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Solitons emerge in various non-linear systems as stable localized configurations, behaving in many ways like particles, from non-linear optics and condensed matter to nuclear physics, cosmology and supersymmetric theories. This book provides an introduction to integrable and non-integrable scalar field models with topological and non-topological soliton solutions. Focusing on both topological and non-topological solitons, it brings together debates around solitary waves and construction of soliton solutions in various models and provides a discussion of solitons using simple model examples. These include the Kortenweg-de-Vries system, sine-Gordon model, kinks and oscillons, and skyrmions and hopfions. The classical field theory of scalar field in various spatial dimensions is used throughout the book in presentation of related concepts, both at the technical and conceptual level. Providing a comprehensive introduction to the description and construction of solitons, this book is ideal for researchers and graduate students in mathematics and theoretical physics.

Solitons. --- Scalar field theory. --- Nonlinear systems. --- Systems, Nonlinear --- System theory --- Scalar fields --- Scalars (Mathematics) --- Calculus of tensors --- Mathematical physics --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of

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This set of lectures, which had its origin in a mini course delivered at the Summer Program of IMPA (Rio de Janeiro), is an introduction to intrinsic scaling, a powerful method in the analysis of degenerate and singular PDEs. In the first part, the theory is presented from scratch for the model case of the degenerate p-Laplace equation. This approach brings to light what is really essential in the method, leaving aside technical refinements needed to deal with more general equations, and is entirely self-contained. The second part deals with three applications of the theory to relevant models arising from flows in porous media and phase transitions. The aim is to convince the reader of the strength of the method as a systematic approach to regularity for this important class of equations.

Differential equations, Partial. --- Scalar field theory. --- Equations aux dérivées partielles --- Champs scalaires --- Scalar field theory --- Differential equations, Partial --- Calculus --- Operations Research --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Partial differential equations --- Scalar fields --- Scalars (Mathematics) --- Mathematics. --- Partial differential equations. --- Partial Differential Equations. --- Math --- Science --- Calculus of tensors --- Mathematical physics --- Differential equations, partial.

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The essential reference book on matrices-now fully updated and expanded, with new material on scalar and vector mathematicsSince its initial publication, this book has become the essential reference for users of matrices in all branches of engineering, science, and applied mathematics. In this revised and expanded edition, Dennis Bernstein combines extensive material on scalar and vector mathematics with the latest results in matrix theory to make this the most comprehensive, current, and easy-to-use book on the subject.Each chapter describes relevant theoretical background followed by specialized results. Hundreds of identities, inequalities, and facts are stated clearly and rigorously, with cross-references, citations to the literature, and helpful comments. Beginning with preliminaries on sets, logic, relations, and functions, this unique compendium covers all the major topics in matrix theory, such as transformations and decompositions, polynomial matrices, generalized inverses, and norms. Additional topics include graphs, groups, convex functions, polynomials, and linear systems. The book also features a wealth of new material on scalar inequalities, geometry, combinatorics, series, integrals, and more.Now more comprehensive than ever, Scalar, Vector, and Matrix Mathematics includes a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index.Fully updated and expanded with new material on scalar and vector mathematicsCovers the latest results in matrix theoryProvides a list of symbols and a summary of conventions for easy and precise useIncludes an extensive bibliography with back-referencing plus an author index

Matrices. --- Linear systems. --- Scalar field theory. --- Vector analysis. --- Algebra, Universal --- Mathematics --- Numbers, Complex --- Quaternions --- Spinor analysis --- Vector algebra --- Scalar fields --- Scalars (Mathematics) --- Calculus of tensors --- Mathematical physics --- Systems, Linear --- Differential equations, Linear --- System theory --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract

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This thesis presents a study of the scalar sector in the standard model (SM), as well as various searches for an extended scalar sector in theories beyond the SM (BSM). The first part of the thesis details the search for an SM Higgs boson decaying to taus, and produced by gluon fusion, vector boson fusion, or associated production with a vector boson, leading to evidence for decays of the Higgs boson to taus. In turn, the second part highlights several searches for an extended scalar sector, with scalar boson decays to taus. In all of the analyses presented, at least one scalar boson decays to a pair of taus. The results draw on data collected by the Compact Muon Solenoid (CMS) detector during proton–proton collisions with a center-of-mass energy of 7 or 8 TeV.

Physics. --- Elementary particles (Physics). --- Quantum field theory. --- Elementary Particles, Quantum Field Theory. --- Theoretical, Mathematical and Computational Physics. --- Decay schemes (Radioactivity) --- Scalar field theory. --- Bosons. --- Bose-Einstein particles --- Particles (Nuclear physics) --- Interacting boson-fermion models --- Interacting boson models --- Scalar fields --- Scalars (Mathematics) --- Calculus of tensors --- Mathematical physics --- Energy levels (Quantum mechanics) --- Radioactive decay --- Quantum theory. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Mathematical physics. --- Physical mathematics --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics --- Mathematics