Format: height x width: 235x155 mm, Approx. 400 p., 1 Paperback / softback
Pub. Date: 13-Dec-2023
ISBN-13: 9783031435096
The book is the second volume of a collection which consists of surveys that focus on important topics in geometry which are at the heart of current research. The topics in the present volume include the conformal and the metric geometry of surfaces, Teichmuller spaces, immersed surfaces of prescribed extrinsic curvature in 3-dimensional manifolds, symplectic geometry, the metric theory of Grassmann spaces, homogeneous metric spaces, polytopes, the higher-dimensional GaussBonnet formula, isoperimetry in finitely generated groups and Coxeter groups.
Each chapter is intended for graduate students and researchers. Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to important topics in geometry.
1 Athanase Papadopoulos: Introduction.- 2 Norbert A'Campo and Athanase
Papadopoulos, Geometry of surfaces.- 3 Kenichi
Ohshika: Teichmuller spaces and their various metrics.- 4 Marc Troyanov:
Double forms, curvature integrals and the Gauss-Bonnet formula.- 5 Graham
Smith, Quaternions, MongeAmpAOre structures and -surfaces.- 6 Peter
Kristel and Eric Schippers, Lagrangian Grassmannians of polarizations.-
7 Arpad Kurusa, Metric characterizations of projective-metric spaces.-
8 Arpad Kurusa, Supplement to Metric Characterization of
Projective-metric Spaces.- 9 Boumediene Et-Taoui, Metric problems in
projective and Grassmann spaces.- 10 Valeri Berestovski and Yuri
Nikonorov, On the geometry of finite homogeneous subsets of
Euclidean spaces.- 11 Gue-Seon Lee and Ludovic Marquis, Discrete Coxeter
groups.- 12 Bruno Luiz Santos Correia and Marc Troyanov, Isoperimetry in
Finitely Generated Groups.
Format: 340 pages, height x width: 235x155 mm, XXII, 340 p., 1 Paperback / softback
Series: Universitext
Pub. Date: 16-Dec-2023
ISBN-13: 9783031450358
This book provides an introduction to the broad topic of the calculus of variations. It addresses the most natural questions on variational problems and the mathematical complexities they present. Beginning with the scientific modeling that motivates the subject, the book then tackles mathematical questions such as the existence and uniqueness of solutions, their characterization in terms of partial differential equations, and their regularity. It includes both classical and recent results on one-dimensional variational problems, as well as the adaptation to the multi-dimensional case. Here, convexity plays an important role in establishing semi-continuity results and connections with techniques from optimization, and convex duality is even used to produce regularity results. This is then followed by the more classical Holder regularity theory for elliptic PDEs and some geometric variational problems on sets, including the isoperimetric inequality and the Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of -convergence. While primarily designed for master's-level and advanced courses, this textbook, based on its author's instructional experience, also offers original insights that may be of interest to PhD students and researchers. A foundational understanding of measure theory and functional analysis is required, but all the essential concepts are reiterated throughout the book using special memo-boxes.
1 One-dimensional variational problems.- 2 Multi-dimensional variational
problems.- 3 Lower semicontinuity.- 4 Convexity and its applications.- 5
Holder regularity.- 6 Variational problems for sets.- 7 -convergence: theory
and examples.
Format: 487 pages, height x width: 235x155 mm, 5 Illustrations, black and white; X, 487 p. 5 illus., 1 Hardback
Series: Operator Theory: Advances and Applications 294
Pub. Date: 24-Dec-2023
ISBN-13: 9783031463860
This monograph presents the solution of the classical moment problem, the construction of Jacobi matrices and corresponding polynomials. The cases of strongly,trigonometric, complex and real two-dimensional moment problems are discussed, and the Jacobi-type matrices corresponding to the trigonometric moment problem are shown. The Berezansky theory of the expansion in generalized eigenvectors for corresponding set of commuting operators plays the key role in the proof of results.
The book is recommended for researchers in fields of functional analysis, operator theory, mathematical physics, and engineers who deal with problems of coupled pendulums.
Introduction.- Some aspects of the spectral theory of unbounded operators.- Jacobi matrices and the classical moment problem.- The strong moment problem.- Block Jacobi type matrices in the complex moment problem.- Unitary block Jacobi type matrices and the trigonometric moment problem.- Block Jacobi type matrices and the complex moment problem in the exponential form.- Block Jacobi type matrices and the two dimensional real moment problem.- Applications of the spectral theory of Jacobi matrices and their generalizations to the integration of nonlinear equations.
Format: 282 pages, height x width: 235x155 mm, 19 Tables, color;
1 Illustrations, black and white; XVII, 282 p. 1 illus., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2341
Pub. Date: 31-Dec-2023
ISBN-13: 9789819966851
The stochastic Maxwell equations play an essential role in many fields, including fluctuational electrodynamics, statistical radiophysics, integrated circuits, and stochastic inverse problems.
This book provides some recent advances in the investigation of numerical approximations of the stochastic Maxwell equations via structure-preserving algorithms. It presents an accessible overview of the construction and analysis of structure-preserving algorithms with an emphasis on the preservation of geometric structures, physical properties, and asymptotic behaviors of the stochastic Maxwell equations. A friendly introduction to the simulation of the stochastic Maxwell equations with some structure-preserving algorithms is provided using MATLAB for the readers convenience.
The objects considered in this book are related to several fascinating mathematical fields: numerical analysis, stochastic analysis, (multi-)symplectic geometry, large deviations principle, ergodic theory, partial differential equation, probability theory, etc. This book will appeal to researchers who are interested in these topics.Table of Contents
Introduction.- Solution Theory of Stochastic Maxwell Equations.- Intrinsic Properties of Stochastic Maxwell Equations.- Structure-Preserving Algorithms for Stochastic Maxwell Equations.- Convergence Analysis of Structure-Preserving Algorithms.- Implementation of Numerical Experiments.- Appendix A: Basic Identities and Inequalities.- Appendix B: Semigroup, Sobolev Space, and Differential Calculus.- Appendix C: Estimates Related to Maxwell Operators.- Appendix D: Some Results of Stochastic Partial Differential Equations.- References.
Format: 361 pages, height x width: 235x155 mm, XVII, 361 p., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2342
Pub. Date: 01-Jan-2024
ISBN-13: 9783031451768
This book describes a novel approach to the study of Siegel modular forms of degree two with paramodular level. It introduces the family of stable Klingen congruence subgroups of GSp(4) and uses this family to obtain new relations between the Hecke eigenvalues and Fourier coefficients of paramodular newforms, revealing a fundamental dichotomy for paramodular representations. Among other important results, it includes a complete description of the vectors fixed by these congruence subgroups in all irreducible representations of GSp(4) over a nonarchimedean local field. Siegel paramodular forms have connections with the theory of automorphic representations and the Langlands program, Galois representations, the arithmetic of abelian surfaces, and algorithmic number theory. Providing a useful standard source on the subject, the book will be of interest to graduate students and researchers working in the above fields.
ntroduction.- Part I Local Theory.- Background.- Stable Klingen
Vectors.- Some Induced Representations.- Dimensions.- Hecke Eigenvalues and
Minimal Levels.- The Paramodular Subspace.- Further Results about Generic
Representations.- Iwahori-spherical Representations.- Part II Siegel Modular
Forms.- Background on Siegel Modular Forms.- Operators on Siegel Modular
Forms.- Hecke Eigenvalues and Fourier Coefficients.
Format: 240 pages, height x width: 235x155 mm, 30 Illustrations, black and white; X, 240 p. 30 illus., 1 Paperback / softback
Series: Lecture Notes in Physics 1021
Pub. Date: 15-Jan-2024
ISBN-13: 9783031469862
Other books in subject:
This open access book bridges a gap between introductory Quantum Field Theory (QFT) courses and state-of-the-art research in scattering amplitudes. It covers the path from basic definitions of QFT to amplitudes, which are relevant for processes in the Standard Model of particle physics. The book begins with a concise yet self-contained introduction to QFT, including perturbative quantum gravity. It then presents modern methods for calculating scattering amplitudes, focusing on tree-level amplitudes, loop-level integrands and loop integration techniques. These methods help to reveal intriguing relations between gauge and gravity amplitudes and are of increasing importance for obtaining high-precision predictions for collider experiments, such as those at the Large Hadron Collider, as well as for foundational mathematical physics studies in QFT, including recent applications to gravitational wave physics.These course-tested lecture notes include numerous exercises with solutions. Requiring only minimal knowledge of QFT, they are well-suited for MSc and PhD students as a preparation for research projects in theoretical particle physics. They can be used as a one-semester graduate level course, or as a self-study guide for researchers interested in fundamental aspects of quantum field theory.
1. Introduction & basics
1.1 Poincare group & representations
1.2. Weyl & Dirac spinors
1.3. Non-abelian gauge theories
1.4. Perturbative quantum gravity
1.5. Feynman-rules
1.6. Spinor helicity formalism for massless particles
1.7. Polarizations
1.8. Color decomposition
1.9. Color ordered amplitudes
1.10. Outlook 1: Massive spinor helicity
1.11. Outlook 2: Momentum twistors
2. Tree-level amplitudes
2.1. BCFW recursion
2.2. 3-point amplitudes
2.3. Factorizations
2.4. Symmetries of scattering amplitudes
2.5. Dualities for gluons & gravitons
2.6. Massive BCFW
2.7. Outlook 1: Scattering eqs. and the CHY Formalism
3. Loop-level integrands and amplitudes
3.1. Introduction
3.2. Unitarity and Cut-Construction
3.3. Generalised Unitarity
3.4. Reduction methods
3.5. General method for one-loop amplitudes
3.5.1. The integral basis
3.5.2. Constructing integrand basis for box, triangle and bubble topologies
3.5.3. D-dimensional integrands and rational terms
3.5.4. Direct construction method (Forde)
3.6. Outlook: multi-loop integrand reduction
4. Loop integration techniques and special functions
4.1. Introduction
4.2. Conventions and Feynman parameter method
4.3. Ultraviolet and infrared divergences
4.4. Mellin-Barnes method
4.5. Feynman integrals and transcedental weights
4.6. Differential equation method
4.7. Functional identities and symbol method
4.8. Other topics
4.9. Exercises
4.10. Outlook, suggested reading for student presentations
5. Exercises with solutions