Towards Higher Categories

Author: John C. Baez,J. Peter May

Publisher: Springer Science & Business Media

ISBN: 1441915249

Category: Mathematics

Page: 283

View: 6555

This IMA Volume in Mathematics and its Applications TOWARDS HIGHER CATEGORIES contains expository and research papers based on a highly successful IMA Summer Program on n-Categories: Foundations and Applications. We are grateful to all the participants for making this occasion a very productive and stimulating one. We would like to thank John C. Baez (Department of Mathematics, University of California Riverside) and J. Peter May (Department of Ma- ematics, University of Chicago) for their superb role as summer program organizers and editors of this volume. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Fadil Santosa, Director of the IMA Markus Keel, Deputy Director of the IMA v PREFACE DEDICATED TO MAX KELLY, JUNE 5 1930 TO JANUARY 26 2007. This is not a proceedings of the 2004 conference “n-Categories: Fo- dations and Applications” that we organized and ran at the IMA during the two weeks June 7–18, 2004! We thank all the participants for helping make that a vibrant and inspiring occasion. We also thank the IMA sta? for a magni?cent job. There has been a great deal of work in higher c- egory theory since then, but we still feel that it is not yet time to o?er a volume devoted to the main topic of the conference.

Higher Structures in Geometry and Physics

In Honor of Murray Gerstenhaber and Jim Stasheff

Author: Alberto S. Cattaneo,Anthony Giaquinto,Ping Xu

Publisher: Springer Science & Business Media

ISBN: 9780817647353

Category: Mathematics

Page: 362

View: 8643

This book is centered around higher algebraic structures stemming from the work of Murray Gerstenhaber and Jim Stasheff that are now ubiquitous in various areas of mathematics— such as algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics— and in theoretical physics such as quantum field theory and string theory. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more. Each contribution in this volume expands on the ideas of Gerstenhaber and Stasheff. The volume is intended for post-graduate students, mathematical and theoretical physicists, and mathematicians interested in higher structures.

Homotopy Type Theory

Univalent Foundations of Mathematics

Author: Univalent Foundations Program

Publisher: Univalent Foundations


Category: Homotopy theory

Page: 589

View: 8158

Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak $\infty$-groupoids. Homotopy type theory brings new ideas into the very foundation of mathematics. On the one hand, there is Voevodsky's subtle and beautiful univalence axiom. The univalence axiom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the "official" doctrines of conventional foundations. On the other hand, we have higher inductive types, which provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory: spheres, cylinders, truncations, localizations, etc. Both ideas are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics - and convenient machine implementations, which can serve as a practical aid tothe working mathematician. This is the Univalent Foundations program. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning - but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the "implicit foundation" for the unformalized mathematics done by most mathematicians.

Category Theory in Context

Author: Emily Riehl

Publisher: Courier Dover Publications

ISBN: 0486820807

Category: Mathematics

Page: 272

View: 3121

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

Analytic and Algebraic Geometry

Author: Anilatmaja Aryasomayajula,Indranil Biswas,Archana S. Morye,A. J. Parameswaran

Publisher: Springer

ISBN: 981105648X

Category: Mathematics

Page: 292

View: 3960

This volume is an outcome of the International conference held in Tata Institute of Fundamental Research and the University of Hyderabad. There are fifteen articles in this volume. The main purpose of the articles is to introduce recent and advanced techniques in the area of analytic and algebraic geometry. This volume attempts to give recent developments in the area to target mainly young researchers who are new to this area. Also, some research articles have been added to give examples of how to use these techniques to prove new results.

Amorphous Polymers and Non-Newtonian Fluids

Author: Constantine Dafermos,J.L. Ericksen,David Kinderlehrer

Publisher: Springer Science & Business Media

ISBN: 146121064X

Category: Science

Page: 202

View: 5586

This IMA Volume in Mathematics and its Applications AMORPHOUS POLYMERS AND NON-NEWTONIAN FLUIDS is in part the proceedings of a workshop which was an integral part of the 1984-85 IMA program on CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS We are grateful to the Scientific Committee: Haim Brezis Constantine Dafermos Jerry Ericksen David Kinderlehrer for planning and implementing an exciting and stimulating year-long program. We espe cially thank the Program Organizers, Jerry Ericksen, David Kinderlehrer, Stephen Prager and Matthew Tirrell for organizing a workshop which brought together scientists and mathematicians in a variety of areas for a fruitful exchange of ideas. George R. Sell Hans Weinberger Preface Experiences with amorphous polymers have supplied much of the motivation for developing novel kinds of molecular theory, to try to deal with the more significant features of systems involving very large molecules with many degrees offreedom. Similarly, the observations of many unusual macroscopic phenomena has stimulated efforts to develop linear and nonlinear theories of viscoelasticity to describe them. In either event, we are confronted not with a well-established, specific set of equations, but with a variety of equations, conforming to a loose pattern and suggested by general kinds of reasoning. One challenge is to devise techniques for finding equations capable of delivering definite and reliable predictions. Related to this is the issue of discovering ways to better grasp the nature of solutions ofthose equations showing some promise.

Mathematical Morphology and Its Applications to Image and Signal Processing

Author: Petros Maragos,Ronald W. Schafer,Muhammad Akmal Butt

Publisher: Springer Science & Business Media

ISBN: 9780792397335

Category: Computers

Page: 476

View: 9903

Mathematical morphology (MM) is a powerful methodology for the quantitative analysis of geometrical structures. It consists of a broad and coherent collection of theoretical concepts, nonlinear signal operators, and algorithms aiming at extracting, from images or other geometrical objects, information related to their shape and size. Its mathematical origins stem from set theory, lattice algebra, and integral and stochastic geometry. MM was initiated in the late 1960s by G. Matheron and J. Serra at the Fontainebleau School of Mines in France. Originally it was applied to analyzing images from geological or biological specimens. However, its rich theoretical framework, algorithmic efficiency, easy implementability on special hardware, and suitability for many shape- oriented problems have propelled its widespread diffusion and adoption by many academic and industry groups in many countries as one among the dominant image analysis methodologies. The purpose of Mathematical Morphology and its Applications to Image and Signal Processing is to provide the image analysis community with a sampling from the current developments in the theoretical (deterministic and stochastic) and computational aspects of MM and its applications to image and signal processing. The book consists of the papers presented at the ISMM'96 grouped into the following themes: Theory Connectivity Filtering Nonlinear System Related to Morphology Algorithms/Architectures Granulometries, Texture Segmentation Image Sequence Analysis Learning Document Analysis Applications

Differential Geometry and Its Applications

Author: John Oprea

Publisher: MAA

ISBN: 9780883857489

Category: Mathematics

Page: 469

View: 8239

Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only “see” geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.

Was ist Mathematik?

Author: Richard Courant,Herbert Robbins

Publisher: Springer-Verlag

ISBN: 3662000539

Category: Mathematics

Page: N.A

View: 9691

47 brauchen nur den Nenner n so groß zu wählen, daß das Intervall [0, IJn] kleiner wird als das fragliche Intervall [A, B], dann muß mindestens einer der Brüche m/n innerhalb des Intervalls liegen. Also kann es kein noch so kleines Intervall auf der Achse geben, das von rationalen Punkten frei wäre. Es folgt weiterhin, daß es in jedem Intervall unendlich viele rationale Punkte geben muß; denn wenn es nur eine endliche Anzahl gäbe, so könnte das Intervall zwischen zwei beliebigen benachbarten Punkten keine rationalen Punkte enthalten, was, wie wir eben sahen, unmöglich ist. § 2. Inkommensurable Strecken, irrationale Zahlen und der Grenzwertbegriff 1. Einleitung Vergleicht man zwei Strecken a und b hinsichtlich ihrer Größe, so kann es vor kommen, daß a in b genau r-mal enthalten ist, wobei r eine ganze Zahl darstellt. In diesem Fall können wir das Maß der Strecke b durch das von a ausdrücken, indem wir sagen, daß die Länge von b das r-fache der Länge von a ist.

Clifford Algebras and their Applications in Mathematical Physics

Clifford analysis. Volume 2

Author: John Ryan

Publisher: Springer Science & Business Media

ISBN: 9780817641832

Category: Mathematics

Page: 320

View: 2636

The second part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras. from applications such as complex-distance potential theory, supersymmetry, and fluid dynamics to Fourier analysis, the study of boundary value problems, and applications, to mathematical physics and Schwarzian derivatives in Euclidean space. Among the mathematical topics examined are generalized Dirac operators, holonomy groups, monogenic and hypermonogenic functions and their derivatives, quaternionic Beltrami equations, Fourier theory under Mobius transformations, Cauchy-Reimann operators, and Cauchy type integrals.

Introduction to Higher-Order Categorical Logic

Author: J. Lambek,P. J. Scott

Publisher: Cambridge University Press

ISBN: 9780521356534

Category: Mathematics

Page: 304

View: 8547

Part I indicates that typed-calculi are a formulation of higher-order logic, and cartesian closed categories are essentially the same. Part II demonstrates that another formulation of higher-order logic is closely related to topos theory.

Linear Algebra for Signal Processing

Author: Adam Bojanczyk,George Cybenko

Publisher: Springer Science & Business Media

ISBN: 9780387944913

Category: Technology & Engineering

Page: 184

View: 5401

Signal processing applications have burgeoned in the past decade. During the same time, signal processing techniques have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This trend will continue as many new signal processing applications are opening up in consumer products and communications systems. In particular, signal processing has been making increasingly sophisticated use of linear algebra on both theoretical and algorithmic fronts. This volume gives particular emphasis to exposing broader contexts of the signal processing problems so that the impact of algorithms and hardware can be better understood; it brings together the writings of signal processing engineers, computer engineers, and applied linear algebraists in an exchange of problems, theories, and techniques. This volume will be of interest to both applied mathematicians and engineers.

Zeitschrift für angewandte Mathematik und Mechanik

Author: N.A

Publisher: N.A


Category: Engineering

Page: N.A

View: 4218

Vols. 41-44, 46-50 include Sonderheft: Vorträge der wissenschaftlichen Jahrestagung der Gesellschaft für Angewandte Mathematik und Mechanik (v. 44, Vorträge der Tagung für Angewandte Mathematik und Mechanik).

Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences

Author: I. Grattan-Guinness

Publisher: JHU Press

ISBN: 9780801873966

Category: Mathematics

Page: 1806

View: 3058

Mathematics is one of the most basic -- and most ancient -- types of knowledge. Yet the details of its historical development remain obscure to all but a few specialists. The two-volume Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences recovers this mathematical heritage, bringing together many of the world's leading historians of mathematics to examine the history and philosophy of the mathematical sciences in a cultural context, tracing their evolution from ancient times to the twentieth century. In 176 concise articles divided into twelve parts, contributors describe and analyze the variety of problems, theories, proofs, and techniques in all areas of pure and applied mathematics, including probability and statistics. This indispensable reference work demonstrates the continuing importance of mathematics and its use in physics, astronomy, engineering, computer science, philosophy, and the social sciences. Also addressed is the history of higher education in mathematics. Carefully illustrated, with annotated bibliographies of sources for each article, The Companion Encyclopedia is a valuable research tool for students and teachers in all branches of mathematics. Contents of Volume 1: •Ancient and Non-Western Traditions •The Western Middle Ages and the Renaissance •Calculus and Mathematical Analysis •Functions, Series, and Methods in Analysis •Logic, Set Theories, and the Foundations of Mathematics •Algebras and Number Theory Contents of Volume 2: •Geometries and Topology •Mechanics and Mechanical Engineering •Physics, Mathematical Physics, and Electrical Engineering •Probability, Statistics, and the Social Sciences •Higher Education and Institutions •Mathematics and Culture •Select Bibliography, Chronology, Biographical Notes, and Index

Amstat News

Author: N.A

Publisher: N.A


Category: Statistics

Page: N.A

View: 3959

Mathematical Reviews

Author: N.A

Publisher: N.A


Category: Mathematics

Page: N.A

View: 4167

Diagnosis and Management of Lameness in the Horse - E-Book

Author: Michael W. Ross,Sue J. Dyson

Publisher: Elsevier Health Sciences

ISBN: 1437711766

Category: Medical

Page: 1424

View: 7870

Covering many different diagnostic tools, this essential resource explores both traditional treatments and alternative therapies for conditions that can cause gait abnormalities in horses. Broader in scope than any other book of its kind, this edition describes equine sporting activities and specific lameness conditions in major sport horse types, and includes up-to-date information on all imaging modalities. This title includes additional digital media when purchased in print format. For this digital book edition, media content may not be included. Cutting-edge information on diagnostic application for computed tomography and magnetic resonance imaging includes the most comprehensive section available on MRI in the live horse. Coverage of traditional treatment modalities also includes many aspects of alternative therapy, with a practical and realistic perspective on prognosis. An examination of the various types of horses used in sports describes the lameness conditions to which each horse type is particularly prone, as well as differences in prognosis. Guidelines on how to proceed when a diagnosis cannot easily be reached help you manage conditions when faced with the limitations of current diagnostic capabilities. Clinical examination and diagnostic analgesia are given a special emphasis. Practical, hands-on information covers a wide range of horse types from around the world. A global perspective is provided by a team of international authors, editors, and contributors. A full-color insert shows thermography images. Updated chapters include the most current information on topics such as MRI, foot pain, stem cell therapy, and shock wave treatment. Two new chapters include The Biomechanics of the Equine Limb and its Effect on Lameness and Clinical Use of Stem Cells, Marrow Components, and Other Growth Factors. The chapter on the hock has been expanded substantially, and the section on lameness associated with the foot has been completely rewritten to include state-of-the-art information based on what has been learned from MRI. Many new figures appear throughout the book.

Leibel and Phillips Textbook of Radiation Oncology - E-Book

Expert Consult

Author: Richard Hoppe,Theodore L. Phillips,Mack Roach

Publisher: Elsevier Health Sciences

ISBN: 1437737757

Category: Medical

Page: 1664

View: 5691

Stay on top of the latest scientific and therapeutic advances with the new edition of Leibel and Phillips Textbook of Radiation Oncology. Dr. Theodore L. Phillips, in collaboration with two new authors, Drs. Richard Hoppe and Mack Roach, offers a multidisciplinary look at the presentation of uniform treatment philosophies for cancer patients emphasizing the "treat for cure" philosophy. You can also explore the implementation of new imaging techniques to locate and treat tumors, new molecularly targeted therapies, and new types of treatment delivery. Supplement your reading with online access to the complete contents of the book, a downloadable image library, and more at Gather step-by-step techniques for assessing and implementing radiotherapeutic options with this comprehensive, full-color, clinically oriented text. Review the basic principles behind the selection and application of radiation as a treatment modality, including radiobiology, radiation physics, immobilization and simulation, high dose rate, and more. Use new imaging techniques to anatomically locate tumors before and during treatment. Apply multidisciplinary treatments with advice from experts in medical, surgical, and radiation oncology. Explore new treatment options such as proton therapy, which can facilitate precise tumor-targeting and reduce damage to healthy tissue and organs. Stay on the edge of technology with new chapters on IGRT, DNA damage and repair, and molecularly targeted therapies.