Sums of Generalized Harmonic Series (NOT Multiple Zeta Values)

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Sums of Generalized Harmonic Series (NOT Multiple Zeta Values)"

Transcription

1 Sums of Series (NOT Multiple Zeta Values) Michael E. Hoffman USNA Basic Notions 6 February 2014

2 1 2 3

3 harmonic series The generalized harmonic series we will be concerned with are H(a 1, a 2,..., a k ) = n=1 1 n a 1 (n + 1) a 2 (n + k 1) a k, where a 1,..., a k are nonnegative integers whose sum is at least 2. We call this quantity an H-series of length k and weight a a k. Of course H-series of length 1 are just values of the Riemann zeta function. We will prove that H(a 1,..., a k ) = kζ(m), (1) a 1 + +a k =m where m 2 and the sum is over k-tuples of nonnegative integers. This was conjectured by C. Moen last spring, and proved by both of us over the summer.

4 First cases Equation (1) is only interesting if k > 1. The first case is H(2, 0) + H(1, 1) + H(0, 2) = 2ζ(2), which is pretty trivial: evidently H(2, 0) = ζ(2) and H(0, 2) = ζ(2) 1, and the usual telescoping-sum argument shows 1 H(1, 1) = n(n + 1) = 1. n=1 More generally, equation (1) for k = 2 is H(m, 0) + H(m 1, 1) + + H(1, m 1) + H(0, m) = 2ζ(m), which reduces to H(m 1, 1) + + H(1, m 1) = 1. (2)

5 Multiple zeta values Equation (1) is superficially similar to the sum theorem for multiple zeta values (conjectured by C. Moen 1988, proved by A. Granville 1997): for m 2, where a 1 + +a k =m, a i 1, a 1 2 ζ(a 1, a 2..., a k ) = ζ(a 1, a 2,..., a k ) = ζ(m) (3) n 1 >n 2 > >n k 1 1 n a 1 1 na 2 2 na k k. But note that the multiple zeta value ζ(a 1,..., a k ) is a k-fold sum, while H(a 1,..., a k ) is a simple sum. Also, the left-hand side of (3) is a sum over k-tuples of positive integers, while that of (1) is a sum over k-tuples of nonnegative integers.

6 Another sum of H-series We could ask what happens if we sum H-series over all k-tuples of positive integers with sum m 2. There is a nice formula in this case, though by no means as simple as equation (1): a 1 + +a k =m, a i 1 H(a 1,..., a k ) = 1 (k 1)! k 2 ( ) k 2 i i=0 ( 1) i. (4) (i + 1) m k+1 Note that the right-hand side is a rational number. The case k = 2 is equation (2).

7 Telescoping We shall build up a chain of lemmas on H-series. Using 1 n(n+1) = 1 n 1 n+1, note that for a, b 1, H(a, b) = [ n=1 Hence n=1 1 n a (n + 1) b = 1 n a (n + 1) b 1 1 n a 1 (n + 1) b ] = H(a, b 1) H(a 1, b). H(m 1, 1) + H(m 2, 2) + + H(1, m 1) = H(m 1, 0) H(m 2, 1) + H(m 2, 1) H(m 3, 2) + +H(1, m 2) H(0, m 1) = H(m 1, 0) H(0, m 1) = 1, proving equation (2).

8 Sum over indices of fixed sum, fixed positions This generalizes as follows. Lemma (1) Let 1 i < j p, and let a 1,..., a i 1, a i+1,..., a j 1, a j+1,..., a p be fixed nonnegative integers. Then m 1 k=1 H(a 1,..., a i 1, k, a i+1,..., a j 1, m k, a j+1,..., a p ) = 1 j i [H(a 1,..., a i 1, m 1, a i+1,..., a j 1, 0, a j+1,..., a p ) H(a 1,..., a i 1, 0, a i+1,..., a j 1, m 1, a j+1,..., a p )].

9 Sum over indices with fixed nonzero positions We can use Lemma (1) to obtain a formula for the sum of all H-series of fixed length and weight and nonzero entries at specified positions. Let H p,q (r) = q j=p j r for p q. Lemma (2) For k 1 and 1 i 0 < i 1 < < i k n, let P(i 0,..., i k ) be the set of strings (a 1,..., a n ) of nonnegative integers with a a n = m and a j 0 iff j = i q for some q. Then (a 1,...,a n) P(i 0,...,i k ) k j=1 H(a 1,..., a n ) = ( 1) j 1 H (m k) i 0,i j 1 (i j i 0 )(i j i 1 ) (i j i j 1 )(i j+1 i j ) (i k i j ).

10 Sum over indices with fixed number of nonzero positions For 0 k n 1, let C(k, n; m) be the sum of all H-series H(a 1..., a n ) of length n, weight m, and exactly k + 1 of the a i nonzero. If k 1, C(k, n; m) is the sum over all sequences 1 i 0 < < i k n of the sums given by the preceding result; since each H (m k) i 0,i j 1 in that result is a sum of 1 with p m k 1 i 0 p i j 1 n 1, we have C(k, n; m) = n 1 j=1 c (n) k,j j m k (5) with c (n) k,j rational. We can deduce results about the c (n) k,j Lemma (2). from

11 An (anti)symmetry property Lemma (3) For 1 k, j n 1, c (n) k,n j = ( 1)k 1 c (n) k,j. Proof. For 1 i 0 < < i k n, let S m n (i 0,..., i k ) be the left-hand side of the equation in Lemma (2). Then if S m n (i 0,..., i k ) = p 1 + p 2 2 m k + + p n 1 (n 1) m k, it follows from Lemma (2) that ( 1) k 1 S m n (n + 1 i k,..., n + 1 i 0 ) = p n 1 + p n 2 2 m k + + p 1 (n 1) m k.

12 Some explicit coefficients Here are some of the coefficients c (n) k,j. c (3) 1,1 = 3 2 c (3) 1,2 = 3 2 c (3) 2,1 = 1 2 c (3) 2,2 = 1 2 c (4) 1,1 = 11 6 c (4) 1,2 = 7 3 c (4) 1,3 = 11 6 c (4) 2,1 = 1 c(4) 2,2 = 0 c(4) 2,3 = 1 c (4) 3,1 = 1 6 c (4) 3,2 = 2 6 c (4) 3,3 = 1 6 1,1 = ,2 = ,3 = ,4 = ,1 = ,2 = 5 8 2,3 = 5 8 2,4 = ,1 = ,2 = ,3 = ,4 = ,1 = ,2 = ,3 = ,4 = 1 24

13 More explicit coefficients For brevity, we give the matrices [(n 1)!c (n) i,j ] for n = 6, Note that all row sums except the first are 0.

14 The first columns the of the matrices above turn out to be well-known. Let [ n k] be the number of permutations of {1, 2,..., n} with exactly k disjoint cycles; this is the unsigned number of. We set [ 0 0] = 1, and for n 1 the number [ n k] is only nonzero if 1 k n. Some useful facts are [ ] [ ] [ ] ( ) n n n n = (n 1)!, = 1, =. 1 n n 1 2 One has the generating function x(x + 1) (x + n 1) = n k=1 [ ] n x k, n 1. (6) k

15 cont d The following result relates the numbers to the c (n) k,1. Lemma (4) For 1 k n 1, Proof. c (n) k,1 = ( 1)k 1 c k,n 1 = [ ] 1 n. (n 1)! k + 1 By Lemma (3), it suffices to prove the last equality. Lemma (2) says c (n) k,n 1 = ( 1) k 1 (n i 0 ) (n i k 1 ) 1 i 0 < <i k 1 <n

16 cont d Proof cont d. so it s enough to show 1 i 0 < <i k 1 <n (n 1)! (n i 0 ) (n i k 1 ) = [ ] n. k + 1 Now the left-hand side is evidently the sum of all products of n k 1 distinct factors from the set {1, 2..., n 1}, and that this is [ n k+1] follows from consideration of the generating function (6).

17 A general formula The preceding result generalizes as follows. Lemma (5) For 1 k, j n 1, c (n) k,j = q q=1 p=1 ( 1) p 1[ ][ q n+1 q p k+2 p (q 1)!(n q)! We only sketch the proof. The lemma amounts to c (n) k,j c (n) k,j 1 = ( 1) p 1[ ][ j n+1 j ] p k+2 p. (7) (q 1)!(n q)! p=1 ]. By Lemma (3), we can make this about c (n) k,n j c(n) k,n j+1.

18 A general formula cont d Now think of the c (n) k,i as sums of terms via Lemma (2). If j k, then all terms contributing to c (n) k,n j+1, and in addition there are j classes of terms that c (n) k,n j also contribute to contribute to c (n) k,n j and not to c (n) k,n j+1 : these j classes of terms can be matched up with the j terms on the right-hand side of equation (7). If j > k, then there are also terms that contribute to c (n) k,n j+1 and not to c(n) k,n j : one can use Lemmas (2) and (4) to match up their sum with a term on the right-hand side of equation (7).

19 second sum formula Now we ll prove the (1) and (4). As a warm-up, we ll do the second of these first. Replacing k by n, equation (4) is a 1 + +a n=m, a i 1 H(a 1,..., a n ) = 1 (n 1)! n 1 j=1 ( ) n 2 ( 1) j 1 j 1 j m n+1. The left-hand side is C(n 1, n; m), so it suffices to show that ( ) c (n) n 1,j = ( 1)j 1 n 2. (n 1)! j 1

20 second sum formula cont d Since [ n k] = 0 when k > n and [ n n] = 1, this turns out to be an easy consequence of Lemma (5): c (n) n 1,j = = q q=1 p=1 q=1 = 1 (n 1)! = ( 1)j 1 (n 1)! ( 1) p 1[ ][ q n+1 q p n+1 p (q 1)!(n q)! ] ( 1) q 1[ q q ][ n+1 q n+1 q (q 1)!(n q)! ( ) n 1 ( 1) q 1 q 1 q=1 ( ) n 2. j 1 ]

21 first sum formula Next we attack equation (1). We have a 1 + +a n=m Now C(0, n; m) is n 1 H(a 1,..., a n ) = C(0, n; m) + C(k, n; m). k=1 H(m, 0,..., 0) + H(0, m, 0,..., 0) + + H(0,..., 0, m) = ζ(m) + ζ(m) ζ(m) m 1 (n 1) m = nζ(m) (n 1) n 2 2 m 1 (n 1) m n 1 = nζ(m) j=1 n j j m

22 first sum formula cont d so (using equation (5)) a 1 + +a n=m n 1 H(a 1,..., a n ) = nζ(m) Hence, to prove the result requires n 1 j k c (n) k,j k=1 j=1 = n j n j j m n 1 n 1 + k=1 j=1 c (n) k,j j m k. for 1 j n 1. In view of Lemma (5), this is n 1 q k=1 q=1 p=1 j k ( 1)p 1[ q p ][ n+1 q k+2 p (q 1)!(n q)! ] = n j. (8)

23 first sum formula cont d The left-hand side of equation (8) is q q=1 p=1 If p = 1, the inner sum is ( 1) p 1[ ] q p (q 1)!(n q)! n 1 k=1 [ ] n + 1 q j k. k + 2 p n 1 [ ] n + 1 q n [ ] [ ] n + 1 q n + 1 q j k = j k 1 = k + 1 k 1 k=1 k=1 [ ] [( ) ] (n + j q)! n + 1 q n + j q = (n q)! 1, j(j 1)! 1 j where we have used the generating function (6).

24 first sum formula cont d If p 2, the inner sum is [ n + 1 q k k 1 ] j k+p 2 = j again using equation (6), and so k=1 p 2 (n + j q)!, (j 1)! n 1 [ ] ( ) n + 1 q n + j q j k = (n q)!j p 1. k + 2 p j Thus the left-hand side of equation (8) is ( ) n + j q j + j q=1 q q=2 p=2 ( 1) p 1[ q p (q 1)! ]( n + j q j ) j p 1.

25 first sum formula cont d The double sum can be rewritten as ( ) n + j q 1 j (q 1)! q=2 ( ) n + j q q=2 = j q=2 1 (q 1)! ( ) n + j q + j q p=2 [ ] q ( j) p 1 = p [ (1 j)(2 j) (q 1 j) ( )( ) n + j q j 1 j q 1 q=2 ( 1) q 1 [ ]] q 1 using equation (6) again, so the left-hand side of equation (8) is ( )( ) n + j q j 1 j + ( 1) q 1. j q 1 q=1

26 One more lemma This means we need one last lemma. Lemma For positive integers n and j, Proof. ( )( ) n + j q j 1 ( 1) q 1 = n. j q 1 q=1 Use Wilf s snake oil method : let F (x) = n=1 q=1 ( n + j q j )( j 1 q 1 ) ( 1) q 1 x n.

27 One more lemma cont d Proof cont d. Then F (x) = = q=1 q=1 ( j 1 q 1 = ( ) j 1 ( 1) q 1 q 1 n=1 ) ( 1) q 1 x q ( n + j q (1 x) j+1 = x (1 x) j+1 j ) x n j 1 ( j 1 p p=0 ) ( x) p x(1 x)j 1 (1 x) j+1 = x (1 x) 2 = x + 2x 2 + 3x 3 + and the result follows.

Refined enumerations of alternating sign matrices. Ilse Fischer. Universität Wien

Refined enumerations of alternating sign matrices. Ilse Fischer. Universität Wien Refined enumerations of alternating sign matrices Ilse Fischer Universität Wien 1 Central question Which enumeration problems have a solution in terms of a closed formula that (for instance) only involves

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information

Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

More information

FACTORING SPARSE POLYNOMIALS

FACTORING SPARSE POLYNOMIALS FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

Factoring Algorithms

Factoring Algorithms Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

Computing exponents modulo a number: Repeated squaring

Computing exponents modulo a number: Repeated squaring Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems. Matrix Factorization Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

Homework 5 Solutions

Homework 5 Solutions Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0. Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

More information

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03 EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

More information

Sample Induction Proofs

Sample Induction Proofs Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

P (A) = lim P (A) = N(A)/N,

P (A) = lim P (A) = N(A)/N, 1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

More information

5. Orthogonal matrices

5. Orthogonal matrices L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal

More information

arxiv:math/0202219v1 [math.co] 21 Feb 2002

arxiv:math/0202219v1 [math.co] 21 Feb 2002 RESTRICTED PERMUTATIONS BY PATTERNS OF TYPE (2, 1) arxiv:math/0202219v1 [math.co] 21 Feb 2002 TOUFIK MANSOUR LaBRI (UMR 5800), Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

Solution to Homework 2

Solution to Homework 2 Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

More information

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors. 3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

More information

11 Ideals. 11.1 Revisiting Z

11 Ideals. 11.1 Revisiting Z 11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

Math 115 Spring 2011 Written Homework 5 Solutions

Math 115 Spring 2011 Written Homework 5 Solutions . Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

More information

Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

The Determinant: a Means to Calculate Volume

The Determinant: a Means to Calculate Volume The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

More information

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +... 6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

More information

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse. 1 Inverse Definition Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

More information

5.3 Determinants and Cramer s Rule

5.3 Determinants and Cramer s Rule 290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

More information

Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set. Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

More information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

More information

Notes on counting finite sets

Notes on counting finite sets Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

More information

DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH

DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

MA106 Linear Algebra lecture notes

MA106 Linear Algebra lecture notes MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector

More information

Linear Algebra Notes

Linear Algebra Notes Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note

More information

The Rational Numbers

The Rational Numbers Math 3040: Spring 2011 The Rational Numbers Contents 1. The Set Q 1 2. Addition and multiplication of rational numbers 3 2.1. Definitions and properties. 3 2.2. Comments 4 2.3. Connections with Z. 6 2.4.

More information

1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems 1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

More information

LS.6 Solution Matrices

LS.6 Solution Matrices LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

The Chinese Remainder Theorem

The Chinese Remainder Theorem The Chinese Remainder Theorem Evan Chen evanchen@mit.edu February 3, 2015 The Chinese Remainder Theorem is a theorem only in that it is useful and requires proof. When you ask a capable 15-year-old why

More information

Explicit inverses of some tridiagonal matrices

Explicit inverses of some tridiagonal matrices Linear Algebra and its Applications 325 (2001) 7 21 wwwelseviercom/locate/laa Explicit inverses of some tridiagonal matrices CM da Fonseca, J Petronilho Depto de Matematica, Faculdade de Ciencias e Technologia,

More information

Finite Fields and Error-Correcting Codes

Finite Fields and Error-Correcting Codes Lecture Notes in Mathematics Finite Fields and Error-Correcting Codes Karl-Gustav Andersson (Lund University) (version 1.013-16 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents

More information

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12 CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

More information

1 A duality between descents and connectivity.

1 A duality between descents and connectivity. The Descent Set and Connectivity Set of a Permutation 1 Richard P. Stanley 2 Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA rstan@math.mit.edu version of 16 August

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method

10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method 578 CHAPTER 1 NUMERICAL METHODS 1. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after

More information

2 Polynomials over a field

2 Polynomials over a field 2 Polynomials over a field A polynomial over a field F is a sequence (a 0, a 1, a 2,, a n, ) where a i F i with a i = 0 from some point on a i is called the i th coefficient of f We define three special

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

6.2 Permutations continued

6.2 Permutations continued 6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

More information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S. Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a

More information

Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850

Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850 Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do

More information

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES 1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

More information

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

More information

Lecture 4: Partitioned Matrices and Determinants

Lecture 4: Partitioned Matrices and Determinants Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

*.I Zolotareff s Proof of Quadratic Reciprocity

*.I Zolotareff s Proof of Quadratic Reciprocity *.I. ZOLOTAREFF S PROOF OF QUADRATIC RECIPROCITY 1 *.I Zolotareff s Proof of Quadratic Reciprocity This proof requires a fair amount of preparations on permutations and their signs. Most of the material

More information

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14 4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

More information

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

More information

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes) NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

More information

PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

More information

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

More information

Solution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d.

Solution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d. [Chap. ] Pythagorean Triples 6 (b) The table suggests that in every primitive Pythagorean triple, exactly one of a, b,orc is a multiple of 5. To verify this, we use the Pythagorean Triples Theorem to write

More information

LINEAR SYSTEMS. Consider the following example of a linear system:

LINEAR SYSTEMS. Consider the following example of a linear system: LINEAR SYSTEMS Consider the following example of a linear system: Its unique solution is x +2x 2 +3x 3 = 5 x + x 3 = 3 3x + x 2 +3x 3 = 3 x =, x 2 =0, x 3 = 2 In general we want to solve n equations in

More information

Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

Prime numbers and prime polynomials. Paul Pollack Dartmouth College

Prime numbers and prime polynomials. Paul Pollack Dartmouth College Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)

More information

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 425, PRACTICE FINAL EXAM SOLUTIONS. MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

More information

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.

Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity. 7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,

More information

THE SIGN OF A PERMUTATION

THE SIGN OF A PERMUTATION THE SIGN OF A PERMUTATION KEITH CONRAD 1. Introduction Throughout this discussion, n 2. Any cycle in S n is a product of transpositions: the identity (1) is (12)(12), and a k-cycle with k 2 can be written

More information

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013 Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

More information

Sets and Counting. Let A and B be two sets. It is easy to see (from the diagram above) that A B = A + B A B

Sets and Counting. Let A and B be two sets. It is easy to see (from the diagram above) that A B = A + B A B Sets and Counting Let us remind that the integer part of a number is the greatest integer that is less or equal to. It is denoted by []. Eample [3.1] = 3, [.76] = but [ 3.1] = 4 and [.76] = 6 A B Let A

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

More information

Five fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms

Five fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms UC Berkeley Trinity University March 31, 2008 This talk is about counting, and it s about

More information

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

6. Cholesky factorization

6. Cholesky factorization 6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

More information