6.4 Binomial Coefficients
|
|
- Godwin Phelps
- 6 years ago
- Views:
Transcription
1 64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter for the study of coutig ad listig probles Theore 1 (Pascal s Forula Let ad r be positive itegers ad suppose r The ( ( ( 1 r r 1 r Proof (algebraic versio Let ad r be positive itegers ad suppose r By the forula for C(, r, ( (! r 1 r (r 1!( r 1!! r!( r!! r r(r 1!( r 1!!( r 1 r!( r!( r 1! r!! r! r!( r 1! r!( r 1!! r!! r! r!( r 1!!! r!( r 1!!( 1 r!( r 1! ( 1! r!( r 1! ( 1 r Proof (cobiatorial versio Let ad r be positive itegers ad suppose r Suppose S is a set with 1 eleets The uber of subsets of S of size r ca be calculated by thiig of S as cosistig of two pieces: oe with eleets {x 1, x 2,, x } ad the other with oe eleet {x 1 } Ay subset of S with r eleets either cotais x 1 or it does ot If it cotais x 1, the it cotais r 1 eleets fro the set {x 1, x 2,, x } If it does ot cotai x 1, the it cotais r eleets fro the set {x 1, x 2,, x } Subsets of size r of {x 1, x 2,, x 1 }: subsets of size r subsets of size r that cosist etirely that cotai x 1 of eleets fro ad r 1 eleets {x 1, x 2,, x } fro {x 1, x 2,, x } There ( are There are ( r of these r 1 of these By the additio rule, uber of subsets of uber of subsets of {x 1, x 2,, x 1 } {x 1, x 2,, x } of size r of size r 1 uber of subsets of {x 1, x 2,, x } of size r
2 By cobiatio forula, the set {x 1, x 2,, x 1 } has ( 1 ( r subsets of size r, the set {x1, x 2,, x } has r 1 subsets of size r 1, ad the set {x1, x 2,, x } has ( r subsets of size r Thus ( ( ( 1 r r 1 r The Bioial Theore I algebra a su of two ters, such as a b, is called a bioial The bioial theore gives a expressio for the powers of a bioial (a b, for each positive iteger ad all real ubers a ad b Cosider what happes whe you calculate the first few powers of a b Accordig to the distributive law of algebra, you tae the su of the products of all cobiatios of idividual ters: (a b 2 (a b(a b aa ab ba bb (a b 3 (a b(a b(a b aaa aab aba abb baa bab bba bbb (a b 4 (a b(a b(a b(a b aaaa aaab aaba aabb abaa abab abba abbb baaa baab baba babb bbaa bbab bbba bbbb Now focus o the expasio of (a b 4 (It is cocrete, ad yet it has all the features of the geeral case A typical ter of this expasio is obtaied by ultiplyig oe of the two ters fro the first factor ties oe of the two ters fro the secod factor ties oe of the two ters fro the third factor ties oe of the two ters fro the fourth factor Sice there are two possible values - a or b - for each ter selected fro oe of the four factors, there are ters i the expasio of (a b 4 Now soe ters i the expasio are lie ters ad ca be cobied Cosider all possible orderigs of three a s ad oe b, for exaple By the techiques of the Sectio 63, there are ( of the Ad each of the four occurs as a ter i the expasio of (a b 4 : aaab aaba abaa baaa By the coutative ad associative laws of algebra, each such ter equals a 3 b, so all four are lie ters Whe the lie ters are cobied, therefore, the coefficiet of a 3 b equals ( 4 1 Siilarly, the expasio of (a b 4 cotais the ( differet orderigs of two a s ad two b s, aabb abab abba baab baba bbaa, all of which equal a 2 b 2, so the coefficiet of a 2 b 2 equals ( 4 2 By a siilar aalysis, the coefficiet of ab 3 equals ( 4 3 Also, sice there is oly oe way to order four a s, the coefficiet of a 4 is 1 (which equals ( 4 0, ad sice there is oly oe way to order four b s, the coefficiet of b 4 is 1 (which equals ( 4 4 Thus, whe all of the lie ters are cobied, ( ( ( ( ( (a b 4 a 4 a 3 b a 2 b 2 ab 3 b a 4 4a 3 b 6a 2 b 2 4ab 3 b 4 The bioial theore geeralizes this forula to a arbitrary oegative iteger Theore 2 (The Bioial Theore Give ay real ubers a ad b ad ay oegative iteger, (a b 0 ( a b a ( a 1 b 1 ( ( a 2 b 2 a 1 b 1 b 2 1
3 Note that the secod expressio equals the first because ( 0 1 ad ( 1, for all oegative itegers, provided that b 0 1 ad a 1 It is istructive to see two proofs of the bioial theore: a algebraic proof ad a cobiatorial proof Both require a precise defiitio of iteger power Defiitio 1 For ay real uber a ad ay oegative iteger, the oegative iteger powers of a are defied as follows: { a 1 if 0 a a 1 if > 0 I soe atheatical cotexts, 0 0 is left udefied Defiig it to be 1, as is doe here, aes it possible to write geeral forulas such as i0 xi 1 1 x without havig to exclude values of the variables that result i the expressio 0 0 The algebraic versio of the bioial theore uses atheatical iductio ad calls upo Pascal s forula at a crucial poit However, algebraic versio of the proof of the bioial theore is rather too log to fit here Please refer to the ed of this lecture otes to see the algebraic proof Proof of the Bioial Theore (cobiatorial versio [The cobiatorial arguet used here to prove the bioial theore wors oly for 1 If we were givig oly this cobiatorial proof, we would have to prove the case 0 separately Sice we have already give a coplete algebraic proof that icludes the case 0, we do ot prove it agai here] Let a ad b be real ubers ad a iteger that is at least 1 The expressio (a b ca be expaded ito products of letters, where each letter is either a or b For each 0, 1,,, the product a b a } a {{ a a } factors b b b }{{} factors occurs as a ter i the su the sae uber of ties as there are orderigs of ( a s ad b s But this uber is (, the uber of ways to choose positios ito which to place the b s [The other positios ( will be filled by a s] Hece, whe lie ters are cobied, the coefficiet of a b i the su is Thus ( (a b a b This is what was to be proved Corollary Let be a oegative iteger The 0 0 ( 2 Proof Usig the bioial theore with a 1 ad b 1, we see that This is the desired result 2 (1 1 0 ( ( Theore 3 (Vaderode s Idetity Let,, ad r be oegative itegers with r ot exceedig either or The r ( r r 0
4 Proof Suppose that there are ites i oe set ad ites i a secod set The the total uber of ways to pic r eleets fro the uio of these sets is ( r Aother way to pic r eleets fro the uio is to pic eleets fro the secod set ad the r eleets fro the first set, where is a iteger with 0 r Because there are ( ways to choose eleets fro the secod set ad ( r ways to choose r eleets fro the first set, the product rule tells us that this ca be doe i ( r ( ways Hece, the total uber of ways to pic r eleets fro the uio also equals r ( 0 r ( We have foud two expressios for the uber of ways to pic r eleets fro the uio of a set with ites ad a set with ites Equatig the gives us Vaderode s idetity Proof (of the bioial theore, algebraic versio Suppose a ad b are real ubers We use atheatical iductio ad let the property P ( be the equatio (a b 0 BASIS STEP Whe 0, the bioial theore states that (a b ( a b ( 0 a 0 b But the left-had side is (a b 0 1 (by defiitio of power ad the right-had side is 0 0 ( 0 a 0 b ( 0 a 0 0 b 0 0 0! 0! (0 0! , sice 0! 1, a 0 1, ad b 0 1 Hece P (0 is true INDUCTIVE STEP We wat to show that for all itegers 0, if P ( is true, the P ( 1 is also true Let 0 be a iteger, ad suppose P ( is true That is, suppose We eed to show that P ( 1 is true: (a b (a b Now, by defiitio of the ( 1st power, 0 0 a b ( 1 a ( b (a b (a b (a b,
5 so by substitutio fro the iductive hypothesis, (a b (a b a b 0 a a b b a b 0 0 a b a b 1 0 We trasfor the secod suatio o the right-had side by aig the chage of variable j 1 Whe 0, the j 1 Whe, the j 1 Ad sice, the geeral ter is ( ( ( a b 1 a (j 1 b j a j b j 0 Hece the secod suatio o the right-had side above is a j b j j1 j1 But the j i this suatio is a duy variable; it ca be replaced by the letter, as log as the replaceet is ade everywhere the j occurs: ( a j b j a b 1 Substitutig bac, we get (a b 0 1 a b 1 a b 1 [The reaso for the above aeuvers was to ae the powers of a ad b agree so that we ca add the suatios together ter by ter, except for the first ad the last ters, which we ust write separately] Thus (a b ( 0 ( 0 a a b a 0 b 0 1 [( a b 1 1 [( ( ] 1 ( 1 ( a b 1 ] a b b ( 1 1 But by Pascal s forula, [( ( ] ( 1 1 Hece 1 (a b a a b b 1 1 a b, 0 which is what we eeded to show Note that ( ( 0 1 a ( b
Binomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationDiscrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009
Discrete Matheatics: Lectures 8 ad 9 Priciple of Iclusio ad Exclusio Istructor: Arijit Bishu Date: August ad 3, 009 As you ca observe by ow, we ca cout i various ways. Oe such ethod is the age-old priciple
More informationCOMP 2804 Solutions Assignment 1
COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig
More informationdistinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)
THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject
More informationMath 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths
More informationBernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes
Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3
More informationJacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
More informationName Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions
Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve
More information9.7 Pascal s Formula and the Binomial Theorem
592 Chapte 9 Coutig ad Pobability Example 971 Values of 97 Pascal s Fomula ad the Biomial Theoem I m vey well acquaited, too, with mattes mathematical, I udestad equatios both the simple ad quadatical
More informationBertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationProblem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0
GENERATING FUNCTIONS Give a ifiite sequece a 0,a,a,, its ordiary geeratig fuctio is A : a Geeratig fuctios are ofte useful for fidig a closed forula for the eleets of a sequece, fidig a recurrece forula,
More informationMathematical Induction
Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a
More informationSome results on the Apostol-Bernoulli and Apostol-Euler polynomials
Soe results o the Apostol-Beroulli ad Apostol-Euler polyoials Weipig Wag a, Cagzhi Jia a Tiaig Wag a, b a Departet of Applied Matheatics, Dalia Uiversity of Techology Dalia 116024, P. R. Chia b Departet
More informationMa/CS 6a Class 22: Power Series
Ma/CS 6a Class 22: Power Series By Ada Sheffer Power Series Mooial: ax i. Polyoial: a 0 + a 1 x + a 2 x 2 + + a x. Power series: A x = a 0 + a 1 x + a 2 x 2 + Also called foral power series, because we
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationFACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a =
FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of -bioial coefficiets. Dedicated to the eory of
More informationGenerating Functions and Their Applications
Geeratig Fuctios ad Their Applicatios Agustius Peter Sahaggau MIT Matheatics Departet Class of 2007 18.104 Ter Paper Fall 2006 Abstract. Geeratig fuctios have useful applicatios i ay fields of study. I
More informationThe Binomial Theorem.
The Binomial Theorem RajeshRathod42@gmail.com The Problem Evaluate (A+B) N as a polynomial in powers of A and B Where N is a positive integer A and B are numbers Example: (A+B) 5 = A 5 +5A 4 B+10A 3 B
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationProblem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
More informationMT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 1 Coutig subsets I this sectio, we cout the subsets of a -elemet set. The coutig umbers are the biomial coefficiets, familiar objects but there are some ew thigs to say about
More informationModern Algebra 1 Section 1 Assignment 1. Solution: We have to show that if you knock down any one domino, then it knocks down the one behind it.
Moder Algebra 1 Sectio 1 Assigmet 1 JOHN PERRY Eercise 1 (pg 11 Warm-up c) Suppose we have a ifiite row of domioes, set up o ed What sort of iductio argumet would covice us that ocig dow the first domio
More informationOptimal Estimator for a Sample Set with Response Error. Ed Stanek
Optial Estiator for a Saple Set wit Respose Error Ed Staek Itroductio We develop a optial estiator siilar to te FP estiator wit respose error tat was cosidered i c08ed63doc Te first 6 pages of tis docuet
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
More information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationq-fibonacci polynomials and q-catalan numbers Johann Cigler [ ] (4) I don t know who has observed this well-known fact for the first time.
-Fiboacci polyoials ad -Catala ubers Joha Cigler The Fiboacci polyoials satisfy the recurrece F ( x s) = s x = () F ( x s) = xf ( x s) + sf ( x s) () with iitial values F ( x s ) = ad F( x s ) = These
More informationPerturbation Theory, Zeeman Effect, Stark Effect
Chapter 8 Perturbatio Theory, Zeea Effect, Stark Effect Ufortuately, apart fro a few siple exaples, the Schrödiger equatio is geerally ot exactly solvable ad we therefore have to rely upo approxiative
More information(1 x n ) 1, (1 + x n ). (1 + g n x n ) r n
COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS HGINGOLD, WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, USA, GINGOLD@MATHWVUEDU JOCELYN QUAINTANCE, UNIVERSITY OF
More information(ii) Two-permutations of {a, b, c}. Answer. (B) P (3, 3) = 3! (C) 3! = 6, and there are 6 items in (A). ... Answer.
SOLUTIONS Homewor 5 Due /6/19 Exercise. (a Cosider the set {a, b, c}. For each of the followig, (A list the objects described, (B give a formula that tells you how may you should have listed, ad (C verify
More informationAutomated Proofs for Some Stirling Number Identities
Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;
More informationA Stirling Encounter with Harmonic Numbers
VOL 75, NO, APRIL 00 95 A Stirlig Ecouter with Haroic Nubers ARTHUR T BENJAMIN GREGORY O PRESTON Harvey Mudd College Clareot, CA 91711 bejai@hcedu gpresto@hcedu JENNIFER J QUINN Occidetal College 1600
More informationCombinatorics and Newton s theorem
INTRODUCTION TO MATHEMATICAL REASONING Key Ideas Worksheet 5 Combiatorics ad Newto s theorem This week we are goig to explore Newto s biomial expasio theorem. This is a very useful tool i aalysis, but
More informationChapter 4. Regular Expressions. 4.1 Some Definitions
Chapter 4 Regular Expressions 4.1 Some Definitions Definition: If S and T are sets of strings of letters (whether they are finite or infinite sets), we define the product set of strings of letters to be
More informationObservations on Derived K-Fibonacci and Derived K- Lucas Sequences
ISSN(Olie): 9-875 ISSN (Prit): 7-670 Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 Observatios o Derived K-iboacci
More informationA canonical semi-deterministic transducer
A canonical semi-deterministic transducer Achilles A. Beros Joint work with Colin de la Higuera Laboratoire d Informatique de Nantes Atlantique, Université de Nantes September 18, 2014 The Results There
More informationCombinatorially Thinking
Combiatorially Thiig SIMUW 2008: July 4 25 Jeifer J Qui jjqui@uwashigtoedu Philosophy We wat to costruct our mathematical uderstadig To this ed, our goal is to situate our problems i cocrete coutig cotexts
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More informationMATH10040 Chapter 4: Sets, Functions and Counting
MATH10040 Chapter 4: Sets, Fuctios ad Coutig 1. The laguage of sets Iforally, a set is ay collectio of objects. The objects ay be atheatical objects such as ubers, fuctios ad eve sets, or letters or sybols
More informationContinuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationAVERAGE MARKS SCALING
TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I
More informationA GENERALIZED BERNSTEIN APPROXIMATION THEOREM
Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: 10.2478/v10127-011-0029-x Tatra Mt. Math. Publ. 49 2011, 99 109 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Miloslav Duchoň ABSTRACT. The preset paper is cocered with soe
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More information~W I F
A FIBONACCI PROPERTY OF WYTHOFF PAIRS ROBERT SILBER North Carolia State Uiversity, Raleigh, North Carolia 27607 I this paper we poit out aother of those fasciatig "coicideces" which are so characteristically
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More information8.3 Perturbation theory
8.3 Perturbatio theory Slides: Video 8.3.1 Costructig erturbatio theory Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (u to First order erturbatio theory ) Perturbatio theory Costructig
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationA Pair of Operator Summation Formulas and Their Applications
A Pair of Operator Suatio Forulas ad Their Applicatios Tia-Xiao He 1, Leetsch C. Hsu, ad Dogsheg Yi 3 1 Departet of Matheatics ad Coputer Sciece Illiois Wesleya Uiversity Blooigto, IL 6170-900, USA Departet
More information) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii
square atrix is oe that has the sae uber of rows as colus; that is, a atrix. he idetity atrix (deoted by I, I, or [] I ) is a square atrix with the property that for ay atrix, the product I equals. he
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationCSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo
Coutig Methods CSE 191, Class Note 05: Coutig Methods Computer Sci & Eg Dept SUNY Buffalo c Xi He (Uiversity at Buffalo CSE 191 Discrete Structures 1 / 48 Need for Coutig The problem of coutig the umber
More information#A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I
#A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I Miloud Mihoubi 1 Uiversité des Scieces et de la Techologie Houari Bouediee Faculty
More informationIntroductions to LucasL
Itroductios to LucasL Itroductio to the Fiboacci ad Lucas ubers The sequece ow ow as Fiboacci ubers (sequece 0,,,, 3,, 8, 3...) first appeared i the wor of a aciet Idia atheaticia, Pigala (40 or 00 BC).
More informationIntegrals of Functions of Several Variables
Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio
More informationThe Binomial Multi-Section Transformer
4/15/2010 The Bioial Multisectio Matchig Trasforer preset.doc 1/24 The Bioial Multi-Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where:
More informationSome Basic Counting Techniques
Some Basic Coutig Techiques Itroductio If A is a oempty subset of a fiite sample space S, the coceptually simplest way to fid the probability of A would be simply to apply the defiitio P (A) = s A p(s);
More informationThe Binomial Theorem
The Biomial Theorem Lecture 47 Sectio 9.7 Robb T. Koether Hampde-Sydey College Thu, Apr 8, 03 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 / 7 Combiatios Pascal s Triagle 3
More information5.6 Binomial Multi-section Matching Transformer
4/14/21 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-25 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.
More informationDouble Derangement Permutations
Ope Joural of iscrete Matheatics, 206, 6, 99-04 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri
More information(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1
ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like
More informationTHE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION
MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia
More informationCIS Spring 2018 (instructor Val Tannen)
CIS 160 - Sprig 2018 (istructor Val Tae) Lecture 5 Thursday, Jauary 25 COUNTING We cotiue studyig how to use combiatios ad what are their properties. Example 5.1 How may 8-letter strigs ca be costructed
More informationChapter 2. Asymptotic Notation
Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It
More informationFABER Formal Languages, Automata. Lecture 2. Mälardalen University
CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 2 Mälardalen University 2010 1 Content Languages, g Alphabets and Strings Strings & String Operations Languages & Language Operations
More informationA PROBABILITY PROBLEM
A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,
More information19.1 The dictionary problem
CS125 Lecture 19 Fall 2016 19.1 The dictioary proble Cosider the followig data structural proble, usually called the dictioary proble. We have a set of ites. Each ite is a (key, value pair. Keys are i
More informationThe Binomial Multi- Section Transformer
4/4/26 The Bioial Multisectio Matchig Trasforer /2 The Bioial Multi- Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where: ( ω ) = + e +
More informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationMath 104: Homework 2 solutions
Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationChapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients:
Chapter 7 COMBINATIONS AND PERMUTATIONS We have see i the previous chapter that (a + b) ca be writte as 0 a % a & b%þ% a & b %þ% b where we have the specific formula for the biomial coefficiets: '!!(&)!
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm
More informationThe multiplicative structure of finite field and a construction of LRC
IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationContents Two Sample t Tests Two Sample t Tests
Cotets 3.5.3 Two Saple t Tests................................... 3.5.3 Two Saple t Tests Setup: Two Saples We ow focus o a sceario where we have two idepedet saples fro possibly differet populatios. Our
More informationHomework 1 Solutions. The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
Homewor 1 Solutios Math 171, Sprig 2010 Hery Adams The exercises are from Foudatios of Mathematical Aalysis by Richard Johsobaugh ad W.E. Pfaffeberger. 2.2. Let h : X Y, g : Y Z, ad f : Z W. Prove that
More informationBernoulli Numbers and a New Binomial Transform Identity
1 2 3 47 6 23 11 Joural of Iteger Sequece, Vol. 17 2014, Article 14.2.2 Beroulli Nuber ad a New Bioial Trafor Idetity H. W. Gould Departet of Matheatic Wet Virgiia Uiverity Morgatow, WV 26506 USA gould@ath.wvu.edu
More informationChapter 4 Postulates & General Principles of Quantum Mechanics
Chapter 4 Postulates & Geeral Priciples of Quatu Mechaics Backgroud: We have bee usig quite a few of these postulates already without realizig it. Now it is tie to forally itroduce the. State of a Syste
More informationPermutations, Combinations, and the Binomial Theorem
Permutatios, ombiatios, ad the Biomial Theorem Sectio Permutatios outig methods are used to determie the umber of members of a specific set as well as outcomes of a evet. There are may differet ways to
More informationMathematical Preliminaries
Matheatical Preliiaries I this chapter we ll review soe atheatical cocepts that will be used throughout this course. We ll also lear soe ew atheatical otatios ad techiques that are iportat for aalysis
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationCombinatorial and Automated Proofs of Certain Identities
Cobiatorial ad Autoated Proofs of Certai Idetities The MIT Faculty has ade this article opely available Please share how this access beefits you Your story atters Citatio As Published Publisher Brereto,
More information1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND
NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE 1.2.1 Iclusio-Exclusio Forula I the followig sectio we will discuss
More informationBinomial Notations Traditional name Traditional notation Mathematica StandardForm notation Primary definition
Bioial Notatios Traditioal ae Bioial coefficiet Traditioal otatio Matheatica StadardFor otatio Bioial, Priary defiitio 06.03.0.0001.01 1 1 1 ; 0 For Ν, Κ egative itegers with, the bioial coefficiet Ν caot
More informationIs mathematics discovered or
996 Chapter 1 Sequeces, Iductio, ad Probability Sectio 1. Objectives Evaluate a biomial coefficiet. Expad a biomial raised to a power. Fid a particular term i a biomial expasio. The Biomial Theorem Galaxies
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationOrthogonal Function Solution of Differential Equations
Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple,
More informationCurve Sketching Handout #5 Topic Interpretation Rational Functions
Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials
More informationWorksheet on Generating Functions
Worksheet o Geeratig Fuctios October 26, 205 This worksheet is adapted from otes/exercises by Nat Thiem. Derivatives of Geeratig Fuctios. If the sequece a 0, a, a 2,... has ordiary geeratig fuctio A(x,
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More information