# Permutations, the Parity Theorem, and Determinants

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 2 of the set i a ew order. I this way, a permutatio specifies a reorderig of the elemets of a fiite set. Without loss of geerality, it suffices to take as our fiite set {1,...,} for some positive, fiite iteger. A permutatio ϕ o {1,...,} ca be described explicitly with the 2 matrix ( 1 ϕ1 ϕ The top row lists the first itegers i their usual order, ad the bottom row lists them i a ew order. Example 1. O {1,2,3,4,5}, is the permutatio such that ϕ := ). ( ) ϕ1 = 4, ϕ2 = 3, ϕ3 = 5, ϕ4 = 1, ad ϕ5 = 2. Similarly, is the permutatio such that ψ := ( ) ψ1 = 4, ψ2 = 5, ψ3 = 3, ψ4 = 1, ad ψ5 = Cycles A cycle is a especially simple kid of permutatio. Give k distict elemets i {1,...,}, say x 1,...,x k, we write ϕ = (x 1,...,x k ) if ϕ takes x 1 to x 2, x 2 to x 3,..., x k 1 to x k, ad x k to x 1, while leavig all other iputs uchaged. I other words, ad (x 1,...,x i,...,x k )x i = x i+1, where x k+1 := x 1, (1) (x 1,...,x i,...,x k )x = x, if x / {x 1,...,x k }. (2) Such a permutatio is called a k-cycle. The support of a k-cycle ϕ = (x 1,...,x k ) is suppϕ := {x 1,...,x k }. Formula (2) says that all other values of x are fixed poits of the cycle. Remarks. (i) From (1), it is apparet that for x suppϕ, ϕ k x = x. Of course, for x / suppϕ, (2) also implies ϕ k x = x. Hece, for a k-cycle ϕ, we always have that ϕ k

3 3 is the idetity, which we deote by I. I particular, for k = 2, (x 1,x 2 )(x 1,x 2 ) = I, which says that a 2-cycle is its ow iverse. (ii) The cycle otatio is ot uique i the sese that ay circular shift of the sequece x 1,...,x k yields the same permutatio; i.e., (x 1,...,x k ), (x 2,...,x k,x 1 ),..., ad (x k,x 1,...,x k 1 ) all defie the same permutatio. I particular, ote that (x 1,x 2 ) = (x 2,x 1 ). Example 2. O {1,2,3,4,5,6,7,8,9}, cosider the 3-cycle ψ := (1,2,6) ad the 4-cycle ϕ := (3, 5, 7, 6). The followig calculatios illustrate how to compute with cycle otatio: ϕ7 = (3,5,7,6)7 = 6 ϕ6 = (3,5,7,6)6 = 3 ψ6 = (1,2,6)6 = 1 ψϕ6 = (1,2,6)(3,5,7,6)6 = (1,2,6)3 = 3 ϕψ6 = (3,5,7,6)(1,2,6)6 = (3,5,7,6)1 = 1 ϕ4 = (3,5,7,6)4 = 4 ψ4 = (1,2,6)4 = 4 ψϕ4 = (1,2,6)(3,5,7,6)4 = (1,2,6)4 = 4 ϕψ4 = (3,5,7,6)(1,2,6)4 = (3,5,7,6)4 = 4. I particular, otice that ψϕ6 = 3 1 = ϕψ6. So i geeral, cycles do ot commute. Propositio 3. If cycles ψ 1,...,ψ m have pairwise disjoit supports, the ψ i x, x suppψ i, ψ 1 ψ m x = m x x / suppψ i. Furthermore, the ψ i commute ad ca be applied i ay order. Proof. Fix a value of i i the rage from 1 to m. If x suppψ i, the ψ i x also belogs to suppψ i. Therefore, x ad ψ i x do ot belog to the supports of the other cycles; i.e., x ad ψ i x are fixed poits of the other cycles. Hece, we ca write ψ 1 ψ m x = ψ 1 ψ m 1 x, sice x is a fixed poit of ψ m, i=1 = ψ 1 ψ m 2 x, sice x is a fixed poit of ψ m 1, =. = ψ 1 ψ i 1 ψ i x

4 4 = ψ 1 ψ i 2 ψ i x, sice ψ i x is a fixed poit of ψ i 1, = ψ 1 ψ i 3 ψ i x, sice ψ i x is a fixed poit of ψ i 2, =. = ψ 1 ψ i x = ψ i x. We ca similarly argue that ψ m ψ 1 x = ψ i x. I fact, applyig the cycles i ay order to x suppψ i always results i ψ i x. Now suppose x does ot belog to the support of ay ψ i. The x is a fixed poit of every ψ i ad we ca write ψ 1 ψ m x = ψ 1 ψ m 1 x = = ψ 1 x = x. Agai, we ca apply the cycles i ay order; e.g., ψ m ψ 1 x = x. Propositio 3 is importat, because we will see later that every permutatio ca be decomposed ito a compositio of cycles with pairwise disjoit supports Traspositios A traspositio is a 2-cycle such as (x,y), where x y. Thus, (x,y)x = y ad (x,y)y = x, while for all z x,y, we have (x,y)z = z. As metioed i the Remarks above, a traspositio is its ow iverse, ad (x,y) = (y,x). Lemma 4. Let ψ be a k-cycle, ad let τ be a traspositio whose support is a subset of the support of ψ. The ψτ is equal to the compositio of two cycles with disjoit supports. Proof. Suppose ψ = (x 1,...,x i,...,x j,...,x k ) ad τ = (x i,x j ). It is easy to check that ψτ = (x 1,...,x i,x j+1,...,x k )(x i+1,...,x j ), which is the compositio of two cycles with disjoit supports. Lemma 5. Let ψ ad η be cycles with disjoit supports, ad let τ be a traspositio whose support itersects the supports of both ψ ad η. The ψητ is a sigle cycle whose support is suppψ suppη. Proof. Suppose ψ = (x 1,...,x i,...,x k ), η = (y 1,...,y j,...,y m ), ad τ = (x i,y j ). It is easy to check that ψητ = (x 1,...,x i,y j+1,...,y m,y 1,...,y j,x i+1,...,x k ), which is a sigle cycle whose support is suppψ suppη.

7 7 Example 9. O {1,2,3,4,5}, we ca use (4) to write the 3-cycle (1,2,3) as (1,2,3) = (1,3)(1,2). Sice k = 3 is odd, k 1 = 2 is eve. Hece, it is o surprise that we ca write a 3-cycle as the compositio of 2 (a eve umber) traspositios. However, sice (4, 5)(4, 5) = I, we ca also write (1, 2, 3) = (1, 3)(1, 2)(4, 5)(4, 5), which is aother way to write this 3 cycle as a eve umber of traspositios Decompositio of Permutatios ito Cycles with Disjoit Supports For a arbitrary permutatio ϕ, oce we have idetified its disjoit orbits we ca associate each orbit with a cycle i the followig way. The disjoit orbit decompositio (3) suggests that we put { ϕx, x Oyi, ψ i x := x, otherwise. (5) Note that the ψ i are cycles ad have disjoit supports. We ca further write ϕ = ψ 1 ψ, (6) which expresses ϕ as the compositio of cycles with disjoit supports. The fact that (6) holds follows from Propositio 3 ad the defiitio (5). Parity Theorem. Wheever a eve (resp. odd) permutatio is expressed as a compositio of traspositios, the umber of traspositios must be eve (resp. odd). Proof. Cosider a permutatio ϕ with disjoit orbits ad correspodig represetatio as cycles with disjoit supports as i (6). Suppose also that ϕ = τ 1 τ m, where each τ i is a traspositio. Recallig that a traspositio is its ow iverse, write I = ϕϕ 1 = (ψ 1 ψ )(τ 1 τ m ) 1 = (ψ 1 ψ )(τ m τ 1 ) = ψ 1 ψ τ m τ 1. Cosider the expressio ψ 1 ψ τ m. Sice the supports of the ψ i partitio {1,...,}, the two poits i the support of τ m must belog to the supports of the ψ i. If τ m = (u,v), there are two cases to cosider. First, there is the case that u ad v both belog to the support of a sigle ψ i. Secod, u belogs to the support of some ψ i, while v belogs to the support of some other ψ j with j > i. 2 I the first case, we use Propositio 3 ad Lemma 4 to write ψ 1 ψ τ m = ψ 1 ψ i 1 ψ i+1 ψ (ψ i τ m ), 2 There is o loss of geerality i assumig j > i because (u,v) is equal to (v,u).

10 10 = sg(ϕ)a 1 (τϕ1) a (τϕ) ϕ = sg(τ 1 ψ)a 1 (ττ 1 ψ) a (ττ 1 ψ) ψ = sg(τ 1 ψ)a 1 (ψ) a (ψ) ψ = sg(τ 1 )sg(ψ)a 1 (ψ) a (ψ) ψ = sg(ψ)a 1 (ψ) a (ψ) ψ = deta, where the third equality follows by substitutig ϕ = τ 1 ψ; the fifth equality follows because the sig of a compositio of permutatios is the product of their sigs by Corollary 10; ad the sixth equality follows because traspositios are odd. Now suppose A has two equal colums, ad B is obtaied by iterchagig those two colums. The by the precedig paragraph, det B = det A. But sice B = A, detb = deta. Therefore, deta = 0. We showed above that if b( j) = a(τ j) for some traspositio τ, the detb = deta. What if b( j) = a(ϕ j) for some arbitrary permutatio ϕ? Writig ϕ i terms of traspositios, say ϕ = τ 1 τ k, we see that detb = ( 1) k deta = sg(ϕ)deta. Theorem 12. detba = detb deta. Proof. It is coveiet to write deta i terms of its colums. We use the otatio (a(1),...,a()) = deta. If B is aother matrix with colums b(1),...,b(), the the colums of BA are Ba(1),...,Ba(), ad detba = (Ba(1),...,Ba()). Now recall that Write a( j) = i=1 ([ detba = B = i=1 i=1 a i ( j)e(i), j = 1,...,. ] ) a i (1)e(i),Ba(2),...,Ba() a i (1) (Be(i),Ba(2),...,Ba()).

11 11 Repeatig this calculatio for a(2),...,a() yields detba = = i 1 =1 i 1 =1 i =1 i =1 a i1 (1) a i () (Be(i 1 ),...,Be(i )) a i1 (1) a i () (b(i 1 ),...,b(i )). Amog the -tuples (i 1,...,i ), if the i k are ot distict, the (b(i 1 ),...,b(i )) is the determiat of a matrix with two or more equal colums ad is therefore zero. Otherwise, (i 1,...,i ) = (ϕ1,...,ϕ) for some permutatio ϕ. Hece, detba = a ϕ1 (1) a ϕ () (b(ϕ1),...,b(ϕ)) ϕ = a ϕ1 (1) a ϕ ()sg(ϕ)detb ϕ = detbsg(ϕ)a 1 (ϕ 1 1) a (ϕ 1 ) ϕ = detbsg(ψ 1 )a 1 (ψ1) a (ψ) ψ = detbsg(ψ)a 1 (ψ1) a (ψ) ψ = detb deta. Corollary 13. deta = 0 if ad oly if A is osigular. Proof. If A is osigular, the 1 = deti = det(a 1 A) = (deta 1 )(deta) implies deta 0. Now suppose A is sigular. The the colums of A are liearly depedet. Without loss of geerality, suppose a() + 1 j=1 c ja( j) = 0 for some coefficiets c j. Hece, 0 = ( a(1),...,a( 1),0 ) ( 1 ) = a(1),...,a( 1),a() + c j a( j) j=1 = ( a(1),...,a( 1),a() ) 1 + = deta + 0, j=1 c j ( a(1),...,a( 1),a( j) ) sice i the sum over j, the jth term ivolves the determiat of a matrix whose colum j ad colum are the same. Such a determiat is zero.

12 12 Refereces [1] T. L. Barlow, A historical ote o the parity of permutatios, The America Mathematical Mothly, vol. 79, o. 7, pp , [2] W. Phillips, O the defiitio of eve ad odd permutatios, The America Mathematical Mothly, vol. 74, o. 10, pp , [3] W. Rudi, Priciples of Mathematical Aalysis, 3rd ed. New York: McGraw-Hill, 1976.

### 1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

### FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

### Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

### CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

### Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

### 3. Greatest Common Divisor - Least Common Multiple

3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

### Convexity, Inequalities, and Norms

Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

### 5.3. Generalized Permutations and Combinations

53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

### CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8

CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive

### A probabilistic proof of a binomial identity

A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

### Properties of MLE: consistency, asymptotic normality. Fisher information.

Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

### Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

### Chapter 5: Inner Product Spaces

Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

### SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

### THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

### Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

### Notes on exponential generating functions and structures.

Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a -elemet set, (2) to fid for each the

### Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

### Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

### S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + \$ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

### Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

### Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

### Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

### .04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

### CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

### Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>

(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

### The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

### Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

### where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

### WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

### 0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

### Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

### Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

### THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### Factors of sums of powers of binomial coefficients

ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

### Lecture 4: Cheeger s Inequality

Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

### BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

### Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

### 1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

### Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

### A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

### Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

### 5 Boolean Decision Trees (February 11)

5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

### Overview of some probability distributions.

Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

### Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

### Confidence Intervals for One Mean

Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

### Class Meeting # 16: The Fourier Transform on R n

MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

### 3 Basic Definitions of Probability Theory

3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio

### Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

### SEQUENCES AND SERIES

Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

### 2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS

2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 1. The group rig k[g] The mai idea is that represetatios of a group G over a field k are the same as modules over the group rig k[g]. First I

### Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

### Irreducible polynomials with consecutive zero coefficients

Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

### Solutions to Exercises Chapter 4: Recurrence relations and generating functions

Solutios to Exercises Chapter 4: Recurrece relatios ad geeratig fuctios 1 (a) There are seatig positios arraged i a lie. Prove that the umber of ways of choosig a subset of these positios, with o two chose

### THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

### Part - I. Mathematics

Part - I Mathematics CHAPTER Set Theory. Objectives. Itroductio. Set Cocept.. Sets ad Elemets. Subset.. Proper ad Improper Subsets.. Equality of Sets.. Trasitivity of Set Iclusio.4 Uiversal Set.5 Complemet

### MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

### Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

### Tangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem

116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio

### A Note on Sums of Greatest (Least) Prime Factors

It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423-432 HIKARI Ltd, www.m-hikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos

### Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

### PERMUTATIONS AND COMBINATIONS

Chapter 7 PERMUTATIONS AND COMBINATIONS Every body of discovery is mathematical i form because there is o other guidace we ca have DARWIN 7.1 Itroductio Suppose you have a suitcase with a umber lock. The

### Maximum Likelihood Estimators.

Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

### Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### 2-3 The Remainder and Factor Theorems

- The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

### Section 11.3: The Integral Test

Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

### Descriptive Statistics

Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

### CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

### 1 Correlation and Regression Analysis

1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

### 4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph.

4. Trees Oe of the importat classes of graphs is the trees. The importace of trees is evidet from their applicatios i various areas, especially theoretical computer sciece ad molecular evolutio. 4.1 Basics

### Section 8.3 : De Moivre s Theorem and Applications

The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

### 1.3. VERTEX DEGREES & COUNTING

35 Chapter 1: Fudametal Cocepts Sectio 1.3: Vertex Degrees ad Coutig 36 its eighbor o P. Note that P has at least three vertices. If G x v is coected, let y = v. Otherwise, a compoet cut off from P x v

### The Stable Marriage Problem

The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

### NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

### How Euler Did It. In a more modern treatment, Hardy and Wright [H+W] state this same theorem as. n n+ is perfect.

Amicable umbers November 005 How Euler Did It by Ed Sadifer Six is a special umber. It is divisible by, ad 3, ad, i what at first looks like a strage coicidece, 6 = + + 3. The umber 8 shares this remarkable

### PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

### THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe

### THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for

### Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

### Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis

Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.

### Exploratory Data Analysis

1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

### A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY

J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON

### Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

### , a Wishart distribution with n -1 degrees of freedom and scale matrix.

UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that

### Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

### Overview on S-Box Design Principles

Overview o S-Box Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA -721302 What is a S-Box? S-Boxes are Boolea

### The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity