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Math 121 Homework #4 Spring 2007 Due Monday May 7 Page 82-- # 8, 9, 10 Also: 1. A compact exhaustion of a ...

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UCLA - MATH - 121

Mathematics 121 Final Terence Tao June 10, 1997Problem 1. A set X Rn is said to be star-shaped at the origin if for every x X, the line segment {tx : 0 t 1} is also contained in X. Show that if X is star-shaped at the origin, then X is simply c

UCLA - MATH - 121

Math 121 Homework Assignment #1 Spring 2007 Due Friday April 13From your textbook: Page 8 # 7,8,11Page 12-13 #7Also: 1. Show that the completion of Q in the sense of problem 7 (on page 12-13) is R. 2. Show that the irrational numbers in R are n

UCLA - MATH - 131b

MATH 131B 2ND PRACTICE MIDTERM Problem 1. State the books denition of: (a) A complete metric space (b) and (c) Convergence of a series of real numbers (d) Normed vector space; Banach space Solution. See book. Problem 2. Let be a metric space with a m

UCLA - MATH - 131b

UCLA - MATH - 131bh

"Te " $ g (Qq CTC e "T$"TC" f f R V nD CISS e "T$"TC" f @8aY @ AbTR v qe T$"Tg" ~ v f @ B6 gWu'{ 2 1 k H g f V f V D X a Y X Y

UCLA - MATH - 135

Philosophy 135Spring 2008Tony MartinIntroduction to Metalogic1 The semantics of sentential logic.The language L of sentential logic. Symbols of L: (i) (ii) (iii) Remarks: (a) We shall pay little or no attention to the use-mention distinction

UCLA - MATH - 146

UCLA NS 172/272/Psych 213 Brain Mapping and NeuroimagingqInstructor:Ivo Dinov,Asst. Prof. In Statistics and NeurologyUniversity of California, Los Angeles, Winter 2006http:/www.stat.ucla.edu/~dinov/http:/www.loni.ucla.edu/CCB/Training/Cours

UCLA - MATH - 146

UCLA NS 172/272/Psych 213 Brain Mapping and NeuroimagingInstructor:Ivo Dinov,Asst. Prof. In Statistics and NeurologyNeuroimaging and MRI PhysicsUniversity of California, Los Angeles, Winter 2006http:/www.stat.ucla.edu/~dinov/http:/www.lon

UCLA - MATH - 146

UCLA NS 172/272/Psych 213 Brain Mapping and NeuroimagingInstructor:Ivo Dinov,Asst. Prof. In Statistics and Neurology University of California, Los Angeles, Winter 2006Statistical Methodshttp:/www.stat.ucla.edu/~dinov/http:/www.loni.ucla.edu

UCLA - MATH - 155

Math 155, Vese Reminder: Midterm exam, Monday, May 15, time 1-1.50pm (closed-note and closed-book exam, no calculators will be allowed). Sections covered for the midterm: 2.3.4, 2.4 (except 2.4.4), 3.1, 3.2, 3.3, 3.4.1, 3.4.2, 3.5, 3.6, 3.7, Chapter

UCLA - MATH - 155

Math 155, Vese Homework # 8, due on Friday, June 2 [1] Start with equation (5.4-19) and derive equation (5.4-21). [2] Consider the motion blur in the frequency domain given by H(u, v) =T 0e2i[ux0 (t)+vy0 (t)] dt.bt TFor uniform motion given by

UCLA - MATH - 155

Math 155, Vese REMINDER: One hour nal written exam on Friday, June 9, time 1-2pm. All sections are covered for the nal, but more questions will be given from the second part of the course. Sections covered for the midterm: 2.3.4, 2.4 (except 2.4.4),

UCLA - MATH - 164

Brief List of Study Topics for Math 164 Midterm Spring 07 DISCLAIMER: This list is in no way claimed to be comprehensive. You are responsible for knowing all relevant material from the lectures and the book. In other words, dont come to me and argue

UCLA - MATH - 167

Math 167 (Winter 2007) MATHEMATICAL GAME THEORY Instructor: Roberto Schonmann www.math.ucla.edu/rhs Time/Place: MWF 9:00-9:50 in MS 6229. Text: Fun and Games by Ken Binmore. Oce hours: M 10:00-10:50, W 11:00-11:50 in MS 6156. Examinations: There will

UCLA - MATH - 167

UCLA - MATH - 167

UCLA - MATH - 170a

UCLA - MATH - 170b

UCLA - MATH - 180

Class Information for Calculus I Math 180, Fall 2006MWF 1:001:50pm, Lecture Center BInstructor: Matthias Aschenbrenner E-mail: maschenb@math.uic.edu Course webpage: http:/www.math.uic.edu/maschenb (follow the link to Math 180) Oce: 417 SEO Oce pho

UCLA - MATH - 181

Course Syllabus Math 181: THE MATHEMATICS OF FINANCEFall 20011Background in Finance and Probability1. Introduction and Course Description 2. Review of probability 3. Discrete Random Walks 4. Random walks with Gaussian increments 5. Equity model

UCLA - MATH - 181

Course Information Math 181: THE MATHEMATICS OF FINANCEWinter 20031Instructor: Prof. Russel Caisch Oce: 7619D Math Sciences Bldg Phone: 310-206-0200 Email: caisch@math.ucla.edu Web site: http:/www.math.ucla.edu/caisch/181.1.03w/ Oce hours:

UCLA - MATH - 181

Figure 1: Payo for a straddle option, as a function of S = S(T ).Math 181 Exotic OptionsLecture 17Derivatives that are more complicated than standard European and American calls and puts are called exotic options. Here we describe only a few of

UCLA - MATH - 181

Math 181Lecture 9Pricing Options for the Random Walk with Discrete StepsConsider a single step in a random walk S0 given at t = 0 uS0 probability p S1 = dS0 probability q = 1-p in whichu = edt+dt d = edt dt . Also consider an option where val

UCLA - MATH - 191

UCLA - MATH - 191

UCLA - MATH - 191

Project Information Algorithms for Elementary Algebraic Geometry Math 191, Fall Quarter 2007The purpose of the project is to provide you with an opportunity to learn new mathematics and to improve your mathematical exposition and verbal explanations

UCLA - MATH - 191

Course Announcement Algorithms for Elementary Algebraic Geometry Math 191, Fall Quarter 2007MWF 11:50pm, Dodd Hall 162Instructor. Matthias Aschenbrenner E-mail. matthias@math.ucla.edu Course webpage. http:/www.math.ucla.edu/matthias/191.1.07f Oce.

UCLA - MATH - 191h

UCLA - MATH - 191h

Mathematics 191, Honors Seminar: Information Theory.MWF 10-11, MS 6201 Dimitri Shlyakhtenko, MS 7901, shlyakht@math.ucla.eduCourse description.Textbook: Robert B. Ash, Information Theory, Dover Publ., New York, 1990 (available in the bookstore).

UCLA - MATH - 191h

A CONTINUOUS MOVEMENT VERSION OF THE BANACHTARSKI PARADOX: A SOLUTION TO DE GROOTS PROBLEM.TREVOR M. WILSON Abstract. In 1924 Banach and Tarski demonstrated the existence of a paradoxical decomposition of the 3-ball B, i.e., a piecewise isometry fro

UCLA - MATH - 2

Math 2 (Spring 2001) Instructor: Roberto Schonmann rhs@math.ucla.edu Time: MWF 2:00 to 2:50pm in WGYOUNG CS24 Text: Essentials of Finite Mathematics by Robert F. Brown and Brenda W. Brown. We will cover most of Chapters 1 and 3 and parts of Chapters

UCLA - MATH - 207c

p-ADIC ANALYTIC FAMILIES OF MODULAR FORMS28ReferencesBooks [AFC] [BCM] [CGP] [CPI] [CRT] [GME] [IAT] [ICF] [LEC] [LFE] [MFM] [MFG] K. Iwasawa, Algebraic functions. Translations of Mathematical Monographs, 118. American Mathematical Society, Prov

UCLA - MATH - 207c

p-ADIC ANALYTIC FAMILIES OF MODULAR FORMSHARUZO HIDAContents 1. Introduction 1.1. p-Adic L-functions as a power series 1.2. Eisenstein series 1.3. Eisenstein family 1.4. Hecke operators 1.5. Modular forms of level N 1.6. Slope of modular forms 2.

UCLA - MATH - 207c

p-ADIC ANALYTIC FAMILIES OF MODULAR FORMS21. Introduction In this introduction, rst, without going into technical details, we describe a prototypical example of a p-adic analytic families of modular forms. Starting with the third week (or slightl

UCLA - MATH - 214a

PROBLEMS, MATH 214AAffine and quasi-affine varieties k is an algebraically closed eld Basic notions 1. Let X A2 be dened by x2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset in A2 consisting of all points of the form (t2 , t3 )

UCLA - MATH - 215b

Problems, 215B Do 15 problems. Due Dec 5. 1. Let A = A0 A1 . . . be a graded commutative ring. Prove that A is Noetherian if and only if A0 is Noetherian and the A0 -algebra A is nitely generated. 2. Let R be a local ring, f R[t1 , . . . , tn ] a

UCLA - MATH - 220c

Additional exercises for Math 220C.Ex. 5A. Let A be the standard model of arithmetic. Let be a 1 sentence. Prove that PA i A |= . Hint: Remember that PA proves a sentence i holds in all models of PA. Remember also that all models of PA are (isom

UCLA - MATH - 220c

closed under negation, conjunction and bounded quantication. Closure of 0 under bounded quantication means that 0 (x)(x<t ) 0 ; (x)(xt ) 0 ,for any term t not containing x. The 1 formulas of LPAE are those of the form (x1 ) (xn ) , where

UCLA - MATH - 223b

Mathematics 223BWinter 05Solutions to Exercises 4.1 and 4.2.Exercise 4.1. To see that G0 G1 is a lter, note that p0 , p1 G0 G1 p0 , p1 q0 , q1 p0 G0 p1 G1 p0 0 q0 p1 1 q1 q0 G 0 q1 G 1 and that p0 , p1 G0 G1 q0 , q1 G0 G1 p

UCLA - MATH - 223b

Borel equivalence relationsGreg Hjorth March 30, 2006This chapter is setting out to achieve an impossibility, namely to survey the rapidly exploding eld of Borel equivalence relations as found in descriptive set theory and the connections with area

UCLA - MATH - 223b

Math 223BWinter 2005D.A. MartinSet TheoryThis course will be an introduction to independence proofs by forcing. Our basic treatment will be close to that in Kenneth Kunens Set Theory: an Introduction to Independence Proofs, North-Holland, 1980

UCLA - MATH - 223b

Mathematics 223BFall 2003Homework Problems 11 and 12H11. Let U witness that is measurable and let j : V M = Ult(V ; U) be the canonical embedding. Prove that j(2 )+ ) = (2 )+ . Hint. If f : (2 )+ then the range of f is bounded. H12. In the fo

UCLA - MATH - 223d

223D: Descriptive Set TheoryInstructor: Greg Hjorth, MSB 7340 Oce hours: Tentatively planned for Mon: 3:45-4:45, Wed. 11-1. Please feel free to grab me after lectures especially something quick which I can answer in ve minutes. Time: Mon, Wed: 2-3:

UCLA - MATH - 223d

223D: HW3; due Monday, December 1, 2pmQ1: Let T ( < )n be a recursively enumerable tree. Show there is a recursive tree S with p[S] = p[T ].The point of this observation: Using a universal r.e. set we can obtain a light faced 1 set A which is

UCLA - MATH - 223d

223D: HW2; due Wednesday, November 12, 2pmLet T < < . For each x and < 1 dene Tx, by induction: Tx,0 = {t : (t, x|h(t) ) T }; Tx,+1 = {t Tx, : t0 , t1 t(t0 t1 , t0 Tx, , t1 Tx, }; (Footnote below ) and at a limit, Tx, =< 1Tx, .Q1

UCLA - MATH - 223d

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UCLA - MATH - 225a

MATH 225A HOMEWORK ASSIGNMENTSSECTION 1.1: 1.2: 1.3: 1.4: 1.5: 1.6: 1.7: 1.8: 2.1: 2.2: 2.3: 2.4: 2.5: 2.6: 3.2: 3.3: 3.4: 3.5: 3.6: 3.7:EXERCISES 8, 4, 1, 2, 2, 4, 7, 3, 2, 1, 4, 3, 1, 1, 5, 2, 3, 5, 10, 1, 10, 5, 2, 6, 5, 6, 5, 7, 6, 7, 13, 7,

UCLA - MATH - 234

Applied Mathematical Modelling 29 (2005) 215234www.elsevier.com/locate/apmOpen-surface MHD ow over a curved wall in the 3-D thin-shear-layer approximationS. Smolentsev *, M. AbdouDepartment of Mechanical and Aerospace Engineering, Fusion Science

UCLA - MATH - 234

Homework # 3 ChE234Chemical Engineering Principles of Plasma ProcessingUCLA, Spring 2004 Due: April 29, 2004 (From HW#2) 5. Determine the gas flow regimes and calculate the mean free paths of the N2 gas molecules at (1) 0.1 mtorr, 300K and (2) 10

UCLA - MATH - 245a

Math 245A Fall 2007 Midterm: Solutions 1. (a) Show that, for any set E with m (E) < , there exists a measurable set A exists such that E A and m (E) = m(A). Proof We know that (1) m (E) = inf{m(O) : O is open and contains E}.Hence for one can choo

UCLA - MATH - 245b

PicP0R e WY hHsih1`Yzy ~ t er T }Hsih1`Yzy | t er T t er T { Hsih1`YzyQ& I ) Ag0h0Ax u QwH#' u vF ) 3 I QDshhhr t Q Dshhr q e aY #i!sX'Y p u p T u T W t YU YU i tc c T Wi Uc t Y oaViW'YDn`i1YGVmtVHViasjTh1dW'Pc 1lkjP!ihyhfd`i'Hgvbf ai

UCLA - MATH - 245b

h~pb'ag1CYRERhXR0 u Vh } | ` uV h~pb'ag1CY8{kad!pVEz u Vh } | r `q i h~pb'ag1CY8{ead!pVEz u Vh } | r `q i h~pb'ag1CY8{ad!pVEz u Vh } | r `q i h~pb'ag1CY8{gad!pVEz u Vh } | r `q i h~pb'ag1CY8{ead!pVEz u Vh } | r `q i h~pb'ag1CY8{ad!pVEz u Vh } |

UCLA - MATH - 246a

MATH 246A - Spring 2008 Complex Analysis MWF 1:00 MS 5117 and Thurs 1:00 MS 5117 Oce hours: John Garnett: MWF 4:00 in MS 7941; William Meyerson: ThF 2:00 in MS 2961. Texts: 1) L. Ahlfors, Complex Analysis, 3rd. Edition, (0-07-000657-1) (required) 2)

UCLA - MATH - 246b

August 22, 2008 MATH 246B - Fall 2008 - Complex Analysis Time and Place: MS 6221, MWF 11:00, starting Sept. 26. Oce hours: John Garnett MWF 1:30 in MS 7941. Texts: 1) L. Ahlfors, Complex Analysis, 3rd. Edition, (0-07-000657-1) (required) 2) D. Saraso

UCLA - MATH - 246b

u 9' r 99'9)'v wt tut t t '9) tt yy l U U a ` ` G } ~ } X 6 ` i ! ypd l ~ l v} y'P 9E f 1bp j $P j gy'P G @bb E 'P E # }w t tw } d t l y1y l } l v~W~ l ~ l F o w xQ' t 8 6a P 6 t yy l vt sb l y l

UCLA - MATH - 247a

247A Homework. The two sources for notes are http:/www.math.ubc.ca/~ilaba/wolff/ and http:/www.its.caltech.edu/~schlag/notes_033002.pdf 1. Let us dene J0 (x) = Show that f (x) F () = 2 J0 (2x)f (x)x dx1 2 0 2cos x sin() ddenes a unitary map fro

UCLA - MATH - 247a

HARMONIC ANALYSISTERENCE TAOAnalysis in general tends to revolve around the study of general classes of functions (often real-valued or complex-valued) and operators (which take one or more functions as input, and return some other function as out

UCLA - MATH - 251b

d v|y 'u | In 4d epr x yr | d b ` U u b d W p x d d W Ivy6IeU cf q d epr r x q~d d d l u v| q'n x d U b x d X ht h b ut ` ot ` z z 8mGiykyayvkvW (d epr u x rhb` dhU yI

UCLA - MATH - 251b

TW ~ | eR TW ~ | eR IW @I @H b v TB C R R @ H y`} | rp z '1 | rp wa `xQof(GC vu (DdU"(fY CB @ I F F C T W H F T S(Qt%dt(yy U(QGPPadt(QGEU(US%aDhQiSQGUayQQP(QA%G'SV p R b e I H F c C F @ I R I C v T Rg F Y CB W g W F e F tB IB C

UCLA - MATH - 251b

46The Cauchy problem for the parabolic equations.u(x, t) + A(x, t, D)u(x, t) = f, t > 0, x Rn , tConsider a dierential equation in Rn+1 = {t > 0, x Rn } of the form: + (46.1)where A(x, t, ) = A0 (x, t, ) + A1 (x, t, ), A0 (x, t, ) = |k|=m ak

UCLA - MATH - 251b

56The heat trace asymptotics.nConsider the Laplace-Beltrami operator in Rn (56.1) A(x, D)u = j,k=1 g(x) xj1g(x)g jk (x)u xk,where ([g jk (x)]n )1 is the matrix tensor, g(x) = (det[g jk ])1 , g jk (x) j,k=1 C (). Let GD (x.x(0) ,