DOCSLIB.ORG

Geometric Number Theory Lenny Fukshansky

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometric Number Theory Lenny Fukshansky Geometric Number Theory Lenny Fukshansky Minkowki's creation of the geometry of numbers was likened to the story of Saul, who set out to look for his father's asses and discovered a Kingdom. J. V. Armitage Contents Chapter 1. Geometry of Numbers 1 1.1. Introduction 1 1.2. Lattices 2 1.3. Theorems of Blichfeldt and Minkowski 10 1.4. Successive minima 13 1.5. Inhomogeneous minimum 18 1.6. Problems 21 Chapter 2. Discrete Optimization Problems 23 2.1. Sphere packing, covering and kissing number problems 23 2.2. Lattice packings in dimension 2 29 2.3. Algorithmic problems on lattices 34 2.4. Problems 38 Chapter 3. Quadratic forms 39 3.1. Introduction to quadratic forms 39 3.2. Minkowski's reduction 46 3.3. Sums of squares 49 3.4. Problems 53 Chapter 4. Diophantine Approximation 55 4.1. Real and rational numbers 55 4.2. Algebraic and transcendental numbers 57 4.3. Dirichlet's Theorem 61 4.4. Liouville's theorem and construction of a transcendental number 65 4.5. Roth's theorem 67 4.6. Continued fractions 70 4.7. Kronecker's theorem 76 4.8. Problems 79 Chapter 5. Algebraic Number Theory 82 5.1. Some field theory 82 5.2. Number fields and rings of integers 88 5.3. Noetherian rings and factorization 97 5.4. Norm, trace, discriminant 101 5.5. Fractional ideals 105 5.6. Further properties of ideals 109 5.7. Minkowski embedding 113 5.8. The class group 116 5.9. Dirichlet's unit theorem 119 v vi CONTENTS 5.10. Problems 120 Appendices Appendix A. Some properties of abelian groups 124 Appendix B. Maximum Modulus Principle and Fundamental Theorem of Algebra 127 Appendix. Bibliography 129 CHAPTER 1 Geometry of Numbers 1.1. Introduction The foundations of the Geometry of Numbers were laid down by Hermann Minkowski in his monograph \Geometrie der Zahlen", which was published in 1910, a year after his death. This subject is concerned with the interplay of compact convex 0-symmetric sets and lattices in Euclidean spaces. A set K ⊂ Rn is compact if it is closed and bounded, and it is convex if for any pair of points x; y 2 K the line segment connecting them is entirely contained in K, i.e. for any 0 ≤ t ≤ 1, tx + (1 − t)y 2 K. Further, K is called 0-symmetric if for any x 2 K, −x 2 K. Given such a set K in Rn, one can ask for an easy criterion to determine if K contains any nonzero points with integer coordinates. While for an arbitrary set K such a criterion can be rather difficult, in case of K as above a criterion purely in terms of its volume is provided by Minkowski's fundamental theorem. It is not difficult to see that K must in fact be convex and 0-symmetric for a criterion like this purely in terms of the volume of K to be possible. Indeed, the rectangle 2 R = (x; y) 2 R : 1=3 ≤ x ≤ 2=3; −t ≤ y ≤ t is convex for every t, but not 0-symmetric, and its area is 2t=3, which can be arbitrarily large depending on t while it still contains no integer points at all. On the other hand, the set R+ [ −R+ where R+ = f(x; y) 2 R : y ≥ 0g and −R+ = f(−x; −y):(x; y) 2 R+g is 0-symmetric, but not convex, and again can have arbitrarily large area while containing no integer points. Minkowski's theory applies not only to the integer lattice, but also to more general lattices. Our goal in this chapter is to introduce Minkowski's powerful theory, starting with the basic notions of lattices. 1 2 1. GEOMETRY OF NUMBERS 1.2. Lattices We start with an algebraic definition of lattices. Let a1;:::; ar be a collection of linearly independent vectors in Rn. n Definition 1.2.1. A lattice Λ of rank r, 1 ≤ r ≤ n, spanned by a1;:::; ar in R is the set of all possible linear combinations of the vectors a1;:::; ar with integer coefficients. In other words, ( r ) X Λ = span fa ;:::; a g := n a : n 2 for all 1 ≤ i ≤ r : Z 1 r i i i Z i=1 The set a1;:::; ar is called a basis for Λ. There are usually infinitely many different bases for a given lattice. Notice that in general a lattice in Rn can have any rank 1 ≤ r ≤ n. We will often however talk specifically about lattices of rank n, that is of full rank. The most obvious example of a lattice is the set of all points with integer coordinates in Rn: n Z = fx = (x1; : : : ; xn): xi 2 Z for all 1 ≤ i ≤ ng: Notice that the set of standard basis vectors e1;:::; en, where ei = (0;:::; 0; 1; 0;:::; 0); with 1 in i-th position is a basis for Zn. Another basis is the set of all vectors ei + ei+1; 1 ≤ i ≤ n − 1: n If Λ is a lattice of rank r in R with a basis a1;:::; ar and y 2 Λ, then there exist m1; : : : ; mr 2 Z such that r X y = miai = Am; i=1 where 0 1 m1 B . C r m = @ . A 2 Z ; mr and A is an n × r basis matrix for Λ of the form A = (a1 ::: ar), which has rank r. In other words, a lattice Λ of rank r in Rn can always be described as Λ = AZr, where A is its m×r basis matrix with real entries of rank r. As we remarked above, bases are not unique; as we will see later, each lattice has bases with particularly nice properties. An important property of lattices is discreteness. To explain what we mean more notation is needed. First notice that Euclidean space Rn is clearly not com- pact, since it is not bounded. It is however locally compact: this means that for every point x 2 Rn there exists an open set containing x whose closure is compact, for instance take an open unit ball centered at x. More generally, every subspace V of Rn is also locally compact. A subset Γ of V is called discrete if for each x 2 Γ there exists an open set S ⊆ V such that S \ Γ = fxg. For instance Zn is a discrete subset of Rn: for each point x 2 Zn the open ball of radius 1=2 centered at x contains no other points of Zn. We say that a discrete subset Γ is co-compact 1.2. LATTICES 3 in V if there exists a compact 0-symmetric subset U of V such that the union of translations of U by the points of Γ covers the entire space V , i.e. if [ V = fU + x : x 2 Γg: Here U + x = fu + x : u 2 Ug. Recall that a subset G is a subgroup of the additive abelian group Rn if it satisfies the following conditions: (1) Identity: 0 2 G, (2) Closure: For every x; y 2 G, x + y 2 G, (3) Inverses: For every x 2 G, −x 2 G. By Problems 1.3 and 1.4 a lattice Λ of rank r in Rn is a discrete co-compact subgroup of V = span Λ. In fact, the converse is also true. R Theorem 1.2.1. Let V be an r-dimensional subspace of Rn, and let Γ be a discrete co-compact subgroup of V . Then Γ is a lattice of rank r in Rn. Proof. In other words, we want to prove that Γ has a basis, i.e. that there exists a collection of linearly independent vectors a1;:::; ar in Γ such that Γ = span fa ;:::; a g. We start by inductively constructing a collection of vectors Z 1 r a1;:::; ar, and then show that it has the required properties. Let a1 6= 0 be a point in Γ such that the line segment connecting 0 and a1 contains no other points of Γ. Now assume a1;:::; ai−1, 2 ≤ i ≤ r, have been selected; we want to select ai. Let H = span fa ;:::; a g; i−1 R 1 i−1 and pick any c 2 Γ n Hi−1: such c exists, since Γ 6⊆ Hi−1 (otherwise Γ would not be co-compact in V ). Let Pi be the closed parallelotope spanned by the vectors a1;:::; ai−1; c. Notice that since Γ is discrete in V ,Γ \ Pi is a finite set. Moreover, since c 2 Pi,Γ \ Pi 6⊆ Hi−1. Then select ai such that d(ai;Hi−1) = min fd(y;Hi−1)g; y2(Pi\Γ)nHi−1 where for any point y 2 Rn, d(y;Hi−1) = inf fd(y; x)g: x2Hi−1 Let a1;:::; ar be the collection of points chosen in this manner. Then we have a 6= 0; a 2= span fa ;:::; a g 8 2 ≤ i ≤ r; 1 i Z 1 i−1 which means that a1;:::; ar are linearly independent. Clearly, span fa ;:::; a g ⊆ Γ: Z 1 r We will now show that Γ ⊆ span fa ;:::; a g: Z 1 r First of all notice that a1;:::; ar is certainly a basis for V , and so if x 2 Γ ⊆ V , then there exist c1; : : : ; cr 2 R such that r X x = ciai: i=1 4 1. GEOMETRY OF NUMBERS Notice that r X x0 = [c ]a 2 span fa ;:::; a g ⊆ Γ; i i Z 1 r i=1 where [ ] stands for the integer part function (i.e.
Load more

Recommended publications
  • Kissing Numbers: Surprises in Dimension Four Günter M
    Kissing numbers: Surprises in Dimension Four Günter M. Ziegler, TU Berlin The “kissing number problem” is a basic geometric problem that got its name from billards: Two balls “kiss” if they touch. The kissing number problem asks how many other balls can touch one given ball at the same time. If you arrange the balls on a pool ta- ble, it is easy to see that the answer is exactly six: Six balls just perfectly surround a given ball. Graphic: Detlev Stalling, ZIB Berlin and at the same time Vladimir I. Levenstein˘ in Russia proved that the correct, exact maximal numbers for the kissing number prob- lem are 240 in dimension 8, and 196560 in dimension 24. This is amazing, because these are also the only two dimensions where one knows a precise answer. It depends on the fact that mathemati- cians know very remarkable configurations in dimensions 8 and 24, which they call the E8 lattice and the Leech lattice, respectively. If, however, you think about this as a three-dimensional problem, So the kissing number problem remained unsolved, in particular, the question “how many balls can touch a given ball at the same for the case of dimension four. The so-called 24-cell, a four- time” becomes much more interesting — and quite complicated. In dimensional “platonic solid” of remarkable beauty (next page), fact, The 17th century scientific genius Sir Isaac Newton and his yields a configuration of 24 balls that would touch a given one in colleague David Gregory had a controversy about this in 1694 — four-dimensional space.
    [Show full text]
  • Minkowski's Geometry of Numbers
    Geometry of numbers MAT4250 — Høst 2013 Minkowski’s geometry of numbers Preliminary version. Version +2✏ — 31. oktober 2013 klokken 12:04 1 Lattices Let V be a real vector space of dimension n.Byalattice L in V we mean a discrete, additive subgroup. We say that L is a full lattice if it spans V .Ofcourse,anylattice is a full lattice in the subspace it spans. The lattice L is discrete in the induced topology, meaning that for any point x L 2 there is an open subset U of V whose intersection with L is just x.Forthesubgroup L to be discrete it is sufficient that this holds for the origin, i.e., that there is an open neigbourhood U about 0 with U L = 0 .Indeed,ifx L,thenthetranslate \ { } 2 U 0 = U + x is open and intersects L in the set x . { } Aoftenusefulpropertyofalatticeisthatithasonlyfinitelymanypointsinany bounded subset of V .Toseethis,letS V be a bounded set and assume that L S is ✓ \ infinite. One may then pick a sequence xn of different elements from L S.SinceS { } \ is bounded, its closure is compact and xn has a subsequence that converges, say to { } x. Then x is an accumulation point in L;everyneigbourhoodcontainselementsfrom L distinct from x. The lattice associated to a basis If = v1,...,vn is any basis for V ,onemayconsiderthesubgroupL = Zvi of B { } B i V consisting of all linear combinations of the vi’s with integral coefficients. Clearly L P B is a discrete subspace of V ;IfonetakesU to be an open ball centered at the origin and having radius less than mini vi ,thenU L = 0 .HenceL is a full lattice in k k \ { } B V .
    [Show full text]
  • Gazette Des Mathématiciens
    JUILLET 2018 – No 157 la Gazette des Mathématiciens • Autour du Prix Fermat • Mathématiques – Processus ponctuels déterminantaux • Raconte-moi... le champ libre gaussien • Tribune libre – Quelle(s) application(s) pour le plan Torossian-Villani la GazetteComité de rédaction des Mathématiciens Rédacteur en chef Sébastien Gouëzel Boris Adamczewski Université de Nantes Institut Camille Jordan, Lyon [email protected] [email protected] Sophie Grivaux Rédacteurs Université de Lille [email protected] Thomas Alazard École Normale Supérieure de Paris-Saclay [email protected] Fanny Kassel IHÉS Maxime Bourrigan [email protected] Lycée Saint-Geneviève, Versailles [email protected] Pauline Lafitte Christophe Eckès École Centrale, Paris Archives Henri Poincaré, Nancy [email protected] [email protected] Damien Gayet Romain Tessera Institut Fourier, Grenoble Université Paris-Sud [email protected] [email protected] Secrétariat de rédaction : smf – Claire Ropartz Institut Henri Poincaré 11 rue Pierre et Marie Curie 75231 Paris cedex 05 Tél. : 01 44 27 67 96 – Fax : 01 40 46 90 96 [email protected] – http://smf.emath.fr Directeur de la publication : Stéphane Seuret issn : 0224-8999 À propos de la couverture. Cette image réalisée par Alejandro Rivera (Institut Fourier) est le graphe de la partie positive du champ libre gaussien sur le tore plat 2=(2á2) projeté sur la somme des espaces propres du laplacien de valeur propre plus petite que 500. Les textures et couleurs ajoutées sur le graphe sont pure- ment décoratives. La fonction a été calculée avec C++ comme une somme de polynômes trigonométriques avec des poids aléatoires bien choisis, le graphe a été réalisé sur Python et les textures et couleurs ont été réalisées avec gimp.
    [Show full text]
  • Basic Concepts of Linear Algebra by Jim Carrell Department of Mathematics University of British Columbia 2 Chapter 1
    1 Basic Concepts of Linear Algebra by Jim Carrell Department of Mathematics University of British Columbia 2 Chapter 1 Introductory Comments to the Student This textbook is meant to be an introduction to abstract linear algebra for first, second or third year university students who are specializing in math- ematics or a closely related discipline. We hope that parts of this text will be relevant to students of computer science and the physical sciences. While the text is written in an informal style with many elementary examples, the propositions and theorems are carefully proved, so that the student will get experince with the theorem-proof style. We have tried very hard to em- phasize the interplay between geometry and algebra, and the exercises are intended to be more challenging than routine calculations. The hope is that the student will be forced to think about the material. The text covers the geometry of Euclidean spaces, linear systems, ma- trices, fields (Q; R, C and the finite fields Fp of integers modulo a prime p), vector spaces over an arbitrary field, bases and dimension, linear transfor- mations, linear coding theory, determinants, eigen-theory, projections and pseudo-inverses, the Principal Axis Theorem for unitary matrices and ap- plications, and the diagonalization theorems for complex matrices such as the Jordan decomposition. The final chapter gives some applications of symmetric matrices positive definiteness. We also introduce the notion of a graph and study its adjacency matrix. Finally, we prove the convergence of the QR algorithm. The proof is based on the fact that the unitary group is compact.
    [Show full text]
  • Local Covering Optimality of Lattices: Leech Lattice Versus Root Lattice E8
    Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8 Achill Sch¨urmann, Frank Vallentin ∗ 10th November 2004 Abstract We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the formerly ∗ best known given by the lattice A8. Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense. 1 Introduction The Leech lattice is the exceptional lattice in dimension 24. Soon after its discovery by Leech [Lee67] it was conjectured that it is extremal for several geometric problems in R24: the kissing number problem, the sphere packing problem and the sphere covering problem. In 1979, Odlyzko and Sloane and independently Levenshtein solved the kissing number problem in dimension 24 by showing that the Leech lattice gives an optimal solution. Two years later, Bannai and Sloane showed that it gives the unique solution up to isometries (see [CS88], Ch. 13, 14). Unlike the kissing number problem, the other two problems are still open. Recently, Cohn and Kumar [CK04] showed that the Leech lattice gives the unique densest lattice sphere packing in R24.
    [Show full text]
  • Sharp Finiteness Principles for Lipschitz Selections: Long Version
    Sharp finiteness principles for Lipschitz selections: long version By Charles Fefferman Department of Mathematics, Princeton University, Fine Hall Washington Road, Princeton, NJ 08544, USA e-mail: [email protected] and Pavel Shvartsman Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, Israel e-mail: [email protected] Abstract Let (M; ρ) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from M into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness number. Contents 1. Introduction. 2 1.1. Main definitions and main results. 2 1.2. Main ideas of our approach. 3 2. Nagata condition and Whitney partitions on metric spaces. 6 arXiv:1708.00811v2 [math.FA] 21 Oct 2017 2.1. Metric trees and Nagata condition. 6 2.2. Whitney Partitions. 8 2.3. Patching Lemma. 12 3. Basic Convex Sets, Labels and Bases. 17 3.1. Main properties of Basic Convex Sets. 17 3.2. Statement of the Finiteness Theorem for bounded Nagata Dimension. 20 Math Subject Classification 46E35 Key Words and Phrases Set-valued mapping, Lipschitz selection, metric tree, Helly’s theorem, Nagata dimension, Whitney partition, Steiner-type point. This research was supported by Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF). The first author was also supported in part by NSF grant DMS-1265524 and AFOSR grant FA9550-12-1-0425.
    [Show full text]
  • The BEST WRITING on MATHEMATICS
    The BEST WRITING on MATHEMATICS 2012 The BEST WRITING on MATHEMATICS 2012 Mircea Pitici, Editor FOREWORD BY DAVID MUMFORD P RI NC E TO N U N IVER S I T Y P RE SS P RI NC E TO N A N D OX FORD Copyright © 2013 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved ISBN 978- 0- 691-15655-2 This book has been composed in Perpetua Printed on acid- free paper. ∞ Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 For my parents Contents Foreword: The Synergy of Pure and Applied Mathematics, of the Abstract and the Concrete DAVID MUMFORD ix Introduction MIRCEA PITICI xvii Why Math Works MARIO LIVIO 1 Is Mathematics Discovered or Invented? TIMOTHY GOWERS 8 The Unplanned Impact of Mathematics PETER ROWLETT 21 An Adventure in the Nth Dimension BRIAN HAYES 30 Structure and Randomness in the Prime Numbers TERENCE TAO 43 The Strangest Numbers in String Theory JOHN C. BAEZ AND JOHN HUERTA 50 Mathematics Meets Photography: The Viewable Sphere DAVID SWART AND BRUCE TORRENCE 61 Dancing Mathematics and the Mathematics of Dance SARAH- MARIE BELCASTRO AND KARL SCHAFFER 79 Can One Hear the Sound of a Theorem? ROB SCHNEIDERMAN 93 Flat- Unfoldability and Woven Origami Tessellations ROBERT J. LANG 113 A Continuous Path from High School Calculus to University Analysis TIMOTHY GOWERS 129 viii Contents Mathematics Teachers’ Subtle, Complex Disciplinary Knowledge BRENT DAVIS 135 How to Be a Good Teacher Is an Undecidable Problem ERICA FLAPAN 141 How Your Philosophy of Mathematics Impacts Your Teaching BONNIE GOLD 149 Variables in Mathematics Education SUSANNA S.
    [Show full text]
  • Mathematical Circus & 'Martin Gardner
    MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3.
    [Show full text]
  • Chapter 2 Geometry of Numbers
    Chapter 2 Geometry of numbers Literature: W.M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics 785, Springer Verlag 1980, Chap.II, xx1,2, Chap. IV, x1 J.W.S. Cassels, An Introduction to the Geometry of Numbers, Springer Verlag 1997, Classics in Mathematics series, reprint of the 1971 edition C.L. Siegel, Lectures on the Geometry of Numbers, Springer Verlag 1989 2.1 Introduction Geometry of numbers is concerned with the study of lattice points in certain bodies n in R , where n > 2. We discuss Minkowski's theorems on lattice points in central symmetric convex bodies. In this introduction we give the necessary definitions. Lattices. A (full) lattice in Rn is an additive group L = fz1v1 + ··· + znvn : z1; : : : ; zn 2 Zg n where fv1;:::; vng is a basis of R , i.e., fv1;:::; vng is linearly independent. We call fv1;:::; vng a basis of L. The determinant of L is defined by d(L) := j det(v1;:::; vn)j; that is, the absolute value of the determinant of the matrix with columns v1;:::; vn. 7 We show that the determinant of a lattice does not depend on the choice of the basis. Recall that GL(n; Z) is the multiplicative group of n×n-matrices with entries in Z and determinant ±1. Lemma 2.1. Let L be a lattice, and fv1;:::; vng, fw1;:::; wng two bases of L. Then there is a matrix U = (uij) 2 GL(n; Z) such that n X (2.1) wi = uijvj for i = 1; : : : ; n: j=1 Consequently, j det(v1;:::; vn)j = j det(w1;:::; wn)j.
    [Show full text]
  • Chapter 8. Eigenvalues: Further Applications and Computations
    8.1 Diagonalization of Quadratic Forms 1 Chapter 8. Eigenvalues: Further Applications and Computations 8.1. Diagonalization of Quadratic Forms Note. In this section we define a quadratic form and relate it to a vector and ma- trix product. We define diagonalization of a quadratic form and give an algorithm to diagonalize a quadratic form. The fact that every quadratic form can be diago- nalized (using an orthogonal matrix) is claimed by the “Principal Axis Theorem” (Theorem 8.1). It is applied in Section 8.2 to address quadratic surfaces. Definition. A quadratic form f(~x) = f(x1, x2,...,xn) in n variables is a polyno- mial that can be written as n f(~x)= u x x X ij i j i,j=1; i≤j where not all uij are zero. Note. For n = 2, a quadratic form is f(x, y)= ax2 + bxy + xy2. The graph of such an equation in the xy-plane is a conic section (ellipse, hyperbola, or parabola). For n = 3, we have 2 2 2 f(x1, x2, x3)= u11x1 + u12x1x2 + u13x1x3 + u22x2 + u23x2x3 + u33x2. We’ll see in the next section that the graph of f(x1, x2, x3) is a quadric surface. We 8.1 Diagonalization of Quadratic Forms 2 can easily verify that u11 u12 u13 x1 [f(x1, x2, x3)]= [x1, x2, x3] 0 u22 u23 x2 . 0 0 u33 x3 Definition. In the quadratic form ~xT U~x, the matrix U is the upper-triangular coefficient matrix of the quadratic form. Example. Page 417 Number 4(a).
    [Show full text]
  • Introductory Number Theory Course No
    Introductory Number Theory Course No. 100 331 Spring 2006 Michael Stoll Contents 1. Very Basic Remarks 2 2. Divisibility 2 3. The Euclidean Algorithm 2 4. Prime Numbers and Unique Factorization 4 5. Congruences 5 6. Coprime Integers and Multiplicative Inverses 6 7. The Chinese Remainder Theorem 9 8. Fermat’s and Euler’s Theorems 10 × n × 9. Structure of Fp and (Z/p Z) 12 10. The RSA Cryptosystem 13 11. Discrete Logarithms 15 12. Quadratic Residues 17 13. Quadratic Reciprocity 18 14. Another Proof of Quadratic Reciprocity 23 15. Sums of Squares 24 16. Geometry of Numbers 27 17. Ternary Quadratic Forms 30 18. Legendre’s Theorem 32 19. p-adic Numbers 35 20. The Hilbert Norm Residue Symbol 40 21. Pell’s Equation and Continued Fractions 43 22. Elliptic Curves 50 23. Primes in arithmetic progressions 66 24. The Prime Number Theorem 75 References 80 2 1. Very Basic Remarks The following properties of the integers Z are fundamental. (1) Z is an integral domain (i.e., a commutative ring such that ab = 0 implies a = 0 or b = 0). (2) Z≥0 is well-ordered: every nonempty set of nonnegative integers has a smallest element. (3) Z satisfies the Archimedean Principle: if n > 0, then for every m ∈ Z, there is k ∈ Z such that kn > m. 2. Divisibility 2.1. Definition. Let a, b be integers. We say that “a divides b”, written a | b , if there is an integer c such that b = ac. In this case, we also say that “a is a divisor of b” or that “b is a multiple of a”.
    [Show full text]
  • Proved for Real Hilbert Spaces. Time Derivatives of Observables and Applications
    AN ABSTRACT OF THE THESIS OF BERNARD W. BANKSfor the degree DOCTOR OF PHILOSOPHY (Name) (Degree) in MATHEMATICS presented on (Major Department) (Date) Title: TIME DERIVATIVES OF OBSERVABLES AND APPLICATIONS Redacted for Privacy Abstract approved: Stuart Newberger LetA andH be self -adjoint operators on a Hilbert space. Conditions for the differentiability with respect totof -itH -itH <Ae cp e 9>are given, and under these conditionsit is shown that the derivative is<i[HA-AH]e-itHcp,e-itHyo>. These resultsare then used to prove Ehrenfest's theorem and to provide results on the behavior of the mean of position as a function of time. Finally, Stone's theorem on unitary groups is formulated and proved for real Hilbert spaces. Time Derivatives of Observables and Applications by Bernard W. Banks A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1975 APPROVED: Redacted for Privacy Associate Professor of Mathematics in charge of major Redacted for Privacy Chai an of Department of Mathematics Redacted for Privacy Dean of Graduate School Date thesis is presented March 4, 1975 Typed by Clover Redfern for Bernard W. Banks ACKNOWLEDGMENTS I would like to take this opportunity to thank those people who have, in one way or another, contributed to these pages. My special thanks go to Dr. Stuart Newberger who, as my advisor, provided me with an inexhaustible supply of wise counsel. I am most greatful for the manner he brought to our many conversa- tions making them into a mutual exchange between two enthusiasta I must also thank my parents for their support during the earlier years of my education.Their contributions to these pages are not easily descerned, but they are there never the less.
    [Show full text]