# THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p

Save this PDF as:

Size: px
Start display at page:

Download "THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p"

## Transcription

1 THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS PETE L CLARK Circ 1870, Zolotrev observed tht the Legendre symbol ( p ) cn be interpreted s the sign of multipliction by viewed s permuttion of the set Z/pZ He used this observtion to give strikingly originl proof of qudrtic reciprocity [2] We shll not discuss Zolotrev s proof per se, but rther 2005 pper of W Duke nd K Hopkins which explores the connection between permuttions nd qudrtic symbols in more mbitious wy En route, we explore qudrtic reciprocity s expressed in terms of the Kronecker symbol 1 The Kronecker Symbol The Jcobi symbol ( n ) is n extension of the Legendre symbol ( p ) which is defined for ny positive odd integer n by ( 1 ) = 1 for ll Z; if n = r i=1 p i, then ( r ( ) n) i=1 For n integer, define ( ) 0 0 (mod 2) = 1 1, 7 (mod 8) 2, 1 3, 5 (mod 8) ( ) 0 = 0 = 1 > 0 1, 1 < 0 ( ) { } 0 1 = 0 1 = 1 With these dditionl rules there is unique extension of the Jcobi symbol to symbol ( n ) defined for ny n, Z such tht for ll integers n,, b, we hve ( n b ) = ( n )( n b b ) One lso hs ( n ) = ( n )( b n ), ie, the symbol is bi-multiplictive This extension of the Jcobi symbol is known s the Kronecker symbol When n is not odd nd positive, some uthors (eg [1]) define ( n ) only when 0, 1 (mod 4) It is not worth our time to discuss these two conventions, but we note tht ll of our results involve only this restricted Kronecker symbol For odd n Z +, define n = ( 1) n 1 2 n Full qudrtic reciprocity ie, the usul QR lw together with its First nd Second Supplements is equivlent to one elegnt identity: for Z nd n odd positive n Z, (1) p i n n) 1

2 2 PETE L CLARK 2 The Duke-Hopkins Reciprocity Lw Let be finite commuttive group (written multiplictively) of order n define n ction of (Z/nZ) on, by ( mod n) g := g By Lgrnge s Theorem, g n = 1, so tht g = g if (mod n) nd is well defined It is immedite tht ech gives homomorphism from to ; moreover, since (Z/nZ), there exists b (Z/nZ) such tht b 1 (mod n), nd then b = b = Id, so tht ech is n utomorphism of As for ny group ction on set, this determines homomorphism from (Z/nZ) to the group Sym( of permuttions of, the ltter group being isomorphic to S n, the symmetric group on n elements Recll tht there is unique homomorphism from S n to the cyclic group Z 2 given by the sign of the permuttion Therefore we hve composite homomorphism which we will denote by (Z/nZ) Sym( Z 2 (mod n) ( Exmple 21 (Zolotrev): Let p be n odd prime nd ( = is the cyclic group of order p The mpping (Z/pZ) Z 2 given by is nothing else thn the usul Legendre symbol ( p ) Indeed, the group (Z/pZ) is cyclic of even order, so dmits unique surjective homomorphism to the group Z 2 = {±1}: if g is primitive root mod p, we send g to 1 nd hence every odd power of g to 1 nd every even power of g to +1 This precisely describes the Legendre ( ) symbol ( p ) Thus it suffices to see tht for some (Z/pZ) we hve = 1, ie, the sign of the permuttion n n is 1 To see this, switch to dditive nottion, viewing s the isomorphic group (Z/pZ, +); the ction in question is now just multipliction by nonzero element If g is primitive root modulo p, multipliction by g fixes 0 nd cycliclly permutes ll p 1 nonzero elements, so is cycle of even order nd hence n odd permuttion: thus ( g ) = 1 ) The next result shows tht the symbol ( ) is lso bi-multiplictive Proposition 1 For i = 1, 2 let i be finite commuttive group of order n i nd (Z/n 1 n 2 Z) Then ( ) ( ) ( ) (mod n1 ) (mod n2 ) = Proof: If (Z/n 1 n 2 Z), then (mod n2) (g 1, g 2 ) = (g1, g2) (mod n1) = (g1, g2 ) After identifying 1 (resp 2 ) with the subset 1 e 2 (resp e 1 2 ) of 1 2, the permuttion tht induces on 1 2 is the product of the permuttion tht (mod n) 1 induces on 1 with the permuttion tht (mod n) 2 induces on 2 Let us now consider the ction of 1 on Sym( Let r 1 be the number of fixed We

3 THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS 3 points of 1 More concretely, 1 g = g 1 = g iff g hs order 1 or 2 Note tht r 1 1 becuse of the identity element The n r 1 other elements of re ll distinct from their multiplictive inverses, so there exists positive integer r 2 such tht n r 1 = 2r 2 Definition: We put = ( 1) r2 r1 = ( 1) r2 n r1 Lemm 2 For ny finite commuttive group, we hve 0 or 1 (mod 4) Proof: Let n = If n is odd, then by Lgrnge the only g with g 1 = g is the identity, so tht r 1 = 1 nd r 2 = n 1 2 In this cse = = ( 1) n 1 2 n 1 (mod 4) If n is even, then n r 1 = 2r 2 0 (mod 2), so r 1 is even nd hence is t lest 2, so = ( 1) r2 n r1 0 (mod 4) ( ) So the Kronecker symbol is lwys defined (even in the restricted sense) Theorem 3 (Duke-Hopkins Reciprocity Lw) For finite commuttive group nd n integer, we hve The proof will be given in the next section Corollry 4 ) Suppose hs odd order n Then for ny (Z/nZ), we hve n b) Tking = Z n we recover (1) c) We hve ( ) = 1 for ll (Z/nZ) iff n is squre Proof of Corollry 4: In the proof of Lemm 2 we sw tht = n ; prt ) then follows immeditely from the reciprocity lw By prt ), the symbol ( ) cn be computed using ny group of order n, so fctor n into product p 1 p r of not necessrily distinct primes nd pply Exmple 21: we get ( ) = r i=1 ( p i ) = ( n ) This gives prt b) Finlly, using the Chinese Reminder Theorem it is esy to see tht there is some such tht ( n ) = 1 iff n is not squre 3 The Proof Enumerte the elements of s g 1,, g n nd the chrcters of s χ 1,, χ n Let M be the n n mtrix whose (i, j) entry is χ i (g j ) Since ny chrcter χ X( hs vlues on the unit circle in C, we hve χ 1 = χ Therefore the number r 1 of fixed points of 1 on is the sme s the number of chrcters χ such tht χ = χ, ie, rel-vlued chrcters Thus the effect of complex conjugtion on the chrcter mtrix M is to fix ech row corresponding to rel-vlued chrcter nd to otherwise swp the ith row with the jth row where χ j = χ i In ll r 2 pirs of rows get swpped, so Moreover, with M = (M) t, we hve det(m) = det(m) ( 1) r2 MM = ni n,

4 4 PETE L CLARK so tht so det(m) det(m) = n n, (2) det(m) 2 = ( 1) r2 n n = ( 1) r2 n r1 n 2r2 = l 2, where l = n r2 (In prticulr det(m) 2 is positive integer Note tht det(m) itself lies in Q( ), nd is not rtionl if n is odd) So for ny Z, we hve ( ) ( ) det(m) 2 (3) The chrcter mtrix M hs vlues in the cyclotomic field Q(ζ n ), which is lois extension of Q, with lois group isomorphic to (wht concidence!) (Z/nZ), n explicit isomorphism being given by mking (Z/nZ) correspond to the unique utomorphism σ of Q(ζ n ) stisfying σ (ζ n ) = ζ n (All of this is elementry lois theory except for the more number-theoretic fct tht the cyclotomic polynomil Φ n is irreducible over Q) In prticulr the group (Z/nZ) lso cts by permuttions on the chrcter group X(, nd indeed in exctly the sme wy it cts on : g, ( χ)(g) = χ(g ) = (χ(g)) = χ (g), so χ = χ This hs the following beutiful consequence: For (Z/nZ), pplying the lois utomorphism σ to the chrcter mtrix M induces permuttion of the rows which is the sme s the permuttion of In prticulr the signs re the sme, so ( (4) det(σ M) = det(m) Combining (2) nd (4), we get tht for ll (Z/nZ), σ ( ( ) ) = Now, by the multiplictivity on both sides it is enough to prove Theorem 3 when = p is prime not dividing n nd when = 1 Proposition 5 Let p be prime not dividing n TFAE: ) σ p ( ) = b) p splits in Q( ) c) ( p ) = 1 The proof of this stndrd result in lgebric number theory is omitted for now We deduce tht ( ) ( p p Finlly, when = 1, σ 1 is simply complex conjugtion, so ( ) 1 = σ { } ( ) 1( ) = > 0 =, < 0 1 so ( ) ( ) 1 1 This completes the proof of Theorem 3

5 THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS 5 4 In fct the rel Duke-Hopkins reciprocity lw is n ssertion bout group of order n which is not necessrily commuttive In this cse, the mp g g need not be n utomorphism of, so more sophisticted pproch is needed Rther, one considers the ction of (Z/nZ) on the conjugcy clsses {C 1,, C m } of : if g = xhx 1 then g = xh x 1, so this mkes sense We further define r 1 to be the number of rel conjugcy clsses C = C 1 nd ssume tht in our lbelling C 1,, C r1 re ll rel nd define r 2 by the eqution m = r 1 + 2r 2 Then in plce of our (nottion which is not used in [1]), one hs the discriminnt d( = ( 1) r2 n r1 r 1 j=1 C j 1 The Duke-Hopkins reciprocity lw sserts tht for (Z/nZ), d( The proof is very similr, except the group X( of one-dimensionl chrcters gets replced by the set {χ 1,, χ m } of chrcters (ie, trce functions) of the irreducible complex representtions of Perhps surprisingly, the only prt of the proof which looks truly deeper is the clim tht d( 0, 1 (mod 4) which is required, ccording to the conventions of [1], for the Kronecker symbol ( d( ) cn be defined Duke nd Hopkins suggest this s n nlogue of Stickelberger s theorem in lgebric number theory which sserts tht the discriminnt of ny number field is n integer which is 0 or 1 modulo 4; moreover they dpt 1928 proof of tht theorem due to Issi Schur References [1] W Duke nd K Hopkins, Qudrtic reciprocity in finite group Amer Mth Monthly 112 (2005), no 3, [2] Zolotrev, Nouvelle démonstrtion de l loi de réciprocité de Legendre Nouvelles Ann Mth (2) 11 (1872)

### MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35

MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.

nd Interntionl Mthemticl Olympid Wshington, DC, United Sttes of Americ July 8 9, 001 Problems Ech problem is worth seven points. Problem 1 Let ABC be n cute-ngled tringle with circumcentre O. Let P on

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

### MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

### Semigroup of generalized inverses of matrices

Semigroup of generlized inverses of mtrices Hnif Zekroui nd Sid Guedjib Abstrct. The pper is divided into two principl prts. In the first one, we give the set of generlized inverses of mtrix A structure

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### 8 Laplace s Method and Local Limit Theorems

8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

### Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

### Lecture 3. Limits of Functions and Continuity

Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

### UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that

Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de

### The Leaning Tower of Pingala

The Lening Tower of Pingl Richrd K. Guy Deprtment of Mthemtics & Sttistics, The University of Clgry. July, 06 As Leibniz hs told us, from 0 nd we cn get everything: Multiply the previous line by nd dd

### USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

### 440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam

440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

### Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

### How to simulate Turing machines by invertible one-dimensional cellular automata

How to simulte Turing mchines by invertible one-dimensionl cellulr utomt Jen-Christophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex

### SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

### MTH 505: Number Theory Spring 2017

MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of \$ nd \$ s two denomintions of coins nd \$c

### State Minimization for DFAs

Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

### #A29 INTEGERS 17 (2017) EQUALITY OF DEDEKIND SUMS MODULO 24Z

#A29 INTEGERS 17 (2017) EQUALITY OF DEDEKIND SUMS MODULO 24Z Kurt Girstmir Institut für Mthemtik, Universität Innsruck, Innsruck, Austri kurt.girstmir@uik.c.t Received: 10/4/16, Accepted: 7/3/17, Pulished:

### Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### Summary of Elementary Calculus

Summry of Elementry Clculus Notes by Wlter Noll (1971) 1 The rel numbers The set of rel numbers is denoted by R. The set R is often visulized geometriclly s number-line nd its elements re often referred

### Matrices and Determinants

Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

### arxiv: v1 [math.gt] 10 Sep 2007

SURGERY DESCRIPTION OF COLORED KNOTS R.A. LITHERLAND AND STEVEN D. WALLACE rxiv:0709.1507v1 [mth.gt] 10 Sep 2007 Abstrct. The pir (K, ρ) consisting of knot K S 3 nd surjective mp ρ from the knot group

### a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.

Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

### A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions

Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### 1 From NFA to regular expression

Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

### On the degree of regularity of generalized van der Waerden triples

On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### Diophantine Steiner Triples and Pythagorean-Type Triangles

Forum Geometricorum Volume 10 (2010) 93 97. FORUM GEOM ISSN 1534-1178 Diophntine Steiner Triples nd Pythgoren-Type Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

### NWI: Mathematics. Various books in the library with the title Linear Algebra I, or Analysis I. (And also Linear Algebra II, or Analysis II.

NWI: Mthemtics Literture These lecture notes! Vrious books in the librry with the title Liner Algebr I, or Anlysis I (And lso Liner Algebr II, or Anlysis II) The lecture notes of some of the people who

### 1.3 Regular Expressions

56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued

### Chapter 2. Determinants

Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

### Matrix Eigenvalues and Eigenvectors September 13, 2017

Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

### Numerical integration

2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### Acta Universitatis Carolinae. Mathematica et Physica

Act Universittis Croline. Mthemtic et Physic Thoms N. Vougiouklis Cyclicity in specil clss of hypergroups Act Universittis Croline. Mthemtic et Physic, Vol. 22 (1981), No. 1, 3--6 Persistent URL: http://dml.cz/dmlcz/142458

### REVIEW Chapter 1 The Real Number System

Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole

### Math 211A Homework. Edward Burkard. = tan (2x + z)

Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =

### expression simply by forming an OR of the ANDs of all input variables for which the output is

2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

### Precalculus Spring 2017

Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

### A new algorithm for generating Pythagorean triples 1

A new lgorithm for generting Pythgoren triples 1 RH Dye 2 nd RWD Nicklls 3 The Mthemticl Gzette (1998; 82 (Mrch, No. 493, pp. 86 91 http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf 1 Introduction

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### Homework Solution - Set 5 Due: Friday 10/03/08

CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

### MA Handout 2: Notation and Background Concepts from Analysis

MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

### New Expansion and Infinite Series

Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

### ECON 331 Lecture Notes: Ch 4 and Ch 5

Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve

### Introduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices

Introduction to Determinnts Remrks The determinnt pplies in the cse of squre mtrices squre mtrix is nonsingulr if nd only if its determinnt not zero, hence the term determinnt Nonsingulr mtrices re sometimes

### MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

### Introduction to Numerical Analysis

Introduction to Numericl Anlysis Doron Levy Deprtment of Mthemtics nd Center for Scientific Computtion nd Mthemticl Modeling (CSCAMM) University of Mrylnd June 14, 2012 D. Levy CONTENTS Contents 1 Introduction

### Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

### 1 Online Learning and Regret Minimization

2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in

### dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

### Math 324 Course Notes: Brief description

Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd

### Intuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras

Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore-632014, TN, Indi

Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

### ON THE ENTRY SUM OF CYCLOTOMIC ARRAYS. Don Coppersmith IDACCR. John Steinberger UC Davis

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 ON THE ENTRY SUM OF CYCLOTOMIC ARRAYS Don Coppersmith IDACCR John Steinberger UC Dvis Received: 3/29/05, Revised: 0/8/06, Accepted:

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### New Integral Inequalities for n-time Differentiable Functions with Applications for pdfs

Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute

### Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations

18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following

### Arithmetic & Algebra. NCTM National Conference, 2017

NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

### CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

### Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

### 1 Error Analysis of Simple Rules for Numerical Integration

cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

### UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

### Sturm-Liouville Theory

LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

### Boolean Algebra. Boolean Algebra

Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with: - two inry opertions, commonly denoted y + nd, - unry opertion, usully denoted y ˉ or ~ or, - two elements usully clled zero nd

### Counting intersections of spirals on a torus

Counting intersections of spirls on torus 1 The problem Consider unit squre with opposite sides identified. For emple, if we leve the centre of the squre trveling long line of slope 2 (s shown in the first

### CSCI FOUNDATIONS OF COMPUTER SCIENCE

1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not

### S. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:

FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy

### Practice final exam solutions

University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If

### STRAND B: NUMBER THEORY

Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,

### Chapter 4. Lebesgue Integration

4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.

### Linear Systems with Constant Coefficients

Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

### Some examples of Mahler measures as multiple polylogarithms

Some exmples of Mhler mesures s multiple polylogrithms Mtilde N. Llín, University of Texs t Austin. Deprtment of Mthemtics. University Sttion C. Austin, TX 787, USA Abstrct The Mhler mesures of certin

### 4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

### Variational Techniques for Sturm-Liouville Eigenvalue Problems

Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment

### 2 Definitions and Basic Properties of Extended Riemann Stieltjes Integrals

2 Definitions nd Bsic Properties of Extended Riemnn Stieltjes Integrls 2.1 Regulted nd Intervl Functions Regulted functions Let X be Bnch spce, nd let J be nonempty intervl in R, which my be bounded or

### u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.

Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex

### NOTES AND PROBLEMS: INTEGRATION THEORY

NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFS-I to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents

### Math 4200: Homework Problems

Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,

### Triangles The following examples explore aspects of triangles:

Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

### Parameterized Norm Form Equations with Arithmetic progressions

Prmeterized Norm Form Equtions with Arithmetic progressions A. Bérczes,1 A. Pethő b, V. Ziegler c, Attil Bérczes Institute of Mthemtics, University of Debrecen Number Theory Reserch Group, Hungrin Acdemy