On Classification And Superposition Principles For Nonlinear Operators

Superpositions (ways of expressing a solution of an equation as a function of other solutions of the equation) for nonlinear operators are discussed. For operators having certain kinds of superpositions, a unique representation involving their superposition and additively linear operators is given. Equivalence classes of operators are obtained, each class being composed of operators having a given superposition. Generalized translations of operators are defined in terms of the superpositions. The classification of operators is further extended, and a number of well-known nonlinear differential operators are shown to belong to certain of these extended classes. (Author).

The book presents a comprehensive guide to the study of Lie systems from the fundamentals of differential geometry to the development of contemporary research topics. It embraces several basic topics on differential geometry and the study of geometric structures while developing known applications in the theory of Lie systems. The book also includes a brief exploration of the applications of Lie systems to superequations, discrete systems, and partial differential equations.Offering a complete overview from the topic's foundations to the present, this book is an ideal resource for Physics and Mathematics students, doctoral students and researchers.

This book is the first volume of proceedings from the joint conference X International Symposium “Quantum Theory and Symmetries” (QTS-X) and XII International Workshop “Lie Theory and Its Applications in Physics” (LT-XII), held on 19–25 June 2017 in Varna, Bulgaria. The QTS series was founded on the core principle that symmetries underlie all descriptions of quantum systems. It has since evolved into a symposium at the forefront of theoretical and mathematical physics. The LT series covers the whole field of Lie theory in its widest sense, together with its applications in many areas of physics. As an interface between mathematics and physics, the workshop serves as a meeting place for mathematicians and theoretical and mathematical physicists. In dividing the material between the two volumes, the Editor has sought to select papers that are more oriented toward mathematics for the first volume, and those focusing more on physics for the second. However, this division is relative, since many papers are equally suitable for either volume. The topics addressed in this volume represent the latest trends in the fields covered by the joint conferences: representation theory, integrability, entanglement, quantum groups, number theory, conformal geometry, quantum affine superalgebras, noncommutative geometry. Further, they present various mathematical results: on minuscule modules, symmetry breaking operators, Kashiwara crystals, meta-conformal invariance, the superintegrable Zernike system.

Many problems in partial differential equations which arise from physical models can be considered as ordinary differential equations in appropriate infinite dimensional spaces, for which elegant theories and powerful techniques have recently been developed. This book gives a detailed account of the current state of the theory of nonlinear differential equations in a Banach space, and discusses existence theory for differential equations with continuous and discontinuous right-hand sides. Of special importance is the first systematic presentation of the very important and complex theory of multivalued discontinuous differential equations.

This book is a self-contained account of knowledge of the theory of nonlinear superposition operators: a generalization of the notion of functions. The theory developed here is applicable to operators in a wide variety of function spaces, and it is here that the modern theory diverges from classical nonlinear analysis. The purpose of this book is to collect the basic facts about the superposition operator, to present the main ideas which are useful in studying its properties and to provide a comparison of its behaviour in different function spaces. Some applications are also considered, for example to control theory and optimization. Much of the work here has only appeared before in research literature, which itself is catalogued in detail here.

Proceedings of the International Symposium organized by the Mathematical Institute of the Polish Academy of Sciences, The Institute for Nuclear Research, Warsaw, 25-30 March 1974