9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.

Size: px
Start display at page:

Download "9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic."

Transcription

1 Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this is the only case we ae inteested in. But since the poof woks equally well fo any finite fld we pove the moe geneal esult. Poof. The exponent of a finite goup G is the smallest numbe e > 0 such that g e = e fo all g G. By Lagange s Theoem, if G is of ode n g n = e fo all g G. Hence e n. In fact it is easy to see that e n. Fo suppose d = gcd(e, n). Then It follows that d = e + ns. g d = (g e ) (g n ) s = e. We assume in the est of the poof that F is a finite fld, containing q elements. Lemma 9.1. The exponent of F is q

2 9.1. THE MULTIPLICATIVE GROUP CHAPTER OF A FINITE 9. PRIMITIVE FIELD ROOTS Poof. Each of the q 1 elements x F ( all the elements of F except 0) satisfs the equation x e 1 = 0 ove the fld F. But this equation has at most e oots. It follows that Since e q 1 it follows that q 1 e. e = q 1. Lemma 9.2. If A is a finite abelian goup, and a, b A have copime odes, s then ode(ab) = s. Poof. Suppose ode(ab) = n. Then (ab) s = 1 = n s. On the othe hand, since, s ae copime we can find x, y Z such that But then x + sy = 1. (ab) sy = a sy = a 1 x = a. It follows that n. Similaly s n. Since gcd(, s) = 1 this impls that s n. Hence n = s. Lemma 9.3. Suppose A is a finite abelian goup of exponent e. Then A has an element of ode e

3 9.2. PRIMITIVE ROOTS CHAPTER 9. PRIMITIVE ROOTS Poof. Let e = p e 1 1 p e, whee p 1,..., p ae distinct pimes. Suppose i [1, ]. Thee must be an element a i whose ode is divisible by p e i i ; fo othewise we could take e/p i as exponent in place of e. Let Then has ode p e i i. Let ode(a i ) = p e i i q i. b i = a q i i a = b 1 b. Since the odes p e 1 1,..., p e of b 1,..., b ae mutually copime it follows fom the last Lemma that that the ode of a is p e 1 1 p e = e. It follows fom the fist and last of these 3 Lemmas that we can find an element a F of ode q 1. In othe wods, F is cyclic. 9.2 Pimitive oots Definition 9.1. A geneato of (Z/p) is called a pimitive oot mod p. Example: Take p = 7. Then mod 7; so 2 has ode 3 mod 7, and is not a pimitive oot. Howeve, mod 7, mod 7. Since the ode of an element divides the ode of the goup, which is 6 in this case, it follows that 3 has ode 6 mod 7, and so is a pimitive oot. If g geneates the cyclic goup G then so does g 1. Hence is also a pimitive oot mod mod

4 9.2. PRIMITIVE ROOTS CHAPTER 9. PRIMITIVE ROOTS Poposition 9.1. If a is a pimitive oot mod p then a is a pimitive oot if and only if gcd(, p 1) = 1. Poof. This is eally a esult fom elementay goup theoy: If G is a cyclic goup of ode n geneated by g, then g is also a geneato if and only if gcd(, n) = 1. Fo suppose gcd(, n) = 1. If g has ode d then But since gcd(, n) = 1 Hence (g ) d = e, g d = e. d = n = d = n. Convesely, suppose g geneates the goup. Then g is a powe of g, say and in paticula gcd(, n) = 1. g = (g ) s = g s. s 1 mod n, Coollay 9.1. Thee ae φ(p 1) pimitive oots mod p. Example: Suppose p = 11. Then (Z/11) has ode 10, so its elements have odes 1,2,5 o 10. Now 2 5 = 32 1 mod 11. So 2 must be a pimitive oot mod 11. Thee ae φ(10) = 4 pimitive oots mod 11, namely 2, 2 3, 2 7, 2 9 mod 11, 2, 8, 7,

5 9.3. PRIME POWER MODULI CHAPTER 9. PRIMITIVE ROOTS 9.3 Pime powe moduli Suppose Then n = p e p e. (Z/n) = (Z/p e 1 1 ) (Z/p e 1 1 ). Thus the stuctue of the multiplicative goups (Z/n) will be completely detemined once we know the stuctue of (Z/p e ) fo each pime powe p e. It tuns out that we have aleady done most of the wok in detemining the stuctue of (Z/p). Poposition 9.2. If p is an odd pime numbe then the multiplicative goup is cyclic fo all e 1. (Z/p e ) Poof. We have poved the esult fo e = 1. We deive the esult fo e > 1 in the following way. The goup (Z/p e ) has ode φ(p e ) = p e 1 (p 1). By the Theoem, thee exists an element a with Evidently ode(a mod p) = p 1. ode(a mod p) ode(a mod p e ). Thus the ode of a mod p e is divisible by p 1, say Then ode(a mod p e ) = (p 1). ode(a mod p e ) = p 1. It is theefoe sufficnt by Lemma 9.2 to show that thee exists an element of ode p e 1 in the goup. The elements in (Z/p e ) of the fom x = 1 + py fom a subgoup S = {x (Z/p e ) : x 1 mod p} of ode p e 1. It suffices to show that this subgoup is cyclic

6 9.3. PRIME POWER MODULI CHAPTER 9. PRIMITIVE ROOTS That is elatively staightfowad. Each element of the goup has ode p j fo some j. We have to show that some element x = 1 + py has ode p e 1, (1 + py) pe 2 1 mod p e. By the binomial theoem, (1 + py) pe 2 = 1 + p e 2 e 2 py + p 2 y ( p e 2 3 ) p 3 y 3 +. We claim that all the tems afte the fist two ae divisible by p e, p e e 2 p y fo 2. Fo e 2 = pe 2 (p e 2 1) (p e 2 + 1) 1 2 = pe 2 (pe 2 1) (p e 2 + 1) 1 2 ( 1) = pe 2 ( p e ). Thus if ( p f but p f+1 ) then p f p e 2 f e 2. Hence We must show that p e 2 f+ e 2 p y. e 2 f + e, f

7 9.3. PRIME POWER MODULI CHAPTER 9. PRIMITIVE ROOTS Now p f (since p f ), so it is sufficnt to show that p f f + 2, which is moe o less obvious. (If f = 1 then p 3 since p is an odd pime, and each time we incease f we multiply the left by p and add 1 to the ight.) It follows that (1 + py) pe p e 1 y mod p e. Thus any element of the fom 1 + py whee y is not divisible by p (fo example, 1 + p) must have multiplicative ode p e 1, and so must geneate S. In paticula the subgoup S is cyclic, and so (Z/p e ) is cyclic. Tuning to p = 2, it is evident that (Z/2) is tivial, while (Z/4) = C 2. Poposition 9.3. If e 3 then Poof. Since (Z/2 e ) = C2 C 2 e 2. φ(2 e ) = 2 e 1, (Z/2 e ) contains 2 e 1 elements. We ague as we did fo odd p, except that now we take the elements in (Z/2 e ) of the fom x = y, foming the subgoup o ode 2 e 2. By the binomial theoem, S = {x (Z/2 e ) : x 1 mod 4} ( 2 ( y) 2e 3 = e e 3 y + 2 ) 2 4 y 2 + ( 2 e 3 3 ) 2 6 y 3 +. As befoe, all the tems afte the fist two ae divisible by 2 e, 2 e e y fo 2. Fo ( ) ( ) 2 e 3 = 2e 3 2 e Thus if 2 f it is sufficnt to show that e 3 f + 2 e,

8 9.3. PRIME POWER MODULI CHAPTER 9. PRIMITIVE ROOTS 2 f + 3, which follows easily fom the fact that 2 f. Thus any element of the fom y with y odd (fo example, 5) must have multiplicative ode 2 e 2. So the subgoup S is cyclic of this ode. Now let C = {±1 mod 2 e }. This is a subgoup of ode 2. Also it is clea that C S = {1}. It follows that (Z/2 e ) = C S = C 2 C 2 e 2, as equd. Example: Conside (Z/8) = {1, 3, 5, 7}. All the elements except 1 have ode 2, so (Z/8) = C 2 C 2. Concetely, (Z/8) = {±1} {1, 5}. As we said, this allows us to detemine the stuctue of any (Z/n). Example: Suppose n = 48. Then (Z/48) = (Z/16) (Z/3) = (C 2 C 8 ) C 2 = C 2 C 2 C

9 9.4. CARMICHAEL NUMBERS, AGAIN CHAPTER 9. PRIMITIVE ROOTS 9.4 Camichael numbes, again We can now complete the poof of ou Poposition on Camichael numbes in the last Chapte: Poposition 9.4. The numbe n is a Camichael numbe if and only if it is squae-fee, and n = p 1 p 2 p whee 2 and p i 1 n 1 fo i = 1, 2,...,. Poof. Suppose is a Camichael numbe, n = p e 1 1 p e x n x mod n fo all x. Note fist that n must be odd; fo othewise ( 1) n 1 1 mod n. Fist we show that n is squae-fee. Fo suppose p e n, whee e > 1. Then (Z/p e ), and so (Z/n), contains an element x of ode p. But p n. Hence x n 1 x mod n. Now suppose p n. Then (Z/p), and so (Z/n), contains an element x of ode p 1. This element must be copime to n, so x n x mod n = x n 1 1 mod n = p 1 n

Chapter 3: Theory of Modular Arithmetic 38

Chapter 3: Theory of Modular Arithmetic 38 Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences

More information

arxiv: v1 [math.nt] 12 May 2017

arxiv: v1 [math.nt] 12 May 2017 SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking

More information

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0}, ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability

More information

Method for Approximating Irrational Numbers

Method for Approximating Irrational Numbers Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations

More information

Divisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that

Divisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that Divisibility DEFINITION: If a and b ae integes with a 0, we say that a divides b if thee is an intege c such that b = ac. If a divides b, we also say that a is a diviso o facto of b. NOTATION: d n means

More information

A Bijective Approach to the Permutational Power of a Priority Queue

A Bijective Approach to the Permutational Power of a Priority Queue A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation

More information

Introduction Common Divisors. Discrete Mathematics Andrei Bulatov

Introduction Common Divisors. Discrete Mathematics Andrei Bulatov Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes

More information

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics

More information

Lecture 28: Convergence of Random Variables and Related Theorems

Lecture 28: Convergence of Random Variables and Related Theorems EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An

More information

6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS 6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to

More information

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The

More information

Chapter Eight Notes N P U1C8S4-6

Chapter Eight Notes N P U1C8S4-6 Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that

More information

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let

More information

Math Section 4.2 Radians, Arc Length, and Area of a Sector

Math Section 4.2 Radians, Arc Length, and Area of a Sector Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic

More information

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!

More information

arxiv: v1 [math.co] 4 May 2017

arxiv: v1 [math.co] 4 May 2017 On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has

More information

(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.

(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2. Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed

More information

arxiv: v1 [math.co] 1 Apr 2011

arxiv: v1 [math.co] 1 Apr 2011 Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and

More information

arxiv: v2 [math.ag] 4 Jul 2012

arxiv: v2 [math.ag] 4 Jul 2012 SOME EXAMPLES OF VECTOR BUNDLES IN THE BASE LOCUS OF THE GENERALIZED THETA DIVISOR axiv:0707.2326v2 [math.ag] 4 Jul 2012 SEBASTIAN CASALAINA-MARTIN, TAWANDA GWENA, AND MONTSERRAT TEIXIDOR I BIGAS Abstact.

More information

arxiv: v1 [math.co] 6 Mar 2008

arxiv: v1 [math.co] 6 Mar 2008 An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,

More information

Secret Exponent Attacks on RSA-type Schemes with Moduli N = p r q

Secret Exponent Attacks on RSA-type Schemes with Moduli N = p r q Secet Exponent Attacks on RSA-type Schemes with Moduli N = p q Alexande May Faculty of Compute Science, Electical Engineeing and Mathematics Univesity of Padebon 33102 Padebon, Gemany alexx@uni-padebon.de

More information

When two numbers are written as the product of their prime factors, they are in factored form.

When two numbers are written as the product of their prime factors, they are in factored form. 10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The

More information

3.1 Random variables

3.1 Random variables 3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated

More information

C/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22

C/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22 C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.

More information

Vanishing lines in generalized Adams spectral sequences are generic

Vanishing lines in generalized Adams spectral sequences are generic ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal

More information

Lecture 16 Root Systems and Root Lattices

Lecture 16 Root Systems and Root Lattices 1.745 Intoduction to Lie Algebas Novembe 1, 010 Lectue 16 Root Systems and Root Lattices Pof. Victo Kac Scibe: Michael Cossley Recall that a oot system is a pai (V, ), whee V is a finite dimensional Euclidean

More information

SPECTRAL SEQUENCES. im(er

SPECTRAL SEQUENCES. im(er SPECTRAL SEQUENCES MATTHEW GREENBERG. Intoduction Definition. Let a. An a-th stage spectal (cohomological) sequence consists of the following data: bigaded objects E = p,q Z Ep,q, a diffeentials d : E

More information

New problems in universal algebraic geometry illustrated by boolean equations

New problems in universal algebraic geometry illustrated by boolean equations New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic

More information

Goodness-of-fit for composite hypotheses.

Goodness-of-fit for composite hypotheses. Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

Physics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!

Physics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!! Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time

More information

Do Managers Do Good With Other People s Money? Online Appendix

Do Managers Do Good With Other People s Money? Online Appendix Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth

More information

Suborbital graphs for the group Γ 2

Suborbital graphs for the group Γ 2 Hacettepe Jounal of Mathematics and Statistics Volume 44 5 2015, 1033 1044 Subobital gaphs fo the goup Γ 2 Bahadı Özgü Güle, Muat Beşenk, Yavuz Kesicioğlu, Ali Hikmet Değe Keywods: Abstact In this pape,

More information

On a generalization of Eulerian numbers

On a generalization of Eulerian numbers Notes on Numbe Theoy and Discete Mathematics Pint ISSN 1310 513, Online ISSN 367 875 Vol, 018, No 1, 16 DOI: 10756/nntdm018116- On a genealization of Euleian numbes Claudio Pita-Ruiz Facultad de Ingenieía,

More information

On the Quasi-inverse of a Non-square Matrix: An Infinite Solution

On the Quasi-inverse of a Non-square Matrix: An Infinite Solution Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J

More information

Compactly Supported Radial Basis Functions

Compactly Supported Radial Basis Functions Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically

More information

Stanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012

Stanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012 Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,

More information

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506 A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia 26506 Almost eveyone

More information

Enumerating permutation polynomials

Enumerating permutation polynomials Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, 70013 Heaklion, Geece Abstact We conside thoblem

More information

Lecture 18: Graph Isomorphisms

Lecture 18: Graph Isomorphisms INFR11102: Computational Complexity 22/11/2018 Lectue: Heng Guo Lectue 18: Gaph Isomophisms 1 An Athu-Melin potocol fo GNI Last time we gave a simple inteactive potocol fo GNI with pivate coins. We will

More information

Brief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis

Brief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis Bief summay of functional analysis APPM 5440 Fall 014 Applied Analysis Stephen Becke, stephen.becke@coloado.edu Standad theoems. When necessay, I used Royden s and Keyzsig s books as a efeence. Vesion

More information

10/04/18. P [P(x)] 1 negl(n).

10/04/18. P [P(x)] 1 negl(n). Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the

More information

Practice Integration Math 120 Calculus I Fall 2015

Practice Integration Math 120 Calculus I Fall 2015 Pactice Integation Math 0 Calculus I Fall 05 Hee s a list of pactice eecises. Thee s a hint fo each one as well as an answe with intemediate steps... ( + d. Hint. Answe. ( 8 t + t + This fist set of indefinite

More information

k. s k=1 Part of the significance of the Riemann zeta-function stems from Theorem 9.2. If s > 1 then 1 p s

k. s k=1 Part of the significance of the Riemann zeta-function stems from Theorem 9.2. If s > 1 then 1 p s 9 Pimes in aithmetic ogession Definition 9 The Riemann zeta-function ζs) is the function which assigns to a eal numbe s > the convegent seies k s k Pat of the significance of the Riemann zeta-function

More information

AQI: Advanced Quantum Information Lecture 2 (Module 4): Order finding and factoring algorithms February 20, 2013

AQI: Advanced Quantum Information Lecture 2 (Module 4): Order finding and factoring algorithms February 20, 2013 AQI: Advanced Quantum Infomation Lectue 2 (Module 4): Ode finding and factoing algoithms Febuay 20, 203 Lectue: D. Mak Tame (email: m.tame@impeial.ac.uk) Intoduction In the last lectue we looked at the

More information

Internet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks

Internet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks Intenet Appendix fo A Bayesian Appoach to Real Options: The Case of Distinguishing Between Tempoay and Pemanent Shocks Steven R. Genadie Gaduate School of Business, Stanfod Univesity Andey Malenko Gaduate

More information

2-Monoid of Observables on String G

2-Monoid of Observables on String G 2-Monoid of Obsevables on Sting G Scheibe Novembe 28, 2006 Abstact Given any 2-goupoid, we can associate to it a monoidal categoy which can be thought of as the 2-monoid of obsevables of the 2-paticle

More information

Practice Integration Math 120 Calculus I D Joyce, Fall 2013

Practice Integration Math 120 Calculus I D Joyce, Fall 2013 Pactice Integation Math 0 Calculus I D Joyce, Fall 0 This fist set of indefinite integals, that is, antideivatives, only depends on a few pinciples of integation, the fist being that integation is invese

More information

ON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida

ON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida #A8 INTEGERS 5 (205) ON SPARSEL SCHEMMEL TOTIENT NUMBERS Colin Defant Depatment of Mathematics, Univesity of Floida, Gainesville, Floida cdefant@ufl.edu Received: 7/30/4, Revised: 2/23/4, Accepted: 4/26/5,

More information

Galois points on quartic surfaces

Galois points on quartic surfaces J. Math. Soc. Japan Vol. 53, No. 3, 2001 Galois points on quatic sufaces By Hisao Yoshihaa (Received Nov. 29, 1999) (Revised Ma. 30, 2000) Abstact. Let S be a smooth hypesuface in the pojective thee space

More information

Classical Mechanics Homework set 7, due Nov 8th: Solutions

Classical Mechanics Homework set 7, due Nov 8th: Solutions Classical Mechanics Homewok set 7, due Nov 8th: Solutions 1. Do deivation 8.. It has been asked what effect does a total deivative as a function of q i, t have on the Hamiltonian. Thus, lets us begin with

More information

A proof of the binomial theorem

A proof of the binomial theorem A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative

More information

Fractional Zero Forcing via Three-color Forcing Games

Fractional Zero Forcing via Three-color Forcing Games Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that

More information

The evolution of the phase space density of particle beams in external fields

The evolution of the phase space density of particle beams in external fields The evolution of the phase space density of paticle beams in extenal fields E.G.Bessonov Lebedev Phys. Inst. RAS, Moscow, Russia, COOL 09 Wokshop on Beam Cooling and Related Topics August 31 Septembe 4,

More information

16 Modeling a Language by a Markov Process

16 Modeling a Language by a Markov Process K. Pommeening, Language Statistics 80 16 Modeling a Language by a Makov Pocess Fo deiving theoetical esults a common model of language is the intepetation of texts as esults of Makov pocesses. This model

More information

THE NUMBER OF TWO CONSECUTIVE SUCCESSES IN A HOPPE-PÓLYA URN

THE NUMBER OF TWO CONSECUTIVE SUCCESSES IN A HOPPE-PÓLYA URN TH NUMBR OF TWO CONSCUTIV SUCCSSS IN A HOPP-PÓLYA URN LARS HOLST Depatment of Mathematics, Royal Institute of Technology S 100 44 Stocholm, Sweden -mail: lholst@math.th.se Novembe 27, 2007 Abstact In a

More information

On the ratio of maximum and minimum degree in maximal intersecting families

On the ratio of maximum and minimum degree in maximal intersecting families On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting

More information

PHYS 301 HOMEWORK #10 (Optional HW)

PHYS 301 HOMEWORK #10 (Optional HW) PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2

More information

Functions Defined on Fuzzy Real Numbers According to Zadeh s Extension

Functions Defined on Fuzzy Real Numbers According to Zadeh s Extension Intenational Mathematical Foum, 3, 2008, no. 16, 763-776 Functions Defined on Fuzzy Real Numbes Accoding to Zadeh s Extension Oma A. AbuAaqob, Nabil T. Shawagfeh and Oma A. AbuGhneim 1 Mathematics Depatment,

More information

Berkeley Math Circle AIME Preparation March 5, 2013

Berkeley Math Circle AIME Preparation March 5, 2013 Algeba Toolkit Rules of Thumb. Make sue that you can pove all fomulas you use. This is even bette than memoizing the fomulas. Although it is best to memoize, as well. Stive fo elegant, economical methods.

More information

On a quantity that is analogous to potential and a theorem that relates to it

On a quantity that is analogous to potential and a theorem that relates to it Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,

More information

CERFACS 42 av. Gaspard Coriolis, Toulouse, Cedex 1, France. Available at Date: April 2, 2008.

CERFACS 42 av. Gaspard Coriolis, Toulouse, Cedex 1, France. Available at   Date: April 2, 2008. ON THE BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES IAIN S. DUFF and BORA UÇAR Technical Repot: No: TR/PA/08/26 CERFACS 42 av. Gaspad Coiolis, 31057 Toulouse, Cedex 1, Fance. Available at http://www.cefacs.f/algo/epots/

More information

arxiv: v1 [math.ca] 31 Aug 2009

arxiv: v1 [math.ca] 31 Aug 2009 axiv:98.4578v [math.ca] 3 Aug 9 On L-convegence of tigonometic seies Bogdan Szal Univesity of Zielona Góa, Faculty of Mathematics, Compute Science and Econometics, 65-56 Zielona Góa, ul. Szafana 4a, Poland

More information

Miskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp

Miskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp Miskolc Mathematical Notes HU e-issn 8- Vol. (0), No, pp. 5- DOI 0.85/MMN.0.5 Tibonacci numbes with indices in aithmetic pogession and thei sums Nuettin Imak and Muat Alp Miskolc Mathematical Notes HU

More information

RECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA

RECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts

More information

Semicanonical basis generators of the cluster algebra of type A (1)

Semicanonical basis generators of the cluster algebra of type A (1) Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:

More information

Some RSA-based Encryption Schemes with Tight Security Reduction

Some RSA-based Encryption Schemes with Tight Security Reduction Some RSA-based Encyption Schemes with Tight Secuity Reduction Kaou Kuosawa 1 and Tsuyoshi Takagi 2 1 Ibaaki Univesity, 4-12-1 Nakanausawa, Hitachi, Ibaaki, 316-8511, Japan kuosawa@cis.ibaaki.ac.jp 2 Technische

More information

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the

More information

THE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.

THE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space. THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset

More information

COLLAPSING WALLS THEOREM

COLLAPSING WALLS THEOREM COLLAPSING WALLS THEOREM IGOR PAK AND ROM PINCHASI Abstact. Let P R 3 be a pyamid with the base a convex polygon Q. We show that when othe faces ae collapsed (otated aound the edges onto the plane spanned

More information

MAC Module 12 Eigenvalues and Eigenvectors

MAC Module 12 Eigenvalues and Eigenvectors MAC 23 Module 2 Eigenvalues and Eigenvectos Leaning Objectives Upon completing this module, you should be able to:. Solve the eigenvalue poblem by finding the eigenvalues and the coesponding eigenvectos

More information

Supplementary information Efficient Enumeration of Monocyclic Chemical Graphs with Given Path Frequencies

Supplementary information Efficient Enumeration of Monocyclic Chemical Graphs with Given Path Frequencies Supplementay infomation Efficient Enumeation of Monocyclic Chemical Gaphs with Given Path Fequencies Masaki Suzuki, Hioshi Nagamochi Gaduate School of Infomatics, Kyoto Univesity {m suzuki,nag}@amp.i.kyoto-u.ac.jp

More information

Last time: S n xt y where T tpijq 1 i j nu.

Last time: S n xt y where T tpijq 1 i j nu. Lat time: Let G ü A. ( ) The obit of an element a P A i O a tg a g P Gu. ( ) The tabilize of an element a P A i G a tg P G g a au, and i a ubgoup of G. ( ) The kenel of the action i ke tg P G g a a fo

More information

On the ratio of maximum and minimum degree in maximal intersecting families

On the ratio of maximum and minimum degree in maximal intersecting families On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting

More information

Physics 121 Hour Exam #5 Solution

Physics 121 Hour Exam #5 Solution Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given

More information

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES #A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet

More information

Related Rates - the Basics

Related Rates - the Basics Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the

More information

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating

More information

QUANTUM ALGORITHMS IN ALGEBRAIC NUMBER THEORY

QUANTUM ALGORITHMS IN ALGEBRAIC NUMBER THEORY QUANTU ALGORITHS IN ALGEBRAIC NUBER THEORY SION RUBINSTEIN-SALZEDO Abstact. In this aticle, we discuss some quantum algoithms fo detemining the goup of units and the ideal class goup of a numbe field.

More information

The Strain Compatibility Equations in Polar Coordinates RAWB, Last Update 27/12/07

The Strain Compatibility Equations in Polar Coordinates RAWB, Last Update 27/12/07 The Stain Compatibility Equations in Pola Coodinates RAWB Last Update 7//7 In D thee is just one compatibility equation. In D polas it is (Equ.) whee denotes the enineein shea (twice the tensoial shea)

More information

5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS

5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS 5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons,

More information

A Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction

A Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić

More information

Subject : MATHEMATICS

Subject : MATHEMATICS CCE RF 560 00 KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE 560 00 05 S. S. L. C. EXAMINATION, MARCH/APRIL, 05 : 06. 04. 05 ] MODEL ANSWERS : 8-E Date : 06. 04. 05 ] CODE NO.

More information

Journal of Algebra 323 (2010) Contents lists available at ScienceDirect. Journal of Algebra.

Journal of Algebra 323 (2010) Contents lists available at ScienceDirect. Journal of Algebra. Jounal of Algeba 33 (00) 966 98 Contents lists available at ScienceDiect Jounal of Algeba www.elsevie.com/locate/jalgeba Paametes fo which the Lawence Kamme epesentation is educible Claie Levaillant, David

More information

Graphs of Sine and Cosine Functions

Graphs of Sine and Cosine Functions Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the

More information

f h = u, h g = v, we have u + v = f g. So, we wish

f h = u, h g = v, we have u + v = f g. So, we wish Answes to Homewok 4, Math 4111 (1) Pove that the following examples fom class ae indeed metic spaces. You only need to veify the tiangle inequality. (a) Let C be the set of continuous functions fom [0,

More information

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS ESMERALDA NĂSTASE MATHEMATICS DEPARTMENT XAVIER UNIVERSITY CINCINNATI, OHIO 4507, USA PAPA SISSOKHO MATHEMATICS DEPARTMENT ILLINOIS STATE UNIVERSITY

More information

F-IF Logistic Growth Model, Abstract Version

F-IF Logistic Growth Model, Abstract Version F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth

More information

We give improved upper bounds for the number of primitive solutions of the Thue inequality

We give improved upper bounds for the number of primitive solutions of the Thue inequality NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES WITH POSITIVE DISCRIMINANT N SARADHA AND DIVYUM SHARMA Abstact Let F(X, Y) be an ieducible binay cubic fom with intege coefficients and positive disciminant

More information

PROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.

PROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr. POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and

More information

The Chromatic Villainy of Complete Multipartite Graphs

The Chromatic Villainy of Complete Multipartite Graphs Rocheste Institute of Technology RIT Schola Wos Theses Thesis/Dissetation Collections 8--08 The Chomatic Villainy of Complete Multipatite Gaphs Anna Raleigh an9@it.edu Follow this and additional wos at:

More information

Bounds for Codimensions of Fitting Ideals

Bounds for Codimensions of Fitting Ideals Ž. JOUNAL OF ALGEBA 194, 378 382 1997 ATICLE NO. JA966999 Bounds fo Coensions of Fitting Ideals Michał Kwiecinski* Uniwesytet Jagiellonski, Instytut Matematyki, ul. eymonta 4, 30-059, Kakow, Poland Communicated

More information

2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8

2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8 5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute

More information

Several new identities involving Euler and Bernoulli polynomials

Several new identities involving Euler and Bernoulli polynomials Bull. Math. Soc. Sci. Math. Roumanie Tome 9107 No. 1, 016, 101 108 Seveal new identitie involving Eule and Benoulli polynomial by Wang Xiaoying and Zhang Wenpeng Abtact The main pupoe of thi pape i uing

More information

Suggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004

Suggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004 Suggested Solutions to Homewok #4 Econ 5b (Pat I), Sping 2004. Conside a neoclassical gowth model with valued leisue. The (epesentative) consume values steams of consumption and leisue accoding to P t=0

More information

q i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by

q i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by CSISZÁR f DIVERGENCE, OSTROWSKI S INEQUALITY AND MUTUAL INFORMATION S. S. DRAGOMIR, V. GLUŠČEVIĆ, AND C. E. M. PEARCE Abstact. The Ostowski integal inequality fo an absolutely continuous function is used

More information

Gap modules for direct product groups

Gap modules for direct product groups J. Math. Soc. Japan Vol. 53, No. 4, 001 Gap modules fo diect poduct goups Dedicated to Pofesso Masayoshi Kamata on his 60th bithday By Toshio Sumi (Received Nov. 4, 1999) (Revised Jun. 7, 000) Abstact.

More information

Syntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)

Syntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland) Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, 02-097 Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes,

More information