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Math /home/m262f99/KOEPF/worksheetsV.4/hyperdemo.mws 262a, Fall 1999, Glenn Tesler Polynomial/Hypergeometric solutions of recurrences 11/8/99 > read hsum.mpl ; Copyright 1998 Wolfram Koepf, Konrad-Zuse-Zentrum Berlin Example 1. Polynomial solutions of a recurrence equation. Since (n*(n+1)*(n+2,8))^2 - (n,8)*(n+8)*(n+9))^2 = 0, we must have f(n) = pochhammer(n,8)^2 = (n(n+1)...(n+7))^2 as a solution of (n(n+1))^2 f(n+2) - ((n+8)(n+9))^2 f(n) = 0 > recpoly((n*(n+1))^2*f(n+2) - ((n+8)*(n+9))^2*f(n), f(n)); 0 n2 ( n + 7 )2 ( n + 6 )2 ( n + 5 )2 ( n + 4 )2 ( n + 3 )2 ( n + 2 )2 ( n + 1 )2 delta_0 is an arbitrary constant. Example 2. Hypergeometric solutions of a recurrence equation. > f1 := hyperterm([a,b],[c],x,n); pochhammer( c, n ) n! > simpcomb(subs(n=n+2,f1)/f1); ( a + n ) ( a + n + 1 ) ( b + n ) ( b + n + 1 ) x2 (c + n) (c + n + 1) (n + 1) (n + 2) > eq := numer(")*f(n) - denom(")*f(n+2); eq := ( a + n ) ( a + n + 1 ) ( b + n ) ( b + n + 1 ) x2 f( n ) ( c + n ) ( c + n + 1 ) ( n + 1 ) ( n + 2 ) f( n + 2 ) By construction, this is solved by the hypergeometric term 2F1(a,b;c;x)_n > ratios := rechyper(eq,f(n)); x (b + n) (a + n) x (b + n) (a + n) , } (n + 1) (c + n) (n + 1) (c + n) It doesn t return f(n), but rather the ratio f(n+1)/f(n). Also, it found a SECOND hypergeometric solution: 2F1(a,b;c; -x)_n ratios := { Page 1 f1 := pochhammer( a, n ) pochhammer( b, n ) xn which in retrospect isn t surprising. > ratiosk := subs(n=k,ratios): > product(ratiosk[1],k=0..n-1); product(ratiosk[2],k=0..n-1); xn ( b + n ) ( a + n ) ( c ) ( n + 1 ) ( c + n ) ( b ) ( a ) ( -1 )n xn ( b + n ) ( a + n ) ( c ) ( n + 1 ) ( c + n ) ( b ) ( a ) Application to factorization of operators Given a root x0 in C of a polynomial f(x) over Q, then f(x)=g(x)*(x-x0) for some polynomial g(x) over C. If the minimal polynomial of x0 over Q is h(x), then f(x)=p(x)h(x) for some polynomial p(x) over Q. Now consider a differential operator L = sum a_i(x) Dx^i, each a_i(x) in Q(x), that annihilates a function f(x), or a recurrence operator that annihilates a function f(n), etc. The first order operator R = f(x) Dx - f (x) annihilates f(x) in the differential case, and R = f(n) En - f(n+1) annihilates f(n) in the shift case. f is the analogue of a root x0 in an extension field; R is the analogue of the linear factor x-x0. Using non-commutative right division of L by R we obtain a factorization L = G R, with G, R operators (over the extension, usually not over Q(x)). There is a minimal order operator H over Q(x) that annihilates f. Using non-commutative right division of L by H gives an P operator over Q(x) with L = P H. There is software available for these divisions that we will use later. Non-commutative factorization in general is only a partially solved problem, and I don t have anyone s software for it. > L := subs(f(n)=1,f(n+1)=En,f(n+2)=En^2,eq); L := ( a + n ) ( a + n + 1 ) ( b + n ) ( b + n + 1 ) x2 ( c + n ) ( c + n + 1 ) ( n + 1 ) ( n + 2 ) En2 > R_op := En - ratios[1]; Rf := f(n+1) - ratios[1]*f(n); R_op := En + Rf := f( n + 1 ) + x (b + n) (a + n) (n + 1) (c + n) x ( b + n ) ( a + n ) f( n ) (n + 1) (c + n) > G_op := A*En - B; GRf := A*subs(n=n+1,Rf) - B*Rf; Page 2 G_op := A En B x ( b + n + 1 ) ( a + n + 1 ) f( n + 1 ) GRf := A f( n + 2 ) + (n + 2) (c + n + 1) x ( b + n ) ( a + n ) f( n ) B f( n + 1 ) + (n + 1) (c + n) > fs := {f(n),f(n+1),f(n+2)}: collect(GRf - eq, fs): coeffs(",fs): sols := factor(solve({"},{A,B})); sols := { A = ( c + n ) ( c + n + 1 ) ( n + 1 ) ( n + 2 ), B = ( a + n + 1 ) ( b + n + 1 ) x ( n + 1 ) ( c + n ) } > map(collect,factor(subs(sols,G_op)),En,factor); ( n + 1 ) ( c + n ) ( ( n + 2 ) ( c + n + 1 ) En + x ( b + n + 1 ) ( a + n + 1 ) ) The factored form of L obtained by using the first ratio to generate a right factor is > factorL := " * R_op; factorL := ( n + 1 ) ( c + n ) ( ( n + 2 ) ( c + n + 1 ) En + x ( b + n + 1 ) ( a + n + 1 ) ) x (b + n) (a + n) En + (n + 1) (c + n) Example 3. Failure to find a hypergeometric solution. Now test the recurrence obtained on my handout "Creative telescoping for a triple integral": The sequence a(n) = binomial(n,n/2)^2 satisfies a recurrence: > rechyper(16*(n+1)^2*a(n) - (n+2)^2*a(n+2), a(n)); {} There s no hypergeometric solutions because binomial(n,n/2) is 2-fold hypergeometric, not 1-fold. This can be detected automatically, but not with this software. (Even that can fail if there s no k-fold solutions for any k, which is possible.) Substitute n=2m, b(m)=a(n/2): > rechyper(16*(2*m+1)^2 * b(m) - (2*m+2)^2 * b(m+1), b(m)); } ( m + 1 )2 > subs(m=m0,"[1]): product(",m0=0..m-1); 1 2 4 ( 2 ) m + 2 m m2 {4 ( 2 m + 1 )2 ( m + 1 )2 > an1 := simplify(subs(m=n/2,")); Page 3 2 an1 := (2 n) 1 1 2 n+ 2 2 1 2 n + 1 2 This is the same as binomial(n,n/2)^2 (identifying binomial coefficients with their def. in terms of factorials, and factorials in terms of Gamma functions, so that non-nonnegative integer arguments are valid). > simplify( an1 / binomial(n,n/2)^2); 1 > Page 4
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dfin_closure.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: dfin_closure.mws Math 262a, Fall 1999, Glenn Tesler Sum/product/. of D-finite functions is D-finite 11/17/99 We have two functions defined by differential equations: > 5*diff(f(x),x$2)+f(x)=0; f_sol := dsolve(\",f(x); 3*diff(g(x),x)+g(x)=0; g_sol := ...
gbsample1.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: /home/m262f99/CHYZAK/worksheetsV.4/gbsample1.mws Math 262a, Fall 1999, Glenn Tesler Using commutative Grobner bases to solve a system of polynomial equations via elimination ideals 12/2/99 > restart; > with(Groebner): with(Mgfun): with(Holonomy): wi...
hw7ans.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: Math 262a Topics in Combinatorics Fall 1999 Glenn Tesler Homework 7 answers November 19, 1999 12. See Maple worksheet. Res(f, f , x). 3. (a) Let D1 = a2n2 i<j (i j )2 and D2 = n an View D1 and D2 as polynomials in the roots i of f and j of f . L...
hw6ansm.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: /home/m262f99/KOEPF/worksheetsV.4/hw6ansm.mws Math 262a, Fall 1999, Glenn Tesler Homework 6 > read hsum.mpl; Copyright 1998 Wolfram Koepf, Konrad-Zuse-Zentrum Berlin Koepf # 9.1 > recpoly(n*s(n+1)-(n+20)*s(n),s(n); 0 n ( n + 19 ) ( n + 18 ) ( n + 1...
hw6ans.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: Math 262 Topics in Combinatorics Glenn Tesler Homework 6 answers November 12, 1999 1 2. Koepf # 9.11. We do the order d case: d Now v(n + i) v(n) = (1 + )i 1 v(n) Pi (n)s(n + i) = 0 i=0 i = j=1 We found that the choices for q(n) are the ...
WZdemo.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: /home/m262f99/KOEPF/worksheetsV.4/WZdemo.mws Math 262a, Fall 1999, Glenn Tesler WZ method demo 10/29/99 > read hsum.mpl; Copyright 1998 Wolfram Koepf, Konrad-Zuse-Zentrum Berlin Example 1. Prove > Sum(binomial(n,k),k)=2^n; k binomial( n, k ) = 2 ...
hw7ansm.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: /home/m262f99/CHYZAK/worksheetsV.4/hw7ansm.mws Math 262a, Fall 1999, Glenn Tesler Homework 7 11/19/99 > with(Ore_algebra): cleanpol := (f,Sn) -> sort(collect(expand(f),Sn,factor),[Sn,x,n]): Problem 1. Set up an algebra. > An := shift_algebra([Sn,n]...
hw2.pdf
Path: UCSD >> MATH >> 262a Fall, 2008
Description: Math 262a Topics in Combinatorics Fall 1999 Glenn Tesler Homework 2 October 13, 1999 1. Facts about q-hypergeometric series. (They are all easy to prove.) 1+sr k . Prove the conuence process (a) Heres a reason for the factor (1)k q (2) r lim r...
matlab.pdf
Path: UCSD >> MATH >> 270b Winter, 2008
Description: Math 270AB. Numerical Mathematics Introduction to Matlab Instructor: Philip E. Gill September 20, 2006 Matlab is an interactive matrix manipulation program which allows the user to perform standard matrix operations such as multiplication, inversion...
280Ahw9.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Problem Set #9 (due Wednesday, November 26, in class) 1. (Method of Moments) Suppose X1 , X2 , . . . are i.i.d. random variables such that E[|Xi |k ] < . For j = 1, . . . , k, let j = E[Xij ] and j,n = 1 n n Xij . i=1 Suppose g : Rk R is continuo...
280Ahw2.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Problem Set #2 (due Wednesday, October 8, in class) 1. Let (, F, P ) be a probability space. Suppose A F and P (A) > 0. Let = A, and let G = {B F : B A}. For all B G, let Q(B) = P (B)/P (A). Prove that ( , G, Q) is a probability space. 2. Let d...
280Asyllabus.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Math 280A, Probability Theory, Fall 2008 Lectures: Instructor: Oce Hours: TA: TA Oce Hours: Web Page: Textbook: 9:00-9:50 AM, Mondays, Wednesdays, and Fridays in APM 5402 Jason Schweinsberg (jschwein@math.ucsd.edu) Mondays 10:00-11:30 AM and Tuesdays...
280Ahw10.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Problem Set #10 (due Friday, December 5, in class) Note: Problems 1-3 can be done using material covered in class through Wednesday, November 26. Problems 4 and 5 require material from section 1.9, which will be discussed on Monday, December 1. 1. (s...
280Ahw5.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Problem Set #5 (due Wednesday, October 29, in class) 1. (Durrett, Exercise 4.2, p. 22) Suppose A and B are events. a) Show that if A and B are independent, then so are A and B c , Ac and B, and Ac and B c . b) Show that A and B are independent if and...
280Ahw4.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Problem Set #4 (due Wednesday, October 22, in class) 1. Suppose X is a nonnegative random variable and P (X > 0) > 0. Show that E[X] > 0. 2. Suppose X is a random variable such that E[|X|] < . Prove that for all > 0, there exists a simple random var...
280Ahw6.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Problem Set #6 (due Wednesday, November 5, in class) 1. (Resnick, Exercise 3, p. 110) Suppose A1 , A2 , . . . is an innite sequence of independent events. Prove that P i=1 Ai = i=1 P (Ai ). 2. (Durrett, Exercise 4.14, p. 33) Let = {1, 2, 3, 4...
hw2sol.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Math 280A, Fall 2007 Homework 2 Solutions (Section 2.6) 6. The inclusion-exclusion formula expresses P (n Ak ) as k=1 vanish because of the assumed almost disjointness. Thus n n k=1 P (Ak ) plus a linear combination of probabilities of intersection...
hw7sol.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Math 280A, Fall 2007 Homework 7 Solutions (Section 6.7) 4. If 0 < m < n, then E[(Sn Sm )2 ] = 2 n k=m+1 a2 , which tends to 0 as m and n k tend to innity if and only if k=1 a2 < . That is, {Sn } is Cauchy in L2 if and only if k 2 k=1 ak < . Sin...
mct.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Math 280A Supplement on the Monotone Class Theorem Fall 2004 In dealing with integrals, the following functional form of the Monotone Class Theorem is often useful. (1) Theorem. Let K be a collection of bounded real-valued functions on that is clo...
hw5sol.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Math 280A, Fall 2007 Homework 5 Solutions (Sections 4.6 and 5.10) 27. As hinted, we know from the theory of power series that the radius of convergence R() of the power series (27.1) n=0 Xn ()z n is given by R()1 = lim sup |Xn ()|1/n . n The limit...
hw1sol.pdf
Path: UCSD >> MATH >> 280a Fall, 2008
Description: Math 280A, Fall 2007 Homework 1 Solutions (Section 1.9) 18. [Note: The displayed condition should read A, B A implies A \\ B = AB c A.] Suppose A A. Then Ac = \\ A A as well, because A. Thus A is closed under the formation of complements. Next, ...
doob.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2005 Doobs Inequalities Everything that follows takes place on a probability space (, F, P ) equipped with a ltration {Fn : n = 0, 1, 2, . . .}, with Fn F for all n. 1. Submartingale maximal inequality. Let {Xn } be a non-negative ...
martslln.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2005 SLLN for Martingales Everything that follows takes place on a probability space (, F, P) equipped with a ltration {Fn : n = 0, 1, 2, . . .}, with Fn F for all n. 1. Square-integrable martingales. A martingale M = (Mn ) is said...
normal.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2005 Conditioning and the Bivariate Normal Distribution In what follows, X and Y are random variables dened on a probability space (, B, P ), and G is a sub-eld of B. 1. Regular Conditional Distributions. The conditional probability...
hw7sol.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2008 Homework 7 Solutions (Section 10.17) 27. If is B -measurable and if B is a Borel subset of R, then { B} B ; this means that for each n {0, 1, 2, . . . , }, { B} { = n} Bn . In this case, {1{=n} B} = { B} { = n} Bn if...
hw4sol.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2008 Homework 4 Solutions (Section 9.9) 4. Notice that E[Uk ] = 0 and Var(Uk ) = a2 /3. k n (a) If k a2 = , then s2 := k=1 a2 increases to as n . In particular, given > 0, we can nd n k k n0 so large that sn0 > M/. If n n0 then ...
hw6sol.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2008 Homework 6 Solutions (Section 10.17) 14. We shall take the ltration (Fn ) to be that generated by {Yn }. Using the abbreviation Sn := we have Xn+1 = (Sn + Yn+1 )2 (n + 1) 2 (14.1) 2 2 = Sn + 2Sn Yn+1 + Yn+1 (n + 1) 2 2 = Xn +...
normal.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2008 Conditioning and the Bivariate Normal Distribution In what follows, X and Y are random variables dened on a probability space (, B, P ), and G is a sub-eld of B. 1. Regular Conditional Distributions. The conditional probability...
hw3sol.pdf
Path: UCSD >> MATH >> 280b Winter, 2008
Description: Math 280B, Winter 2008 Homework 3 Solutions (Section 9.9) 5. Let n denote the characteristic function of Xn + Yn , n 0. Then, for t R, n (t) = E[exp(it(Xn + Yn )] (5.1) = E[exp(itXn ) exp(itYn )] = E[exp(itXn )] E[exp(itYn )]. right side of (5.1) ...
hw5.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Homework 5 Due May 23 In all of the problems that follow, B = (Bt )t0 is a standard Brownian motion with B0 = 0, dened on some probability space (, F , P). 1. The hyperbolic functions sinh and cosh are dened for real x by sinh...
hw3.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Homework 3 Due April 30 In the problems below, X = (Xn ) is a Markov chain with countable state space S and transition n=0 probability matrix P = {p(x, y)}x,yS . We suppose that X has been constructed on the sequence space = ...
hw5sol.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Homework 5 Solutions 1. The hyperbolic functions sinh and cosh are dened for real x by sinh(x) := Show that St := e 2 ex ex , 2 2 cosh(x) := t/2 ex + ex . 2 t/2 sinh(Bt ) and Ct := e cosh(Bt ) are martingales. Solution...
syllabus.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C Topics Spring 2008 1. Concluding discussion of Chapter 10 in the Resnick text 2. Markov Chains (a) Strong Markov property (b) Transience and recurrence (c) Limit theorems and invariant measures 3. Brownian Motion (a) Construction and basic ...
stochint.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Stochastic Integral In what follows, (, F , P)is the canonical sample space of the Brownian motion (Bt )t0 with B0 = 0; other notation is that used in class. We dene the stochastic integral with respect to Brownian motion in se...
hw1.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Homework 1 Due April 9 1. Exercise 45, p. 438. 2. Exercise 54, p. 440. 3. Let {1 , 2 , . . .} be an iid sequence, and let f : R2 R be a symmetric (measurable) function of two variables such that E|f (1 , 2 )| < . Dene Un := an...
levy.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Lvys Theorem e Let (, F , P) be a complete probability space endowed with a right-continuous* ltra- tion (Ft )t0 such that F0 contains all the P-null sets in F and t Ft = F . Let M = (Mt )t0 be a real-valued stochastic process...
mct.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Functional Form of the Monotone Class Theorem In dealing with integrals, the following form of the Monotone Class Theorem is often useful. (1) Theorem. Let K be a collection of bounded real-valued functions on that is closed u...
bmall.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: An Invitation to Sample Paths of Brownian Motion Yuval Peres Lecture notes edited by Balint Virag, Elchanan Mossel and Yimin Xiao Version of November 1, 2001 1 2 Preface. These notes record lectures I gave at the Statistics Department, Universit...
integral.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 An Integral The following integral arises in the calculation of the Laplace transform of the density function of the rst passage time Tb . Proposition. For > 0 and 0, (1) 0 et/t t1/2 dt = 2 . e Proof. Fix > 0 and view...
hw4.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Homework 4 Due May 9 1. (CLT for Markov chains) In this problem X = (Xn )n=0 is an irreducible Markov chain with nite state space S and transition probability matrix P = {p(x, y)}x,yS . Because S is nite, X is necessarily pos...
expmart.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Exponential Martingales In what follows, (, F , P) is the canonical sample space of the Brownian motion (Bt )t0 with B0 = 0; other notation is that used in class. Given H L2 let M denote the associated local martingale: loc t ...
exchange.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Exchangeability Let X = (X1 , X2 , . . .) be an exchangeable sequence of random variables. As discussed in class, there is no loss of generality (and some gain of convenience) in assuming that the sample space is the sequence ...
hw4sol.pdf
Path: UCSD >> MATH >> 280c Spring, 2008
Description: Math 280C, Spring 2008 Homework 4 Solutions 1. (CLT for Markov chains) In this problem X = (Xn ) is an irreducible Markov chain with nite n=0 state space S and transition probability matrix P = {p(x, y)}x,yS . Because S is nite, X is necessarily pos...
539.pdf
Path: UCSD >> MATH >> 283 Spring, 2008
Description: The New England Journal of Medicine C o py r ig ht 2 0 0 1 by t he Ma s s ac h u s e t t s Me d ic a l S o c ie t y VOLUME 344 F E B R U A R Y 22, 2001 NUMBER 8 GENE-EXPRESSION PROFILES IN HEREDITARY BREAST CANCER INGRID HEDENFALK, M.S., DAVID D...
gm-65.pdf
Path: UCSD >> MATH >> 283 Spring, 2008
Description: EXPERIMENTS IN PLANT HYBRIDIZATION (1865) GREGOR MENDEL Read at the February 8th, and March 8th, 1865, meetings of the Brnn Natural History Society Mendel, Gregor. 1866. Versuche ber Plflanzenhybriden. Verhandlungen des naturforschenden Vereines in ...
539.pdf
Path: UCSD >> MATH >> 283 Spring, 2008
Description: The New England Journal of Medicine C o py r ig ht 2 0 0 1 by t he Ma s s ac h u s e t t s Me d ic a l S o c ie t y VOLUME 344 F E B R U A R Y 22, 2001 NUMBER 8 GENE-EXPRESSION PROFILES IN HEREDITARY BREAST CANCER INGRID HEDENFALK, M.S., DAVID D...
hw1.pdf
Path: UCSD >> MATH >> 283 Spring, 2008
Description: Math 283, Winter 2009, Prof. Tesler Homework #1, Due Wednesday January 14, 2009 Ewens & Grant 2nd edition: Problems 1.1, 1.3 (geometric distribution only, skip Poisson), 1.8, 1.31, 1.32, 1.34 and the problems below. Problem H-1. A value is determined...
SDE.course.pdf
Path: UCSD >> MATH >> 286 Fall, 2008
Description: AN INTRODUCTION TO STOCHASTIC DIFFERENTIAL EQUATIONS VERSION 1.2 Lawrence C. Evans Department of Mathematics UC Berkeley Chapter 1: Introduction Chapter 2: A crash course in basic probability theory Chapter 3: Brownian motion and white noise Chapte...
integral.pdf
Path: UCSD >> MATH >> 286 Fall, 2008
Description: Math 286, Fall 2006 An Integral The following integral arises in the calculation of the Laplace transform of the density function of the rst passage time Tb . Proposition. For > 0 and 0, (1) 0 et/t t1/2 dt = 2 e . Proof. Fix > 0 and view th...
hw_1.pdf
Path: UCSD >> MATH >> 287a Winter, 2008
Description: ECE 287A: Convex Optimization and Applications Fall 2006 Homework 1 Lecturer: Gert Lanckriet (please indicate whether you are taking the class for letter grade or S/U grade) Due: Friday 10/13/06 (in class) 1. In this problem we examine the geometr...
hw_1.pdf
Path: UCSD >> ECE >> 287a Fall, 2008
Description: ECE 287A: Convex Optimization and Applications Fall 2006 Homework 1 Lecturer: Gert Lanckriet (please indicate whether you are taking the class for letter grade or S/U grade) Due: Friday 10/13/06 (in class) 1. In this problem we examine the geometr...
CvxOptTutPaper.pdf
Path: UCSD >> MATH >> 287a Winter, 2008
Description: A Tutorial on Convex Optimization Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California email: hhindi@parc.com Abstract In recent years, convex optimization has become a computational tool of central importance in engineering, thanks ...
CvxOptTutPaper.pdf
Path: UCSD >> ECE >> 287a Fall, 2008
Description: A Tutorial on Convex Optimization Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California email: hhindi@parc.com Abstract In recent years, convex optimization has become a computational tool of central importance in engineering, thanks ...
hw_3.pdf
Path: UCSD >> MATH >> 287a Winter, 2008
Description: ECE 287A: Convex Optimization and Applications Fall 2006 Homework 3 Lecturer: Gert Lanckriet Due: Thursday 11/09/06 (in class) 1. Consider the following quadratic programming (QP) problem 1 min xT P x + q T x + r, x2 where 13 12 2 P = 12 17 6 ...
hw_3.pdf
Path: UCSD >> ECE >> 287a Fall, 2008
Description: ECE 287A: Convex Optimization and Applications Fall 2006 Homework 3 Lecturer: Gert Lanckriet Due: Thursday 11/09/06 (in class) 1. Consider the following quadratic programming (QP) problem 1 min xT P x + q T x + r, x2 where 13 12 2 P = 12 17 6 ...
hindiTutorial2.pdf
Path: UCSD >> MATH >> 287a Winter, 2008
Description: A Tutorial on Convex Optimization II: Duality and Interior Point Methods Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California 94304 email: hhindi@parc.com Abstract In recent years, convex optimization has become a computational tool ...
hindiTutorial2.pdf
Path: UCSD >> ECE >> 287a Fall, 2008
Description: A Tutorial on Convex Optimization II: Duality and Interior Point Methods Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California 94304 email: hhindi@parc.com Abstract In recent years, convex optimization has become a computational tool ...
levy.pdf
Path: UCSD >> MATH >> 294 Winter, 2008
Description: Math 294, Winter 2004 Lvys Theorem e Let (, F, P) be a complete probability space endowed with a right-continuous* ltration (Ft )t0 such that F0 contains all the P-null sets in F and t Ft = F. Let M = (Mt )t0 be a real-valued stochastic process adapt...
SMT1_ans.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Practice Midterm 1 No Books, Notes, Calculators, Electronic Devices, or Talking. Show ALL work to receive full credit. Note: this practice midterm has 7 questions, but the actual midterm will have 4-6 questions. 1 1. Assume that the functi...
lecture05.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 3, 2008 1 Section 2.2 - Domain and Range f (x) = x2 x+5 More examples: 100 80 60 40 20 0 y 20 40 60 80 100 10 8 6 4 2 0 x 2 4 6 8 10 The domain is all real number...
lecture14.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 20, 2008 1 Last Time At the end of last time, we were going over a problem where we had a function of the form f (t) = 2e0.4621t , and we had gured out that the half-life was 1.5 years, ...
finalreview.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Final Exam Review Outline Fall 2008 Daniel P. McAllaster Functional notation You should understand the meaning of f (x) and be able to evaluate a function using tables, graphs, and formulas. What is a function? What is not a function? You...
mt2review.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Midterm 2 Review Outline Fall 2008 Daniel P. McAllaster November 17, 2008 The Logarithm Function Be able to recognize and graph logarithmic functions. Be able to identify important features of the graphs. Logarithmic scales Function Trans...
lecture11.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 15, 2008 (Afternoon) 1 Section 3.4 - Continuous Growth and e Suppose you have $100 in the bank, and you get 10% interest annually. At the end of the year, you will get $10 in interest. W...
lecture10.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 13, 2008 1 Section 3.3 - Graphs of Exponential Functions Recall that all exponential functions can be written as Q = f (t) = abt . With linear functions, we know that they looked like y ...
lecture20.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 30, 2008 1 Section 6.3 - Radians We already know what degrees are, but dividing the circle in to 360 parts was quite arbitrary. Another system which many people use is to label angles by...
lecture18.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 25, 2008 1 2 Section 5.5 - The Family of Quadratic Functions Section 5 Tools - Completing the Square Often, especially in graphing functions, it is advantageous to have a quadratic funct...
lecture06.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 6, 2008 1 Comments from grader re: group work About half of the groups thought that 44 = 256. If it is not (4)4 then you do the power rst, and then the negative. A LOT of groups thought...
lecture23.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster November 5, 2008 (Evening) 1 Section 6.7 - Inverse Trigonometric Functions A couple of lectures ago, we found a function for your height on the ferris wheel to be h(t) = 225 sin 2 (t 7.5) + 225...
lecture04.pdf
Path: UCSD >> MATH >> 3c Fall, 2008
Description: Math 3C Lecture Notes Fall 2008 Daniel P. McAllaster October 1, 2008 (Evening) 1 1.1 Section 2.1 - Input and Output Evaluating Functions We had w = f (d) given by w = 3 d + 32. Suppose we want to know how much 2 water we should drink if we bicycle...
