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Tōhoku Rick Jardine
INFERENCE / Vol. 1, No. 3 Tōhoku Rick Jardine he publication of Alexander Grothendieck’s learning led to great advances: the axiomatic description paper, “Sur quelques points d’algèbre homo- of homology theory, the theory of adjoint functors, and, of logique” (Some Aspects of Homological Algebra), course, the concepts introduced in Tōhoku.5 Tin the 1957 number of the Tōhoku Mathematical Journal, This great paper has elicited much by way of commen- was a turning point in homological algebra, algebraic tary, but Grothendieck’s motivations in writing it remain topology and algebraic geometry.1 The paper introduced obscure. In a letter to Serre, he wrote that he was making a ideas that are now fundamental; its language has with- systematic review of his thoughts on homological algebra.6 stood the test of time. It is still widely read today for the He did not say why, but the context suggests that he was clarity of its ideas and proofs. Mathematicians refer to it thinking about sheaf cohomology. He may have been think- simply as the Tōhoku paper. ing as he did, because he could. This is how many research One word is almost always enough—Tōhoku. projects in mathematics begin. The radical change in Gro- Grothendieck’s doctoral thesis was, by way of contrast, thendieck’s interests was best explained by Colin McLarty, on functional analysis.2 The thesis contained important who suggested that in 1953 or so, Serre inveigled Gro- results on the tensor products of topological vector spaces, thendieck into working on the Weil conjectures.7 The Weil and introduced mathematicians to the theory of nuclear conjectures were certainly well known within the Paris spaces. -
Scarica Scarica
Archivi, Biblioteche, Musei L’archivio e la biblioteca di Francesco G. Tricomi ∗ ERIKA LUCIANO - LUISA ROSSO 1. Francesco Giacomo Tricomi1 Nato a Napoli il 5 maggio 1897 in un’agiata famiglia borghese, Tricomi frequenta l’Istituto tecnico locale, dove concepisce una passione per gli studi scientifici grazie all’influenza del suo insegnante di Matematica Alfredo Perna. Conseguito il diploma all’età di appena sedici anni, s’iscrive al corso di laurea in Chimica dell’Università di Bologna. L’anno successivo passa a Fisica e infine, nel 1915-16, torna a Napoli e si immatricola al terz’anno del corso di studi in Matematica. Arruolato nell’autunno del 1916, segue un corso per allievi ufficiali di complemento presso l’Accademia militare di Torino e il 1 aprile 1917, fresco di nomina, è inviato al fronte, dapprima sul Carso e poi nella zona del monte Grappa e del Piave. Nonostante la guerra, riesce comunque a portare avanti gli studi scientifici e consegue la laurea in Matematica a Napoli il 16 aprile 1918, durante una licenza. Terminata la Grande Guerra, Tricomi torna a Napoli all’inizio del 1920, decorato di due croci al merito, e riallaccia i rapporti con alcuni suoi ex docenti, fra cui R. Marcolongo e G. Torelli che lo indirizzano nelle prime ricerche e lo mettono in contatto con G. Fubini. Nel febbraio del 1921, su suggerimento di U. Amaldi, è assunto da F. Severi quale assistente alla cattedra di Geometria analitica presso l’Università di Padova. L’esperienza alla ‘Scuola’ di Severi è tuttavia di breve durata, poiché nei primi mesi del 1922 Tricomi è chiamato a Roma dove consegue la libera docenza in Analisi algebrica e infinitesimale e dove ha modo di inserirsi in un ambiente accademico di eccezionale levatura, che vanta in quegli anni la presenza di illustri matematici, fra cui V. -
Multiple Mixing from Weak Hyperbolicity by the Hopf Argument Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy
,pdfcreator=HAL,pdfproducer=PDFLaTeX,pdfsubject=Mathematics [math]/Dynamical Systems [math.DS] Multiple mixing from weak hyperbolicity by the Hopf argument Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy To cite this version: Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy. Multiple mixing from weak hyperbolicity by the Hopf argument. 2014. hal-01006451v2 HAL Id: hal-01006451 https://hal.archives-ouvertes.fr/hal-01006451v2 Submitted on 16 Jun 2014 (v2), last revised 15 Sep 2015 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MULTIPLE MIXING FROM WEAK HYPERBOLICITY BY THE HOPF ARGUMENT YVES COUDÈNE, BORIS HASSELBLATT AND SERGE TROUBETZKOY ABSTRACT. We show that using only weak hyperbolicity (no smoothness, com- pactness or exponential rates) the Hopf argument produces multiple mixing in an elementary way. While this recovers classical results with far simpler proofs, the point is the broader applicability implied by the weak hypothe- ses. Some of the results can also be viewed as establishing “mixing implies multiple mixing” outside the classical hyperbolic context. 1. INTRODUCTION The origins of hyperbolic dynamical systems are connected with the efforts by Boltzmann and Maxwell to lay a foundation under statistical mechanics. In today’s terms their ergodic hypothesis was that the mechanical system defined by molecules in a container is ergodic, and the difficulties of establishing this led to the search for any mechanical systems with this property. -
On the Uniqueness of Gibbs States in the Pirogov-Sinai Theory?
Commun. Math. Phys. 189, 311 – 321 (1997) Communications in Mathematical Physics c Springer-Verlag 1997 On the Uniqueness of Gibbs States in the Pirogov-Sinai Theory? J.L. Lebowitz1, A.E. Mazel2 1 Department of Mathematics and Physics, Rutgers University, New Brunswick, NJ 08903, USA 2 International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow 113556, Russia Received: 4 June 1996 / Accepted: 30 October 1996 Dedicated to the memory of Roland Dobrushin Abstract: We prove that, for low-temperature systems considered in the Pirogov-Sinai theory, uniqueness in the class of translation-periodic Gibbs states implies global unique- ness, i.e. the absence of any non-periodic Gibbs state. The approach to this infinite volume state is exponentially fast. 1. Introduction The problem of uniqueness of Gibbs states was one of R.L.Dobrushin’s favorite subjects in which he obtained many classical results. In particular when two or more translati- on-periodic states coexist, it is natural to ask whether there might also exist other, non translation-periodic, Gibbs states, which approach asymptotically, in different spatial directions, the translation periodic ones. The affirmative answer to this question was given by R.L.Dobrushin with his famous construction of such states for the Ising model, using boundary conditions, in three and higher dimensions [D]. Here we consider the opposite± situation: we will prove that in the regions of the low-temperature phase diagram where there is a unique translation-periodic Gibbs state one actually has global uniqueness of the limit Gibbs state. Moreover we show that, uniformly in boundary conditions, the finite volume probability of any local event tends to its infinite volume limit value exponentially fast in the diameter of the domain. -
Diagonalizable Flows on Locally Homogeneous Spaces and Number
Diagonalizable flows on locally homogeneous spaces and number theory Manfred Einsiedler and Elon Lindenstrauss∗ Abstract.We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous spaces, particularly their invariant measures, and present some number theoretic and spectral applications. Entropy plays a key role in the study of theses invariant measures and in the applications. Mathematics Subject Classification (2000). 37D40, 37A45, 11J13, 81Q50 Keywords. invariant measures, locally homogeneous spaces, Littlewood’s conjecture, quantum unique ergodicity, distribution of periodic orbits, ideal classes, entropy. 1. Introduction Flows on locally homogeneous spaces are a special kind of dynamical systems. The ergodic theory and dynamics of these flows are very rich and interesting, and their study has a long and distinguished history. What is more, this study has found numerous applications throughout mathematics. The spaces we consider are of the form Γ\G where G is a locally compact group and Γ a discrete subgroup of G. Typically one takes G to be either a Lie group, a linear algebraic group over a local field, or a product of such. Any subgroup H < G acts on Γ\G and this action is precisely the type of action we will consider here. One of the most important examples which features in numerous number theoretical applications is the space PGL(n, Z)\ PGL(n, R) which can be identified with the space of lattices in Rn up to homothety. Part of the beauty of the subject is that the study of very concrete actions can have meaningful implications. For example, in the late 1980s G. -
Network Map of Knowledge And
Humphry Davy George Grosz Patrick Galvin August Wilhelm von Hofmann Mervyn Gotsman Peter Blake Willa Cather Norman Vincent Peale Hans Holbein the Elder David Bomberg Hans Lewy Mark Ryden Juan Gris Ian Stevenson Charles Coleman (English painter) Mauritz de Haas David Drake Donald E. Westlake John Morton Blum Yehuda Amichai Stephen Smale Bernd and Hilla Becher Vitsentzos Kornaros Maxfield Parrish L. Sprague de Camp Derek Jarman Baron Carl von Rokitansky John LaFarge Richard Francis Burton Jamie Hewlett George Sterling Sergei Winogradsky Federico Halbherr Jean-Léon Gérôme William M. Bass Roy Lichtenstein Jacob Isaakszoon van Ruisdael Tony Cliff Julia Margaret Cameron Arnold Sommerfeld Adrian Willaert Olga Arsenievna Oleinik LeMoine Fitzgerald Christian Krohg Wilfred Thesiger Jean-Joseph Benjamin-Constant Eva Hesse `Abd Allah ibn `Abbas Him Mark Lai Clark Ashton Smith Clint Eastwood Therkel Mathiassen Bettie Page Frank DuMond Peter Whittle Salvador Espriu Gaetano Fichera William Cubley Jean Tinguely Amado Nervo Sarat Chandra Chattopadhyay Ferdinand Hodler Françoise Sagan Dave Meltzer Anton Julius Carlson Bela Cikoš Sesija John Cleese Kan Nyunt Charlotte Lamb Benjamin Silliman Howard Hendricks Jim Russell (cartoonist) Kate Chopin Gary Becker Harvey Kurtzman Michel Tapié John C. Maxwell Stan Pitt Henry Lawson Gustave Boulanger Wayne Shorter Irshad Kamil Joseph Greenberg Dungeons & Dragons Serbian epic poetry Adrian Ludwig Richter Eliseu Visconti Albert Maignan Syed Nazeer Husain Hakushu Kitahara Lim Cheng Hoe David Brin Bernard Ogilvie Dodge Star Wars Karel Capek Hudson River School Alfred Hitchcock Vladimir Colin Robert Kroetsch Shah Abdul Latif Bhittai Stephen Sondheim Robert Ludlum Frank Frazetta Walter Tevis Sax Rohmer Rafael Sabatini Ralph Nader Manon Gropius Aristide Maillol Ed Roth Jonathan Dordick Abdur Razzaq (Professor) John W. -
Professor AO Kuku
CURRICULUM VITAE Professor A.O. Kuku I. Personal Details Date of Birth: March 20, 1941 Marital Status: Married with four children Nationality: Nigerian Sex: Male U.S.A. Permanent Resident (Green card) since March 2002 II CURRENT POSITION: Professor of Mathematics, Grambling State University, Grambling, LA, USA. Since August 2008 III. Position held in the last five years (a) Member, Institute for Advanced Study Princeton, NJ, USA. Sept. 2003-Aug. 2004 (b) Visiting Research Professor, MSRI Berkeley, CA, USA. Aug-Dec 2004 (c) Visiting Professor, OSU (Ohio State Univ.) Columbus, OH, USA 2005 (d) Distinguished Visiting Professor, Miami 2005 – 2006 University, Oxford, OH, USA (e) Visiting Professor, Universitat Bielefeld, Germany ,USA. 2006 (f) Visiting Professor, IHES, Paris, France 2006 (g) Visiting Professor, Max Planck Inst. Fur Mathematik, Bonn, Germany 2007 (h) Visiting Professor, National Mathema- tical Centre, Abuja, Nigeria. 2007 (i) Visiting professor, The University of Iowa, Iowa-City, USA 2007-2008 (j) Visiting Professor, National Mathema- Tical Centre, Abuja, Nigeria. 2008 IV. Educational Institutions Attended (University Education) 1.Makerere University College, Kampala, Uganda (then under special relationship with the University of London) 1962-1965 2.University of Ibadan, Nigeria 1966-1971 1 3Columbia University, New York City, USA (To write my Ph.D thesis) (Thesis written under Professor Hyman Bass) 1970-1971 V. Academic Qualification (with dates and granting bodies) 1.B. Sc (Special- Honours) Mathematics, University of London 1965 2.M. Sc. (Mathematics), University of Ibadan, Nigeria. 1968 3.Ph. D. (Mathematics), University of Ibadan, Nigeria 1971 (Thesis written under Professor Hyman Bass of Columbia Univerisity, New York). -
Curriculum Vitae
CURRICULUM VITAE Professor Aderemi Oluyomi Kuku Ph.D, FAMS (USA), FTWAS, FAAS, FAS (Nig), FNMS, FMAN, FASI, OON, NNOM I. Personal Details Date of Birth: March 20, 1941 Marital Status: Married with four children Nationality: USA/Nigeria. Sex: Male II CONTACT ADDRESSES: Email [email protected] Website: www.aderemikuku.com MAILING ADDRESS: USA: 307 Penny Lane, Apt 5, Ruston, LA 71270, USA. NIGERIA(a) 2 Amure Street, Kongi-NewBodija, Ibadan, Oyo State, Nigeria. (b) Univerdityof Ibadan Post office Box 22574 Ibadan, Oyo State , Nigeria. PHONE NUMBERS: USA: +1-318-255-6433 Cell: +1-224-595-4854 NIGERIA: +234-70-56871969; +234-80-62329855 III. Positions held in the last 14 years (a) Member, Institute for Advanced Study Princeton, NJ, USA. Sept. 2003-Aug. 2004 (b) Visiting Research Professor, MSRI-- (Math. Sci. Research Inst) Berkeley, CA, USA. Aug-Dec, 2004 (c) Visiting Professor, OSU (Ohio State Univ.) Columbus, OH, USA 2005 (d) Distinguished Visiting Professor, Miami 2005 – 2006 University, Oxford, OH, USA (e) Visiting Professor, Universitat Bielefeld, Germany 2006 (f) Visiting Professor, IHES, Paris, France 2006 (g) Visiting Professor, Max Planck Inst. Fur Mathematik, Bonn, Germany 2007 1 (h) Distinguished Visiting Professor, National Mathematical Centre, Abuja, Nigeria. 2007 (i) Visiting Professor, The University of Iowa, Iowa-City, USA 2007-2008 (j) Professor of Mathematics, Grambling State University, Grambling, LA 71245, USA 2008-2009 (k) William W. S. Claytor Endowed Professor of Mathematics Grambling State University, Grambling, LA 71245, USA. 2009-2014 (l) Distinguished Visiting Professor, National Mathematical Centre, Abuja, Nigeria. Summer 2009, 2010,2011, 2012, 2013, 2014 (m) Distinguished Visiting Professor of Mathematics, IMSP—Institut demathematiques etde Sciences Physiques, Porto Novo, BeninRepublic,Nov/Dec,2015. -
Geometry of Noncommutative Three-Dimensional Flat Manifolds
Philosophiae Doctor Dissertation GEOMETRY OF NONCOMMUTATIVE THREE-DIMENSIONAL FLAT MANIFOLDS Piotr Olczykowski Krakow´ 2012 Dla moich Rodzic´ow Contents Introduction ix 1 Preliminaries 1 1.1 C∗−algebras . .1 1.2 Gelfand-Naimark-Seagal Theorem . .3 1.2.1 Commutative Case . .4 1.2.2 Noncommutative Case . .4 1.3 C∗-dynamical systems . .7 1.3.1 Fixed Point Algebras of C(M)..............8 1.4 K−theory in a Nutshell . .8 1.5 Fredholm Modules in a Nutshell . 10 1.5.1 Pairing between K−theory and K−homology . 12 1.5.2 Unbounded Fredholm Modules . 12 2 Spectral Triples 15 2.1 Spin Structures . 15 2.1.1 Clifford Algebras . 15 2.1.2 SO(n) and Spin(n) Groups . 16 2.1.3 Representation of the Clifford Algebra . 17 2.1.4 Spin Structures and Bundles . 18 2.2 Classical Dirac Operator . 20 2.3 Real Spectral Triple { Definition . 23 2.3.1 Axioms . 24 2.3.2 Commutative Real Spectral Triples . 27 3 Noncommutative Spin Structures 29 3.1 Noncommutative Spin Structure . 29 3.2 Equivariant Spectral Triples - Definition . 31 3.3 Noncommutati Tori . 32 3.3.1 Algebra . 33 3.3.2 Representation . 34 v vi CONTENTS T3 3.3.3 Equivariant real spectral triples over A( Θ)...... 35 3.4 Quotient Spaces . 36 3.4.1 Reducible spectral triples . 36 Z 3.4.2 Spectral Triples over A(T1) N .............. 38 3.4.3 Summary . 40 4 Noncommutative Bieberbach Manifolds 41 4.1 Classical Bieberbach Manifolds . 42 4.2 Three-dimensional Bieberbach Manifolds . 44 4.2.1 Spin structures over Bieberbach manifolds . -
Dirac Spectra, Summation Formulae, and the Spectral Action
Dirac Spectra, Summation Formulae, and the Spectral Action Thesis by Kevin Teh In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2013 (Submitted May 2013) i Acknowledgements I wish to thank my parents, who have given me their unwavering support for longer than I can remember. I would also like to thank my advisor, Matilde Marcolli, for her encourage- ment and many helpful suggestions. ii Abstract Noncommutative geometry is a source of particle physics models with matter Lagrangians coupled to gravity. One may associate to any noncommutative space (A; H; D) its spectral action, which is defined in terms of the Dirac spectrum of its Dirac operator D. When viewing a spin manifold as a noncommutative space, D is the usual Dirac operator. In this paper, we give nonperturbative computations of the spectral action for quotients of SU(2), Bieberbach manifolds, and SU(3) equipped with a variety of geometries. Along the way we will compute several Dirac spectra and refer to applications of this computation. iii Contents Acknowledgements i Abstract ii 1 Introduction 1 2 Quaternionic Space, Poincar´eHomology Sphere, and Flat Tori 5 2.1 Introduction . 5 2.2 The quaternionic cosmology and the spectral action . 6 2.2.1 The Dirac spectra for SU(2)=Q8.................... 6 2.2.2 Trivial spin structure: nonperturbative spectral action . 7 2.2.3 Nontrivial spin structures: nonperturbative spectral action . 9 2.3 Poincar´ehomology sphere . 10 2.3.1 Generating functions for spectral multiplicities . 10 2.3.2 The Dirac spectrum of the Poincar´esphere . -
Nominations for President
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society September 2013 Volume 60, Number 8 The Calculus Concept Inventory— Measurement of the Effect of Teaching Methodology in Mathematics page 1018 DML-CZ: The Experience of a Medium- Sized Digital Mathematics Library page 1028 Fingerprint Databases for Theorems page 1034 A History of the Arf-Kervaire Invariant Problem page 1040 About the cover: 63 years since ENIAC broke the ice (see page 1113) Solve the differential equation. Solve the differential equation. t ln t dr + r = 7tet dt t ln t dr + r = 7tet dt 7et + C r = 7et + C ln t ✓r = ln t ✓ WHO HAS THE #1 HOMEWORK SYSTEM FOR CALCULUS? THE ANSWER IS IN THE QUESTIONS. When it comes to online calculus, you need a solution that can grade the toughest open-ended questions. And for that there is one answer: WebAssign. WebAssign’s patent pending grading engine can recognize multiple correct answers to the same complex question. Competitive systems, on the other hand, are forced to use multiple choice answers because, well they have no choice. And speaking of choice, only WebAssign supports every major textbook from every major publisher. With new interactive tutorials and videos offered to every student, it’s not hard to see why WebAssign is the perfect answer to your online homework needs. It’s all part of the WebAssign commitment to excellence in education. Learn all about it now at webassign.net/math. 800.955.8275 webassign.net/math WA Calculus Question ad Notices.indd 1 11/29/12 1:06 PM Notices 1051 of the American Mathematical Society September 2013 Communications 1048 WHAT IS…the p-adic Mandelbrot Set? Joseph H. -
The Problem of Distributed Consensus: a Survey Stephen Wolfram*
The Problem of Distributed Consensus: A Survey Stephen Wolfram* A survey is given of approaches to the problem of distributed consensus, focusing particularly on methods based on cellular automata and related systems. A variety of new results are given, as well as a history of the field and an extensive bibliography. Distributed consensus is of current relevance in a new generation of blockchain-related systems. In preparation for a conference entitled “Distributed Consensus with Cellular Automata and Related Systems” that we’re organizing with NKN (= “New Kind of Network”) I decided to explore the problem of distributed consensus using methods from A New Kind of Science (yes, NKN “rhymes” with NKS) as well as from the Wolfram Physics Project. A Simple Example Consider a collection of “nodes”, each one of two possible colors. We want to determine the majority or “consensus” color of the nodes, i.e. which color is the more common among the nodes. A version of this document with immediately executable code is available at writings.stephenwolfram.com/2021/05/the-problem-of-distributed-consensus Originally published May 17, 2021 *Email: [email protected] 2 | Stephen Wolfram One obvious method to find this “majority” color is just sequentially to visit each node, and tally up all the colors. But it’s potentially much more efficient if we can use a distributed algorithm, where we’re running computations in parallel across the various nodes. One possible algorithm works as follows. First connect each node to some number of neighbors. For now, we’ll just pick the neighbors according to the spatial layout of the nodes: The algorithm works in a sequence of steps, at each step updating the color of each node to be whatever the “majority color” of its neighbors is.