Math 261 Exercise sheet 2
|
|
- Merry McGee
- 5 years ago
- Views:
Transcription
1 Math 261 Exercse sheet 2 Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse 2.1 Fnd all solutons x, y Z to the followng equatons: 1. 3x + 5y = 2, 2. 18x + 30y = 2016, 3. 18x + 30y = Soluton Snce 3 and 5 are coprme, ths equaton has solutons. An obvous one s x = 1, y = 1, and we know that we get all the other ones by addng a multple of 3 5 on one sde, and compensatng on the other sde. So the solutons are exactly the x = 1 + 5t, y = 1 3t (t Z). 2. Ths tme 18 and 30 are not coprme, but ther gcd, whch s 6, dvdes 2016, so we can get an equvalent equaton by dvdng everythng by 6: 3x + 5y = 336. We spot a soluton, for nstance y = 60, x = 12; and the other solutons dffer by multples of 3 5. So fnally the solutons are x = t, y = 60 3t (t Z). 3. Ths tme, gcd(18, 30) = 6 does not dvde 2017, so the equaton has no solutons. 1
2 Exercse 2.2: The lcm 1. Let I and J be two deals of Z. Apply the defnton of an deal to prove that the ntersecton I J s also an deal of Z. 2. Let a and b be postve ntegers. By the prevous queston, az bz s an deal of Z. Prove that ths deal s not the zero deal. Why does ths mply that there exsts a postve nteger c such that az bz = cz? 3. The nteger c defned n the prevous queston s called the lowest common multple of a and b, and s denoted by lcm(a, b). Explan ths name. 4. Let p 1,, p n be the prmes that dvde ether a or b, so that we may wrte n a = p u, b = wth non-negatve ntegers u 1,, u n and v 1,, v n. Express lcm(a, b) n terms of the p, the u and the v. 5. Deduce that gcd(a, b) lcm(a, b) = ab. Remark: Ths means that t s possble to compute the lcm by computng the gcd through the Eucldan algorthm. When a and b are large, ths s much more effcent than computng the factorzaton of a and b. Soluton 2.2: 1. We know that every deal of Z contans at least 0. So 0 I, and 0 J, so that 0 I J; as a result, I J s not empty. Let x and y be elements of I J. Then snce x and y are n I, x + y s n I; smlarly, snce x and y are n J, x + y s also n J. So x + y I J. Fnally, let n Z, and let x I J. Then nx I, and nx J, so nx I J. Ths shows that I J s an deal of Z. 2. We have ab az bz, so az bz {0}. Snce every deal of Z s of the form cz for some c Z, and snce az bz s an deal by the prevous queston, there exsts c Z such that az bz = cz. But snce ab az bz, we cannot have c = 0, and snce cz = ( c)z, we may assume that c > Clearly, c s the smallest postve element of the deal cz. But cz = az bz s the set of ntegers that are both a multple of a and of b. The number c s thus the smallest such common multple, whence ths name. 4. The formula az bz = cz says that the common multples of a and b are precsely the multples of c. Now ab = n pu +v s such a common multple, so c dvdes t; therefore c = n pw for some ntegers w u + v. In order for c to be a multple both of a and b, we need w to be both u and v for each. Then c wll be the smallest common multple precsely when w = max(u, v ), so that c = p max(u,v ). 2 p v
3 5. Recall that gcd(a, b) = n all u and v, so that pmn(u,v ). Now, mn(u, v) + max(u, v) = u + v for gcd(a, b) lcm(a, b) = p mn(u,v )+max(u,v ) = p u +v = ab. Exercse Use the Eucldan algorthm to determne f 47 s nvertble mod 111, and to fnd ts nverse f t s. 2. Solve the equaton 47x 5 (mod 111) n Z/111Z. Soluton We know that 47 s nvertble mod 111 f and only f 47 and 111 are coprme. If they are, we need to look for u and v Z such that 47u + 111v = 1; ndeed, u wll then be an nverse of 47 (mod 111). To fnd u and v, we ether spot them drectly 1, or we use the Eucldan algorthm. Ths algorthm wll also tell us f the gcd of 47 and 111 s not 1, so let s apply t: 111 = = = = So the gcd s 1, so 47 s nvertble mod 111. To fnd t, we wrte 1 = = 13 3(17 13) = = 4( ) 3 17 = = ( ) = , whence 26 mod 111 s the nverse of 47 mod Snce 47 s nvertble mod 111, the only soluton s x = = 5 26 = (mod 111). 1 It can happen sometmes, but here there are no obvous canddates 3
4 Exercse 2.4: An unsolvable dophantne equaton Prove that the equaton x 3 + y 3 + z 3 = 31 has no soluton wth x, y, z Z. Hnt: try solvng the equaton mod 9. Soluton 2.4: Let us make a table of the cubes n Z/9Z: x x So the only cubes n Z/9Z are 1,0 and 1. As a result, a sum of 3 cubes can be 3, 2, 1, 0, 1, 2 or 3 (mod 9), but t can never be 4 nor 4. Snce 31 4 (mod 9), the equaton x 3 + y 3 + z 3 = 31 has no soluton n Z/9Z; therefore, t has no soluton n Z ether. Exercse 2.5: Prmes mod 6 1. Let p be a prme number whch s nether 2 nor 3. Prove that ether p 1 (mod 6) or p 1 (mod 6). 2. Prove that there are nfntely many prmes p such that p 1 (mod 6). Hnt: Suppose on the contrary that there are fntely many, say p 1,, p k. Let N = 6p 1 p k 1, and consder a prme dvsor of N. 3. Why does the same proof fal to show that there are nfntely may prmes p such that p 1 (mod 6)? 4. Drchlet s theorem on prmes n arthmetc progressons, whch s way beyond the scope of ths course, states that for all coprme postve ntegers a and b, there are nfntely many prmes p such that p a (mod b); n partcular, there are n fact nfntely many prmes p such that p 1 (mod 6). Why, n the statement of ths theorem, s t necessary to assume that a and b are coprme? Soluton 2.5: 1. If p s nether 2 nor 3, then p 6, so p and 6 are coprme. But the only nvertbles n Z/6Z are ±1 (ether see t by nspecton of all 6 elements, or use the fact that φ(6) = 2), so we must have p ±1 (mod 6). 2. Suppose that p 1,, p k are the only such prmes, and let N = 6p 1 p k 1. Clearly, nether 2 nor 3 dvde N (snce N 1 mod 6), so the prmes dvdng N are all ±1 (mod 6) by the prevous queston. If they were all +1 (mod 6), then N, ther product, would also be +1 (mod 6), whch s not the case. So at least one of them, say p, s 1 (mod 6). But ths p cannot be one of p 1,, p k, else we would have p (6p 1 p k N) = 1. We therefore have reached a contradcton. 3. We could suppose by contradcton that p 1,, p k are the only prmes 1 (mod 6), and consder a prme dvsor of N = 6p 1 p k 1 (or N = 6p 1 p k + 1). Such a prme could not be any of the p for the same reason as above, but there s no reason why t would have to be 1 (mod 6); ndeed, nothng prevents the dvsors of N from beng all 1 (mod 6). So we are stuck. 4
5 4. If p a (mod b), then p = bx + a for some x Z, so gcd(a, b) p; and obvously, f gcd(a, b) > 1, ths can only happen for at most one prme p (exactly one f gcd(a, b) = p s tself prme, and none else). The exercse below has been added for practce. not mandatory. It s not mandatory, and not worth any ponts. The soluton wll be made avalable wth the solutons to the other exercses. Exercse 2.6: Let n N. Dvsblty crtera 1. Prove that 3 n ff. 3 dvdes the sum of dgts of n. 2. Prove that 9 n ff. 9 dvdes the sum of dgts of n. 3. Fnd a smlar crteron to test whether 11 n. Soluton 2.6: The key s the followng observaton: f n 0, n 1, n 2, are the dgts of n from rght to left, so that n = n n n 2 + = n 10, and snce 10 1 (mod 9), we have n = n 10 n 1 = n (mod 9); n other words, n s congruent to the sum of ts dgts (mod 9). In partcular, ths congruence also holds mod 3 snce 3 9. So we have and 9 n n 0 (mod 9) n 0 (mod 9) 9 n 3 n n 0 (mod 3) n 0 (mod 3) 3 n. For dvsblty by 11, we notce that 10 1 (mod 11), so that n = n 10 n ( 1) = n 0 n 1 + n 2 n 3 + (mod 11). As a result, 11 n f and only f the expresson n 0 n 1 + n 2 n 3 +, whch s called the alternate sum of dgts of n, s dvsble by 11. Examples: For n = 261, we have = 9, so For n = 1452, we have = 0, so We also see that and that but
Foundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationREDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].
REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß
More informationProblem Solving in Math (Math 43900) Fall 2013
Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:
More information18.781: Solution to Practice Questions for Final Exam
18.781: Soluton to Practce Questons for Fnal Exam 1. Fnd three solutons n postve ntegers of x 6y = 1 by frst calculatng the contnued fracton expanson of 6. Soluton: We have 1 6=[, ] 6 6+ =[, ] 1 =[,, ]=[,,
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationarxiv: v6 [math.nt] 23 Aug 2016
A NOTE ON ODD PERFECT NUMBERS JOSE ARNALDO B. DRIS AND FLORIAN LUCA arxv:03.437v6 [math.nt] 23 Aug 206 Abstract. In ths note, we show that f N s an odd perfect number and q α s some prme power exactly
More informationSome Consequences. Example of Extended Euclidean Algorithm. The Fundamental Theorem of Arithmetic, II. Characterizing the GCD and LCM
Example of Extended Eucldean Algorthm Recall that gcd(84, 33) = gcd(33, 18) = gcd(18, 15) = gcd(15, 3) = gcd(3, 0) = 3 We work backwards to wrte 3 as a lnear combnaton of 84 and 33: 3 = 18 15 [Now 3 s
More informationComplex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)
Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For
More informationInternational Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions
Internatonal Mathematcal Olympad Prelmnary Selecton ontest Hong Kong Outlne of Solutons nswers: 7 4 7 4 6 5 9 6 99 7 6 6 9 5544 49 5 7 4 6765 5 6 6 7 6 944 9 Solutons: Snce n s a two-dgt number, we have
More informationChristian Aebi Collège Calvin, Geneva, Switzerland
#A7 INTEGERS 12 (2012) A PROPERTY OF TWIN PRIMES Chrstan Aeb Collège Calvn, Geneva, Swtzerland chrstan.aeb@edu.ge.ch Grant Carns Department of Mathematcs, La Trobe Unversty, Melbourne, Australa G.Carns@latrobe.edu.au
More information(2mn, m 2 n 2, m 2 + n 2 )
MATH 16T Homewk Solutons 1. Recall that a natural number n N s a perfect square f n = m f some m N. a) Let n = p α even f = 1,,..., k. be the prme factzaton of some n. Prove that n s a perfect square f
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More informationPRIMES 2015 reading project: Problem set #3
PRIMES 2015 readng project: Problem set #3 page 1 PRIMES 2015 readng project: Problem set #3 posted 31 May 2015, to be submtted around 15 June 2015 Darj Grnberg The purpose of ths problem set s to replace
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationTHERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q.
THERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q. IAN KIMING We shall prove the followng result from [2]: Theorem 1. (Bllng-Mahler, 1940, cf. [2]) An ellptc curve defned over Q does not have a
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationPolynomials. 1 What is a polynomial? John Stalker
Polynomals John Stalker What s a polynomal? If you thnk you already know what a polynomal s then skp ths secton. Just be aware that I consstently wrte thngs lke p = c z j =0 nstead of p(z) = c z. =0 You
More informationand problem sheet 2
-8 and 5-5 problem sheet Solutons to the followng seven exercses and optonal bonus problem are to be submtted through gradescope by :0PM on Wednesday th September 08. There are also some practce problems,
More informationThe Ramanujan-Nagell Theorem: Understanding the Proof By Spencer De Chenne
The Ramanujan-Nagell Theorem: Understandng the Proof By Spencer De Chenne 1 Introducton The Ramanujan-Nagell Theorem, frst proposed as a conjecture by Srnvasa Ramanujan n 1943 and later proven by Trygve
More informationBernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationNo-three-in-line problem on a torus: periodicity
arxv:1901.09012v1 [cs.dm] 25 Jan 2019 No-three-n-lne problem on a torus: perodcty Mchael Skotnca skotnca@kam.mff.cun.cz Abstract Let τ m,n denote the maxmal number of ponts on the dscrete torus (dscrete
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationDirichlet s Theorem In Arithmetic Progressions
Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationProblem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?
Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2
More informationRandomness and Computation
Randomness and Computaton or, Randomzed Algorthms Mary Cryan School of Informatcs Unversty of Ednburgh RC 208/9) Lecture 0 slde Balls n Bns m balls, n bns, and balls thrown unformly at random nto bns usually
More informationinv lve a journal of mathematics 2008 Vol. 1, No. 1 Divisibility of class numbers of imaginary quadratic function fields
nv lve a journal of mathematcs Dvsblty of class numbers of magnary quadratc functon felds Adam Merberg mathematcal scences publshers 2008 Vol. 1, No. 1 INVOLVE 1:1(2008) Dvsblty of class numbers of magnary
More informationMTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i
MTH 819 Algebra I S13 Homework 1/ Solutons Defnton A. Let R be PID and V a untary R-module. Let p be a prme n R and n Z +. Then d p,n (V) = dm R/Rp p n 1 Ann V (p n )/p n Ann V (p n+1 ) Note here that
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationfind (x): given element x, return the canonical element of the set containing x;
COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationDiscussion 11 Summary 11/20/2018
Dscusson 11 Summary 11/20/2018 1 Quz 8 1. Prove for any sets A, B that A = A B ff B A. Soluton: There are two drectons we need to prove: (a) A = A B B A, (b) B A A = A B. (a) Frst, we prove A = A B B A.
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationBasic Number Theory. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. Last revision: June 11, 2001
Basc Number Theory Instructor: Laszlo Baba Notes by Vncent Lucarell and the nstructor Last revson: June, 200 Notaton: Unless otherwse stated, all varables n ths note are ntegers. For n 0, [n] = {, 2,...,
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More informationComplex Numbers Alpha, Round 1 Test #123
Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test
More informationOn the irreducibility of a truncated binomial expansion
On the rreducblty of a truncated bnomal expanson by Mchael Flaseta, Angel Kumchev and Dmtr V. Pasechnk 1 Introducton For postve ntegers k and n wth k n 1, defne P n,k (x = =0 ( n x. In the case that k
More informationREGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction
REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997
More informationA combinatorial problem associated with nonograms
A combnatoral problem assocated wth nonograms Jessca Benton Ron Snow Nolan Wallach March 21, 2005 1 Introducton. Ths work was motvated by a queston posed by the second named author to the frst named author
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationA 2D Bounded Linear Program (H,c) 2D Linear Programming
A 2D Bounded Lnear Program (H,c) h 3 v h 8 h 5 c h 4 h h 6 h 7 h 2 2D Lnear Programmng C s a polygonal regon, the ntersecton of n halfplanes. (H, c) s nfeasble, as C s empty. Feasble regon C s unbounded
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More informationExercises. 18 Algorithms
18 Algorthms Exercses 0.1. In each of the followng stuatons, ndcate whether f = O(g), or f = Ω(g), or both (n whch case f = Θ(g)). f(n) g(n) (a) n 100 n 200 (b) n 1/2 n 2/3 (c) 100n + log n n + (log n)
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationLecture 20: Lift and Project, SDP Duality. Today we will study the Lift and Project method. Then we will prove the SDP duality theorem.
prnceton u. sp 02 cos 598B: algorthms and complexty Lecture 20: Lft and Project, SDP Dualty Lecturer: Sanjeev Arora Scrbe:Yury Makarychev Today we wll study the Lft and Project method. Then we wll prove
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationSome basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C
Some basc nequaltes Defnton. Let V be a vector space over the complex numbers. An nner product s gven by a functon, V V C (x, y) x, y satsfyng the followng propertes (for all x V, y V and c C) (1) x +
More informationarxiv: v1 [math.nt] 12 Sep 2018
ON p-adic VALUATIONS OF COLORED p-ary PARTITIONS arxv:180904628v1 [mathnt] 12 Sep 2018 MACIEJ ULAS AND B LAŻEJ ŻMIJA Abstract Letm N 2 andforgven k N + consderthesequence (A m,k (n n N defned by the power
More informationTHE CLASS NUMBER THEOREM
THE CLASS NUMBER THEOREM TIMUR AKMAN-DUFFY Abstract. In basc number theory we encounter the class group (also known as the deal class group). Ths group measures the extent that a rng fals to be a prncpal
More informationk(k 1)(k 2)(p 2) 6(p d.
BLOCK-TRANSITIVE 3-DESIGNS WITH AFFINE AUTOMORPHISM GROUP Greg Gamble Let X = (Z p d where p s an odd prme and d N, and let B X, B = k. Then t was shown by Praeger that the set B = {B g g AGL d (p} s the
More informationMultiplicative Functions and Möbius Inversion Formula
Multplcatve Functons and Möbus Inverson Forula Zvezdelna Stanova Bereley Math Crcle Drector Mlls College and UC Bereley 1. Multplcatve Functons. Overvew Defnton 1. A functon f : N C s sad to be arthetc.
More informationDynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)
/24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler sub-problems. It s applcable to problems exhbtng the propertes
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More information( ) 2 ( ) ( ) Problem Set 4 Suggested Solutions. Problem 1
Problem Set 4 Suggested Solutons Problem (A) The market demand functon s the soluton to the followng utlty-maxmzaton roblem (UMP): The Lagrangean: ( x, x, x ) = + max U x, x, x x x x st.. x + x + x y x,
More informationOn quasiperfect numbers
Notes on Number Theory and Dscrete Mathematcs Prnt ISSN 1310 5132, Onlne ISSN 2367 8275 Vol. 23, 2017, No. 3, 73 78 On quasperfect numbers V. Sva Rama Prasad 1 and C. Suntha 2 1 Nalla Malla Reddy Engneerng
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationHomework 9 Solutions. 1. (Exercises from the book, 6 th edition, 6.6, 1-3.) Determine the number of distinct orderings of the letters given:
Homework 9 Solutons PROBLEM ONE 1 (Exercses from the book, th edton,, 1-) Determne the number of dstnct orderngs of the letters gven: (a) GUIDE Soluton: 5! (b) SCHOOL Soluton:! (c) SALESPERSONS Soluton:
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationOn the Divisibility of Binomial Coefficients
On the Dvsblty of Bnomal Coeffcents Sílva Casacuberta Pug Abstract We analyze an open problem n number theory regardng the dvsblty of bnomal coeffcents. It s conjectured that for every nteger n there exst
More informationSolutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010
Solutons to the 7st Wllam Lowell Putnam Mathematcal Competton Saturday, December 4, 2 Kran Kedlaya and Lenny Ng A The largest such k s n+ 2 n 2. For n even, ths value s acheved by the partton {,n},{2,n
More informationFor all questions, answer choice E) NOTA" means none of the above answers is correct.
0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationCharacter Degrees of Extensions of PSL 2 (q) and SL 2 (q)
Character Degrees of Extensons of PSL (q) and SL (q) Donald L. Whte Department of Mathematcal Scences Kent State Unversty, Kent, Oho 444 E-mal: whte@math.kent.edu July 7, 01 Abstract Denote by S the projectve
More informationALGEBRA HW 7 CLAY SHONKWILER
ALGEBRA HW 7 CLAY SHONKWILER 1 Whch of the followng rngs R are dscrete valuaton rngs? For those that are, fnd the fracton feld K = frac R, the resdue feld k = R/m (where m) s the maxmal deal), and a unformzer
More informationSociété de Calcul Mathématique SA
Socété de Calcul Mathématque SA Outls d'ade à la décson Tools for decson help Probablstc Studes: Normalzng the Hstograms Bernard Beauzamy December, 202 I. General constructon of the hstogram Any probablstc
More informationMULTIPLICATIVE FUNCTIONS: A REWRITE OF ANDREWS CHAPTER 6
MULTIPLICATIVE FUNCTIONS: A REWRITE OF ANDREWS CHAPTER 6 In these notes we offer a rewrte of Andrews Chapter 6. Our am s to replace some of the messer arguments n Andrews. To acheve ths, we need to change
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More informationMath 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
More information28 Finitely Generated Abelian Groups
8 Fntely Generated Abelan Groups In ths last paragraph of Chapter, we determne the structure of fntely generated abelan groups A complete classfcaton of such groups s gven Complete classfcaton theorems
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationExhaustive Search for the Binary Sequences of Length 2047 and 4095 with Ideal Autocorrelation
Exhaustve Search for the Bnary Sequences of Length 047 and 4095 wth Ideal Autocorrelaton 003. 5. 4. Seok-Yong Jn and Hong-Yeop Song. Yonse Unversty Contents Introducton Background theory Ideal autocorrelaton
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationPolynomial PSet Solutions
Polynomal PSet Solutons Note: Problems A, A2, B2, B8, C2, D2, E3, and E6 were done n class. (A) Values and Roots. Rearrange to get (x + )P (x) x = 0 for x = 0,,..., n. Snce ths equaton has roots x = 0,,...,
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationLinear Feature Engineering 11
Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19
More informationUNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction
Prerequste Sklls Ths lesson requres the use of the followng sklls: understandng that multplyng the numerator and denomnator of a fracton by the same quantty produces an equvalent fracton multplyng complex
More informationMATH Homework #2
MATH609-601 Homework #2 September 27, 2012 1. Problems Ths contans a set of possble solutons to all problems of HW-2. Be vglant snce typos are possble (and nevtable). (1) Problem 1 (20 pts) For a matrx
More informationU.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan 2/21/2008. Notes for Lecture 8
U.C. Berkeley CS278: Computatonal Complexty Handout N8 Professor Luca Trevsan 2/21/2008 Notes for Lecture 8 1 Undrected Connectvty In the undrected s t connectvty problem (abbrevated ST-UCONN) we are gven
More informationNOTES ON SIMPLIFICATION OF MATRICES
NOTES ON SIMPLIFICATION OF MATRICES JONATHAN LUK These notes dscuss how to smplfy an (n n) matrx In partcular, we expand on some of the materal from the textbook (wth some repetton) Part of the exposton
More informationOn the Nilpotent Length of Polycyclic Groups
JOURNAL OF ALGEBRA 203, 125133 1998 ARTICLE NO. JA977321 On the Nlpotent Length of Polycyclc Groups Gerard Endmon* C.M.I., Unerste de Proence, UMR-CNRS 6632, 39, rue F. Jolot-Cure, 13453 Marselle Cedex
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationFeature Selection: Part 1
CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More information