, probability theory
* Appendix on the use of "Mathematica" for analyzing difference equaitons
* Exponential generating functions
* Many new examples and exercises

This book explains the essentials of fractional calculus and demonstrates its application in control system modeling, analysis and design. It presents original research to find high-precision solutions to fractional-order differentiations and differential equations. Numerical algorithms and their implementations are proposed to analyze multivariable fractional-order control systems. Through high-quality MATLAB programs, it provides engineers and applied mathematicians with theoretical and numerical tools to design control systems. Contents Introduction to fractional calculus and fractional-order control Mathematical prerequisites Definitions and computation algorithms of fractional-order derivatives and Integrals Solutions of linear fractional-order differential equations Approximation of fractional-order operators Modelling and analysis of multivariable fractional-order transfer function Matrices State space modelling and analysis of linear fractional-order Systems Numerical solutions of nonlinear fractional-order differential Equations Design of fractional-order PID controllers Frequency domain controller design for multivariable fractional-order Systems Inverse Laplace transforms involving fractional and irrational Operations FOTF Toolbox functions and models Benchmark problems for the assessment of fractional-order differential equation algorithms

This monograph is devoted to covering the main results in the qualitative theory of symplectic difference systems, including linear Hamiltonian difference systems and Sturm-Liouville difference equations, with the emphasis on the oscillation and spectral theory. As a pioneer monograph in this field it contains nowadays standard theory of symplectic systems, as well as the most current results in this field, which are based on the recently developed central object - the comparative index. The book contains numerous results and citations, which were till now scattered only in journal papers. The book also provides new applications of the theory of matrices in this field, in particular of the Moore-Penrose pseudoinverse matrices, orthogonal projectors, and symplectic matrix factorizations. Thus it brings this topic to the attention of researchers and students in pure as well as applied mathematics.

Conventional methods of financial modeling are often overly exact, to the point that their purpose--to aid in financial decision making--is easily lost. Tarrazo's approach, the use of approximation, gives professionals in finance, economics, and portfolio management a sound and sophisticated way to improve their decision making, particularly in such tasks as economic prediction, financial planning, and portfolio management. Tarrazo reviews how to build models, especially those with simultaneous equation systems, then provides a simple way to use approximate equation systems to solve them. Down to earth, readable, and meticulously explained throughout, the book is not only an important tool in practical problem solving situations, but it also provides valuable methods and guidance for upper level students and their instructors.

Among the book's important contributions is its chapter on portfolio optimization. Tarrazo helps clarify the theory and application of modern portfolio theory, especially in regard to its implementation with commonly available information management tools (such as EXCEL). He also provides innovative ways to optimize portfolios under realistic conditions and a method to obtain optimal weights in interval form that does not rely on probability; instead, it relies on the mathematical quality of the matrix in the optimization. Another chapter shows that approximate equations are a general-purpose optimization tool, one that subsumes all other known optimization tools such as classical and mathematical programming. Tarrazo closes with an unusually full bibliography, containing more than 200 references spanning several areas of analysis and various disciplines.

This book provides a unique survey displaying the power of Riccati equations to describe reversible and irreversible processes in physics and, in particular, quantum physics. Quantum mechanics is supposedly linear, invariant under time-reversal, conserving energy and, in contrast to classical theories, essentially based on the use of complex quantities. However, on a macroscopic level, processes apparently obey nonlinear irreversible evolution equations and dissipate energy. The Riccati equation, a nonlinear equation that can be linearized, has the potential to link these two worlds when applied to complex quantities. The nonlinearity can provide information about the phase-amplitude correlations of the complex quantities that cannot be obtained from the linearized form. As revealed in this wide ranging treatment, Riccati equations can also be found in many diverse fields of physics from Bose-Einstein-condensates to cosmology. The book will appeal to graduate students and theoretical physicists interested in a consistent mathematical description of physical laws.

The present volume contains selected papers issued from the sixth edition of the International Conference "Numerical methods for hyperbolic problems" that took place in 2019 in Malaga (Spain). NumHyp conferences, which began in 2009, focus on recent developments and new directions in the field of numerical methods for hyperbolic partial differential equations (PDEs) and their applications. The 11 chapters of the book cover several state-of-the-art numerical techniques and applications, including the design of numerical methods with good properties (well-balanced, asymptotic-preserving, high-order accurate, domain invariant preserving, uncertainty quantification, etc.), applications to models issued from different fields (Euler equations of gas dynamics, Navier-Stokes equations, multilayer shallow-water systems, ideal magnetohydrodynamics or fluid models to simulate multiphase flow, sediment transport, turbulent deflagrations, etc.), and the development of new nonlinear dispersive shallow-water models. The volume is addressed to PhD students and researchers in Applied Mathematics, Fluid Mechanics, or Engineering whose investigation focuses on or uses numerical methods for hyperbolic systems. It may also be a useful tool for practitioners who look for state-of-the-art methods for flow simulation.

The focus of this book is on the Schatten-von Neumann properties and the product formulas of localization operators defined in terms of infinite-dimensional and square-integrable representations of locally compact and Hausdorff groups. Wavelet transforms, which are the building blocks of localization operators, are also studied in their own right. Daubechies operators on the Weyl-Heisenberg group, localization operators on the affine group, and wavelet multipliers on the Euclidean space are investigated in detail. The study is carried out in the perspective of pseudo-differential operators, quantization and signal analysis. Although the emphasis is put on locally compact and Hausdorff groups, results in the context of homogeneous spaces are given in order to unify the various localization operators into a single theory. Several new spectral results on pseudo-differential operators in the setting of localization operators are presented for the first time. The book is accessible to graduate students and mathematicians who have a basic knowledge of measure theory and functional analysis and wish to have a fast track to the frontier of research at the interface of pseudo-differential operators, quantization and signal analysis.

The Handbook of Mathematical Fluid Dynamics is a compendium of essays that provides a survey of the major topics in the subject. Each article traces developments, surveys the results of the past decade, discusses the current state of knowledge and presents major future directions and open problems. Extensive bibliographic material is provided. The book is intended to be useful both to experts in the field and to mathematicians and other scientists who wish to learn about or begin research in mathematical fluid dynamics. The Handbook illuminates an exciting subject that involves rigorous mathematical theory applied to an important physical problem, namely the motion of fluids.

This book is devoted to recent developments concerning linear operators, covering topics such as the Cauchy problem, Riesz basis, frames, spectral theory and applications to the Gribov operator in Bargmann space. Also, integral and integro-differential equations as well as applications to problems in mathematical physics and mechanics are discussed. Contents Introduction Linear operators Basic notations and results Bases Semi-groups Discrete operator and denseness of the generalized eigenvectors Frames in Hilbert spaces Summability of series -convergence operators -hypercyclic set of linear operators Analytic operators in Bela Szoekefalvi-Nagy's sense Bases of the perturbed operator T( ) Frame of the perturbed operator T( ) Perturbation method for sound radiation by a vibrating plate in a light fluid Applications to mathematical models Reggeon field theory

This book discusses the Tauberian conditions under which convergence follows from statistical summability, various linear positive operators, Urysohn-type nonlinear Bernstein operators and also presents the use of Banach sequence spaces in the theory of infinite systems of differential equations. It also includes the generalization of linear positive operators in post-quantum calculus, which is one of the currently active areas of research in approximation theory. Presenting original papers by internationally recognized authors, the book is of interest to a wide range of mathematicians whose research areas include summability and approximation theory. One of the most active areas of research in summability theory is the concept of statistical convergence, which is a generalization of the familiar and widely investigated concept of convergence of real and complex sequences, and it has been used in Fourier analysis, probability theory, approximation theory and in other branches of mathematics. The theory of approximation deals with how functions can best be approximated with simpler functions. In the study of approximation of functions by linear positive operators, Bernstein polynomials play a highly significant role due to their simple and useful structure. And, during the last few decades, different types of research have been dedicated to improving the rate of convergence and decreasing the error of approximation.

The book contains a collection of 21 original research papers which report on recent developments in various fields of nonlinear analysis. The collection covers a large variety of topics ranging from abstract fields such as algebraic topology, functional analysis, operator theory, spectral theory, analysis on manifolds, partial differential equations, boundary value problems, geometry of Banach spaces, measure theory, variational calculus, and integral equations, to more application-oriented fields like control theory, numerical analysis, mathematical physics, mathematical economy, and financial mathematics. The book is addressed to all specialists interested in nonlinear functional analysis and its applications, but also to postgraduate students who want to get in touch with this important field of modern analysis. It is dedicated to Alfonso Vignoli who has essentially contributed to the field, on the occasion of his sixtieth birthday.

This book highlights a concise and readable introduction to typical treatments of partial differential equations in mathematical physics. Mathematical physics is regarded by many as a profound discipline. In conventional textbooks of mathematical physics, the known and the new pieces of knowledge often intertwine with each other. The book aims to ease readers' struggle by facilitating a smooth transition to new knowledge. To achieve so, the author designs knowledge maps before each chapter and provides comparative summaries in each chapter whenever appropriate. Through these unique ways, readers can clarify the underlying structures among different equations and extend one's vision to the big picture. The book also emphasizes applications of the knowledge by providing practical examples. The book is intended for all those interested in mathematical physics, enabling them to develop a solid command in using partial differential equations to solve physics and engineering problems in a not-so-painful learning experience.

The purpose of the book is to discuss the latest advances in the theory of unitary representations and harmonic analysis for solvable Lie groups. The orbit method created by Kirillov is the most powerful tool to build the ground frame of these theories. Many problems are studied in the nilpotent case, but several obstacles arise when encompassing exponentially solvable settings. The book offers the most recent solutions to a number of open questions that arose over the last decades, presents the newest related results, and offers an alluring platform for progressing in this research area. The book is unique in the literature for which the readership extends to graduate students, researchers, and beginners in the fields of harmonic analysis on solvable homogeneous spaces.

The conference has an interdisciplinary focus and aims to bring together scientists - mathematicians, electrical engineers, computer scientists, and physicists, from universities and industry - to have in-depth discussions of the latest scientific results in Computational Science and Engineering relevant to Electrical Engineering and to stimulate and inspire active participation of young researchers.

This book presents in a detailed and self-contained way a new and important density result in the analysis of fractional partial differential equations, while also covering several fundamental facts about space- and time-fractional equations.

This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their corresponding proofs. The volume consists of eight chapters that provide detailed insight into the representation of monomials of the operators Ln , direct and inverse estimates for a broad class of positive linear operators, and case studies involving finite and unbounded intervals of real and complex functions. Strong converse inequalities of Type A in terminology of Ditzian-Ivanov for linear combinations of Bernstein and Bernstein-Kantorovich operators and various Voronovskaja-type estimates for some linear combinations are analyzed and explained. Graduate students and researchers in approximation theory will find the list of open problems in approximation of linear combinations useful. The book serves as a reference for graduate and postgraduate courses as well as a basis for future study and development.

Branches of mathematics and advanced mathematical algorithms can help solve daily problems throughout various fields of applied sciences. Domains like economics, mechanical engineering, and multi-person decision making benefit from the inclusion of mathematics to maximize utility and cooperation across disciplines. There is a need for studies seeking to understand the theories and practice of using differential mathematics to increase efficiency and order in the modern world. Emerging Applications of Differential Equations and Game Theory is a collection of innovative research that examines the recent advancements on interdisciplinary areas of applied mathematics. While highlighting topics such as artificial neuron networks, stochastic optimization, and dynamical systems, this publication is ideally designed for engineers, cryptologists, economists, computer scientists, business managers, mathematicians, mechanics, academicians, researchers, and students.