الصفحة 1
الصفحة 1
Vortices in the Magnetic Ginzburg-Landau Model
This text presents complete and mathematically rigorous versions of both results either already known by physicists or applied mathematicians, or entirely new. It begins by introducing mathematical tools such as the vortex balls construction and Jacobian estimates. Among the applications presented are: the determination of the vortex densities and vortex locations for energy minimizers in a wide range of regimes of applied fields, the precise expansion of the so-called first critical field in a bounded domain, the existence of branches of solutions with given numbers of vortices, and the derivation of a criticality condition for vortex densities of non-minimizing solutions. Thus, this book retraces in an almost entirely self-contained way many results that are scattered in series of articles, while containing a number of previously unpublished results as well.
Value Distribution Theory Related to Number Theory
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.
Stochastic Control of Hereditary Systems and Applications
This research monograph develops the Hamilton-Jacobi-Bellman (HJB) theory through dynamic programming principle for a class of optimal control problems for stochastic hereditary differential systems. It is driven by a standard Brownian motion and with a bounded memory or an infinite but fading memory. The optimal control problems treated in this book include optimal classical control and optimal stopping with a bounded memory and over finite time horizon.
Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control
Semiconcavity is a natural generalization of concavity that retains most of the good properties known in convex analysis, but arises in a wider range of applications. This text is the first comprehensive exposition of the theory of semiconcave functions, and of the role they play in optimal control and Hamilton–Jacobi equations.
Pro JSF and Ajax : Building Rich Internet Components
Pro JSF and Ajax shows you how to leverage the full potential of JavaServer Faces (JSF) and Ajax. This is not an entry-level tutorial, but a book about building Ajax-enabled JSF components for sophisticated, enterprise-level rich Internet applications. Written by JSF experts and verified by established community figures including Adam Winer (member of the JSF Expert Group, Java Champion), Kito D. Mann (JSFCentral.com and JSF in Action), and Matthias Weßendorf (MyFaces), this JSF 5-compatible book provides reliable and groundbreaking JSF components to help you exploit the power of JSF in your Java web applications.
Optimal Control with Engineering Applications
Because the theoretical part of the book is based on the calculus of variations, the exposition is very transparent and requires mostly a trivial mathematical background. In the case of open-loop optimal control, this leads to Pontryagin’s Minimum Principle and, in the case of closed-loop optimal control, to the Hamilton-Jacobi-Bellman theory which exploits the principle of optimality. Many optimal control problems are solved completely in the body of the text. Furthermore, all of the exercise problems which appear at the ends of the chapters are sketched in the appendix. The book also covers some material that is not usually found in optimal control text books, namely, optimal control problems with non-scalar-valued performance criteria (with applications to optimal filtering) and Lukes’ method of approximatively-optimal control design.
Operator Theory, Analysis and Mathematical Physics
Contains lectures delivered by the participants of the international conference "Operator Theory, Analysis and Mathematical Physics" (OTAMP 2004), held at the Mathematical Research and Conference Center in Bedlewo near Poznan, Poland on July 6-11, 2004. The idea behind these lectures was to present interesting ramifications of operator methods in current research of mathematical physics. The main topics are functional models of non-selfadjoint operators, spectral properties of Dirac and Jacobi matrices, Dirichlet-to-Neumann techniques, Lyapunov exponents methods, and inverse spectral problems for quantum graphs.
Numerical Methods for General and Structured Eigenvalue Problems
The purpose of this book is to describe recent developments in solving eig- value problems, in particular with respect to the QR and QZ algorithms as well as structured matrices. Outline Mathematically speaking, the eigenvalues of a square matrix A are the roots of its characteristic polynomial det(A??I). An invariant subspace is a linear subspace that stays invariant under the action of A. In realistic applications, it usually takes a long process of simpli?cations, linearizations and discreti- tions before one comes up with the problem of computing the eigenvalues of a matrix. In some cases, the eigenvalues have an intrinsic meaning, e.g., for the expected long-time behavior of a dynamical system; in others they are just meaningless intermediate values of a computational method. The same applies to invariant subspaces, which for example can describe sets of initial states for which a dynamical system produces exponentially decaying states. Computing eigenvalues has a long history, dating back to at least 1846 when Jacobi [172] wrote his famous paper on solving symmetric eigenvalue problems. Detailed historical accounts of this subject can be found in two papers by Golub and van der Vorst [140, 327].
Nonsmooth Vector Functions and Continuous Optimization
A recent significant innovation in mathematical sciences has been the progressive use of nonsmooth calculus, an extension of the differential calculus, as a key tool of modern analysis in many areas of mathematics, operations research, and engineering. Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems and variational inequalities in finite dimensions.
Nonlinear H2/H-Infinity Constrained Feedback Control : A Practical Design Approach Using Neural Networks
In this book the authors present algorithms for H2 and H-infinity design for nonlinear systems which, unlike earlier theories, provide solution techniques for the core Hamilton–Jacobi equations that yield control systems which can be implemented in real systems; neural networks are used to solve the nonlinear control design equations.
Integrable Systems in Celestial Mechanics
This work presents a unified treatment of three important integrable problems relevant to both Celestial and Quantum Mechanics. Under discussion are the Kepler (two-body) problem and the Euler (two-fixed center) problem, the latter being the more complex and more instructive, as it exhibits a richer and more varied solution structure. Further, because of the interesting investigations by the 20th century mathematical physicist J.P. Vinti, the Euler problem is now recognized as being intimately linked to the Vinti (Earth-satellite) problem. Here the analysis of these problems is shown to follow a definite shared pattern yielding exact forms for the solutions. A central feature is the detailed treatment of the planar Euler problem where the solutions are expressed in terms of Jacobian elliptic functions, yielding analytic representations for the orbits over the entire parameter range.
H-infinity control for nonlinear descriptor systems
The authors present a study of the H-infinity control problem and related topics for descriptor systems, described by a set of nonlinear differential-algebraic equations. They derive necessary and sufficient conditions for the existence of a controller solving the standard nonlinear H-infinity control problem considering both state and output feedback. One such condition for the output feedback control problem to be solvable is obtained in terms of Hamilton–Jacobi inequalities and a weak coupling condition; a parameterization of output feedback controllers solving the problem is also provided. All of these results are then specialized to the linear case. The derivation of state-space formulae for all controllers solving the standard H-infinity control problem for descriptor systems is proposed. Among other important topics covered are balanced realization, reduced-order controller design and mixed H2/H-infinity control.
Geometric mechanics on riemannian manifolds : Applications to partial differential equations
This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, and Schrödinger's, Einstein's and Newton's equations. Historically, problems in these areas were approached using the Fourier transform or path integrals, although in some cases (e.g., the case of quartic oscillators) these methods do not work. New geometric methods are introduced in the work that have the advantage of providing quantitative or at least qualitative descriptions of operators, many of which cannot be treated by other methods. And, conservation laws of the Euler–Lagrange equations are employed to solve the equations of motion qualitatively when quantitative analysis is not possible. It includes : Lagrangian formalism on Riemannian manifolds; energy momentum tensor and conservation laws; Hamiltonian formalism; Hamilton–Jacobi theory; harmonic functions, maps, and geodesics; fundamental solutions for heat operators with potential; and a variational approach to mechanical curves.
Controlled Markov Processes and Viscosity Solutions
This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. Stochastic control problems are treated using the dynamic programming approach. It approachs stochastic control problems by the method of dynamic programming. The fundamental equation of dynamic programming is a nonlinear evolution equation for the value function. For controlled Markov diffusion processes, this becomes a nonlinear partial differential equation of second order, called a Hamilton-Jacobi-Bellman (HJB) equation. Typically, the value function is not smooth enough to satisfy the HJB equation in a classical sense. Viscosity solutions provide framework in which to study HJB equations, and to prove continuous dependence of solutions on problem data. The theory is illustrated by applications from engineering, management science, and financial economics.
Control problems for conservation laws with traffic applications: modeling, analysis, and numerical methods
Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered.
Machine learning for risk calculations : A practitioner's view
Fundamental Approximation Methods. Machine Learning -- Deep Neural Nets -- Chebyshev Tensors -- The toolkit - plugging in approximation methods. Introduction: why is a toolkit needed -- Composition techniques -- Tensors in TT format and Tensor Extension Algorithms -- Sliding Technique -- The Jacobian projection technique -- Hybrid solutions - approximation methods and the toolkit.
Lectures on Algebraic Geometry I : Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces
This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own.In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them.
Applied Stochastic Control of Jump Diffusions
The main purpose of the book is to give a rigorous, yet mostly nontechnical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusionsThe types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods.The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it.The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations.
An Introduction to Quantum and Vassiliev Knot Invariants
Provides an accessible introduction to knot theory, focussing on Vassiliev invariants, quantum knot invariants constructed via representations of quantum groups, and how these two apparently distinct theories come together through the Kontsevich invariant. Consisting of four parts, the book opens with an introduction to the fundamentals of knot theory, and to knot invariants such as the Jones polynomial. The second part introduces quantum invariants of knots, working constructively from first principles towards the construction of Reshetikhin-Turaev invariants and a description of how these arise through Drinfeld and Jimbo's quantum groups. Its third part offers an introduction to Vassiliev invariants, providing a careful account of how chord diagrams and Jacobi diagrams arise in the theory, and the role that Lie algebras play. The final part of the book introduces the Konstevich invariant. This is a universal quantum invariant and a universal Vassiliev invariant, and brings together these two seemingly different families of knot invariants. The book provides a detailed account of the construction of the Jones polynomial via the quantum groups attached to sl(2), the Vassiliev weight system arising from sl(2), and how these invariants come together through the Kontsevich invariant.
A First Course in Modular Forms
This book introduces the theory of modular forms with an eye toward the Modularity Theorem: All rational elliptic curves arise from modular forms. The topics covered include: • elliptic curves as complex tori and as algebraic curves, • modular curves as Riemann surfaces and as algebraic curves, • Hecke operators and Atkin–Lehner theory, • Hecke eigenforms and their arithmetic properties, • the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms, • elliptic and modular curves modulo p and the Eichler–Shimura Relation, • the Galois representations associated to elliptic curves and to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory.
