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Quantum calculus is the modern name for the investigation of calculus without limits. 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 renowned mathematicians Euler and Jacobi. Lately, quantum calculus has aroused a great amount of interest due to the 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 quantum theory sciences, mechanics, and the theory of relativity. Recently, the concept of general quantum difference operators that generalize quantum calculus has been defined. General Quantum Variational Calculus is specially designed for those who wish to understand this important mathematical concept, as the text encompasses recent developments of general quantum variational calculus. The material is presented in a 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 for several disciplines.
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.
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.
DIVHistorical, theoretical survey with many insights, much hard-to-find material. Hamilton’s principle, Hamilton-Jacobi equation, etc. /div
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.
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
This textbook offers a concise yet rigorous introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. Designed specifically for a one-semester course, the book begins with calculus of variations, preparing the ground for optimal control. It then gives a complete proof of the maximum principle and covers key topics such as the Hamilton-Jacobi-Bellman theory of dynamic programming and linear-quadratic optimal control. Calculus of Variations and Optimal Control Theory also traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter, and suggestions for further study. Offers a concise yet rigorous introduction Requires limited background in control theory or advanced mathematics Provides a complete proof of the maximum principle Uses consistent notation in the exposition of classical and modern topics Traces the historical development of the subject Solutions manual (available only to teachers) Leading universities that have adopted this book include: University of Illinois at Urbana-Champaign ECE 553: Optimum Control Systems Georgia Institute of Technology ECE 6553: Optimal Control and Optimization University of Pennsylvania ESE 680: Optimal Control Theory University of Notre Dame EE 60565: Optimal Control
The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).
Quick Calculus 2nd Edition A Self-Teaching Guide Calculus is essential for understanding subjects ranging from physics and chemistry to economics and ecology. Nevertheless, countless students and others who need quantitative skills limit their futures by avoiding this subject like the plague. Maybe that's why the first edition of this self-teaching guide sold over 250,000 copies. Quick Calculus, Second Edition continues to teach the elementary techniques of differential and integral calculus quickly and painlessly. Your "calculus anxiety" will rapidly disappear as you work at your own pace on a series of carefully selected work problems. Each correct answer to a work problem leads to new material, while an incorrect response is followed by additional explanations and reviews. This updated edition incorporates the use of calculators and features more applications and examples. ".makes it possible for a person to delve into the mystery of calculus without being mystified." --Physics Teacher
Physics has long been regarded as a wellspring of mathematical problems. Mathematical Methods in Physics is a self-contained presentation, driven by historic motivations, excellent examples, detailed proofs, and a focus on those parts of mathematics that are needed in more ambitious courses on quantum mechanics and classical and quantum field theory. Aimed primarily at a broad community of graduate students in mathematics, mathematical physics, physics and engineering, as well as researchers in these disciplines.