ACM116 - Winter 2008-2009 - Homework #2
Handed out: 27 Jan 2009, Due: 3 Feb 2009
Please write down your solutions clearly and concisely, put the problems in order, and box your answers. Presentation will be worth a couple of extra points. 1/ If the number

Algebra 3 (2004-05) Solutions to Assignment 3
Instructor: Dr. Eyal Goren
1) If G, H are nite groups such that (|G|, |H |) = 1 then every group homomorphism f : G H is trivial (f (G) = cfw_1). Proof. Let K be the kernel of f . Then |Im(f )| = |G|/|K | divi

Harvard SEAS
ES250 Information Theory
Homework 1 Solution
1. Let p(x, y ) be given by XY 0 1 0 1/3 0 1 1/3 1/3
Evaluate the following expressions: (a) H (X ), H (Y ) (b) H (X |Y ), H (Y |X ) (c) H (X, Y ) (d) H (Y ) H (Y |X ) (e) I (X ; Y ) (f) Draw a Ven

ACM116 - Winter 2008-2009 - Homework #2
Handed out: 27 Jan 2009, Due: 3 Feb 2009
Please write down your solutions clearly and concisely, put the problems in order, and box your answers. Presentation will be worth a couple of extra points. 1/ If the number

ACM116 - Winter 2008-2009 - Homework #1
Handed out: Jan 15, 2009, Due: Jan 22, 2009
Please write down your solutions clearly and concisely, put the problems in order, and box your answers. Presentation will be worth a couple of extra points. 1/ There are

An Introduction to Stochastic Dierential 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 Chapter 4: Stoc

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Handed out: Nov 3, 2009, Due: Nov 17, 2009 (in class). Please write down your solutions clearly and concisely, box your answers, put problems in order and bind pages. Grader: Molei Tao 1. A function (x) is sai

ACM116 - Winter 2008-2009 - Homework #4
Handed out: March 6, 2009, Due: March 18, 2009, in Sheila Shull's office (Firestone 217) by 4pm
Please write down your solutions clearly and concisely, put the problems in order, and box your answers. For questions

ECE 563 Information Theory
Homework 2 Solutions
Instructor: R. Srikant TA: Akshay Kashyap
3.3 Let type a cuts be those that divide the piece in ratio (2/3, 1/3) and type b cuts be those that divide the piece in ratio (3/5, 2/5). Consider a sequence of n c

EE 376A/Stat 376A Prof. T. Weissman
Information Theory Thursday, February 8, 2007
Midterm 1. (35 points) Throwing a die Suppose you have a die with three sides. given as 1, 2, X= 3,
The probability of outcome of each side is w.p. 1/2 w.p. 1/3 w.p. 1/6
You

CHAPTER 2 SOME APPLICATIONS OF INTEGRATION
Chapter 2 is a bunch of somewhat unrelated topics, each of which illustrates an example of how you can give a meaning to an integral. Note I have only written down what I think are the most important denitions an

The introductory chapter to Apostols Calculus 1 is kind of a mish-mash of dierent topics, but it includes the introduction of several vital topics for the study of Calculus. 1. Section I.1: Historical Introduction This is a really introductory section, re

THE ADDITIVITY OF THE INTEGRAL
FOKKO VAN DE BULT
In this note we prove the following Theorem. Theorem 1. Let f and g be integrable functions on the interval [a, b]. Then f + g is also integrable on [a, b] and we have
b b b
(f + g )(x)dx =
a a
f (x)dx +
a

THE INTEGRAL
1 0
xdx
FOKKO VAN DE BULT
We want to calculate the integral 0 xdx, so we have f (x) = x and a = 0 and b = 1. Consider the step function sn (x) = nx /n. Then we clearly have sn (x) = nx /n (nx)/n = x = f (x). Let us moreover dene tn (x) = nx /

33
PART II: LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 15. Basic concepts 15.1. Linear equations. The standard form of a second order linear equation is L[y] y + p(t)y + q(t)y = g(t). The map y L[y] is a dierential operation. If g(t) 0, then the equation

Ma2a (analytical)
Fall 2010
PART I: FIRST ORDER ODEs 1. Introduction 1.1. Fundamental theorem of calculus. The simplest (trivial) d.e.: y (x) = f (x). Examples: (i) Find all (maximal) solutions of the equation y (x) = Answer: y (x) = (ii) Solve the IVP y

Ma 2a (analytical track)
Fall 2010
PROBLEM SET 7 Due on Monday, November 22
I. Conservative systems Problem 34.1(iii) (sketch the phase portrait by hand). (X1) Find (approximately) the periods of small oscillations in Problem 34.1(iii)
II. Dissipative sys

Ma 2a (analytical track)
Fall 2010
PROBLEM SET 6 Due on Monday, November 15
I. Phase portraits of linear systems Problem 28.6 Problem 31.1(i-ix) [counts as 3 problems]
II. Linearization near critical point Problem 32.1
III. Ecological models Problem 33.1(

Ma 2a (analytical track)
Fall 2010
PROBLEM SET 5 Due on Monday, November 8
I. Undetermined coecients Problems 16.1(iv), 22.4(v) II. Oscillations Problems 13.4, 13. 8, 15.4 III. Linear systems with constant coecients Problem 26.1(v) (X1) Rewrite the n-th o

Ma 2a (analytical track)
Fall 2010
PROBLEM SET 4 Due on Monday, October 25
I. Integration of 1st order equations (X1) Solve the equation x= t t2 x + x3 (Hint: s = t2 ).
(X2) Solve the initial value problem y= y3 , 1 2xy 2 y (0) = 1.
II. 2D autonomous syst

Ma 2a (analytical track)
Fall 2010
PROBLEM SET 3 Due on Monday, October 18
I. Methods of integration of 1st order ODEs Problems 9.1(vi), 10.2, 10.4 (i) (X1) Solve the equation x2 y + 2xy y 3 = 0. II. Applications of 1-st order ODEs Problem 9.3 (X2) Find o

Ma 2a (analytical track)
Fall 2010
PROBLEM SET 2 Due on Monday, October 11 I. Direction (slope) elds 1-2. Draw direction eld diagrams for the following equations: (a) y = y 2 x, and (b) y = y/(x2 + y 2 ). (You can use a computer.) II. Approximation of sol

Ma 2a (analytical track)
Fall 2010
PROBLEM SET 1 Due on Monday, October 4
[All references are to J. C. Robinson]
I. 1D autonomous equations (formula solution) 1. [Problem 8.7] Show that for k = 0 the solution of the IVP x = kx x2 , is x(t) = x(0) = x0 ,
k

NOTES The Mathematical Method via One-variable Calculus Ma1a Fall 2010
DINAKAR RAMAKRISHNAN Taussky-Todd-Lonergan Professor of Mathematics Oce: 278 Sloan Extension: 4348
1
0
Proofs in Mathematics
It s not dicult to write mathematical proofs of statements,

60
PART III: QUALITATIVE ANALYSIS OF 2D AUTONOMOUS SYSTEMS 23. Orbits of autonomous systems 23.1. Autonomous systems. x = v ( x) , x(t) D Rn . The domain D (which can be the whole space Rn ) is the phase space of the system. We interpret the given vector-

Ma 2a P: Homework N.6
due Tuesday Nov 16, 12 noon
1. Problem #26 and #27 of 6.1 (Gamma function)1 2. Problem #27 of 6.2 (Laplace transform and Taylor series) 3. Problem #34 and #36 of 6.3 (periodic functions) 4. Find the inverse Laplace transform of the f

Math 2a Practical Fall 2010
M. Marcolli
Ma2a Practical Fall 2010 HW #5 Due November 9 at 12 PM
(1) (2) (3)
#43 section 5.4 (singularities at innity) #13,16 section 5.5 (Laguerre and Bessel equations) #19 section 5.6 (hypergeometric equation) [Warning: num

Ma 2a P: Homework N.4
due Tuesday Nov 2, 12 noon
1. Solve the following dierential equations: y 2y + y = et /(1 + t2 ) y y 2y = 2et y (5) 3y (4) + 3y (3) 3y + 2y = 0 y (4) + 2y + y = sin(t), with y (0) = 2, y (0) = 0, y (0) = 1, y (3) (0) = 1.
2. Problem

Ma 2a P: Homework N.3
due Tuesday Oct 26, 12 noon
1. Determine if the following functions are linearly independent: f1 (t) = cos(2t) 2 cos2 (t) and f2 (t) = cos(2t) + 2 sin2 (t); f1 (t) = e5t and f2 (t) = e5(t+1) . 2. Let y1 (t) and y2 (t) be two solution