= z 20 z n. (k 20) + 4 z k = 4
|
|
- Ezra Dawson
- 6 years ago
- Views:
Transcription
1 Problem Set #7 solutons (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z (z 5 ( + z + z 2 + z ( z 20 z + 5 z 20 z n 5 + z n+20 Ths s an acceptable but not entrely canoncal representaton of the generatng functon: to be standardzed (and useful for answerng the queston asked we d lke to change the exponent on z to be a sngle varable. So let k n + 20, replacng every n above wth k 20: k 200 ( (k 20 + z k ( k 6 z k so the coeffcent of z k s ( k 6 for z 20 (and for z < 20, the coeffcent s zero, as these terms do not appear n the seres expanson. (b Fnd the coeffcent of z k n (z + z + z 5 ( + z n, k 5. We use the known bnomal expanson (+z n ( n z below. Note that the upper bound can be, snce every bnomal coeffcent ( n k wth k > n s zero; ths s not necessary, but actually makes our seres manpulaton easer, snce we need not worry about the upper lmt of the sum: (z + z + z 5 ( + z n (z + z + z 5 z z + + z + + z +5 Now we need to perform ndex-shfts on each sum to get our z-terms to match; so we ntroduce new ndces k +, k +, and k + 5 to the three sums: z + + z + + z +5 z k + z k + z k k k k 5 k k k5 [( ] [( ] n n [( ( ( ] n n n z + nz z + + n z z k 2 k k k k20 k5 Page of 6 March, 2008
2 Problem Set #7 solutons So despte all the specal-casng necessary for k < 5 (whch need not be calculated as explctly as shown here, we see that the coeffcent of z k for k 5 wll always be ( ( n k + n ( k + n k What s the generatng functon for a n, the number of ntegers from 0 to 99, 999, whose dgts sum to n? How many of these ntegers have dgts that sum to 27? The generatng functon for numbers from zero to 99, 999 categorzed by dgt sum s ( + z + z z 9 5 (cf. problem 7..5(a from Problem Set #6. We shall perform expansons of the generatng functon to determne the z 27 term of ths seres: ( z ( + z + + z z ( z 0 5 ( z 5 ( 5z 0 + 0z 20 0z 0 + 5z 0 z 50 + z n To fnd the z 27 term, we consder all possble products of z 27 that can be produced va a product of the 50th-degree polynomal and seres gven. A z 27 term arses va all of the followng products: (27+ z 27, 5z 0 (7+ z 7, and 0z 20 (7+ z 7. Thus, the total z 27 coeffcent n ths product s: ( ( ( Show that the followng formulas hold: (a z ( + z( + z2 ( + z ( + z 8. We can prove by nducton that (+z(+z 2 (+z 2n +z+z 2 + +z 2n+ : t s clearly true for n, and assumng t for a partcular n, we can derve ts truth for n + : ( + z( + z 2 ( + z 2n + z + z z 2n+ ( ( + z( + z 2 ( + z 2n ( + z 2n+ + z + z z 2n+ ( + z 2n+ ( + z + z z 2n+ ( + z 2n+ + z 2n+ + + z 2n z 2n+ +(2 n+ + z + z z 2(2n+ + z + z z (2n+2 We thus have that n ( + z2 2 n+ z n, so n the lmtng case (dong the sorts of thngs I tell MATH 205 students that we are never, ever allowed to do, ( + z 2 z n z Page 2 of 6 March, 2008
3 Problem Set #7 solutons (b z ( + z + z2 ( + z + z 6 ( + z 9 + z 8. Ths proceeds smlarly to the argument above, only wth n nstead of 2 n everywhere Fnd a generatng functon and formula for a n, the number of ways to dstrbute n smlar jugglng balls to four dfferent jugglers so that each juggler receves an odd number of jugglng balls that s larger than or equal to three. Montorng the number of balls dstrbuted, dstrbuton of balls to a sngle juggler under these constrants can be modeled wth the seres z + z 5 + z 7 + z ( + z 2 + z + z z. Dong so for jugglers yelds the generatng functon 2. z 2 ( z 2 To fnd a formula for a n, we need to put ths generatng functon n the form a n z n n order to make determnng ndvdual coeffcents easy. We use the known seres expanson for wth z 2 standng n for y: ( y k z 2 ( z 2 z2 k 60 ( m + ( m + (z 2 m z 2m+2 m0 Now we use ndex-shftng, lettng k m+6 and replacng all occurrences approprately to clean up the exponent on z and gettng the followng smplfed sum: ( k 6 + ( k z 2k z 2k Thus, the coeffcent of z 2k s ( k, and the coeffcent of any odd pwoer of z s zero; thus a n ( k f n 2k, and an 0 f n s odd In how many ways can a con be flpped 25 tmes n a row so that exactly fve heads occur and no more than seven tals occur consecutvely? Let n keep track of the total number of flps n the followng process: flp from zero to seven tals; flp a head then flp from zero to seven tals fve tmes. Ths process s guaranteed to yeld exactly fve heads and no more than 7 consecutve tals. Ths procedure has dscrete steps, but they fall nto two categores: flps of a sngle con yeldng heads, wth generatng functon z; and flps of a con from zero to seven tmes yeldng tals, wth generatng functon + z + z z 7. Thus, our generatng functon n total s k6 m0 ( + z + z z 7 [ z( + z + z z 7] 5 z 5 ( z 8 6 whch can be calculated to be z 5 ( 6z 8 + 5z 6 + z z n 5 ( z 6 We then determne the z 25 term n ths seres by fndng all possble products of terms from the factors yeldng z 25 : z 5 (25 5 z 20, z 5 6z 8 (7 5 z 2, and z 5 5z 6 (9 5 z ; thus the total coeffcent of z 2 5 n the above product s ( ( ( Page of 6 March, 2008
4 Problem Set #7 solutons 7... Fnd a generatng functon for a n, the number of parttons of n nto (a Odd ntegers. A partton of n nto odd numbers can specfcally be consdered as a partton nto x ones, x threes, x 5 fves, etc.; then, these varables conform to the equaton x +x +5x 5 + n. The generatng functon descrbng the number of solutons to ths equaton s the product of the generatng functons for the ndvdual choces of x, x, x 5, etc. Thus, we have the product (+z+z 2 +z + (+z +z 6 +z 9 + (+z 5 +z 0 +z 5 ( z( z ( z 5 (b Dstnct odd ntegers. Ths s as above, but wth each x constraned to be zero or, so we have fnte generatng functons for each ndvdual choce, yeldng: ( + z( + z ( + z 5 ( + z Fnd a product whose expanson can be used to fnd the number of parttons of (a 2 wth even summands. For even summands we are determnng the number of nonnegatve solutons to the equaton 2x 2 + x + 6x 6 + 8x 8 + 0x 0 + 2x 2 2. The generatng functon for each of these can, n the nterest of yeldng a fnte product, be capped at the z 2 term, and ther product s thus (+z 2 +z +z 6 +z 8 +z 0 +z 2 (+z +z 8 +z 2 (+z 6 +z 2 (+z 8 (+z 0 (+z 2 whose product s an enormous 66th-degree polynomal, n whch the only term of nterest to us s z 2 (so there are even-term parttons of 2; unsurprsng snce each s the double of one of the free parttons of 6. (b 0 wth summands greater than 2. We are determnng the number of nonnegatve solutons to x +x +5x x 0 0; we do ths by multplyng the generatng functons for each summand, cappng each seres wth the z 0 term to get polynomals nstead of seres, and thus get ( + z + z 6 + z 9 ( + z + z 8 ( + z 5 + z 0 ( + z 6 ( + z 7 ( + z 8 ( + z 9 ( + z 0 whose product s a 67th-degree polynomal, n whch we are nterested n the term 5z 0, denotng 5 parttons of 0 nto parts or larger; we can, knowng there are so few, actually explctly enumerate them: + +, + 7, + 6, 5 + 5, and 0. (c 9 wth dstnct summands. We are determnng the number of solutons to x + 2x 2 + x + + 9x 9 9 wth each x {0, }. Thus the generatng functons for ndvdual summands are bnomals, and we multply them as such: ( + z( + z 2 ( + z ( + z ( + z 5 ( + z 6 ( + z 7 ( + z 8 ( + z 9 Page of 6 March, 2008
5 Problem Set #7 solutons to get a 5th-degree polynomal whch contans the term 8z 9, ndcatng 8 parttons of 9 nto dstnct summands. The complete lst of these s +2+6, ++5, 2 + +, + 8, 2 + 7, + 6, + 5, and (a Show that the number of parttons of n s exactly equal to the number of parttons of n whose smallest summand s. One could do ths wth generatng functons: we know that ( + z + z 2 + ( + z 2 + z + ( + z + z 6 + p(nz n but f we force our parttons to contan at least one, the frst factor n the above product becomes (z + z 2 + z + z( + z + z 2, yeldng the generatng functon z p(nz n p(nz n+ p(n z n So we see that forcng at least one n a partton of n elements s possble n p(n ways. We could also accomplsh ths wth an explct bjecton. Any partton of n wth a n ts expanson can be mapped to a partton of n by smply removng the ; ths s clearly a bjectve map snce t can be reversed by takng any partton of n and addng a sngle to t. Snce ths s a bjecton, the two sets nvolved must be of equal sze. (b Descrbe the parttons of n that are counted by the expresson p(n p(n. We know p(n counts the parttons of n; and from the above concluson we know that p(n counts the parttons of n whch have a n ther expanson. Thus, ther dfference counts the parttons of n whch do not contan a n ther expansons. Let P n be the set of parttons of n, so that P n p(n. Then p(n p(n would be the sze of P n wth some set bjectvely mapped to P n removed. There s a smple njecton of P n nto P n : smply add a sngle to every partton n P n to get a partton from P n. The set mapped onto by ths operaton s precsely those parttons of P n contanng n ther expansons, snce ths procedure could be reversed by removng a. The set yelded by removng all these mages of P n from P n s thus those parttons of n not usng the number ; and ths s what s counted by p(n p(n Use the Ferrer s graph to show that the number of parttons of n nto exactly k summands equals the number of parttons of n havng ts largest summand equal to k. All one needs to do s consder the transpose. A Ferrer s graph assocated wth a partton nto k summands wll have heght exactly k, and thus, ts transpose wll have wdth exactly k, meanng that one of ts summands s sze k but none exceed k. Snce the transpose s a bjectve map, the parttons satsfyng these two condtons are equnumerous. n Page 5 of 6 March, 2008
6 Problem Set #7 solutons 7... Fnd a generatng functon for the number of parttons of n nto (a Summands no larger than. We want nonnegatve solutons to x +2x 2 +x 6 +x n, whch can be expressed as a product of the four ndvdual generatng functons for decson of x, x 2, x, and x : (+z+z 2 +z + (+z 2 +z +z 6 + (+z +z 6 +z 9 + (+z +z 8 +z 2 + (b Summands the largest of whch s. Ths s as above, except we requre at least one, so we constran x, whch affects the fourth multplcand n our expresson: (+z +z 2 +z + (+z 2 +z +z 6 + (+z +z 6 +z 9 + (z +z 8 +z 2 + (c At most four summands. By the transposton bjecton on Ferrer s dagrams we know ths s an dentcal enumeraton to part (a, and thus has the same generatng functon. (d Exactly four summands. By the transposton bjecton on Ferrer s dagrams we know ths s an dentcal enumeraton to part (a, and thus has the same generatng functon. Page 6 of 6 March, 2008
1 Generating functions, continued
Generatng functons, contnued. Generatng functons and parttons We can make use of generatng functons to answer some questons a bt more restrctve than we ve done so far: Queston : Fnd a generatng functon
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More informationComplex Numbers Alpha, Round 1 Test #123
Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test
More informationChapter 1. Probability
Chapter. Probablty Mcroscopc propertes of matter: quantum mechancs, atomc and molecular propertes Macroscopc propertes of matter: thermodynamcs, E, H, C V, C p, S, A, G How do we relate these two propertes?
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationSection 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More information1 Generating functions, continued
Generatng functons, contnued. Exponental generatng functons and set-parttons At ths pont, we ve come up wth good generatng-functon dscussons based on 3 of the 4 rows of our twelvefold way. Wll our nteger-partton
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationDigital Signal Processing
Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationComplex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)
Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 12 10/21/2013. Martingale Concentration Inequalities and Applications
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 1 10/1/013 Martngale Concentraton Inequaltes and Applcatons Content. 1. Exponental concentraton for martngales wth bounded ncrements.
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationE Tail Inequalities. E.1 Markov s Inequality. Non-Lecture E: Tail Inequalities
Algorthms Non-Lecture E: Tal Inequaltes If you hold a cat by the tal you learn thngs you cannot learn any other way. Mar Twan E Tal Inequaltes The smple recursve structure of sp lsts made t relatvely easy
More informationExercises. 18 Algorithms
18 Algorthms Exercses 0.1. In each of the followng stuatons, ndcate whether f = O(g), or f = Ω(g), or both (n whch case f = Θ(g)). f(n) g(n) (a) n 100 n 200 (b) n 1/2 n 2/3 (c) 100n + log n n + (log n)
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationSELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:
SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationCase A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.
THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the Szemeréd-Trotter theorem. The method was ntroduced n the paper Combnatoral complexty
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
More informationEdge Isoperimetric Inequalities
November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary
More informationChapter 8 SCALAR QUANTIZATION
Outlne Chapter 8 SCALAR QUANTIZATION Yeuan-Kuen Lee [ CU, CSIE ] 8.1 Overvew 8. Introducton 8.4 Unform Quantzer 8.5 Adaptve Quantzaton 8.6 Nonunform Quantzaton 8.7 Entropy-Coded Quantzaton Ch 8 Scalar
More informationCS 2750 Machine Learning. Lecture 5. Density estimation. CS 2750 Machine Learning. Announcements
CS 750 Machne Learnng Lecture 5 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square CS 750 Machne Learnng Announcements Homework Due on Wednesday before the class Reports: hand n before
More informationand problem sheet 2
-8 and 5-5 problem sheet Solutons to the followng seven exercses and optonal bonus problem are to be submtted through gradescope by :0PM on Wednesday th September 08. There are also some practce problems,
More informationRandom Walks on Digraphs
Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More information11 Tail Inequalities Markov s Inequality. Lecture 11: Tail Inequalities [Fa 13]
Algorthms Lecture 11: Tal Inequaltes [Fa 13] If you hold a cat by the tal you learn thngs you cannot learn any other way. Mark Twan 11 Tal Inequaltes The smple recursve structure of skp lsts made t relatvely
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationCommon loop optimizations. Example to improve locality. Why Dependence Analysis. Data Dependence in Loops. Goal is to find best schedule:
15-745 Lecture 6 Data Dependence n Loops Copyrght Seth Goldsten, 2008 Based on sldes from Allen&Kennedy Lecture 6 15-745 2005-8 1 Common loop optmzatons Hostng of loop-nvarant computatons pre-compute before
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationVapnik-Chervonenkis theory
Vapnk-Chervonenks theory Rs Kondor June 13, 2008 For the purposes of ths lecture, we restrct ourselves to the bnary supervsed batch learnng settng. We assume that we have an nput space X, and an unknown
More informationDepartment of Electrical & Electronic Engineeing Imperial College London. E4.20 Digital IC Design. Median Filter Project Specification
Desgn Project Specfcaton Medan Flter Department of Electrcal & Electronc Engneeng Imperal College London E4.20 Dgtal IC Desgn Medan Flter Project Specfcaton A medan flter s used to remove nose from a sampled
More informationBernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
More informationThe KMO Method for Solving Non-homogenous, m th Order Differential Equations
The KMO Method for Solvng Non-homogenous, m th Order Dfferental Equatons Davd Krohn Danel Marño-Johnson John Paul Ouyang March 14, 2013 Abstract Ths paper shows a smple tabular procedure for fndng the
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationAS-Level Maths: Statistics 1 for Edexcel
1 of 6 AS-Level Maths: Statstcs 1 for Edecel S1. Calculatng means and standard devatons Ths con ndcates the slde contans actvtes created n Flash. These actvtes are not edtable. For more detaled nstructons,
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationBeyond Zudilin s Conjectured q-analog of Schmidt s problem
Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs
More informationSolutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010
Solutons to the 7st Wllam Lowell Putnam Mathematcal Competton Saturday, December 4, 2 Kran Kedlaya and Lenny Ng A The largest such k s n+ 2 n 2. For n even, ths value s acheved by the partton {,n},{2,n
More informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationDeterminants Containing Powers of Generalized Fibonacci Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationCS-433: Simulation and Modeling Modeling and Probability Review
CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown
More informationMixed-integer vertex covers on bipartite graphs
Mxed-nteger vertex covers on bpartte graphs Mchele Confort, Bert Gerards, Gacomo Zambell November, 2006 Abstract Let A be the edge-node ncdence matrx of a bpartte graph G = (U, V ; E), I be a subset the
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationProblem Set 9 - Solutions Due: April 27, 2005
Problem Set - Solutons Due: Aprl 27, 2005. (a) Frst note that spam messages, nvtatons and other e-mal are all ndependent Posson processes, at rates pλ, qλ, and ( p q)λ. The event of the tme T at whch you
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationG = G 1 + G 2 + G 3 G 2 +G 3 G1 G2 G3. Network (a) Network (b) Network (c) Network (d)
Massachusetts Insttute of Technology Department of Electrcal Engneerng and Computer Scence 6.002 í Electronc Crcuts Homework 2 Soluton Handout F98023 Exercse 21: Determne the conductance of each network
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationDiscussion 11 Summary 11/20/2018
Dscusson 11 Summary 11/20/2018 1 Quz 8 1. Prove for any sets A, B that A = A B ff B A. Soluton: There are two drectons we need to prove: (a) A = A B B A, (b) B A A = A B. (a) Frst, we prove A = A B B A.
More informationUNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction
Prerequste Sklls Ths lesson requres the use of the followng sklls: understandng that multplyng the numerator and denomnator of a fracton by the same quantty produces an equvalent fracton multplyng complex
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationFor all questions, answer choice E) NOTA" means none of the above answers is correct.
0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More information