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Finite Type Invariants for Knots 3-Manifolds
Pergamon Topology Vol. 37, No. 3. PP. 673-707, 1998 ~2 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0040-9383/97519.00 + 0.00 PII: soo4o-9383(97)00034-7 FINITE TYPE INVARIANTS FOR KNOTS IN 3-MANIFOLDS EFSTRATIA KALFAGIANNI + (Received 5 November 1993; in revised form 4 October 1995; final version 16 February 1997) We use the Jaco-Shalen and Johannson theory of the characteristic submanifold and the Torus theorem (Gabai, Casson-Jungreis)_ to develop an intrinsic finite tvne__ theory for knots in irreducible 3-manifolds. We also establish a relation between finite type knot invariants in 3-manifolds and these in R3. As an application we obtain the existence of non-trivial finite type invariants for knots in irreducible 3-manifolds. 0 1997 Elsevier Science Ltd. All rights reserved 0. INTRODUCTION The theory of quantum groups gives a systematic way of producing families of polynomial invariants, for knots and links in [w3 or S3 (see for example [18,24]). In particular, the Jones polynomial [12] and its generalizations [6,13], can be obtained that way. All these Jones-type invariants are defined as state models on a knot diagram or as traces of a braid group representation. On the other hand Vassiliev [25,26], introduced vast families of numerical knot invariants (Jinite type invariants), by studying the topology of the space of knots in [w3. The compu- tation of these invariants, involves in an essential way the computation of related invariants for special knotted graphs (singular knots). It is known [l-3], that after a suitable change of variable the coefficients of the power series expansions of the Jones-type invariants, are of Jinite type. -
MODULAR MAGIC SUDOKU 1. Introduction 1.1. Terminology And
MODULAR MAGIC SUDOKU JOHN LORCH AND ELLEN WELD Abstract. A modular magic sudoku solution is a sudoku solution with symbols in f0; 1; :::; 8g such that rows, columns, and diagonals of each subsquare add to zero modulo nine. We count these sudoku solutions by using the action of a suitable symmetry group and we also describe maximal mutually orthogonal families. 1. Introduction 1.1. Terminology and goals. Upon completing a newspaper sudoku puzzle one obtains a sudoku solution of order nine, namely, a nine-by-nine array in which all of the symbols f0; 1;:::; 8g occupy each row, column, and subsquare. For example, both 7 2 3 1 8 5 4 6 0 1 8 0 7 5 6 4 2 3 4 0 5 3 2 6 1 8 7 2 3 4 8 0 1 5 6 7 6 1 8 4 0 7 2 3 5 6 7 5 3 4 2 0 1 8 1 7 0 6 3 2 5 4 8 8 4 6 5 1 3 2 7 0 (1) 5 4 6 8 1 0 7 2 3 and 7 0 2 4 6 8 1 3 5 8 3 2 7 5 4 0 1 6 3 5 1 0 2 7 6 8 4 2 6 4 0 7 8 3 5 1 5 1 3 2 7 0 8 4 6 3 5 7 2 6 1 8 0 4 4 6 8 1 3 5 7 0 2 0 8 1 5 4 3 6 7 2 0 2 7 6 8 4 3 5 1 are sudoku solutions. -
List of AOIME Institutions
List of AOIME Institutions CEEB School City State Zip Code 1001510 Calgary Olympic Math School Calgary AB T2X2E5 1001804 ICUC Academy Calgary AB T3A3W2 820138 Renert School Calgary AB T3R0K4 820225 Western Canada High School Calgary AB T2S0B5 996056 WESTMOUNT CHARTER SCHOOL CALGARY AB T2N 4Y3 820388 Old Scona Academic Edmonton AB T6E 2H5 C10384 University of Alberta Edmonton AB T6G 2R3 1001184 Vernon Barford School Edmonton AB T6J 2C1 10326 ALABAMA SCHOOL OF FINE ARTS BIRMINGHAM AL 35203-2203 10335 ALTAMONT SCHOOL BIRMINGHAM AL 35222-4445 C12963 University of Alabama at Birmingham Birmingham AL 35294 10328 Hoover High School Hoover AL 35244 11697 BOB JONES HIGH SCHOOL MADISON AL 35758-8737 11701 James Clemens High School Madison AL 35756 11793 ALABAMA SCHOOL OF MATH/SCIENCE MOBILE AL 36604-2519 11896 Loveless Academic Magnet Program High School Montgomery AL 36111 11440 Indian Springs School Pelham AL 35124 996060 LOUIS PIZITZ MS VESTAVIA HILLS AL 35216 12768 VESTAVIA HILLS HS VESTAVIA HILLS AL 35216-3314 C07813 University of Arkansas - Fayetteville Fayetteville AR 72701 41148 ASMSA Hot Springs AR 71901 41422 Central High School Little Rock AR 72202 30072 BASIS Chandler Chandler AZ 85248-4598 30045 CHANDLER HIGH SCHOOL CHANDLER AZ 85225-4578 30711 ERIE SCHOOL CAMPUS CHANDLER AZ 85224-4316 30062 Hamilton High School Chandler AZ 85248 997449 GCA - Gilbert Classical Academy Gilbert AZ 85234 30157 MESQUITE HS GILBERT AZ 85233-6506 30668 Perry High School Gilbert AZ 85297 30153 Mountain Ridge High School Glendale AZ 85310 30750 BASIS Mesa -
Certified School List MM-DD-YY.Xlsx
Updated SEVP Certified Schools January 26, 2017 SCHOOL NAME CAMPUS NAME F M CITY ST CAMPUS ID "I Am" School Inc. "I Am" School Inc. Y N Mount Shasta CA 41789 ‐ A ‐ A F International School of Languages Inc. Monroe County Community College Y N Monroe MI 135501 A F International School of Languages Inc. Monroe SH Y N North Hills CA 180718 A. T. Still University of Health Sciences Lipscomb Academy Y N Nashville TN 434743 Aaron School Southeastern Baptist Theological Y N Wake Forest NC 5594 Aaron School Southeastern Bible College Y N Birmingham AL 1110 ABC Beauty Academy, INC. South University ‐ Savannah Y N Savannah GA 10841 ABC Beauty Academy, LLC Glynn County School Administrative Y N Brunswick GA 61664 Abcott Institute Ivy Tech Community College ‐ Y Y Terre Haute IN 6050 Aberdeen School District 6‐1 WATSON SCHOOL OF BIOLOGICAL Y N COLD SPRING NY 8094 Abiding Savior Lutheran School Milford High School Y N Highland MI 23075 Abilene Christian Schools German International School Y N Allston MA 99359 Abilene Christian University Gesu (Catholic School) Y N Detroit MI 146200 Abington Friends School St. Bernard's Academy Y N Eureka CA 25239 Abraham Baldwin Agricultural College Airlink LLC N Y Waterville ME 1721944 Abraham Joshua Heschel School South‐Doyle High School Y N Knoxville TN 184190 ABT Jacqueline Kennedy Onassis School South Georgia State College Y N Douglas GA 4016 Abundant Life Christian School ELS Language Centers Dallas Y N Richardson TX 190950 ABX Air, Inc. Frederick KC Price III Christian Y N Los Angeles CA 389244 Acaciawood School Mid‐State Technical College ‐ MF Y Y Marshfield WI 31309 Academe of the Oaks Argosy University/Twin Cities Y N Eagan MN 7169 Academia Language School Kaplan University Y Y Lincoln NE 7068 Academic High School Ogden‐Hinckley Airport Y Y Ogden UT 553646 Academic High School Ogeechee Technical College Y Y Statesboro GA 3367 Academy at Charlemont, Inc. -
Mathematics of Sudoku I
Mathematics of Sudoku I Bertram Felgenhauer Frazer Jarvis∗ January 25, 2006 Introduction Sudoku puzzles became extremely popular in Britain from late 2004. Sudoku, or Su Doku, is a Japanese word (or phrase) meaning something like Number Place. The idea of the puzzle is extremely simple; the solver is faced with a 9 × 9 grid, divided into nine 3 × 3 blocks: In some of these boxes, the setter puts some of the digits 1–9; the aim of the solver is to complete the grid by filling in a digit in every box in such a way that each row, each column, and each 3 × 3 box contains each of the digits 1–9 exactly once. It is a very natural question to ask how many Sudoku grids there can be. That is, in how many ways can we fill in the grid above so that each row, column and box contains the digits 1–9 exactly once. In this note, we explain how we first computed this number. This was the first computation of this number; we should point out that it has been subsequently confirmed using other (faster!) methods. First, let us notice that Sudoku grids are simply special cases of Latin squares; remember that a Latin square of order n is an n × n square containing each of the digits 1,...,n in every row and column. The calculation of the number of Latin squares is itself a difficult problem, with no general formula known. The number of Latin squares of sizes up to 11 × 11 have been worked out (see references [1], [2] and [3] for the 9 × 9, 10 × 10 and 11 × 11 squares), and the methods are broadly brute force calculations, much like the approach we sketch for Sudoku grids below. -
![Arxiv:1911.03589V1 [Math.CV] 9 Nov 2019 1 Hoe 1 Result](https://docslib.b-cdn.net/cover/6646/arxiv-1911-03589v1-math-cv-9-nov-2019-1-hoe-1-result-296646.webp)
Arxiv:1911.03589V1 [Math.CV] 9 Nov 2019 1 Hoe 1 Result
A NOTE ON THE SCHWARZ LEMMA FOR HARMONIC FUNCTIONS MAREK SVETLIK Abstract. In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given. 1. Introduction 1.1. A summary of some results. In this paper we consider some generalizations of the Schwarz lemma for harmonic functions from the unit disk U = {z ∈ C : |z| < 1} to the interval (−1, 1) (or to itself). First, we cite a theorem which is known as the Schwarz lemma for harmonic functions and is considered as a classical result. Theorem 1 ([10],[9, p.77]). Let f : U → U be a harmonic function such that f(0) = 0. Then 4 |f(z)| 6 arctan |z|, for all z ∈ U, π and this inequality is sharp for each point z ∈ U. In 1977, H. W. Hethcote [11] improved this result by removing the assumption f(0) = 0 and proved the following theorem. Theorem 2 ([11, Theorem 1] and [26, Theorem 3.6.1]). Let f : U → U be a harmonic function. Then 1 − |z|2 4 f(z) − f(0) 6 arctan |z|, for all z ∈ U. 1+ |z|2 π As it was written in [23], it seems that researchers have had some difficulties to arXiv:1911.03589v1 [math.CV] 9 Nov 2019 handle the case f(0) 6= 0, where f is harmonic mapping from U to itself. Before explaining the essence of these difficulties, it is necessary to recall one mapping and some of its properties. -
![[The PROOF of FERMAT's LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] the World's Most Famous Math Problem the World's Most Famous Math Problem](https://docslib.b-cdn.net/cover/2903/the-proof-of-fermats-last-theorem-and-other-mathematical-mysteries-the-worlds-most-famous-math-problem-the-worlds-most-famous-math-problem-312903.webp)
[The PROOF of FERMAT's LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] the World's Most Famous Math Problem the World's Most Famous Math Problem
0Eft- [The PROOF of FERMAT'S LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] The World's Most Famous Math Problem The World's Most Famous Math Problem [ THE PROOF OF FERMAT'S LAST THEOREM AND OTHER MATHEMATICAL MYSTERIES I Marilyn vos Savant ST. MARTIN'S PRESS NEW YORK For permission to reprint copyrighted material, grateful acknowledgement is made to the following sources: The American Association for the Advancement of Science: Excerpts from Science, Volume 261, July 2, 1993, C 1993 by the AAAS. Reprinted by permission. Birkhauser Boston: Excerpts from The Mathematical Experience by Philip J. Davis and Reuben Hersh © 1981 Birkhauser Boston. Reprinted by permission of Birkhau- ser Boston and the authors. The Chronicleof Higher Education: Excerpts from The Chronicle of Higher Education, July 7, 1993, C) 1993 Chronicle of HigherEducation. Reprinted by permission. The New York Times: Excerpts from The New York Times, June 24, 1993, X) 1993 The New York Times. Reprinted by permission. Excerpts from The New York Times, June 29, 1993, © 1993 The New York Times. Reprinted by permission. Cody Pfanstieh/ The poem on the subject of Fermat's last theorem is reprinted cour- tesy of Cody Pfanstiehl. Karl Rubin, Ph.D.: The sketch of Dr. Wiles's proof of Fermat's Last Theorem in- cluded in the Appendix is reprinted courtesy of Karl Rubin, Ph.D. Wesley Salmon, Ph.D.: Excerpts from Zeno's Paradoxes by Wesley Salmon, editor © 1970. Reprinted by permission of the editor. Scientific American: Excerpts from "Turing Machines," by John E. Hopcroft, Scientific American, May 1984, (D 1984 Scientific American, Inc. -
ISCHE 2014 Book of Abstracts
i Published 2014 by ISCHE. ISSN 2313-1837 These abstracts are set in Baskerville Old Face, designed in 1757 by John Baskerville in Birmingham, UK. A writing master, businessman, printer and type designer, he conducted experiments to improve legibility which also included paper making and ink manufacturing. In 1758, he was appointed printer to Cambridge University Press, and despite his personal Atheism, printed a folio Bible in 1763. His typefaces were greatly admired for their simplicity and refinement by Pierre Simon Fournier, and Giambattista Bodoni. Benjamin Franklin, printer and fellow member of the Royal Society of Arts, took the designs to the US, where they were adopted for most federal Government publishing. Baskerville type was revived in 1917 by Harvard University Press and may nowadays be found in Microsoft Word. ii Contents Welcome p. iii Acknowledgements p. viii Conference theme p. x Keynotes: biographies and abstracts p. xi Early career bursaries p. xiv Brian Simon bursaries p. xv Guide to using abstract book p. xvi Abstracts of papers p. 1 (In alphabetical order of authors) Synopses of panels p. 385 (In order of sessions presented at conference) Name index / list of presenters p. 422 iii Welcome To all delegates at ISCHE 36 – a very warm welcome to London! We are looking forward very much indeed to hosting this great event, exploring the immense theme of education, war and peace. My thanks go first of all to the ISCHE executive committee for supporting this event, to the UK History of Education Society as the national hosts, and to the Institute of Education at the University of London for the use of its extensive facilities for the conference. -
On Selmer Groups of Geometric Galois Representations Tom Weston
On Selmer Groups of Geometric Galois Representations Tom Weston Department of Mathematics, Harvard University, Cambridge, Mass- chusetts 02140 E-mail address: [email protected] iii Dedicated to the memory of Annalee Henderson and to Arnold Ross Contents Introduction ix Acknowledgements xii Notation and terminology xv Fields xv Characters xv Galois modules xv Schemes xv Sheaves xvi Cohomology xvi K-theory xvi Part 1. Selmer groups and deformation theory 1 Chapter 1. Local cohomology groups 3 1. Local finite/singular structures 3 2. Functorialities 4 3. Local exact sequences 5 4. Examples of local structures 6 5. Ordinary representations 7 6. Cartier dual structures 8 7. Local structures for archimedean fields 9 Chapter 2. Global cohomology groups 11 1. Selmer groups 11 2. Functorialities 13 3. The global exact sequence 13 4. A finiteness theorem for Selmer groups 14 5. The Kolyvagin pairing 16 6. Shafarevich-Tate groups 18 7. The Bockstein pairing 20 Chapter 3. Annihilation theorems for Selmer groups 21 1. Partial geometric Euler systems 21 2. The key lemmas 22 3. The annihilation theorem 25 4. Right non-degeneracy of the Bockstein pairing 27 5. A δ-vanishing result 28 Chapter 4. Flach systems 31 1. Minimally ramified deformations 31 v vi CONTENTS 2. Tangent spaces and Selmer groups 34 3. Good primes 36 4. Flach systems 38 5. Cohesive Flach systems 39 6. Cohesive Flach systems of Eichler-Shimura type 40 Chapter 5. Flach systems of Eichler-Shimura type 43 1. The map on differentials 43 2. The Tate pairing 45 3. A special case 47 4. -
How Do YOU Solve Sudoku? a Group-Theoretic Approach to Human Solving Strategies
How Do YOU Solve Sudoku? A Group-theoretic Approach to Human Solving Strategies Elizabeth A. Arnold 1 James Madison University Harrisonburg, VA 22807 Harrison C. Chapman Bowdoin College Brunswick, ME 04011 Malcolm E. Rupert Western Washington University Bellingham, WA 98225 7 6 8 1 4 5 5 6 8 1 2 3 9 4 1 7 3 3 5 3 4 1 9 7 5 6 3 2 Sudoku Puzzle P How would you solve this Sudoku puzzle? Unless you have been living in a cave for the past 10 years, you are probably familiar with this popular number game. The goal is to fill in the numbers 1 through 9 in such a way that there is exactly one of each digit in each row, column and 3 × 3 block. Like most other humans, you probably look at the first row (or column or 3 × 3 block) and say, \Let's see.... there is no 3 in the first row. The third, 1Corresponding author: [email protected] 1 sixth and seventh cell are empty. But the third column already has a 3, and the top right block already contains a 3. Therefore, the 3 must go in the sixth cell." We call this a Human Solving Strategy (HSS for short). This particular strategy has a name, called the \Hidden Single" Strategy. (For more information on solving strategies see [8].) Next, you may look at the cell in the second row, second column. This cell can't contain 5, 6 or 8 since these numbers are already in the second row. -
Algebraicity Criteria and Their Applications
Algebraicity Criteria and Their Applications The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Tang, Yunqing. 2016. Algebraicity Criteria and Their Applications. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493480 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Algebraicity criteria and their applications A dissertation presented by Yunqing Tang to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts May 2016 c 2016 – Yunqing Tang All rights reserved. DissertationAdvisor:ProfessorMarkKisin YunqingTang Algebraicity criteria and their applications Abstract We use generalizations of the Borel–Dwork criterion to prove variants of the Grothedieck–Katz p-curvature conjecture and the conjecture of Ogus for some classes of abelian varieties over number fields. The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for all but finitely many primes p,hasfinite monodromy. It is known that it suffices to prove the conjecture for differential equations on P1 − 0, 1, . We prove a variant of this conjecture for P1 0, 1, , which asserts that if the equation { ∞} −{ ∞} satisfies a certain convergence condition for all p, then its monodromy is trivial. -
Algebra & Number Theory
Algebra & Number Theory Volume 4 2010 No. 2 mathematical sciences publishers Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale