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AMATH 731: Applied Functional Analysis Lecture Notes
AMATH 731: Applied Functional Analysis Lecture Notes Sumeet Khatri November 24, 2014 Table of Contents List of Tables ................................................... v List of Theorems ................................................ ix List of Definitions ................................................ xii Preface ....................................................... xiii 1 Review of Real Analysis .......................................... 1 1.1 Convergence and Cauchy Sequences...............................1 1.2 Convergence of Sequences and Cauchy Sequences.......................1 2 Measure Theory ............................................... 2 2.1 The Concept of Measurability...................................3 2.1.1 Simple Functions...................................... 10 2.2 Elementary Properties of Measures................................ 11 2.2.1 Arithmetic in [0, ] .................................... 12 1 2.3 Integration of Positive Functions.................................. 13 2.4 Integration of Complex Functions................................. 14 2.5 Sets of Measure Zero......................................... 14 2.6 Positive Borel Measures....................................... 14 2.6.1 Vector Spaces and Topological Preliminaries...................... 14 2.6.2 The Riesz Representation Theorem........................... 14 2.6.3 Regularity Properties of Borel Measures........................ 14 2.6.4 Lesbesgue Measure..................................... 14 2.6.5 Continuity Properties of Measurable Functions................... -
A New Approach to the Minimum Cut Problem
A New Approach to the Minimum Cut Problem DAVID R. KARGER Massachusetts Institute of Technology, Cambridge, Massachusetts AND CLIFFORD STEIN Dartmouth College, Hanover, New Hampshire Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n2log3n) time, a significant improvement over the previous O˜(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in 51# with n2 processors; this gives the first proof that the minimum cut problem can be solved in 51#. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of a of the minimum cut’s in expected O˜(n2a) time, or in 51# with n2a processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected O˜(n2(r21)) time, or in 51# with n2(r21) processors. The “trace” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the more standard cactus representation for minimum cuts. -
List of Theorems from Geometry 2321, 2322
List of Theorems from Geometry 2321, 2322 Stephanie Hyland [email protected] April 24, 2010 There’s already a list of geometry theorems out there, but the course has changed since, so here’s a new one. They’re in order of ‘appearance in my notes’, which corresponds reasonably well to chronolog- ical order. 24 onwards is Hilary term stuff. The following theorems have actually been asked (in either summer or schol papers): 5, 7, 8, 17, 18, 20, 21, 22, 26, 27, 30 a), 31 (associative only), 35, 36 a), 37, 38, 39, 43, 45, 47, 48. Definitions are also asked, which aren’t included here. 1. On a finite-dimensional real vector space, the statement ‘V is open in M’ is independent of the choice of norm on M. ′ f f i 2. Let Rn ⊃ V −→ Rm be differentiable with f = (f 1, ..., f m). Then Rn −→ Rm, and f ′ = ∂f ∂xj f 3. Let M ⊃ V −→ N,a ∈ V . Then f is differentiable at a ⇒ f is continuous at a. 4. f = (f 1, ...f n) continuous ⇔ f i continuous, and same for differentiable. 5. The chain rule for functions on finite-dimensional real vector spaces. 6. The chain rule for functions of several real variables. f 7. Let Rn ⊃ V −→ R, V open. Then f is C1 ⇔ ∂f exists and is continuous, for i =1, ..., n. ∂xi 2 2 Rn f R 2 ∂ f ∂ f 8. Let ⊃ V −→ , V open. f is C . Then ∂xi∂xj = ∂xj ∂xi 9. (φ ◦ ψ)∗ = φ∗ ◦ ψ∗, where φ, ψ are maps of manifolds, and φ∗ is the push-forward of φ. -
Calculus I – Math
Math 380: Low-Dimensional Topology Instructor: Aaron Heap Office: South 330C E-mail: [email protected] Web Page: http://www.geneseo.edu/math/heap Textbook: Topology Now!, by Robert Messer & Philip Straffin. Course Info: We will cover topological equivalence, deformations, knots and links, surfaces, three- dimensional manifolds, and the fundamental group. Topics are subject to change depending on the progress of the class, and various topics may be skipped due to time constraints. An accurate reading schedule will be posted on the website, and you should check it often. By the time you take this course, most of you should be fairly comfortable with mathematical proofs. Although this course only has multivariable calculus, elementary linear algebra, and mathematical proofs as prerequisites, students are strongly encouraged to take abstract algebra (Math 330) prior to this course or concurrently. It requires a certain level of mathematical sophistication. There will be a lot of new terminology you must learn, and we will be doing a significant number of proofs. Please note that we will work on developing your independent reading skills in Mathematics and your ability to learn and use definitions and theorems. I certainly won't be able to cover in class all the material you will be required to learn. As a result, you will be expected to do a lot of reading. The reading assignments will be on topics to be discussed in the following lecture to enable you to ask focused questions in the class and to better understand the material. It is imperative that you keep up with the reading assignments. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Math 025-1,2 List of Theorems and Definitions Fall 2010
Math 025-1,2 List of Theorems and Definitions Fall 2010 Here are the 19 theorems of Math 25: Domination Law (aka Monotonicity Law) If f and g are integrable functions on [a; b] such that R b R b f(x) ≤ g(x) for all x 2 [a; b], then a f(x) dx ≤ a g(x) dx. R b Positivity Law If f is a nonnegative, integrable function on [a; b], then a f(x) dx ≥ 0. Also, if f(x) is R b a nonnegative, continuous function on [a; b] and a f(x) dx = 0, then f(x) = 0 for all x 2 [a; b]. Max-Min Inequality If f(x) is an integrable function on [a; b], then Z b (min value of f(x) on [a; b]) · (b − a) ≤ f(x) dx ≤ (max value of f(x) on [a; b]) · (b − a): a Triangle Inequality (for Integrals) Let f :[a; b] ! R be integrable. Then Z b Z b f(x) dx ≤ jf(x)j dx: a a Mean Value Theorem for Integrals Let f : R ! R be continuous on [a; b]. Then there exists a number c 2 (a; b) such that 1 Z b f(c) = f(x) dx: b − a a Fundamental Theorem of Calculus (one part) Let f :[a; b] ! R be continuous and define F : R x [a; b] ! R by F (x) = a f(t) dt. Then F is differentiable (hence continuous) on [a; b] and F 0(x) = f(x). Fundamental Theorem of Calculus (other part) Let f :[a; b] ! R be continuous and let F be any antiderivative of f on [a; b]. -
Calculus I Teacher(S): Mr
Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming. April 20 - 24, 2020 Course: 11 Calculus I Teacher(s): Mr. Simmons Weekly Plan: Monday, April 20 ⬜ Revise your proof of Fermat’s Theorem. Tuesday, April 21 ⬜ Extreme Value Theorem proof and diagram. Wednesday, April 22 ⬜ Diagrams for Fermat’s Theorem, Rolle’s Theorem, and the MVT Thursday, April 23 ⬜ Prove Rolle’s Theorem. Friday, April 24 ⬜ Prove the MVT. Statement of Academic Honesty I affirm that the work completed from the packet I affirm that, to the best of my knowledge, my is mine and that I completed it independently. child completed this work independently _______________________________________ _______________________________________ Student Signature Parent Signature Monday, April 20 I would like to apologize, because in the list of theorems that I sent you, there was a typo. In the hypotheses for two of the theorems, there were written nonstrict inequalities, but they should have been strict inequalities. I have corrected this in the new version. If you have felt particularly challenged by these proofs, that’s okay! I hope that this is an opportunity for you not to memorize a method and execute it perfectly, but rather to be challenged and to struggle with real mathematical problems. I highly encourage you to come to office hours (virtually) to ask questions about these problems. And feel free to email me as well! This week’s handout is a rewriting of those same theorems, along with one more, the Extreme Value Theorem. I apologize for all the changes, but I - along with the rest of you - am still adjusting to this new setup. -
Formal Proof—Getting Started Freek Wiedijk
Formal Proof—Getting Started Freek Wiedijk A List of 100 Theorems On the webpage [1] only eight entries are listed for Today highly nontrivial mathematics is routinely the first theorem, but in [2p] seventeen formaliza- being encoded in the computer, ensuring a reliabil- tions of the irrationality of 2 have been collected, ity that is orders of a magnitude larger than if one each with a short description of the proof assistant. had just used human minds. Such an encoding is When we analyze this list of theorems to see called a formalization, and a program that checks what systems occur most, it turns out that there are such a formalization for correctness is called a five proof assistants that have been significantly proof assistant. used for formalization of mathematics. These are: Suppose you have proved a theorem and you want to make certain that there are no mistakes proof assistant number of theorems formalized in the proof. Maybe already a couple of times a mistake has been found and you want to make HOL Light 69 sure that that will not happen again. Maybe you Mizar 45 fear that your intuition is misleading you and want ProofPower 42 to make sure that this is not the case. Or maybe Isabelle 40 you just want to bring your proof into the most Coq 39 pure and complete form possible. We will explain all together 80 in this article how to go about this. Although formalization has become a routine Currently in all systems together 80 theorems from activity, it still is labor intensive. -
A Gentle Tutorial for Programming on the ML-Level of Isabelle (Draft)
The Isabelle Cookbook A Gentle Tutorial for Programming on the ML-Level of Isabelle (draft) by Christian Urban with contributions from: Stefan Berghofer Jasmin Blanchette Sascha Bohme¨ Lukas Bulwahn Jeremy Dawson Rafal Kolanski Alexander Krauss Tobias Nipkow Andreas Schropp Christian Sternagel July 31, 2013 2 Contents Contentsi 1 Introduction1 1.1 Intended Audience and Prior Knowledge.................1 1.2 Existing Documentation..........................2 1.3 Typographic Conventions.........................2 1.4 How To Understand Isabelle Code.....................3 1.5 Aaaaargh! My Code Does not Work Anymore..............4 1.6 Serious Isabelle ML-Programming.....................4 1.7 Some Naming Conventions in the Isabelle Sources...........5 1.8 Acknowledgements.............................6 2 First Steps9 2.1 Including ML-Code.............................9 2.2 Printing and Debugging.......................... 10 2.3 Combinators................................ 14 2.4 ML-Antiquotations............................. 21 2.5 Storing Data in Isabelle.......................... 24 2.6 Summary.................................. 32 3 Isabelle Essentials 33 3.1 Terms and Types.............................. 33 3.2 Constructing Terms and Types Manually................. 38 3.3 Unification and Matching......................... 46 3.4 Sorts (TBD)................................. 55 3.5 Type-Checking............................... 55 3.6 Certified Terms and Certified Types.................... 57 3.7 Theorems.................................. 58 3.8 Theorem -
This Is a Paper in Whiuch Several Famous Theorems Are Poven by Using Ths Same Method
This is a paper in whiuch several famous theorems are poven by using ths same method. There formula due to Marston Morse which I think is exceptional. Vector Fields and Famous Theorems Daniel H. Gottlieb 1. Introduction. What do I mean ? Look at the example of Newton’s Law of Gravitation. Here is a mathematical statement of great simplicity which implies logically vast number of phe- nomena of incredible variety. For example, Galileo’s observation that objects of different weights fall to the Earth with the same acceleration, or Kepler’s laws governing the motion of the planets, or the daily movements of the tides are all due to the underlying notion of gravity. And this is established by deriving the the mathematical statements of these three classes of phenomena from Newton’s Law of Gravity. The mathematical relationship of falling objects, planets, and tides is defined by the beautiful fact that they follow from Newton’s Law by the same kind of argument. The derivation for falling objects is much simpler than the derivation of the tides, but the general method of proof is the same. This process of simple laws implying vast numbers of physical theorems repeats itself over and over in Physics, with Maxwell’s Equations and Conservation of Energy and Momentum. To a mathematician, this web of relationships stemming from a few sources is beautiful. But is it unreasonable ? This is the question I ask. Are there mathematical Laws ? That is are there theorems which imply large numbers of known, important, and beautiful results by using the same kind of proofs ? At thirteen I already knew what Mathematics was. -
Introduction to Algorithms, 3Rd
Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein Introduction to Algorithms Third Edition The MIT Press Cambridge, Massachusetts London, England c 2009 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email special [email protected]. This book was set in Times Roman and Mathtime Pro 2 by the authors. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Introduction to algorithms / Thomas H. Cormen ...[etal.].—3rded. p. cm. Includes bibliographical references and index. ISBN 978-0-262-03384-8 (hardcover : alk. paper)—ISBN 978-0-262-53305-8 (pbk. : alk. paper) 1. Computer programming. 2. Computer algorithms. I. Cormen, Thomas H. QA76.6.I5858 2009 005.1—dc22 2009008593 10987654321 Index This index uses the following conventions. Numbers are alphabetized as if spelled out; for example, “2-3-4 tree” is indexed as if it were “two-three-four tree.” When an entry refers to a place other than the main text, the page number is followed by a tag: ex. for exercise, pr. for problem, fig. for figure, and n. for footnote. A tagged page number often indicates the first page of an exercise or problem, which is not necessarily the page on which the reference actually appears. ˛.n/, 574 (set difference), 1159 (golden ratio), 59, 108 pr. jj y (conjugate of the golden ratio), 59 (flow value), 710 .n/ (Euler’s phi function), 943 (length of a string), 986 .n/-approximation algorithm, 1106, 1123 (set cardinality), 1161 o-notation, 50–51, 64 O-notation, 45 fig., 47–48, 64 (Cartesian product), 1162 O0-notation, 62 pr. -
Thomas H. Cormen Current Position Research Interests Education Honors and Awards
Thomas H. Cormen Department of Computer Science 6211 Sudikoff Laboratory Dartmouth College Hanover, NH 03755-3510 (603) 646-2417 [email protected] http://www.cs.dartmouth.edu/thc/ Current Position Professor of Computer Science. Research Interests Algorithm engineering, parallel computing, speeding up computations with high latency, Gray codes. Education Massachusetts Institute of Technology, Cambridge, Massachusetts Ph.D. in Electrical Engineering and Computer Science, February 1993. Thesis: “Virtual Memory for Data-Parallel Computing.” Advisor: Charles E. Leiserson. Minor: Engineering management and entrepreneurship. S.M. in Electrical Engineering and Computer Science, May 1986. Thesis: “Concentrator Switches for Routing Messages in Parallel Computers.” Advisor: Charles E. Leiserson. Princeton University, Princeton, New Jersey B.S.E. summa cum laude in Electrical Engineering and Computer Science, June 1978. Honors and Awards ACM Distinguished Educator, 2009. McLane Family Fellow, Dartmouth College, 2004–2005. Jacobus Family Fellow, Dartmouth College, 1998–1999. Dartmouth College Class of 1962 Faculty Fellowship, 1995–1996. Adopted Member, Dartmouth College Class of 1962, 2015. Professional and Scholarly Publishing Award in Computer Science and Data Processing, Association of American Publishers, 1990. Distinguished Presentation Award, 1987 International Conference on Parallel Processing, St. Charles, Illinois. Best Presentation Award, 1986 International Conference on Parallel Processing, St. Charles, Illinois. National Science Foundation Fellowship. Elected to Phi Beta Kappa, Tau Beta Pi, Eta Kappa Nu. Prepared on March 28, 2020. 1 Professional Experience Dartmouth College, Hanover, New Hampshire Chair, Department of Computer Science, July 2009–July 2015. Professor, Department of Computer Science, July 2004–present. Director of the Dartmouth Institute for Writing and Rhetoric, July 2007–June 2008.