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Berkeley >> CS >> 170 (Fall, 2006)
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Berkeley >> HW >> 170 (Fall, 2009)
CS170 Problem Set 1 Out: Jan 16, 2001 Due: Jan 19, 2001, 4 pm Most homeworks will be handed out on Tuesdays, and be due by 4 pm Friday of the following week (10 days later) to Jenny Gonzalez in 387 Soda Hall, or to the CS 170 box in 283 Soda. This...
Berkeley >> CS >> 258 (Fall, 2008)
Role of Synchronization CS 258 Parallel Computer Architecture Lecture 23 Hardware-Software Trade-offs in Synchronization and Data Layout April 21, 2008 Prof John D. Kubiatowicz http:/www.cs.berkeley.edu/~kubitron/cs258 A parallel computer is a colle...
Berkeley >> CS >> 294 (Fall, 1924)
Implications of Peer-to-Peer Networks on Worm Attacks and Defenses Jayanthkumar Kannan Karthik Lakshminarayanan kjk,karthik @cs.berkeley.edu CS294-4 Project, Fall 2003 Abstract Recently, two trends have emerged in the eld of peer-to-peer networks: w...
Berkeley >> CS >> 294 (Fall, 1924)
Contents Exploiting Routing Redundancy via Structured Peer-to-Peer Overlays Sep. 17, 2003 Byung-Gon Chun Motivation Resilient Overlay Routing Interface with legacy applications Evaluation Comparison 1 2 Motivation Frequent disconnection and ...
Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
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Berkeley >> CS >> 252 (Fall, 2008)
RPSXWHU $UFKLWHFWXUH /HFWXUH 5HYLHZ RI 7HFKQRORJ\\ 7UHQGV DQG 6.XELDWRZLF] /HF RPSXWHU )RRG &KDLQ 0DVVLYHO\\ 3DUDOOHO...
Berkeley >> CS >> 294 (Fall, 1924)
Persistence of Data in a Dynamic Network Hakim Weatherspoon and Rachel Rubin University of California, Berkeley hweather, rrubin @cs.berkeley.edu Abstract We explore the possibility of achieving robust and efcient peer-to-peer storage by differen...
Berkeley >> CS >> 294 (Fall, 1924)
Making Gnutella-like P2P Systems Scalable Presented by: Karthik Lakshminarayanan Yatin Chawathe, Sylvia Ratnasamy, Lee Breslau, Nick Lanham, and Scott Shenker Central philosophy of the work File-sharing is a dominant P2P application DHTs might not...
Berkeley >> CS >> 170 (Fall, 2006)
U.C. Berkeley CS170: Intro to CS Theory Professor Luca Trevisan Handout N19 November 8, 2001 Notes for Lecture 19 1 1.1 Network Flows The problem Suppose that we are given the network of Figure 1 (top), where the numbers indicate capacities, th...
Berkeley >> CS >> 170 (Fall, 2006)
U.C. Berkeley CS170: Intro to CS Theory Professor Luca Trevisan Handout N24 December 4, 2001 Notes for Lecture 24 1 1.1 Some NP-complete Numerical Problems Subset Sum The Subset Sum problem is dened as follows: Given a sequence of integers a1 ,...
Berkeley >> CS >> 278 (Fall, 2008)
U.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan Handout N15 3/20/2008 Notes for Lecture 15 1 Hardness of Approximation We know that a number of important optimization problems are NP-hard to solve exactly. Today we begin t...
Berkeley >> CS >> 170 (Fall, 2006)
U.C. Berkeley CS170: Intro to CS Theory Professor Luca Trevisan Handout N13 October 18, 2001 (corrected Oct. 25) Notes for Lecture 13 1 1.1 Edit Distance Denition When you run a spell checker on a text, and it nds a word not in the dictionary, ...
Berkeley >> MATH >> 1 (Fall, 2008)
Let us nd a particular solution of the dierential equation coming from LCR circuits: dQ d2 Q 1 + Q = E cos t L 2 +R dt dt C Watch how sweetly it goes if we make it complex-well solve instead: () L dQ d2 Q 1 +R + Q = Eeit dt2 dt C and the real part o...
Berkeley >> RICHARD >> 3 (Fall, 2009)
Boyce-DePrima 10.2 4. The given function is sin(x/L); sin(2 + x) = sin(x), so sin(2L + x)(x/L) = sin(2 + x/L) = sin(x/L). The function is periodic with period 2L. 14. The given function is 1 for L < x < 0 and 0 for 0 < x < L, with period 2L. From the...
Berkeley >> MATH >> 121 (Fall, 2009)
Sample midterm. 1. Find x0 5 lim x(x). 2. Express the integral in terms of special functions x1/3(5 x)10/3dx 0 3. Find the circumference of an ellipse with half axes 2 and 1 in term of elliptic integrals. 4. Solve the dierential equation x2y 5...
Berkeley >> MATH >> 55 (Fall, 2008)
Instructor Jared Weinstein 1075 Evans Hall www.math.berkeley.edu/jared jared@math.berkeley.edu Math 55 Discrete Mathematics U.C. Berkeley Dept. of Mathematics Summer 2007 Groupwork: Generating Functions (1) Find generating functions for the follo...
Berkeley >> MATH >> 1 (Fall, 2008)
1 Quiz 15 - Calculus 1A December 10, 2004 Jonathan Dorfman Answer as many questions as you can (20 + 5 points total) in 30 minutes. 1. (10 points) Find the area of the region bounded above by y = ex , bounded below by y = x, and bounded on the side...
Berkeley >> MATH >> 114 (Fall, 2008)
PROBLEM SET # 5 MATH 114 Due April 13. 1. Find the Galois group of the polynomial x4 + 8x + 12 over Q. 2. Find the Galois group of the polynomial x4 + 3x + 3 over Q. 3. Find the Galois group of x6 3x2 + 1 over Q . 4. Assume that a polynomial x4 + ax...
Berkeley >> MATH >> 114 (Fall, 2008)
HOMEWORK SOLUTIONS MATH 114 Problem set 10. 1. Find the Galois group of x4 + 8x + 12 over Q. Solution. The resolvent cubic x3 48x + 64 does not have rational roots. The discriminant 27 84 + 256 123 = 27 (214 212 ) = 81 212 is a perfect square. T...
Berkeley >> MATH >> 1 (Fall, 2008)
Practice Final Exam #4 1. Find the integral (x2 4)dx . x(x2 + 4) 2. Evaluate the integral dx . 8 + 2x x2 3. Evaluate the integral ln x ln(ln x) dx. x 4. Evaluate the integral or show that it is divergent 0 dx . (x + 1)2 (x + 2) 5. Determine whe...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;11. Infinite Sequences and Series; 11.4 The Comparison Tests 1. (a) We cannot say anything about a . If a >b for all n and n n n b is convergent, then n a could n be convergent or divergent. (See the note after ...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;7. Techniques of Integration; 7.2 Trigonometric Integrals 1. sin xcos xdx = sin xcos xsin xdx= = 3. 3 /4 /2 s = 3 2 2 2 ( u2 1) u2 du= 3 /4 /2 2 /2 5 2 ( 1 cos 2x) cos 2xsin xdx= ( 1 u2) u2( du) ( u4 u2) du= 1 u5 ...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;8. Further Applications of Integration; 8.1 Arc Legth 2. Using the arc length formula with y= 4 x 2 2 dy = dx 2 x 4 x 2 , we get t L= 2 1+ 0 dy dx 1 2 dx= 0 t 1+ 1 x 2 2 dx= 0 1 2dx 4 x 2 2 =2lim t 2...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;7. Techniques of Integration; 7.5 Strategy for Integration 1 5. Let u=arctany . Then du= dy 1+y 2 1 e arctany 2 /4 dy= e du= e /4 u u /4 1+y /4 =e /4 e /4 . 13. Let x=sin dx (1 x ) 2 3/2 , where c...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;9. Differential Equations; 9.4 Exponential Growth and Decay 1. The relative growth rate is P(t)=P(0)e 0.7944t =2e 0.7944t 1 dP dP =0.7944 , so =0.7944P and, by Theorem 2, P dt dt 0.7944 ( 6 ) 234.99 or about 235 ...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;8. Further Applications of Integration; 8.2 Area of a Surface of Revolution 1. y=ln x ds= 1+(dy/dx) dx= 1+(1/x) dx 2 2 3 1 ln 2 S= 2 (ln x) 1+(1/x) dx 2 2 4. y=e x ds= 1+(dy/dx) dx= 1+e 2 2x dx S= 0 2 ...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;11. Infinite Sequences and Series; 11.8 Power Series x 3. If a = , then n n a n+1 lim =lim a n n n n x n+1 n+1 n x n =lim n x n+1 / n =lim n x = x . 1+1/n x converges when x <1 , so the radius of convergenc...
Berkeley >> MATH >> 1 (Fall, 2008)
Practice Questions for the Second Midterm (Without Solutions) Math 1B, N. Reshetikhin Series: I. Determine whether the given series converges or diverges. 1. n=1 4n 3 +5 2. n=2 1 n1 sin2 n n3/2 1 n! (1)n n2 (1)n n2 (n + 1)3 n3 3. n=1 4. ...
Berkeley >> MATH >> 2 (Fall, 2008)
Practice Questions for the Second Midterm (Without Solutions) Math 1B, N. Reshetikhin Series: I. Determine whether the given series converges or diverges. 1. n=1 4n 3 +5 2. n=2 1 n1 sin2 n n3/2 1 n! (1)n n2 (1)n n2 (n + 1)3 n3 3. n=1 4. ...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;9. Differential Equations; 9.2 Direction Fields and Eulers Method 1. (a) (b) It appears that the constant functions y=0 , y= 2 , and y=2 are equilibrium solutions. Note that 1 2 / these three values of y satisfy th...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;7. Techniques of Integration; 7.3 Trigonometric Substitution 1. Let x=3sec , where 0 < 2 or < 3 . Then dx=3sec 2 tan d and 2 2 2 2 x 9 = 9sec 9 = 9 sec 1 = 9tan = 3 tan =3tan for the relevant values of ( ...
Berkeley >> MATH >> 1 (Fall, 2008)
Math 1B, Final Examination N.Reshetikhin, May 18, 2004 Students Name: TAs name: Students i.d. number: 1.10 pnts Evaluate the integral x3ex dx 2 1 2.15 pnts Evaluate the integral (t2 1 dx 1)(t 1) 2 3.15 pnts Indicate which of the following state...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;9. Differential Equations; 9.6 Linear Equations 1. y +e y=x y is not linear since it cannot be put into the standard linear form (1), y +P(x) y=Q(x) . ln x / / 2 / 2 , which is in the standard linear form (1), so 3....
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;9. Differential Equations; 9.5 The Logistic Equation 15. (a) dP/dt=kPcos (rt ) (dP)/P=kcos (rt )dt (dP)/P=k (rt )dt ln P=(k/r)sin (rt )+C . (Since this is a growth model, P>0 and we can write ln P instead of ln P .)...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;17. Second-Order Differential Equations; 17.1 Second-Order Linear Equations 1. The auxiliary equation is r 6r+8=0 is y=c e +c e 1 2 4x 2x 2 (r 4)(r 2)=0 r=4 , r=2 . Then by (8) the general solution . 2 3. The a...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;11. Infinite Sequences and Series; 11.6 Absolute Convergence and the Ratio and Root Tests a 1. (a) Since lim n n+1 n a a n+1 n =8>1 , part (b) of the Ratio Test tells us that the series a is divergent. n (b) Si...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;9. Differential Equations; 9.3 Separable Equations 1. dy y = dx x dy dx = y x C dy dx = y x ln y =ln x +C y =e ln x +C =e ln x e =e C C x y=Kx , where K= e is a constant. (In our derivation, K was nonz...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;9. Differential Equations; 9.1 Modeling with Differential Equations 1. y=x x 1 y =1+x . To show that y is a solution of the differential equation, we will substitute / / 2 the expressions for y and y in the lef...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;11. Infinite Sequences and Series; 11.1 Sequences 4. a = n n+1 , so the sequence is 3n 1 n { 2 3 4 5 6 , , , , ,. 2 5 8 11 14 }{ = 1, 3 1 5 3 , , , ,. 5 2 11 7 } . 3( 1) 5. a = n n! 8. a =4 , a 1 , so the ...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;7. Techniques of Integration; 7.1 Integration by Parts 1. Let u=ln x , dv=xdx xln xdx = du=dx/x , v= 1 2 x ln x 2 1 2 1 2 = x ln x x +C 2 4 r/2 1 2 x . Then by Equation 2, udv=uv vdu , 2 1 2 1 2 1 1 2 1 1 2 x ( d...
Berkeley >> MATH >> 1 (Fall, 2008)
Practice Final Exam #1 1. Evaluate the integral exp(tan x) 2. Evaluate the integral 1 0 sin x dx. cos3 x ln 1 + x2 dx. 3. Evaluate the integral if it converges, or show that it is divergent 1 x ln x dx. 0 4. TrueFalse question. If true: explain...
Berkeley >> MATH >> 1 (Fall, 2008)
Practice Final Exam #5 1. Evaluate the integral tan4 x dx. 2. Evaluate the integral 2x x2 dx. dx . x2 (x + 2) 3. (a) Evaluate the integral (b) Evaluate 1 x2 (x dx or show that it is divergent. + 2) x4 1 + 2 for 1 x 2. a 4x 4. Find the leng...
Berkeley >> MATH >> 1 (Fall, 2008)
Practice Final Exam #3 1. Evaluate the integral cos x dx. esin x + 1 2. Evaluate the integral (e2x + ex)ln(ex + 1)dx. 3. Evaluate the integral if it converges or shows that it is divergent e 1 dx x ln x + ln2 x . 4. Evaluate the integral x3 dx . ...
Berkeley >> MATH >> 1 (Fall, 2008)
Practice Final Exam #2 1 1. This is a multiple choice question. Which of the following is within 1/2 of 0 xexdx? (a) 1 (b) e (c) e 2 (d) 3/4 2. This is a multiple choice question. If f (x) > 0 for all x > 0 and 0 f (x)dx diverges then (a) lim...
Berkeley >> MATH >> 1 (Fall, 2008)
Practice Final 1B 1. Evaluate the integral cos2 (tan1 (x)dx. 2. Determine whether the sequence is convergent or not, if it is convergent, nd the limit an = tan1 (n) + n . 1 + n3 3. Solve the dierential equation y y = x + ex . 4. Find power series ...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;11. Infinite Sequences and Series; 11.9 Representations of Functions as Power Series 1. If f (x)= n=0 n c x has radius of convergence 10 , then f (x)= n / n=1 nc x n n 1 also has radius of convergence 10 by...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;11. Infinite Sequences and Series; 11.10 Taylor and Maclaurin Series 3. n 0 1 2 3 4 ( n) f (x) cos x sin x cos x sin x cos x f ( n) ( 0) 1 0 1 0 1 We use Equation 7 with f (x)=cos x . cos x = f (0)+ f (0)x+ x x...
Berkeley >> MATH >> 1 (Fall, 2008)
Stewart Calculus ET 5e 0534393217;11. Infinite Sequences and Series; Review: Exercises 1. { } 2+n 3 1+2n n+1 n 3 converges since lim n 2+n 3 3 =lim n 2/n +1 1/n +2 3 3 = 1+2n 1 . 2 2. a = n 9 =9 10 9 10 n 3 2 n , so lim a =9lim n ...
Berkeley >> MATH >> 121 (Fall, 2009)
PRACTICE MIDTERM # 1 MATH 121B 1. Use Fermats principle to nd the path of a light ray through a medium of index of refraction r1/2 . 2. Write in terms of the following values : 5 7 . , (a) (3.5) ; (b) B 6 6 3. (a) Find the period T of a pendulum (w...
Berkeley >> RT >> 3 (Fall, 2009)
Forum Math. 14 (2002), 209244 ( de Gruyter 2002 Forum Mathematicum Reidemeister torsion in generalized Morse theory Michael Hutchings (Communicated by Andrew Ranicki) Abstract. In two previous papers with Yi-Jen Lee, we dened and computed a notio...
Berkeley >> DB >> 2 (Fall, 2009)
ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Pages 4549 (July 17, 2000) S 1079-6762(00)00079-2 PROOF OF THE DOUBLE BUBBLE CONJECTURE MICHAEL HUTCHINGS, FRANK MORGAN, MANUEL RITORE, AND ANTONIO ROS (Communicated b...
Berkeley >> RT >> 2 (Fall, 2009)
ISSN 1364-0380 369 Geometry & Topology Volume 3 (1999) 369396 Published: 26 October 1999 G T G G TT T T T G G G T T G T G G T G T G T G GG G T TT Circle-valued Morse theory and Reidemeister torsion Michael Hutchings Yi-Jen Lee Dept of Math, Stanf...
Berkeley >> MIDTERM >> 2 (Fall, 2009)
Midterm #2 solutions 1. The series converges. We use the limit comparison test with n + cos n an = , bn = n3/2 . n3 + n4 Then n1/2 1 + cos n 1 + cos n an n + cos n n n = = = . 3/2 n3 + n4 3/2 n2 n1 + 1 1 bn n n 1+n Note that limn cos n/n = 0 be...
Berkeley >> MIDTERM >> 1 (Fall, 2009)
Math 1b Section 2 Midterm #1, 2/14/06, Solutions 1. Substitute u = ex + 1. Then u2 = ex + 1, so 2u du = ex dx = (u2 1) dx, so dx = 2u du/(u2 1). Thus ln 8 ln 3 dx = ex + 1 3 2 2 du . u2 1 To evaluate the integral on the right we use partial...
Berkeley >> MIDTERM >> 2 (Fall, 2009)
Math 1b Section 2 Midterm #2, 3/21/06, 3:40 PM 5:00 PM Please write your solution to each of the 6 questions on a separate sheet of paper with your name, GSI, and SID# on it. Each question is worth 10 points. Please put a box around your nal answer....
Berkeley >> BOOK >> 3 (Fall, 2009)
Rice-15149 book March 13, 2006 9:44 3.8 Problems 109 b. Sketch the joint density. c. Find P(X 2 + Y 2 ) 1 . 2 d. Find the marginal densities of X and Y . Are X and Y independent random variables? e. Find the conditional densities. 16. What is ...
Berkeley >> WEEK >> 4 (Fall, 2009)
Statistics 246 Spring 2006 The Chi-squared model for recombination, and its extensions Week 4, Lecture 1 1 Why model recombination/interference? There are several reasons why we might try to improve on the Poisson model for recombination; here are...
Berkeley >> WEEK >> 12 (Fall, 2009)
Identifying expression differences in cDNA microarray experiments Lecture 20, Statistics 246, April 6, 2004 1 Introduction Many microarray experiments are carried out to find genes which are differentially expressed between two (or more) samples o...
Berkeley >> WEEK >> 15 (Fall, 2009)
Methods for the discovery of cis-regulatory modules, 3 Comparative genomics Statistics 246 Week 15 Spring 2006 Lecture 1 Introduction In this lecture Ill discuss three papers which seek to find TFBSs using evolutionary conservation. When we look for...
Berkeley >> WEEK >> 7 (Fall, 2009)
Gene and species tree reconciliation (-globin and SERA genes) Statistics 246 Week 7 Spring 2006 Lecture 1 Homologs Recall that homologous genes or proteins result from Speciation (orthologs): when separate lineages diverge from a common ancestor and...
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