This book explores the possibility of using azimuthal Walsh filters as an effective tool for manipulating far-field diffraction characteristics near the focal plane of rotationally symmetric imaging systems. It discusses the generation and synthesis of azimuthal Walsh filters, and explores the inherent self-similarity presented in various orders of these filters, classifying them into self-similar groups and sub-groups. Further, it demonstrates that azimuthal Walsh filters possess a unique rotational self-similarity exhibited among adjacent orders. Serving as an atlas of diffraction phenomena with pupil functions represented by azimuthal Walsh filters of different orders, this book describes how orthogonality and self-similarity of these filters could be harnessed to sculpture 2D and 3D light distributions near the focus.

This book follows a conversational approach in five dozen stories that provide an insight into the colorful world of financial mathematics and financial markets in a relaxed, accessible and entertaining form. The authors present various topics such as returns, real interest rates, present values, arbitrage, replication, options, swaps, the Black-Scholes formula and many more. The readers will learn how to discover, analyze, and deal with the many financial mathematical decisions the daily routine constantly demands. The book covers a wide field in terms of scope and thematic diversity. Numerous stories are inspired by the fields of deterministic financial mathematics, option valuation, portfolio optimization and actuarial mathematics. The book also contains a collection of basic concepts and formulas of financial mathematics and of probability theory. Thus, also readers new to the subject will be provided with all the necessary information to verify the calculations.

This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.

This book reports on the latest knowledge concerning critical phenomena arising in fluid-structure interaction due to movement and/or deformation of bodies. The focus of the book is on reporting progress in understanding turbulence and flow control to improve aerodynamic / hydrodynamic performance by reducing drag, increasing lift or thrust and reducing noise under critical conditions that may result in massive separation, strong vortex dynamics, amplification of harmful instabilities (flutter, buffet), and flow -induced vibrations. Theory together with large-scale simulations and experiments have revealed new features of turbulent flow in the boundary layer over bodies and in thin shear layers immediately downstream of separation. New insights into turbulent flow interacting with actively deformable structures, leading to new ways of adapting and controlling the body shape and vibrations to respond to these critical conditions, are investigated. The book covers new features of turbulent flows in boundary layers over wings and in shear layers immediately downstream: studies of natural and artificially generated fluctuations; reduction of noise and drag; and electromechanical conversion topics. Smart actuators as well as how smart designs lead to considerable benefits compared with conventional methods are also extensively discussed. Based on contributions presented at the IUTAM Symposium "Critical Flow Dynamics involving Moving/Deformable Structures with Design applications", held in June 18-22, 2018, in Santorini, Greece, the book provides readers with extensive information about current theories, methods and challenges in flow and turbulence control, and practical knowledge about how to use this information together with smart and bio-inspired design tools to improve aerodynamic and hydrodynamic design and safety.

This textbook provides a comprehensive overview of noncooperative and cooperative dynamic games involving uncertain parameter values, with the stochastic process being described by an event tree. Primarily intended for graduate students of economics, management science and engineering, the book is self-contained, as it defines and illustrates all relevant concepts originally introduced in static games before extending them to a dynamic framework. It subsequently addresses the sustainability of cooperative contracts over time and introduces a range of mechanisms to help avoid such agreements breaking down before reaching maturity. To illustrate the concepts discussed, the book provides various examples of how dynamic games played over event trees can be applied to environmental economics, management science, and engineering.

The book shows how classical field theory, quantum mechanics, and quantum field theory are related. The description is global from the outset. Quantization is explained using the Peierls bracket rather than the Poisson bracket. This allows one to deal immediately with observables, bypassing the canonical formalism of constrained Hamiltonian systems and bigger-than-physical Hilbert (or Fock) spaces. The Peierls bracket leads directly to the Schwinger variational principle and the Feynman functional integral, the latter of which is taken as defining the quantum theory. Also included are the theory of tree amplitudes and conservation laws, which are presented classically and later extended to the quantum level. The quantum theory is developed from the many-worlds viewpoint, and ordinary path integrals and the topological issues to which they give rise are studied in some detail. The theory of mode functions and Bogoliubov coefficients for linear fields is fully developed, and then the quantum theory of nonlinear fields is confronted. The effective action, correlation functions and counter terms all make their appearance at this point, and the S-matrix is constructed via the introduction of asymptotic fields and the LSZ theorem. Gauge theories and ghosts are studied in great detail. Many applications of the formalism are given: vacuum currents, anomalies, black holes, fourth-order systems, higher spin fields, the (lambda phi) to the fourth power model (and spontaneous symmetry breaking), quantum electrodynamics, the Yang-Mills field and its topology, the gravitational field, etc. Special chapters are devoted to Euclideanization and renormalization, space and time inversion, and the closed-time-path or "in-in" formalism. Emphasis is given throughout to the role of the functional-integral measure in the theory. Six helpful appendices, ranging from superanalysis to analytic continuation in dimension, are included at the end.

Photonics has long been considered an attractive substrate for next generation implementations of machine-learning concepts. Reservoir Computing tremendously facilitated the realization of recurrent neural networks in analogue hardware. This concept exploits the properties of complex nonlinear dynamical systems, giving rise to photonic reservoirs implemented by semiconductor lasers, telecommunication modulators and integrated photonic chips.

This book is a description of why and how to do Scientific Computing for fundamental models of fluid flow. It contains introduction, motivation, analysis, and algorithms and is closely tied to freely available MATLAB codes that implement the methods described. The focus is on finite element approximation methods and fast iterative solution methods for the consequent linear(ized) systems arising in important problems that model incompressible fluid flow. The problems addressed are the Poisson equation, Convection-Diffusion problem, Stokes problem and Navier-Stokes problem, including new material on time-dependent problems and models of multi-physics. The corresponding iterative algebra based on preconditioned Krylov subspace and multigrid techniques is for symmetric and positive definite, nonsymmetric positive definite, symmetric indefinite and nonsymmetric indefinite matrix systems respectively. For each problem and associated solvers there is a description of how to compute together with theoretical analysis that guides the choice of approaches and describes what happens in practice in the many illustrative numerical results throughout the book (computed with the freely downloadable IFISS software). All of the numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope for experimentation in the "computational laboratory " provided by the software. Developments in the field since the first edition was published have been represented in three new chapters covering optimization with PDE constraints (Chapter 5); solution of unsteady Navier-Stokes equations (Chapter 10); solution of models of buoyancy-driven flow (Chapter 11). Each chapter has many theoretical problems and practical computer exercises that involve the use of the IFISS software. This book is suitable as an introduction to iterative linear solvers or more generally as a model of Scientific Computing at an advanced undergraduate or beginning graduate level.

This book presents a theoretical study of the generation and conversion of phonon angular momentum in crystals. Recently, rotational motions of lattice vibrations, i.e., phonons, in crystals attract considerable attentions. As such, the book theoretically demonstrate generations of phonons with rotational motions, based on model calculations and first-principle calculations. In systems without inversion symmetry, the phonon angular momentum is shown to be caused by the temperature gradient, which is demonstrated in crystals such as wurtzite gallium nitride, tellurium, and selenium using the first-principle calculations. In systems with neither time-reversal nor inversion symmetries, the phonon angular momentum is shown to be generated by an electric field. Secondly, the book presents the microscopic mechanisms developed by the author and his collaborator on how these microscopic rotations of nuclei are coupled with electron spins. These predictions serve as building blocks for spintronics with phonons or mechanical motions.

This book provides a comprehensive examination of preconditioners for boundary element discretisations of first-kind integral equations. Focusing on domain-decomposition-type and multilevel methods, it allows readers to gain a good understanding of the mechanisms and necessary techniques in the analysis of the preconditioners. These techniques are unique for the discretisation of first-kind integral equations since the resulting systems of linear equations are not only large and ill-conditioned, but also dense. The book showcases state-of-the-art preconditioning techniques for boundary integral equations, presenting up-to-date research. It also includes a detailed discussion of Sobolev spaces of fractional orders to familiarise readers with important mathematical tools for the analysis. Furthermore, the concise overview of adaptive BEM, hp-version BEM, and coupling of FEM-BEM provides efficient computational tools for solving practical problems with applications in science and engineering.

Our original reason for writing this book was the desire to write down in one place a complete summary of the major results in du ality theory pioneered by Ronald W. Shephard in three of his books, Cost and Production Functions (1953), Theory of Cost and Produc tion Functions (1970), and Indirect Production Functions (1974). In this way, newcomers to the field would have easy access to these important ideas. In adg, ition, we report a few new results of our own. In particular, we show the duality relationship between the profit function and the eight equivalent representations of technol ogy that were elucidated by Shephard. However, in planning the book and discussing it with colleagues it became evident that such a book would be more useful if it also provided a number of applications of Shephard's duality theory to economic problems. Thus, we have also attempted to present exam ples of the use of duality theory in areas such as efficiency measure ment, index number theory, shadow pricing, cost-benefit analysis, and econometric estimation. Much of our thinking about duality theory and its uses has been influenced by our present and former collaborators. They include Charles Blackorby, Shawna Grosskopf, Knox Lovell, Robert Russell, and, not surprisingly, Ronald W. Shephard. We have also benefit ted over the years from many discussions with W. Erwin Diewert."

This eighteenth volume in the Poincare Seminar Series provides a thorough description of Information Theory and some of its most active areas, in particular, its relation to thermodynamics at the nanoscale and the Maxwell Demon, and the emergence of quantum computation and of its counterpart, quantum verification. It also includes two introductory tutorials, one on the fundamental relation between thermodynamics and information theory, and a primer on Shannon's entropy and information theory. The book offers a unique and manifold perspective on recent mathematical and physical developments in this field.

This volume provides a detailed description of some of the most active areas in astrophysics from the largest scales probed by the Planck satellite to massive black holes that lie at the heart of galaxies and up to the much awaited but stunning discovery of thousands of exoplanets. It contains the following chapters: * Jean-Philippe UZAN, The Big-Bang Theory: Construction, Evolution and Status * Jean-Loup PUGET, The Planck Mission and the Cosmic Microwave Background * Reinhard GENZEL, Massive Black Holes: Evidence, Demographics and Cosmic Evolution * Arnaud CASSAN, New Worlds Ahead: The Discovery of Exoplanets Reinhard Genzel and Andrea Ghez shared the 2020 Nobel Prize in Physics "for the discovery of a supermassive compact object at the centre of our galaxy'", alongside Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity". The book corresponds to the twentieth Poincare Seminar, held on November 21, 2015, at Institut Henri Poincare in Paris. Originally written as lectures to a broad scientific audience, these four chapters are of high value and will be of general interest to astrophysicists, physicists, mathematicians and historians.

Professor Atiyah is one of the greatest living mathematicians and is renowned in the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still actively involved in the mathematics community. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into seven volumes, with the first five volumes divided thematically and the sixth and seventh arranged by date. This seven volume set of the collected works of Professor Sir Michael Atiyah, includes: Collected Works: Volume 1: Early Papers; General Papers Collected Works: Volume 2: K-Theory Collected Works: Volume 3: Index Theory: 1 Collected Works: Volume 4: Index Theory: 2 Collected Works: Volume 5: Gauge Theories Collected Works: Volume 6: Publications between 1987 and 2002 New for 2014: Collected Works: Volume 7: 2002-2013, including Sir Michael's work on skyrmions; K-theory and cohomology; geometric models of matter; curvature, cones and characteristic numbers; and reflections on the work of Riemann, Einstein and Bott.

How planets form is one of the long-standing questions in astrophysics. In particular, formation scenarios of planetesimals which are kilometer-sized bodies and a precursor of planets are still unclear and under debate although some promising mechanisms have been proposed. This book highlight disk instabilities that have the potential to explain the origin of planetesimals. Using linear analyses and numerical simulations, it addresses how a disk evolves through the development of instabilities, and also presents a new instability driven by dust coagulation. As a result, the simulation demonstrates a scenario of planetesimal formation: A successive development of multiple instabilities triggers planetesimal formation in resulting dusty rings.

This is the first book in a four-part series designed to give a comprehensive and coherent description of Fluid Dynamics, starting with chapters on classical theory suitable for an introductory undergraduate lecture course, and then progressing through more advanced material up to the level of modern research in the field. The present Part 1 consists of four chapters. Chapter 1 begins with a discussion of Continuum Hypothesis, which is followed by an introduction to macroscopic functions, the velocity vector, pressure, density, and enthalpy. We then analyse the forces acting inside a fluid, and deduce the Navier-Stokes equations for incompressible and compressible fluids in Cartesian and curvilinear coordinates. In Chapter 2 we study the properties of a number of flows that are presented by the so-called exact solutions of the Navier-Stokes equations, including the Couette flow between two parallel plates, Hagen-Poiseuille flow through a pipe, and Karman flow above an infinite rotating disk. Chapter 3 is devoted to the inviscid incompressible flow theory, with particular focus on two-dimensional potential flows. These can be described in terms of the "complex potential", allowing the full power of the theory of functions of complex variables to be used. We discuss in detail the method of conformal mapping, which is then used to study various flows of interest, including the flows past Joukovskii aerofoils. The final Chapter 4 is concerned with compressible flows of perfect gas, including supersonic flows. Particular attention is given to the theory of characteristics, which is used, for example, to analyse the Prandtl-Meyer flow over a body surface bend and a corner. Significant attention is also devoted to the shock waves. The chapter concludes with analysis of unsteady flows, including the theory of blast waves.

The book provides the theoretical fundamentals on turbulence and a complete overview of turbulence models, from the simplest to the most advanced ones including Direct and Large Eddy Simulation. It mainly focuses on problems of modeling and computation, and provides information regarding the theory of dynamical systems and their bifurcations. It also examines turbulence aspects which are not treated in most existing books on this subject, such as turbulence in free and mixed convection, transient turbulence and transition to turbulence. The book adopts the tensor notation, which is the most appropriate to deal with intrinsically tensor quantities such as stresses and strain rates, and for those who are not familiar with it an Appendix on tensor algebra and tensor notation are provided.

Everything you need to know in order to manage risk effectively within your organization

You cannot afford to ignore the explosion in mathematical finance in your quest to remain competitive. This exciting branch of mathematics has very direct practical implications: when a new model is tested and implemented it can have an immediate impact on the financial environment.

With risk management top of the agenda for many organizations, this book is essential reading for getting to grips with the mathematical story behind the subject of financial risk management. It will take you on a journey--from the early ideas of risk quantification up to today's sophisticated models and approaches to business risk management.

To help you investigate the most up-to-date, pioneering developments in modern risk management, the book presents statistical theories and shows you how to put statistical tools into action to investigate areas such as the design of mathematical models for financial volatility or calculating the value at risk for an investment portfolio.Respected academic author Simon Hubbert is the youngest director of a financial engineering program in the U.K. He brings his industry experience to his practical approach to risk analysisCaptures the essential mathematical tools needed to explore many common risk management problemsWebsite with model simulations and source code enables you to put models of risk management into practicePlunges into the world of high-risk finance and examines the crucial relationship between the risk and the potential reward of holding a portfolio of risky financial assets

This book is your one-stop-shop for effective risk management.

This book presents a new method for analyzing the structure and function of the biological branching systems of fractal trees, with a focus on microcirculation. Branching systems in humans (vascular and bronchial trees) and those in the natural world (plants, trees, and rivers) are characterized by a fractal nature. To date, fractal studies have tended to concentrate on fractal dimensions, which quantify the complexity of objects, but the applications for practical use have remained largely unexplored. This book breaks new ground with topics that include the human retinal microcirculatory network, oxygen consumption by vascular walls, the F hraeus-Lindqvist effect, the bifurcation exponent, and the asymmetrical microvascular network. Readers are provided with simple formulas to express functions and a simulation graph with in vivo data. The book also discusses the mechanisms regulating blood flow and pressure and how they are related to pathological changes in the human body. Researchers and clinicians alike will find valuable new insights in these pioneering studies.

This fairly self-contained work embraces a broad range of topics in analysis at the graduate level, requiring only a sound knowledge of calculus and the functions of one variable. A key feature of this lively yet rigorous and systematic exposition is the historical accounts of ideas and methods pertaining to the relevant topics. Most interesting and useful are the connections developed between analysis and other mathematical disciplines, in this case, numerical analysis and probability theory.

The text is divided into two parts: The first examines the systems of real and complex numbers and deals with the notion of sequences in this context. After the presentation of natural numbers as a subset of the reals, elements of combinatorics and a discussion of the mathematical notion of the infinite are introduced. The second part is dedicated to discrete processes starting with a study of the processes of infinite summation both in the case of numerical series and of power series.

Introduction to Statistical Analysis of Laboratory Data presents a detailed discussion of important statistical concepts and methods of data presentation and analysis * Provides detailed discussions on statistical applications including a comprehensive package of statistical tools that are specific to the laboratory experiment process * Introduces terminology used in many applications such as the interpretation of assay design and validation as well as fit for purpose procedures including real world examples * Includes a rigorous review of statistical quality control procedures in laboratory methodologies and influences on capabilities * Presents methodologies used in the areas such as method comparison procedures, limit and bias detection, outlier analysis and detecting sources of variation * Analysis of robustness and ruggedness including multivariate influences on response are introduced to account for controllable/uncontrollable laboratory conditions

This book presents the proceedings from ECONOPHYS-2015, an international workshop held in New Delhi, India, on the interrelated fields of "econophysics" and "sociophysics", which have emerged from the application of statistical physics to economics and sociology. Leading researchers from varied communities, including economists, sociologists, financial analysts, mathematicians, physicists, statisticians, and others, report on their recent work, discuss topical issues, and review the relevant contemporary literature. A society can be described as a group of people who inhabit the same geographical or social territory and are mutually involved through their shared participation in different aspects of life. It is possible to observe and characterize average behaviors of members of a society, an example being voting behavior. Moreover, the dynamic nature of interaction within any economic sector comprising numerous cooperatively interacting agents has many features in common with the interacting systems of statistical physics. It is on these bases that interest has grown in the application within sociology and economics of the tools of statistical mechanics. This book will be of value for all with an interest in this flourishing field.

The wealth of recent cosmic microwave background and large-scale structure data has transformed the field of cosmology. These observations have not only become precise enough to answer questions about the universe on the largest scales, but also to address puzzles in the microscopic description of Nature. This thesis investigates new ways of probing the early universe, the properties of neutrinos and the possible existence of other light particles. In particular, based on detailed theoretical insights and novel analyses, new evidence for the cosmic neutrino background is found in the distribution of galaxies and in cosmic microwave background data. This tests the Standard Model of particle physics and the universe back to a time when it was about one second old. Furthermore, it is demonstrated that future observations will be capable of probing physics beyond the Standard Model since they can achieve a particular target which would either allow the detection of any light particles that have ever been in thermal equilibrium or imply strong bounds on their properties.

The book serves as a primary textbook of partial differential equations (PDEs), with due attention to their importance to various physical and engineering phenomena. The book focuses on maintaining a balance between the mathematical expressions used and the significance they hold in the context of some physical problem. The book has wider outreach as it covers topics relevant to many different applications of ordinary differential equations (ODEs), PDEs, Fourier series, integral transforms, and applications. It also discusses applications of analytical and geometric methods to solve some fundamental PDE models of physical phenomena such as transport of mass, momentum, and energy. As far as possible, historical notes are added for most important developments in science and engineering. Both the presentation and treatment of topics are fashioned to meet the expectations of interested readers working in any branch of science and technology. Senior undergraduates in mathematics and engineering are the targeted student readership, and the topical focus with applications to real-world examples will promote higher-level mathematical understanding for undergraduates in sciences and engineering.