Format: Paperback 240 pages, height x width: 235x155 mm, 2 Illustrations, black and white; VIII, 212 p.,
Series: UNITEXT 168
Pub. Date: 12-Jan-2025
ISBN-13: 9783031716591
The main aim of this book is to provide a compact self-contained presentation of the forcing technique devised by Cohen to establish the independence of the continuum hypothesis from the axioms of set theory. The book follows the approach to the forcing technique via Boolean valued semantics independently introduced by Vopenka and Scott/Solovay; it develops out of notes I prepared for several master courses on this and related topics and aims to provide an alternative (and more compact) account of this topic with respect to the available classical textbooks. The aim of the book is to take up a reader with familiarity with logic and set theory at the level of an undergraduate course on both topics (e.g., familiar with most of the content of introductory books on first-order logic and set theory) and bring her/him to page with the use of the forcing method to produce independence (or undecidability results) in mathematics. Familiarity of the reader with general topology would also be quite helpful; however, the book provides a compact account of all the needed results on this matter. Furthermore, the book is organized in such a way that many of its parts can also be read by scholars with almost no familiarity with first-order logic and/or set theory. The book presents the forcing method outlining, in many situations, the intersections of set theory and logic with other mathematical domains. My hope is that this book can be appreciated by scholars in set theory and by readers with a mindset oriented towards areas of mathematics other than logic and a keen interest in the foundations of mathematics.
1. Introduction.- 2. Preliminaries: Preorders, Topologies,
Axiomatizations of Set Theory.- 3. Boolean Algebras.- 4. Complete Boolean
Algebras.- 5. More on Preorders.- 6. Boolean Valued Models.- 7. Forcing.
Format: Hardback, 540 pages, height x width: 235x155 mm, XVIII, 540 p.,
Series: Vladimir I. Arnold - Collected Works 5
Pub. Date: 20-Dec-2024
ISBN-13: 9783031773945
This volume 5 of the Collected Works includes papers written by V.I. Arnold, one of the most outstanding mathematicians of all times, during the period from 1986 to 1991. Arnolds work during this period covers symplectic topology, contact geometry and wave propagation, quasicrystals, dynamics of intersections, bifurcations, and catastrophe theory.
He was seriously concerned with decaying mathematical education in Russia and worldwide one can see this in several articles translated for this volume. Of particular interest are the sets of problems which Arnold collected under the name Mathematical Trivium in his opinion, any math or physics university graduate should be able to solve any problem from that list. The reader will also enjoy perusing several interviews with Arnold, as well as his remarkable warm memories about Ya.B. Zeldovich and his teacher A.N. Kolmogorov. One of Arnolds papers on catastrophe theory translated for this volume also contains a beautiful translation of E.A. Baratynskys poem made by A.B. Givental.
The book will be of interest to the wide audience from college students to professionals in mathematics or physics and in the history of science.
1 The Sturm theorems and symplectic geometry.- 2 First steps of
symplectic topology.- 3 First steps of symplectic topology.- 4 On some
problems in symplectic topology.- 5 Contact structure, relaxation
oscillations and singular points of implicit differential equations.- 6
Contact geometry and wave propagation.- 7 Contact geometry: the geometrical
method of Gibbss thermodynamics.- 8 Hyperbolic polynomials and Vandermonde
mappings.- 9 On surfaces defined by hyperbolic equations.- 10 On the interior
scattering of waves, defined by hyperbolic variational principles.- 11 The
ramified covering CP^2 S^4, hyperbolicity, and projective topology.- 12
Quasicrystals, Penrose tilings, Markov partitions, stochastic web, and
singularity theory.- 13 Remarks on quasicrystallic symmetries.- 14 Dynamics
of complexity of intersections.- 15 Dynamics of intersections.- 16 Cardiac
arrhythmias and circle mappings.- 17 Remarks on Poisson structures on the
plane and on other powers of volume forms.- 18 Convex hulls and increasing
the output of systems under a pulsating load.- 19 Spaces of functions with
moderate singularities.- 20 ??-graded algebras and continued
fractions.- 21 Topological and ergodic properties of closed 1-forms with
incommensurable periods.- 22 Bifurcations and singularities in mathematics
and mechanics.- 23 Singularities and bifurcations of potential flows.- 24
Singularities of the boundaries of spaces of differential equations.- 25 Some
unsolved problems of the theory of differential equations and mathematical
physics.- 26 Ten problems.- 27 Evolution processes and ordinary differential
equations.- 28 The tercentennial of mathematical natural sciences and
celestial mechanics.- 29 Keplers second law and the topology of Abelian
integrals (according to Newton).- 30 The topological proof of transcendence
of Abelian integrals in Newtons Mathematical Principles of Natural
Philosophy.- 31 Newtons Principia read 300 years later.- 32 Meanders.- 33 A
mathematical trivium.- 33a Comments on A mathematical trivium by V.
Arnold.- 34 A mathematical trivium II.- 35 The catastrophe theory and new
opportunities for application of mathematics.- 36 Catastrophe theory.- 37
Catastrophe theory.- 38 Conversation with Vladimir Igorevich Arnold (an
interview with S. Zdravkovska).- 39 Arnold in his own words (an interview
with S.L. Tabachnikov).- 40 Mathematics in the work of Ya.B. Zeldovich.- 41
YaB and mathematics.- 42 Remembering A.N. Kolmogorov.- 43 A few words on
Andrei Nikolaevich Kolmogorov.- 44 A.N. Kolmogorov.- 45 On A.N.
Kolmogorov.- 46 Look for talents!.- 47 Mathematics with a human face.- 48
Encyclopaedia of Mathematical Sciences, or mathematics with a human face.- 49
Preface to the Russian translation of the book by P.A. Griffiths
Exterior Differential Systems and the Calculus of Variations.- 50 Preface
to the Russian translation (in the form of a book) of the article by P. Scott
The geometries of 3-manifolds.- 51 Preface to the Russian translation of
the book by J.W. Bruce and P.J. Giblin Curves and Singularities. A
Geometrical Introduction to Singularity Theory.- Acknowledgements.
Format: Paperback 185 pages, height x width: 235x155 mm, 2 Illustrations, color; 117 Illustrations, black and white; V, 195 p. 93 illus., 2 illus. in color.,
Series: Compact Textbooks in Mathematics
Pub. Date: 16-Jan-2025
ISBN-13: 9783031722806
This textbook combines the history of synthetic geometry, centered on the years 1800-1855, with a theorem-proof exposition of the geometry developed in those years. The book starts with the background needed from Euclids Elements, followed by chapters on transformations, including dilation (similitude), homology, homogeneous coordinates, projective geometry, inversion, the Mobius transformation, and transformation geometry as in French schoolbooks of 1910. Projective geometry is presented by tracing its path through the work of J. V. Poncelet, J. Steiner, and K. G. C. von Staudt. Extensive exercises are included, many from the period studied. The prerequisites for approaching this course are knowledge of high school geometry and enthusiasm for mathematical demonstration.
This textbook is ideal for a college geometry course, for self-study, or as preparation for the study of modern geometry.
Introduction.-
1. Greek Background.- 2. The Dilation Transformation.-
3. Institutional Transformation of Geometry: France.- 4. Affinity and the
List of Transformations by Moebius.- 5. Background for Homology: the Common
Secant, the Cross-Ratio, and Harmonic Sets.- 6. Plane-to-Plane Projection.-
7. Homology as developed by La Hire and Poncelet.- 8. Matrices and
Homogeneous Coordinates.- 9. Projective Geometry: Steiner and von Staudt.-
10. Transformation in German Universities.- 11. Geometric Inversion.-
12. Moebius Transformation.- 13. Topic after 1855: Beltrami-Klein Model.-
14. Topic after 1855: Isometries and Dilations in French Schoolbooks.
Format: Hardback, 460 pages, height x width: 235x155 mm, XX, 418 p.,
Series: Graduate Texts in Mathematics 173
Pub. Date: 13-Jan-2025
ISBN-13: 9783662701065
This standard textbook on modern graph theory combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject, with concise yet complete proofs, while offering glimpses of more advanced methods in each field via one or two deeper results.
This is a major new edition. Among many other improvements, it offers additional tools for applying the regularity lemma, brings the tangle theory of graph minors up to the cutting edge of current research, and addresses new topics such as chi-boundedness in perfect graph theory.
The book can be used as a reliable text for an introductory graduate course and is also suitable for self-study.
The Basics.- Matching Covering and Packing.- Connectivity.- Planar
Graphs.- Colouring.- Flows.- Extremal Graph Theory.- Infinite Graphs.- Ramsey
Theory for Graphs.- Hamilton Cycles.- Random Graphs.- Graph Minors.
Format: Hardback, 183 pages, height x width: 235x155 mm, 80 Illustrations, color; 29 Illustrations, black and white; X, 390 p. 109 illus., 78 illus. in color
Series: Lecture Notes in Computational Science and Engineering 152
Pub. Date: 08-Jan-2025
ISBN-13: 9783031596513
This volume presents a selection of papers presented at the 11th International Conference on Numerical Geometry, Grid Generation, and Scientific Computing held December 1214, 2022 in memory of Sergei Alexandrovich Ivanenko. The conference focuses on Voronoi-Delaunay theory and algorithms for tilings and partitions, mesh deformation and optimization, equidistribution principle, error analysis, discrete differential geometry, duality in mathematical programming and numerical geometry, mesh-based optimization and optimal control methods, iterative solvers for variational problems, as well as algorithm and software development.
The book provides an overview of recent advances in mesh generation and adaptation in terms of mathematical foundations, algorithm and software development, and applications.
Part I: Parameterizations and Structured Meshes.- Coordinate
Transformations and Numerical Grids for Solving Problems with Hybrid Layers.-
Parametrization of 2D Complex Regions using UNAMalla 6: A Block Structured
Mesh Generator.- Texture Space Optimization via Harmonic Map.- Automatic
Shape Design and Optimization of Missiles Based on Hex Mesh Generation and
CFD Simulation.- Mesh Untangling for Problems with Topological
Singularities.- Part II: CAX and Industrial Software.- Automatic Detection of
Manufacturing Issues in CAD Parts for DFM Analysis.- Adaptive Mesh Refinement
for Electromagnetic Simulation.- Part III: Numerical Methods.- A
Semi-analytic Technique for Integration of the Newtonian Potential and its
Gradient over Triangular Surface Grid Cells.- Dynamic Adaptive Moving Mesh
Finite Volume Method for Navier-Stokes Equations.
Format: Hardback, 440 pages, height x width: 235x155 mm, IX, 440 p.,
Series: Springer Series in Computational Mathematics 62
Pub. Date: 15-Dec-2024
ISBN-13: 9783031743696
This book is about the theory of so-called Schwarz methods for solving variational problems in a Hilbert space V arising from linear equations and their associated quadratic minimization problems. Schwarz methods are based on the construction of a sequence of approximate solutions by solving auxiliary variational problems on a set of (smaller, finite-dimensional) Hilbert spaces $V_i$ in a certain order, combining them, and using the combined approximations in an iterative procedure. The spaces $V_i$ form a so-called space splitting for V, they need not necessarily be subspaces of V, and their number can be finite or infinite.
The convergence behavior of Schwarz methods is influenced by certain properties of the space splittings they are based on. These properties are identified, and a detailed treatment of traditional deterministic and more recent greedy and stochastic orderings in the subproblem solution process is given, together with an investigation of accelerated methods. To illustrate the abstract theory, the numerical linear algebra analogs of the iterative methods covered in the book are discussed. Its standard application to the convergence theory of multilevel and domain decomposition methods for solving PDE problems is explained, and links to optimization theory and online learning algorithms are given.
Providing an introduction and overview of iterative methods which are based on problem decompositions and suitable for parallel and distributed computing, the book could serve as the basis for a one- or two-semester course for M.S. and Ph.D. students specializing in numerical analysis and scientific computing. It will also appeal to a wide range of researchers interested in scientific computing in the broadest sense.
1 Introduction.- 2 Hilbert space splittings: Abstract
theory.- References.- 3 Hilbert space splittings: Examples and extensions.-
References.- 4 Schwarz iterative methods: Finite Omega.- References.- 5
Special topics and extensions.- References.- 6 Schwarz approximation methods:
Infinite Omega.- References.- 7 Applications to PDE solvers.- References.- A
Hilbert space basics.- A.1 Spaces: Basic notation, definitions and
properties.- A.2 Operators between Hilbert spaces.- A.3 Linear equations and
variational problems.- A.4 Constructions on Hilbert spaces.- A.5 Sobolev
spaces on domains.- A.6 Reproducing kernel Hilbert spaces
(RKHS).- References.- Index.