Download or read book General Quantum Variational Calculus written by Khaled Zennir and published by CRC Press. This book was released on 2024-12-19 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum calculus is the modern name for the investigation of calculus without limits. The quantum calculus or q-calculus began with FH Jackson in the early twentieth century, but this kind of calculus had already been worked out by Euler and Jacobi. Recently, it arose interest due to high demand of mathematics that model quantum computing. The q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions and other sciences quantum theory, mechanics, and the theory of relativity. Recently, the concept of general quantum difference operators that generalize the quantum calculus are defined. This book is specially designed for those who wish to understand general quantum variational calculus. This book encompasses recent developments of general quantum variational calculus. The text material of this book is presented in highly readable, mathematically solid format. Many practical problems are illustrated displaying a wide variety of solution techniques. This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines
Download or read book Quantum Variational Calculus written by Agnieszka B. Malinowska and published by Springer Science & Business Media. This book was released on 2013-11-29 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Brief puts together two subjects, quantum and variational calculi by considering variational problems involving Hahn quantum operators. The main advantage of its results is that they are able to deal with nondifferentiable (even discontinuous) functions, which are important in applications. Possible applications in economics are discussed. Economists model time as continuous or discrete. Although individual economic decisions are generally made at discrete time intervals, they may well be less than perfectly synchronized in ways discrete models postulate. On the other hand, the usual assumption that economic activity takes place continuously, is nothing else than a convenient abstraction that in many applications is far from reality. The Hahn quantum calculus helps to bridge the gap between the two families of models: continuous and discrete. Quantum Variational Calculus is self-contained and unified in presentation. It provides an opportunity for an introduction to the quantum calculus of variations for experienced researchers but may be used as an advanced textbook by graduate students and even ambitious undergraduates as well. The explanations in the book are detailed to capture the interest of the curious reader, and complete to provide the necessary background material needed to go further into the subject and explore the rich research literature, motivating further research activity in the area.
Download or read book Advanced Calculus and its Applications in Variational Quantum Mechanics and Relativity Theory written by Fabio Silva Botelho and published by CRC Press. This book was released on 2021-07-12 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a rigorous study on manifolds in Rn. Develops in details important standard topics on advanced calculus, such as the differential forms in surfaces in Rn. Presents a proposal to connect classical and quantum mechanics. Presents variational formulations for relativistic mechanics through semi-Riemannian geometry and differential geometry. Develops a rigorous study on causal structures in space-time manifolds.
Download or read book Variational Principles in Dynamics and Quantum Theory written by Wolfgang Yourgrau and published by Courier Corporation. This book was released on 1979-01-01 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: Historical, theoretical survey with many insights, much hard-to-find material. Covers Hamilton's principle, Hamilton-Jacobi equation, relationship to quantum theory and wave mechanics, and more.
Download or read book General Quantum Numerical Analysis written by Svetlin G. Georgiev and published by CRC Press. This book was released on 2024-05-03 with total page 519 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is focused on the qualitative theory of general quantum calculus, the modern name for the investigation of calculus without limits. It centers on designing, analysing and applying computational techniques for general quantum differential equations. The quantum calculus or q-calculus began with F.H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by Euler and Jacobi. Recently, it has aroused interest due to high demand of mathematics that models quantum computing and the connection between mathematics and physics. Quantum calculus has many applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences such as quantum theory, mechanics and the theory of relativity. The authors summarize the most recent contributions in this area. General Quantum Numerical Analysis is intended for senior undergraduate students and beginning graduate students of engineering and science courses. The twelve chapters in this book are pedagogically organized, each concluding with a section of practical problems.
Download or read book Variational Principles in Physics written by Jean-Louis Basdevant and published by Springer Science & Business Media. This book was released on 2007-03-12 with total page 191 pages. Available in PDF, EPUB and Kindle. Book excerpt: Variational principles have proven to be surprisingly fertile. For example, Fermat used variational methods to demonstrate that light follows the fastest route from one point to another, an idea which came to be a cornerstone of geometrical optics. This book explains variational principles and charts their use throughout modern physics. It examines the analytical mechanics of Lagrange and Hamilton, the basic tools of any physicist. The book also offers simple but rich first impressions of Einstein’s General Relativity, Feynman’s Quantum Mechanics, and more that reveal amazing interconnections between various fields of physics.
Download or read book Computational Mathematics and Variational Analysis written by Nicholas J. Daras and published by Springer Nature. This book was released on 2020-06-06 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents a broad discussion of computational methods and theories on various classical and modern research problems from pure and applied mathematics. Readers conducting research in mathematics, engineering, physics, and economics will benefit from the diversity of topics covered. Contributions from an international community treat the following subjects: calculus of variations, optimization theory, operations research, game theory, differential equations, functional analysis, operator theory, approximation theory, numerical analysis, asymptotic analysis, and engineering. Specific topics include algorithms for difference of monotone operators, variational inequalities in semi-inner product spaces, function variation principles and normed minimizers, equilibria of parametrized N-player nonlinear games, multi-symplectic numerical schemes for differential equations, time-delay multi-agent systems, computational methods in non-linear design of experiments, unsupervised stochastic learning, asymptotic statistical results, global-local transformation, scattering relations of elastic waves, generalized Ostrowski and trapezoid type rules, numerical approximation, Szász Durrmeyer operators and approximation, integral inequalities, behaviour of the solutions of functional equations, functional inequalities in complex Banach spaces, functional contractions in metric spaces.
Download or read book Lectures on Quantum Mechanics for Mathematics Students written by L. D. Faddeev and published by American Mathematical Soc.. This book was released on 2009 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: Describes the relation between classical and quantum mechanics. This book contains a discussion of problems related to group representation theory and to scattering theory. It intends to give a mathematically oriented student the opportunity to grasp the main points of quantum theory in a mathematical framework.
Download or read book Calculus of Variations written by Izrailʹ Moiseevich Gelʹfand and published by Prentice Hall. This book was released on 1963 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: Elements of the theory -- Further generalizations -- The general variation of a functional -- The canonical form of the euler equations and related topics -- The second variation : sufficient conditions for a weak extremum -- Fields : sufficient conditions for a strong extremum -- Variational problems involving multiple integrals -- Direct methods in the calculus of variations -- Appendix I. Propagation of disturbances and the canonical equations -- Appendix II. Variational methods in problems of optimal control.
Download or read book Calculus of Variations written by Robert Weinstock and published by Courier Corporation. This book was released on 2012-04-26 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book by Robert Weinstock was written to fill the need for a basic introduction to the calculus of variations. Simply and easily written, with an emphasis on the applications of this calculus, it has long been a standard reference of physicists, engineers, and applied mathematicians. The author begins slowly, introducing the reader to the calculus of variations, and supplying lists of essential formulae and derivations. Later chapters cover isoperimetric problems, geometrical optics, Fermat's principle, dynamics of particles, the Sturm-Liouville eigenvalue-eigenfunction problem, the theory of elasticity, quantum mechanics, and electrostatics. Each chapter ends with a series of exercises which should prove very useful in determining whether the material in that chapter has been thoroughly grasped. The clarity of exposition makes this book easily accessible to anyone who has mastered first-year calculus with some exposure to ordinary differential equations. Physicists and engineers who find variational methods evasive at times will find this book particularly helpful. "I regard this as a very useful book which I shall refer to frequently in the future." J. L. Synge, Bulletin of the American Mathematical Society.
Download or read book Explorations in Mathematical Physics written by Don Koks and published by Springer Science & Business Media. This book was released on 2006-11-30 with total page 549 pages. Available in PDF, EPUB and Kindle. Book excerpt: Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature? This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and three-dimensional rotations. You will see how the accelerated frames of special relativity tell us about gravity. On the journey, you will discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology. The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for one-forms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods. Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.
Download or read book Mathematical Methods in Physics written by Philippe Blanchard and published by Birkhäuser. This book was released on 2015-04-07 with total page 598 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second edition of this textbook presents the basic mathematical knowledge and skills that are needed for courses on modern theoretical physics, such as those on quantum mechanics, classical and quantum field theory, and related areas. The authors stress that learning mathematical physics is not a passive process and include numerous detailed proofs, examples, and over 200 exercises, as well as hints linking mathematical concepts and results to the relevant physical concepts and theories. All of the material from the first edition has been updated, and five new chapters have been added on such topics as distributions, Hilbert space operators, and variational methods. The text is divided into three parts: - Part I: A brief introduction to (Schwartz) distribution theory. Elements from the theories of ultra distributions and (Fourier) hyperfunctions are given in addition to some deeper results for Schwartz distributions, thus providing a rather comprehensive introduction to the theory of generalized functions. Basic properties and methods for distributions are developed with applications to constant coefficient ODEs and PDEs. The relation between distributions and holomorphic functions is considered, as well as basic properties of Sobolev spaces. - Part II: Fundamental facts about Hilbert spaces. The basic theory of linear (bounded and unbounded) operators in Hilbert spaces and special classes of linear operators - compact, Hilbert-Schmidt, trace class, and Schrödinger operators, as needed in quantum physics and quantum information theory – are explored. This section also contains a detailed spectral analysis of all major classes of linear operators, including completeness of generalized eigenfunctions, as well as of (completely) positive mappings, in particular quantum operations. - Part III: Direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators. The authors conclude with a discussion of the Hohenberg-Kohn variational principle. The appendices contain proofs of more general and deeper results, including completions, basic facts about metrizable Hausdorff locally convex topological vector spaces, Baire’s fundamental results and their main consequences, and bilinear functionals. Mathematical Methods in Physics is aimed at a broad community of graduate students in mathematics, mathematical physics, quantum information theory, physics and engineering, as well as researchers in these disciplines. Expanded content and relevant updates will make this new edition a valuable resource for those working in these disciplines.
Download or read book Introduction To The Calculus of Variations And Its Applications Second Edition written by Frederic Wan and published by CRC Press. This book was released on 1995-01-01 with total page 660 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive text provides all information necessary for an introductory course on the calculus of variations and optimal control theory. Following a thorough discussion of the basic problem, including sufficient conditions for optimality, the theory and techniques are extended to problems with a free end point, a free boundary, auxiliary and inequality constraints, leading to a study of optimal control theory.
Download or read book Variational Calculus with Elementary Convexity written by J.L. Troutman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: The calculus of variations, whose origins can be traced to the works of Aristotle and Zenodoros, is now Ii vast repository supplying fundamental tools of exploration not only to the mathematician, but-as evidenced by current literature-also to those in most branches of science in which mathematics is applied. (Indeed, the macroscopic statements afforded by variational principles may provide the only valid mathematical formulation of many physical laws. ) As such, it retains the spirit of natural philosophy common to most mathematical investigations prior to this century. How ever, it is a discipline in which a single symbol (b) has at times been assigned almost mystical powers of operation and discernment, not readily subsumed into the formal structures of modern mathematics. And it is a field for which it is generally supposed that most questions motivating interest in the subject will probably not be answerable at the introductory level of their formulation. In earlier articles,1,2 it was shown through several examples that a complete characterization of the solution of optimization problems may be available by elementary methods, and it is the purpose of this work to explore further the convexity which underlay these individual successes in the context of a full introductory treatment of the theory of the variational calculus. The required convexity is that determined through Gateaux variations, which can be defined in any real linear space and which provide an unambiguous foundation for the theory.
Download or read book Introduction to the Mathematics of Variation written by Taha Sochi and published by Taha Sochi. This book was released on 2022-08-16 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about the calculus of variations which is a subject concerned mainly with optimization of functionals. However, because part of it is based on using ordinary calculus in solving optimization problems, "Calculus of Variations" in its original title is modified to become “Mathematics of Variation”. In fact, the book is essentially a collection of solved problems with rather modest theoretical background and hence it is based on the method of "learning by example and practice" which in our view is the most effective way for learning mathematics and overcoming its difficulties. The main merit of the book is its clarity, intuitive structure and rather inclusiveness as it includes the main topics and applications of this subject. The materials in this book require decent background in general mathematics (mostly in single-variable and multi-variable differential and integral calculus). The book can be used as a text or as a reference for an introductory course on this subject as part of an undergraduate curriculum in physics or engineering or applied mathematics. The book can also be used as a source of supplementary pedagogical materials used in tutorial sessions associated with such a course.
Download or read book Calculus of Variations written by Filip Rindler and published by Springer. This book was released on 2018-06-20 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a comprehensive introduction to the classical and modern calculus of variations, serving as a useful reference to advanced undergraduate and graduate students as well as researchers in the field. Starting from ten motivational examples, the book begins with the most important aspects of the classical theory, including the Direct Method, the Euler-Lagrange equation, Lagrange multipliers, Noether’s Theorem and some regularity theory. Based on the efficient Young measure approach, the author then discusses the vectorial theory of integral functionals, including quasiconvexity, polyconvexity, and relaxation. In the second part, more recent material such as rigidity in differential inclusions, microstructure, convex integration, singularities in measures, functionals defined on functions of bounded variation (BV), and Γ-convergence for phase transitions and homogenization are explored. While predominantly designed as a textbook for lecture courses on the calculus of variations, this book can also serve as the basis for a reading seminar or as a companion for self-study. The reader is assumed to be familiar with basic vector analysis, functional analysis, Sobolev spaces, and measure theory, though most of the preliminaries are also recalled in the appendix.
Download or read book A Handbook of Calculus in Quantum Mechanics written by N.B. Singh and published by N.B. Singh. This book was released on with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: "A Handbook of Calculus in Quantum Mechanics" is a comprehensive introductory guide designed specifically for absolute beginners with little to no mathematical background in quantum mechanics. This concise yet thorough handbook navigates readers through the fundamental concepts of calculus within the context of quantum mechanics, offering clear explanations and practical examples to facilitate understanding. From essential differential and integral calculus formulas to their application in solving problems in quantum mechanics, this book provides a solid foundation for readers to grasp the mathematical tools essential for exploring the intriguing world of quantum phenomena. Whether you're a student, researcher, or enthusiast, this accessible handbook equips you with the necessary knowledge to embark on your quantum journey with confidence and clarity.
