Continuum Physics

Continuum Physics

During this period, generalized continuum mechanics has not been found wanting, for there have been four distinct avenues of work devoted to theories that can account for the effects of interaction forces of nonzero range. One category ...

Author: A. Cemal Eringen

Publisher: Elsevier

ISBN: 9780323140607

Category: Science

Page: 288

View: 750

Continuum Physics, Volume IV: Polar and Nonlocal Field Theories discusses the exposition of field theories for bodies which possess inner structure that can interact with mechanical and electromagnetic fields. This book provides precise presentations of exact continuum theories on materially non-uniform or non-simple bodies that can respond to short- and long-range inter-particle loads and fields. This volume consists of three parts. Part I is devoted to the study of continuum field theories for bodies having inner structure. All materials, to some extent, are composed of particles that behave like small rigid bodies or deformable particles, unlike the geometrical points of the classical continuum theory. The developments of nonlocal theories of nonpolar and polar continua are covered in Parts II and III. This publication is valuable to students and researchers interested in polar and nonlocal field theories.
Categories: Science

Categories in Algebra Geometry and Mathematical Physics

Categories in Algebra  Geometry and Mathematical Physics

Amer. Math. Soc. 83 (1977), 831–879. [35]. K. T. Chen, On differentiable spaces, in Categories in Continuum Physics, Lecture Notes in Mathematics, 1174, Springer, Berlin, 1986, pp. 38-42. [36] A. Pressley and G. Segal, Loop Groups.

Author: Alexei Davydov

Publisher: American Mathematical Soc.

ISBN: 9780821839706

Category: Mathematics

Page: 467

View: 388

Category theory has become the universal language of modern mathematics. This book is a collection of articles applying methods of category theory to the areas of algebra, geometry, and mathematical physics. Among others, this book contains articles on higher categories and their applications and on homotopy theoretic methods. The reader can learn about the exciting new interactions of category theory with very traditional mathematical disciplines.
Categories: Mathematics

Analysis and Continuum Mechanics

Analysis and Continuum Mechanics

NOLL, W, “Continuum Mechanics and Geometric Integration Theory”, Categories in Continuum Physics, Lecture Notes in Mathematics 1174, 17–29, Springer Verlag, 1986. TRUESDELL, C, “A First Course in Rational Continuum Mechanics, Vol.

Author: Stuart S. Antman

Publisher: Springer Science & Business Media

ISBN: 9783642837432

Category: Technology & Engineering

Page: 829

View: 911

The 39 papers in this collection are devoted mostly to the exact mathematical analysis of problems in continuum mechanics, but also to problems of a purely mathematical nature mainly connected to partial differential equations from continuum physics. All the papers are dedicated to J. Serrin and were originally published in the "Archive of Rational Mechanics and Analysis".
Categories: Technology & Engineering

The Rational Spirit in Modern Continuum Mechanics

The Rational Spirit in Modern Continuum Mechanics

W. Noll, Continuum Mechanics and geometric integration theory. In: F.W. Lawvere and S.H. Schnauel (eds), Categories in Continuum Physics, Buffalo, 1982, Springer Lecture Notes in Mathematics, Vol. 1174. Springer, Berlin (1986) pp.

Author: Chi-Sing Man

Publisher: Springer Science & Business Media

ISBN: 9781402023088

Category: Science

Page: 808

View: 806

Through his voluminous and in?uential writings, editorial activities, organi- tional leadership, intellectual acumen, and strong sense of history, Clifford - brose Truesdell III (1919–2000) was the main architect for the renaissance of - tional continuum mechanics since the middle of the twentieth century. The present collection of 42 essays and research papers pays tribute to this man of mathematics, science, and natural philosophy as well as to his legacy. The ?rst ?ve essays by B. D. Coleman, E. Giusti, W. Noll, J. Serrin, and D. Speiser were texts of addresses given by their authors at the Meeting in memory of Clifford Truesdell, which was held in Pisa in November 2000. In these essays the reader will ?nd personal reminiscences of Clifford Truesdell the man and of some of his activities as scientist, author, editor, historian of exact sciences, and principal founding member of the Society for Natural Philosophy. The bulk of the collection comprises 37 research papers which bear witness to the Truesdellian legacy. These papers cover a wide range of topics; what ties them together is the rational spirit. Clifford Truesdell, in his address upon receipt of a Birkhoff Prize in 1978, put the essence of modern continuum mechanics succinctly as “conceptual analysis, analysis not in the sense of the technical term but in the root meaning: logical criticism, dissection, and creative scrutiny.
Categories: Science

Hyperbolic Conservation Laws in Continuum Physics

Hyperbolic Conservation Laws in Continuum Physics

Dafermos: Hyperbolic Conservation Laws in Continuum Physics 326. Waldschmidt: Diophantine Approximation on Linear Algebraic Groups 327. Martinet: Perfect Lattices in Euclidean Spaces 328. Van der Put/Singer: Galois Theory of Linear ...

Author: Constantine M. Dafermos

Publisher: Springer Science & Business Media

ISBN: 9783642040481

Category: Mathematics

Page: 710

View: 184

The 3rd edition is thoroughly revised, applications are substantially enriched, it includes a new account of the early history of the subject (from 1800 to 1957) and a new chapter recounting the recent solution of open problems of long standing in classical aerodynamics. The bibliography comprises now over fifteen hundred titles. From the reviews: "The author is known as one of the leading experts in the field. His masterly written book is, surely, the most complete exposition in the subject of conservations laws." --Zentralblatt MATH
Categories: Mathematics

Categories for the Working Philosopher

Categories for the Working Philosopher

Communications in Mathematical Physics 26 , 271-5 . ... Symmetry as a guide to superfluous structure , in K. Brading , E. Castellani ( eds ) , Symmetries in Physics : Philosophical Reflections ... Categories in Continuum Physics .

Author: Elaine Landry

Publisher: Oxford University Press

ISBN: 9780191065828

Category: Philosophy

Page: 432

View: 944

Often people have wondered why there is no introductory text on category theory aimed at philosophers working in related areas. The answer is simple: what makes categories interesting and significant is their specific use for specific purposes. These uses and purposes, however, vary over many areas, both "pure", e.g., mathematical, foundational and logical, and "applied", e.g., applied to physics, biology and the nature and structure of mathematical models. Borrowing from the title of Saunders Mac Lane's seminal work "Categories for the Working Mathematician", this book aims to bring the concepts of category theory to philosophers working in areas ranging from mathematics to proof theory to computer science to ontology, from to physics to biology to cognition, from mathematical modeling to the structure of scientific theories to the structure of the world. Moreover, it aims to do this in a way that is accessible to non-specialists. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, and in a way that builds on the concepts that are already familiar to philosophers working in these areas.
Categories: Philosophy

Elementary Categories Elementary Toposes

Elementary Categories  Elementary Toposes

Categories in continuum physics. Lecture Notes in Mathematics No. 1174. Springer- Verlag, Berlin. Lawvere, F. W. et al. (eds) (1975). Model theory and topoi. Lecture Notes in Mathematics No. 445. Springer-Verlag, Berlin.

Author: Colin McLarty

Publisher: Clarendon Press

ISBN: 9780191589492

Category:

Page: 278

View: 206

The book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites, and uses categorical methods throughout rather than beginning with set theoretic foundations. It works with key notions such as cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis. - ;Introduction; PART I: CATEGORIES: Rudimentary structures in a category; Products, equalizers, and their duals; Groups; Sub-objects, pullbacks, and limits; Relations; Cartesian closed categories; Product operators and others; PART II: THE CATEGORY OF CATEGORIES: Functors and categories; Natural transformations; Adjunctions; Slice categories; Mathematical foundations; PART III: TOPOSES: Basics; The internal language; A soundness proof for topos logic; From the internal language to the topos; The fundamental theorem; External semantics; Natural number objects; Categories in a topos; Topologies; PART IV: SOME TOPOSES: Sets; Synthetic differential geometry; The effective topos; Relations in regular categories; Further reading; Bibliography; Index. -
Categories:

Mechanics and Thermodynamics of Continua

Mechanics and Thermodynamics of Continua

17–29 of “Categories in Continuum Physics” (Workshop held at SUNY, Buffalo, 1982), Springer, Berlin etc., 1986. ZIEMER, W. P., Cauchy flux and sets of finite perimeter, Arch. Rational Mech. Anal. 84 (1983), 189–201.

Author: Hershel Markovitz

Publisher: Springer Science & Business Media

ISBN: 9783642759758

Category: Technology & Engineering

Page: 578

View: 666

Reprinted from Archive for Rational Mechanics and Analysis edited by C. Truesdell
Categories: Technology & Engineering

New Structures for Physics

New Structures for Physics

Von Neumann algebras as the base category. Int. J. Theor. Phys. ... Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. ... Introduction to Categories in Continuum Physics.

Author: Bob Coecke

Publisher: Springer

ISBN: 9783642128219

Category: Science

Page: 1031

View: 561

This volume provides a series of tutorials on mathematical structures which recently have gained prominence in physics, ranging from quantum foundations, via quantum information, to quantum gravity. These include the theory of monoidal categories and corresponding graphical calculi, Girard’s linear logic, Scott domains, lambda calculus and corresponding logics for typing, topos theory, and more general process structures. Most of these structures are very prominent in computer science; the chapters here are tailored towards an audience of physicists.
Categories: Science

Integrable Systems Topology and Physics

Integrable Systems  Topology  and Physics

References |B1] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, 107. ... [C] K.T. Chen, On differentiable spaces, Categories in continuum physics (Buffalo, N.Y., 1982), 38–42, ...

Author: Joel B Wolfe

Publisher: American Mathematical Soc.

ISBN: 9780821829394

Category: Mathematics

Page: 324

View: 700

Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of Tokyo in July 2000. It brought together scientists in all of the areas influenced by integrable systems. This book is the second of three collections of expository and research articles. This volume focuses on topology and physics. The role of zero curvature equations outside of the traditional context of differential geometry has been recognized relatively recently, but it has been an extraordinarily productive one, and most of the articles in this volume make some reference to it.Symplectic geometry, Floer homology, twistor theory, quantum cohomology, and the structure of special equations of mathematical physics, such as the Toda field equations - all of these areas have gained from the integrable systems point of view and contributed to it. Many of the articles in this volume are written by prominent researchers and will serve as introductions to the topics. It is intended for graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics. The first volume from this conference, also available from the 'AMS', is ""Differential Geometry and Integrable Systems, Volume 308"" in the ""Contemporary Mathematics"" series. The forthcoming third volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the 'AMS' in the ""Advanced Studies in Pure Mathematics"" series.
Categories: Mathematics

Variational Theories for Liquid Crystals

Variational Theories for Liquid Crystals

Noll, W. (1973) Lectures on the foundations of continuum mechanics and thermodynamics, Arch. Rational Mech. Anal, 52, 62-92. Noll, W. (1986) Continuum mechanics and geometric integration theory, in: Categories in continuum physics, ...

Author: E.G. Virga

Publisher: Routledge

ISBN: 9781351405645

Category: Mathematics

Page: 376

View: 294

Essentially there are two variational theories of liquid crystals explained in this book. The theory put forward by Zocher, Oseen and Frank is classical, while that proposed by Ericksen is newer in its mathematical formulation although it has been postulated in the physical literature for the past two decades. The newer theory provides a better explanation of defects in liquid crystals, especially of those concentrated on lines and surfaces, which escape the scope of the classical theory. The book opens the way to the wealth of applications that will follow.
Categories: Mathematics

The Mathematical World of Walter Noll

The Mathematical World of Walter Noll

Lectures on the Foundations of Continuum Mechanics and Thermodynamics, Archive for Rational Mechanics and Analysis 52, 62–92. ... Continuum Mechanics and Geometric Integration Theory, Categories in Continuum Physics (1982), Berlin, ...

Author: Yurie A. Ignatieff

Publisher: Springer Science & Business Media

ISBN: 9783642798337

Category: Science

Page: 254

View: 140

This book is a comprehensive study of the life and mathematics of Walter Noll, who helped to create the mathematical tools of modern rational mechanics and thermodynamics. Noll is one of the brilliant mathematicians of the second part of the 20th century. His contribution is large in both the applied and pure mathematics. The book stresses particularly Noll's method of axiomatization of physical theories, his axiomatics of continuum mechanics, thermodynamics of materials, special relativity theory, his discovery of the neo-classical space-time of mechanics, his theories of inhomogeneities in simple bodies, fit regions, contact interactions, annihilators of linear differential operators, and finite-dimensional spaces. It is a must for every mathematician, physicist, engineer or graduate student as a reference and key to Noll's mathematical heritage.
Categories: Science

Category Theory

Category Theory

The or em : If a small category has the property that every map factors as a split mono followed by a split epi, then, ... Reference: F. W. Lawvere, Introduction to Categories in Continuum Physics, Springer Lecture Notes in Mathematics ...

Author: Aurelio Carboni

Publisher: Springer

ISBN: 9783540464358

Category: Mathematics

Page: 496

View: 205

With one exception, these papers are original and fully refereed research articles on various applications of Category Theory to Algebraic Topology, Logic and Computer Science. The exception is an outstanding and lengthy survey paper by Joyal/Street (80 pp) on a growing subject: it gives an account of classical Tannaka duality in such a way as to be accessible to the general mathematical reader, and to provide a key for entry to more recent developments and quantum groups. No expertise in either representation theory or category theory is assumed. Topics such as the Fourier cotransform, Tannaka duality for homogeneous spaces, braided tensor categories, Yang-Baxter operators, Knot invariants and quantum groups are introduced and studies. From the Contents: P.J. Freyd: Algebraically complete categories.- J.M.E. Hyland: First steps in synthetic domain theory.- G. Janelidze, W. Tholen: How algebraic is the change-of-base functor?.- A. Joyal, R. Street: An introduction to Tannaka duality and quantum groups.- A. Joyal, M. Tierney: Strong stacks andclassifying spaces.- A. Kock: Algebras for the partial map classifier monad.- F.W. Lawvere: Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes.- S.H. Schanuel: Negative sets have Euler characteristic and dimension.-
Categories: Mathematics

The Philosophy of Mathematical Practice

The Philosophy of Mathematical Practice

(2002), 'Categorical Algebra for Continuum Micro Physics', Journal of Pure and Applied Algebra, 175, 267–287. Lawvere, F. William and Schanuel, S. (eds.) (1986), Categories in Continuum Physics. Proceedings of a Workshop held at SUNY, ...

Author: Paolo Mancosu

Publisher: OUP Oxford

ISBN: 9780191559099

Category: Philosophy

Page: 460

View: 715

Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment - such as visualization, explanation, and understanding - can nonetheless be subjected to philosophical analysis. The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representation systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.
Categories: Philosophy

Essays on the Foundations of Mathematics and Logic

Essays on the Foundations of Mathematics and Logic

... F.W. [1986] “Inroduction” to F. W. Lawvere and S. H. Schanuel (Eds) Categories in Continuum Physics, Springer LNM # 1174. Lawvere, F.W. [1994] “Tools for the Advancement of Objective Logic: Closed Categories and Toposes.

Author: Giandomenico Sica

Publisher: Polimetrica s.a.s.

ISBN: 9788876990144

Category: Mathematics

Page: 351

View: 117

Categories: Mathematics