Theory of the Integral
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(Terse) Introduction to Lebesgue Integration
http://dx.doi.org/10.1090/stml/048 A (Terse) Introduction to Lebesgue Integration STUDENT MATHEMATICAL LIBRARY Volume 48 A (Terse) Introduction to Lebesgue Integration John Franks Providence, Rhode Island Editorial Board Gerald B. Folland Brad G. Osgood (Chair) Robin Forman Michael Starbird 2000 Mathematics Subject Classification. Primary 28A20, 28A25, 42B05. The images on the cover are representations of the ergodic transformations in Chapter 7. The figure with the implied cardioid traces iterates of the squaring map on the unit circle. The “spirograph” figures trace iterates of an irrational rotation. The arc of + signs consists of iterates of an irrational rotation. I am grateful to Edward Dunne for providing the figures. For additional information and updates on this book, visit www.ams.org/bookpages/stml-48 Library of Congress Cataloging-in-Publication Data Franks, John M., 1943– A (terse) introduction to Lebesgue integration / John Franks. p. cm. – (Student mathematical library ; v. 48) Includes bibliographical references and index. ISBN 978-0-8218-4862-3 (alk. paper) 1. Lebesgue integral. I. Title. II. Title: Introduction to Lebesgue integra- tion. QA312.F698 2009 515.43–dc22 2009005870 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Arxiv:1505.04809V5 [Math-Ph] 29 Sep 2016
THE PERTURBATIVE APPROACH TO PATH INTEGRALS: A SUCCINCT MATHEMATICAL TREATMENT TIMOTHY NGUYEN Abstract. We study finite-dimensional integrals in a way that elucidates the mathemat- ical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick's theorem allows one to evaluate integrals perturbatively, i.e., as a series expansion in a formal parameter irrespective of convergence properties. We establish invariance properties of such a Wick expansion under coordinate changes and the action of a Lie group of symmetries, and we use this to study essential features of path integral manipulations, including coordinate changes, Ward iden- tities, Schwinger-Dyson equations, Faddeev-Popov gauge-fixing, and eliminating fields by their equation of motion. We also discuss the asymptotic nature of the Wick expansion and the implications this has for defining path integrals perturbatively and nonperturbatively. Contents Introduction 1 1. The Wick Expansion 4 1.1. The Morse-Bott case 8 2. The Wick Expansion and Gauge-Fixing 12 3. The Wick Expansion and Integral Asymptotics 19 4. Remarks on Quantum Field Theory 22 4.1. Change of variables in the path integral 23 4.2. Ward identities and Schwinger-Dyson equations 26 4.3. Gauge-fixing of Feynman amplitudes and path integrals 28 4.4. Eliminating fields by their equation of motion 30 4.5. Perturbative versus constructive QFT 32 5. Conclusion 33 References 33 arXiv:1505.04809v5 [math-ph] 29 Sep 2016 Introduction Quantum field theory often makes use of manipulations of path integrals that are without a proper mathematical definition and hence have only a formal meaning. -
Densities for the Navier-Stokes Equations with Noise
Densities for the Navier–Stokes equations with noise Marco Romito DIPARTIMENTO DI MATEMATICA UNIVERSITÀ DI PISA LARGO BRUNO PONTECORVO 5 I–56127 PISA,ITALIA E-mail address: [email protected] URL: http://www.dm.unipi.it/pages/romito Leonardo Da Vinci, Royal Collection, Windsor, England, UK 2010 Mathematics Subject Classification. Primary 60G30, 35Q30, 35R60; Secondary 76D05, 76D06, 60H15, 60H30, 76M35 Key words and phrases. Density of laws, Navier–Stokes equations, stochastic partial differential equations, Malliavin calculus, Girsanov transformation, Besov spaces, Fokker–Planck equation ABSTRACT. One of the most important open problem for the Navier–Stokes equations concerns regularity of solutions. There is an extensive literature de- voted to the problem, both in the non–random and randomly forced case. Ex- istence of densities for the distribution of the solution is a probabilistic form of regularity and the course addresses some attempts at understanding, charac- terizing and proving existence of such mathematical objects. While the topic is somewhat specific, it offers the opportunity to introduce the mathematical theory of the Navier–Stokes equations, together with some of the most recent results, as well as a good selection of tools in stochastic analysis and in the theory of (stochastic) partial differential equations. In the first part of the course we give a quick introduction to the equations and discuss a few results in the literature related to densities and absolute con- tinuity. We then present four different methods to prove existence of a density with respect to the Lebesgue measure for the law of finite dimensional function- als of the solutions of the Navier-Stokes equations forced by Gaussian noise. -
Real Analysis
Real Analysis William G. Faris May 1, 2006 ii Contents Preface xi I Sets and Functions 1 1 Logical language and mathematical proof 3 1.1 Terms, predicates and atomic formulas . 3 1.2 Formulas . 4 1.3 Restricted variables . 5 1.4 Free and bound variables . 5 1.5 Quantifier logic . 6 1.6 Natural deduction . 8 1.7 Rules for logical operations . 9 1.8 Additional rules for or and exists . 11 1.9 Strategies for natural deduction . 12 1.10 Lemmas and theorems . 14 1.11 Relaxed natural deduction . 15 1.12 Supplement: Templates . 16 1.13 Supplement: Existential hypotheses . 18 2 Sets 23 2.1 Zermelo axioms . 23 2.2 Comments on the axioms . 24 2.3 Ordered pairs and Cartesian product . 27 2.4 Relations and functions . 28 2.5 Number systems . 29 2.6 The extended real number system . 30 2.7 Supplement: Construction of number systems . 31 3 Relations, functions, dynamical Systems 35 3.1 Identity, composition, inverse, intersection . 35 3.2 Picturing relations . 36 3.3 Equivalence relations . 36 3.4 Generating relations . 36 iii iv CONTENTS 3.5 Ordered sets . 37 3.6 Functions . 37 3.7 Relations inverse to functions . 38 3.8 Dynamical systems . 38 3.9 Picturing dynamical systems . 38 3.10 Structure of dynamical systems . 39 3.11 Isomorphism of dynamical systems . 41 4 Functions, cardinal number 43 4.1 Functions . 43 4.2 Picturing functions . 44 4.3 Indexed sums and products . 44 4.4 Cartesian powers . 45 4.5 Cardinality . 45 II Order and Metric 49 5 Ordered sets and order completeness 51 5.1 Ordered sets . -
On the Modular Operator of Mutli-Component Regions in Chiral
On the modular operator of mutli-component regions in chiral CFT Stefan Hollands1∗ 1Institut für Theoretische Physik,Universität Leipzig, Brüderstrasse 16, D-04103 Leipzig, Germany December 24, 2019 Abstract We introduce a new approach to find the Tomita-Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo- Martin-Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann-Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered. 1 Introduction The reduced density matrix of a subsystem induces an intrinsic internal dynamics called the “modular flow”. The flow is non-trivial only for non-commuting observable algebras – i.e., in quantum theory – and depends on both the subsystem and the given state of the total system. It has been subject to much attention in theoretical physics in recent times because it is closely related to information theoretic concepts. As examples for some topics such as Bekenstein bounds, Quantum Focussing Conjecture, c-theorems, arXiv:1904.08201v2 [hep-th] 21 Dec 2019 holography we mention [1, 2, 3, 4, 5]. In mathematics, the modular flow has played an important role in the study of operator algebras through the work of Connes, Takesaki and others, see [6] for an encyclopedic account. It has been known almost from the beginning that the modular flow has a geometric nature in local quantum field theory when the subsystem is defined by a spacetime region of a simple shape such as an interval in chiral conformal field theory (CFT) [7, 8, 9]: it is the 1-parameter group of Möbius transformations leaving the interval fixed. -
Author Index ______
______AUTHOR INDEX ______ a space; Homology group; Infinite Arellano, E. Ramirez de see: Ramirez de form in logarithms; Siegel theorem; dimensional space; Local decompo Arellano, E. Transcendental number sition; Metrizable space; Pontryagin __A __ Argand, J .R. see: Arithmetic; Complex Ball, R. see: Helical calculus duality; Projection spectrum; Suslin number; Imaginary number Balusabramanian, R. see: Waring prob theorem; Topological space; Width Aristotle see: Modal logic; Space-time lem Abel, N.H. see: Abel differential Alexander, J.W. see: Alexander dual Arkhangel'skil, A.V. see: Feathering Banach, S. see: Area; Banach indica equation; Abel-Goncharov prob ity; Alexander invariants; Duality; Arno I'd, V.I. see: Composite function; trix; Banach-Mazur functional; Ba lem; Abel problem; Abel theorem; Homology group; Jordan theorem; Quasi-periodic motion nach space; Banach-Steinhaus the Abel transformation; Abelian group; Knot theory; Manifold; Pontryagin Arsenin, V. Ya. see: Descriptive set the orem; Completely-continuous oper Abelian integral; Abelian variety; duality; Reaction-diffusion equation; ory ator; Contracting-mapping princi Algebra; Algebraic equation; Alge Topology of imbeddings Artin, E. see: Abstract algebraic ge ple; Franklin system; Hahn-Banach braic function; Algebraic geometry; Alexandrov, A.B. see: Boundary prop ometry; Algebra; Algebraic number theorem; Lacunary system; Linear Binomial series; Convergence, types erties of analytic functions theory; Alternative rings and alge operator; Luzin N -property; Non of; Elliptic function; Finite group; Alling, N.L. see: Surreal numbers bras; Commutative algebra; Con differentiable function; Nuclear op Galois theory; Group; Interpola Almgren, F.J. see: Plateau problem, gruence modulo a prime number; erator; Open-mapping theorem; Or tion; Inversion of an elliptic integral; multi-dimensional Divisorial ideal; Foundations of ge thogonal series; Tonelli plane vari Jacobi elliptic functions; Liouville Alvarez-Gaume, L. -
Real Analysis: Part I
Real Analysis: Part I William G. Faris February 2, 2004 ii Contents 1 Mathematical proof 1 1.1 Logical language . 1 1.2 Free and bound variables . 3 1.3 Proofs from analysis . 4 1.4 Natural deduction . 6 1.5 Natural deduction strategies . 13 1.6 Equality . 16 1.7 Lemmas and theorems . 18 1.8 More proofs from analysis . 19 2 Sets 21 2.1 Zermelo axioms . 21 2.2 Comments on the axioms . 22 2.3 Ordered pairs and Cartesian product . 25 2.4 Relations and functions . 26 2.5 Number systems . 28 3 Relations, Functions, Dynamical Systems 31 3.1 Identity, composition, inverse, intersection . 31 3.2 Picturing relations . 32 3.3 Equivalence relations . 32 3.4 Generating relations . 32 3.5 Ordered sets . 33 3.6 Functions . 33 3.7 Relations inverse to functions . 34 3.8 Dynamical systems . 34 3.9 Picturing dynamical systems . 35 3.10 Structure of dynamical systems . 35 4 Functions, Cardinal Number 39 4.1 Functions . 39 4.2 Picturing functions . 40 4.3 Indexed sums and products . 40 4.4 Cartesian powers . 41 iii iv CONTENTS 4.5 Cardinality . 41 5 Ordered sets and completeness 45 5.1 Ordered sets . 45 5.2 Order completeness . 46 5.3 Sequences in a complete lattice . 47 5.4 Order completion . 48 5.5 The Knaster-Tarski fixed point theorem . 49 5.6 The extended real number system . 49 6 Metric spaces 51 6.1 Metric space notions . 51 6.2 Normed vector spaces . 51 6.3 Spaces of finite sequences . 52 6.4 Spaces of infinite sequences . -
Revision Checklist Analysis II 1
Revision Checklist Analysis II 1. What is the (ε, δ)-definition of a limit of a function? What are examples and counterexamples? 2. What is the epsilon-delta definition of a continuous function? 3. When does the quotient of two continuous functions produce a continuous func- tion? What about the sum? The product? 4. What is the relation between the continuity of a function in higher dimension and the continuity of its component functions? 5. Do you know the definition of a bounded function, examples and non-examples? 6. Do you know the statements of the continuity theorems (IVT, Brouwer’s Fixed Point Theorem, EVT)? Do you know how to apply them? Do you know the idea of the proof? 7. Do you know the statements of the theorems of Fermat and Rolle as well as the Mean Value Theorem? Do you know the idea of the proof? 8. Do you know the definition of a differentiable function, examples and non-examples? 9. When is the quotient of two differentiable functions a differentiable function? What about the sum? The product? 10. Do you know the definition of the derivative in higher dimension? 11. Do you know the relation between the derivative in higher dimension and partial derivatives? Do you understand its motivation? 12. Do you know that the existence of partial derivatives of a higher dimensional func- tion does NOT imply that the function is differentiable? Can you give examples? 13. Do you know the notion of a closed set, open set, compact set? Do you know the connection between continuity of functions and open sets? Do you know the relation between compactness and convergence of (sub)sequences? 14. -
Bibliographical and Historical Comments
Bibliographical and Historical Comments One gets a strange feeling having seen the same drawings as if drawn by the same hand in the works of four schol- ars that worked completely independently of each other. An involuntary thought comes that such a striking, myste- rious activity of mankind, lasting several thousand years, cannot be occasional and must have a certain goal. Having acknowledged this, we come by necessity to the question: what is this goal? I.R. Shafarevich. On some tendencies of the develop- ment of mathematics. However, also in my contacts with the American Shake- speare scholars I confined myself to the concrete problems of my research: dating, identification of prototypes, direc- tions of certain allusions. I avoided touching the problem of personality of the Great Bard, the “Shakespeare prob- lem”; neither did I hear those scholars discussing such a problem between themselves. I.M. Gililov. A play about William Shakespeare or the Mystery of the Great Phoenix. The extensive bibliography in this book covers, however, only a small portion of the existing immense literature on measure theory; in particular, many authors are represented by a minimal number of their most character- istic works. Guided by the proposed brief comments and this incomplete list, the reader, with help of modern electronic data-bases, can considerably en- large the bibliography. The list of books is more complete (although it cannot pretend to be absolutely complete). For the reader’s convenience, the bibli- ography includes the collected (or selected) works of A.D. Alexandrov [15], R. Baire [47], S. Banach [56], E. -
Curriculum Vitae
Curriculum Vitae Erik Talvila Mathematics and Statistics [email protected] University of the Fraser Valley 604-504-7441 (office) Abbotsford, British Columbia V2S 7M8 Canada Place of birth: Toronto, Canada. My Erd¨osnumber is 3. Education • Ph.D. University of Waterloo, Applied Mathematics. January, 1997. Supervisor: D. Siegel. Thesis title: Growth estimates and Phragm´en-Lindel¨ofprinciples for half space problems. • M.Sc. University of Western Ontario, Applied Mathematics. September, 1991. Supervisor: D. Naylor. Thesis title: A finite Bessel transform. • B.Sc. University of Toronto, Mathematics and Physics. May, 1988. Employment history • 2014 to present, Associate Professor; 2005-2014, University College Professor; 2003- 2005, Probationary Faculty; Mathematics and Statistics, University of the Fraser Valley. • July 2000 to July 2003, Visiting Assistant Professor, Mathematical and Statistical Sciences, University of Alberta. • August 1998 to July 2000, NSERC Postdoctoral Fellow, Mathematics, University of Illinois at Urbana{Champaign. • September 1997 to July 1998, Lecturer, Applied Mathematics, University of Western Ontario. • January 1997 to September 1997, Assistant Professor, Applied Mathematics, Uni- versity of Waterloo. • January 1993 to December 1997, Part time lecturer: University of Waterloo, Wilfrid Laurier University, Wichita State University. Publications in refereed journals UFV student authors in bold face. 29. Erik Talvila, The continuous primitive integral in the plane, Real Analysis Exchange, 20(2020), 283{326. Curriculum Vitae - Erik Talvila 2 28. Erik Talvila, Fourier transform inversion using an elementary differential equation and a contour integral, American Mathematical Monthly, 126(2019), 717{727. 27. Cameron Grant and Erik Talvila, Elementary numerical methods for double inte- grals, Minnesota Journal of Undergraduate Mathematics, 4(2018-2019), 1{19. -
On the Modular Operator of Mutli-Component Regions in Chiral CFT
Commun. Math. Phys. 384, 785–828 (2021) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-021-04054-6 Mathematical Physics On the Modular Operator of Mutli-component Regions in Chiral CFT Stefan Hollands Institut für Theoretische Physik, Universität Leipzig, Brüderstrasse 16, 04103 Leipzig, Germany. E-mail: [email protected] Received: 5 January 2020 / Accepted: 26 February 2021 Published online: 29 April 2021 – © The Author(s) 2021 Abstract: We introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin– Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered. 1. Introduction The reduced density matrix of a subsystem induces an intrinsic internal dynamics called the “modular flow”. The flow is non-trivial only for non-commuting observable algebras—i.e., in quantum theory—and depends on both the subsystem and the given state of the total system. It has been subject to much attention in theoretical physics in recent times because it is closely related to information theoretic concepts. As examples for some topics such as Bekenstein bounds, Quantum Focussing Conjecture, c-theorems, holography we mention [1–5]. In mathematics, the modular flow has played an important role in the study of operator algebras through the work of Connes, Takesaki and others, see [6] for an encyclopedic account. -
Math 346 Lecture #12 8.1 Multivariable Integration 8.1.1
Math 346 Lecture #12 8.1 Multivariable Integration We extend the construction of the regulated integral for functions of a single variable to functions of several variables by defining the integral for step functions and then applying the continuous linear extension theorem. 8.1.1 Multivariable Step Functions n Definition 8.1.1. For a = (a1; : : : ; an) and b = (b1; : : : ; bn) in R with ai ≤ bi for all i = 1; : : : ; n, the closed n-interval [a; b] is defined to be n [a1; b1] × · · · × [an; bn] ⊂ R : The closed n-interval [a; b] is a compact box or parallelepiped in Rn. Definition 8.1.2. A subdivision P of a closed n-interval [a; b] consists, for each i = 1; : : : ; n, of a subdivsion (0) (1) (ki−1) (ki) Pi = fti = ai < ti < ··· < ti < ti = big of the closed interval [ai; bi] for some ki 2 N. Definition 8.1.3. A subdivision of an n-interval [a; b] gives a decomposition of [a; b] into a disjoint union of an partially open subinterval and closed subinterval as follows. (j) Each ti with 0 < j < ki defines a hyperplane (j) n (j) Hi = fx 2 R : xi = ti g in Rn which divides [a; b] into two regions; a partially open subinterval (j) [a1; b1] × · · · × [ai−1; bi−1] × [ai; ti ) × [ai+1; bi+1] × · · · × [an; bn]; and a closed subinterval (j) [a1; b1] × · · · × [ai−1; bi−1] × [ti ; bi] × [ai+1; bi+1] × · · · × [an; bn]: (Sketch the picture in R2.) Repeating this process for each resulting subinterval (either partially open or closed) and for each hyperplane, we get a decomposition of [a; b] into a pairwise disjoint union of subintervals.