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Course: MAT 208, Fall 2009
School: Paul Quinn
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208 Math Homework 3 due on Thursday 3/1/07 Problem 1. Evaluate the following triple integrals: (i) D zex+y dx dy dz, where D is the box [0, 1] [0, 1] [0, 1]. xy 2 z 3 dx dy dz, where D is the solid in R3 bounded by the surface z = xy and the planes x = y, x = 1, and z = 0. 1 1x2 0 1y D (ii) Problem 2. The iterated integral 1 0 f (x, y, z) dz dy dx is given. If you were to reverse the order of integrations...

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208 Math Homework 3 due on Thursday 3/1/07 Problem 1. Evaluate the following triple integrals: (i) D zex+y dx dy dz, where D is the box [0, 1] [0, 1] [0, 1]. xy 2 z 3 dx dy dz, where D is the solid in R3 bounded by the surface z = xy and the planes x = y, x = 1, and z = 0. 1 1x2 0 1y D (ii) Problem 2. The iterated integral 1 0 f (x, y, z) dz dy dx is given. If you were to reverse the order of integrations to dx dy dz, what would be the appropriate limits of the three Problem integrals? 3. Suppose D1 and D2 are disjoint thin plates in the plane with masses m1 and m2 and centers of mass c1 and c2 . Show that the center of mass c of the union D1 D2 is given by m1 m2 c= c1 + c2 . m1 + m2 m1 + m2 Thus c is the point on the line segment from c1 to c2 whi...

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Paul Quinn - MAT - 208
Math 208 Homework 1due on Thursday 2/8/07 Problem 1. Use Cavalieris principle to prove the well-known formula V = 4 r3 for 3 the volume of a solid sphere of radius r. Problem 2. Compute the iterated integrals1 0 0 /2 /2 1(y cos x + 2) dx dy and v
Paul Quinn - MAT - 208
Math 208 Homework 4due on Thursday 3/22/07 Problem 1. Evaluate the following line integrals: (i)c(x2 2xy) dx + (y 2 2xy) dy, where c is the part of the parabola y = x2 from (1, 1) to (1, 1).(ii)c(2xy) dx + (x2 + z) dy + y dz, where c is th
Paul Quinn - MAT - 208
Math 208 Homework 5due on Thursday 3/29/07 Problem 1. Find the normal vector of the surface described parametrically as x = u2 v 2 y =u+v z = u2 + 4v at an arbitrary point. At what point(s) is this surface regular? Problem 2. A surface S is describ
Virgin Islands - CSC - 20090429
NATURE|Vol 442|27 July 2006|doi:10.1038/nature05063INSIGHT R EVIEWCells on chipsJamil El-Ali1, Peter K. Sorger2 &amp; Klavs F. Jensen1 Microsystems create new opportunities for the spatial and temporal control of cell growth and stimuli by combining
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 5due on Thursday 3/15/07 Problem 1. Find the (formal) solution of the ut = 4 uxx u(0, t) = 1, ux (, t) = 0 u(x, 0) = x heat equation 0 &lt; x &lt; , t &gt; 0 t&gt;0 0&lt;x&lt;Problem 2. Show that the (formal) solution of the heat equation
Paul Quinn - MAT - 328
Math 328 Homework 10due on Thursday 5/12/05 Problem 1. Recall that the solution to the one-dimensional heat equation ut = kuxx &lt; x &lt; , t &gt; 0 u(x, 0) = f (x) is given by (xy)2 1 u(x, t) = f (y)e 4kt dy. 4kt Compute u(x, t) when the initial condi
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 10due on Tuesday 5/8/07 Problem 1. Suppose f (x) has the Fourier transform F (). If a = 0, show that 1 f (ax) has the Fourier transform |a| F ( ). a Problem 2. Consider the function f (x) = ex 0 x0 . x&lt;0 (i) Find the Fourier tr
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 9due on Thursday 4/26/07 Problem 1. Find the Fourier integral of the functions f (x) = |x| + 1 |x| 1 0 |x| &gt; 1 and 1 0&lt;x&lt;1 g(x) = 1 1 &lt; x &lt; 0 0 |x| &gt; 1For what values of x does each of the equalities F I(f )(x) = f (x) hold?
Paul Quinn - MAT - 328
Math 328 Homework 8due on Tuesday 4/24/07 Problem 1. Find the solution of the Laplace equation u = 0 0 r &lt; 1, u(1, ) = sin(3) in the unit disk given by Fourier method (separation of variables). Then compare your answer with the one given by th
Paul Quinn - MAT - 328
Math 328 Homework 6due on Thursday 3/17/05 Problem 1. Find the (formal) solution of the ut = 4uxx u(0, t) = 1, ux (, t) = 0 u(x, 0) = x heat equation 0 &lt; x &lt; , t &gt; 0 t&gt;0 0&lt;x&lt;Problem 2. Show that the (formal) solution of the heat equation 0
Paul Quinn - MAT - 328
Math 328 Homework 2due on Tuesday 2/20/07 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, &lt; x &lt; +. (ii) f (x) = cos x, &lt; x &lt; , f (x + ) = f (x) for all x. 2 2 (iii) f (x) = 0 1 &lt; x &lt; 0 , f (x + 2) = f (x) for al
Paul Quinn - MAT - 328
Math 328 Homework 6due on Thursday 3/29/07 Problem 1. Consider the wave equation 0 &lt; x &lt; , t &gt; 0 utt = uxx u(0, t) = u(, t) = 0 t&gt;0 u(x, 0) = sin(3x), u (x, 0) = sin(2x) 0 &lt; x &lt; t Verify directly that the solution u(x, t) given by the separa
Paul Quinn - MAT - 328
Math 328 Homework 4due on Thursday 3/8/07 Problem 1. Consider the function f (x) = C 0 0&lt;x&lt;L 2 L 2 x &lt; L, that the (formal) solution of the heat 0 &lt; x &lt; L, t &gt; 0 t&gt;0 0&lt;x&lt;L nx . Lwhere C and L &gt; 0 are constants. Show equation ut = k uxx u(0,
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 3due on Thursday 2/24/05 Problem 1. Let f be the 2-periodic function dened by f (x) = 2x + 2 1 &lt; x &lt; 0 x 0&lt;x&lt;1 f (x + 2) = f (x).Sketch the graph of f as well as its Fourier series over 3 x 3. (No need to compute the Fourier se
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 2due on Thursday 2/17/05 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, &lt; x &lt; +. (ii) f (x) = cos x, &lt; x &lt; , extended as a -periodic function on the real line. 2 2 (iii) f (x) = 0 1 &lt; x &lt; 0 ,
Paul Quinn - MAT - 328
Michigan State University - FSC - 421
FederalismThe Separation of the Powers between the States and the Federal GovernmentFederalism Federalism provides for a separation of powers between the state and federal governments Framers of the Constitution rejected both a blind deference t
Michigan State University - FSC - 421
Housekeeping Grading criteria Lecture Schedule / Spring Break No office hours today (Big meeting) Study Abroad ProgramThe Concept of JurisdictionFSC-421Jurisdiction Jurisdiction over persons Police Authority Jurisdictional prerequisite fo
Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Rhode Island College - MT - 010
Homework 1: p. 33 5-8 p. 36 63-66 p. 59 10, 11, 14, 15 pp. 62-63 99-102 Homework 2: p. 64 159-162 p. 96 21-26 p. 111 11-16 p. 113
SUNY Stony Brook - MAT - 542
MAT 542 Complex Analysis IProblem Set 5due Tuesday, March 6Problem 1. Let f be an entire function and let a, b C such that |a|, |b| &lt; R. If is a circle of radius R, evaluate f (z) dz. (z a)(z b) Use this result to give another proof of Liou
SUNY Stony Brook - MAT - 542
MAT 542 Complex Analysis IProblem Set 10due Tuesday, May 1Problem 1. Let f and g be entire functions with f (0) = g(0), and let P and Q be polynomials. Assume ef (z) + P (z) = eg(z) + Q(z) for all z C.Show that f = g and so P = Q. (Hint: Writ
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MAT 542 Complex Analysis IProblem Set 4due Tuesday, February 27Problem 1. Prove the claim that we needed for the Maximum Modulus Principle: Let U C be open, connected, f : U C holomorphic. Suppose that all the values of f lie on the unit circl
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MAT 542 Complex Analysis IProblem Set 6due Tuesday, March 13Problem 1. Let f be bounded and holomorphic in {z C : |z| &gt; R}. (i) Show that f has a Laurent series representation of the form c1 c2 f (z) = c0 + + 2 + z z containing non-positive po
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MAT 542 Complex Analysis IProblem Set 9due Tuesday, April 24Problem 1. Consider a function f (z) in a neighborhood of i given by a branch of z(1 z), so that 1 log (z(1 z) , f (z) = exp 2 where log stands for the principal value of logarithm. S
SUNY Stony Brook - MAT - 542
MAT 542 Complex Analysis IProblem Set 3due Tuesday, February 20Problem 1. Consider an analytic function g(z), given by a series convergence R &gt; 0. Show that if the series g(z) + g (z) + g (z) + + g (n) (z) + . . .an z n with radius of (*)
SUNY Stony Brook - MAT - 542
MAT 542 Complex Analysis IProblem Set 7due Tuesday, March 27Problem 1. (i) Let a be real, 0 a &lt; 1. Let Ua be the open set obtained from the unit disk {|z| &lt; 1} by removing the segment [a, 1] of the real line. Construct a conformal isomorphism b
SUNY Stony Brook - MAT - 542
MAT 542 Complex Analysis IProblem Set 8due Tuesday, April 17Problem 1. We saw that the function z + 1/z establishes a conformal isomorphism between the set V = {z C : |z| &gt; 1, Im z &gt; 0} and the upper half-plane H. Use Mbius transformations to n
SUNY Stony Brook - MAT - 542
MAT 542 Complex Analysis IProblem Set 2due Tuesday, February 13Problem 1. (i) Recall that we dened the exponential map via the power series exp(z) = z j /j! for all z C. Show that exp(z +w) = exp(z)exp(w) for all z, w C. j=0 (Use multiplicati
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MAT 311 Introduction to Number TheoryProblem Set 7Solutions Problems 8 and 10 sec 2.8 are easy, just use Thm 2.37 from the book (look at its proof for question 10). Problem 18 sec 2.8. If p is an odd prime, g and g primitive roots mod p, show that
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MAT 311 Introduction to Number TheoryProblem Set 2Some Solutions Problem 2. Let x &gt; 0 be such that x + 1/x is an integer (x itself doesnt have to be an integer). Prove that xn + 1/xn is an integer for every natural n. Solution. The base of inducti
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MAT 311 Introduction to Number TheoryProblem Set 6SolutionsProblem 4 sec 2.6. Solve x2 + 5x + 24 0 mod 36. Solution. It is important to realize that, even though 36 = 62 , you cannot solve this mod 6 and then lift roots because the lifting root
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 9due Wednesday, April 29Please prove all your answers. Problem 1. (a) For a real number between 0 and 1, describe how to get its binary expansion (i.e. encode it in the form like .0110010011101010
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 1Solutions to a few questionsProblem 2. Prove that for every integer n (a) n2 n is divisible by 2 (b) n3 n is divisible by 6 (c) n2 + 2 is not divisible by 4 Solution. (a) Observe that n2 n = n
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 10Solutions Problem 1 sec 7.4 Lets expand 2. 1 1 1 1 = 1+ 2 = 1+( 2 1) = 1+ = 1+ = 1+ = ., 1 1 1+ 2 2 + ( 2 1) 2+ 21 1+ 2 which gives the answer 2 = 1, 2, 2, 2, 2, . . . since the pattern
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 1due Wednesday, February 4Please prove all your answers. Many (but not all) of the questions below are from Niven ZuckermanMontgomery. Problem 1. (a) Given a|b and c|d, prove that ac|bd. Is the co
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 4Some SolutionsProblem 1. Solve the following systems of congruences. (a) x 3 mod 5 (b) 13x 2 mod 15 (c) x 0 mod 18 x 2 mod 8 16x 3 mod 25 3x 12 mod 20 x 0 mod 7 2x 2 mod 30 Solution. (a)
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 5due Wednesday, March 4Please prove all your answers. Problem 1. Prove that for each natural n there are n consequtive integers each divisible by a square greater than 1. Hint: use the Chinese Rem
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 4due Wednesday, February 25Please prove all your answers. Problem (a) x 3 x2 x0 1. Solve the following systems of congruences. mod 5 (b) 13x 2 mod 15 (c) x 0 mod 18 mod 8 16x 3 mod 25 3x 12 m
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 3due Wednesday, February 18Please prove all your answers. Problem 1. Prove that a square of an integer cannot end by two odd digits (in decimal notation). Problem 2. For n integer, prove that if t
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 3Some solutionsProblem 1. Prove that a square of an integer cannot end by two odd digits (in decimal notation). Solution. Write n = 10a + b, with 0 b &lt; 9, then n2 = 100a2 + 20ab + b2 . The rst te
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 8Solutions Problem 2 sec 3.2 Quadratic reciprocity tell us in this case that x2 q mod p is solvable; the only thing to check is that it has exactly two solutions. It cannot have more because the nu
SUNY Stony Brook - MAT - 311
MAT 311 Introduction to Number TheoryProblem Set 9Solutions Problem 1 sec 7.1 Just divide, reversing the fractions and applying the division algorithm as you go along: 17 1 1 =5+ =5+ ; 3 3/2 1 + 1 /2 other numbers are even easier. Problem 3 sec 7.
SUNY Stony Brook - MAT - 511
MAT 511 Fundamental Concepts of MathProblem Set 4due Thursday, Oct 2Please prove all your answers. Short and elegant proofs are encouraged but not required. Problem 1. There are n straight lines on the plane, such that no two lines are parallel,
SUNY Stony Brook - MAT - 511
MAT 511 Fundamental Concepts of MathProblem Set 12due Thursday, Dec 11Please prove all your answers. Short and elegant proofs are encouraged but not required. Problem 1. Consider the following relation on some collection of sets: A B if there e
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MAT 511 Fundamental Concepts of MathProblem Set 10due Thursday, Nov 20Please prove all your answers. Short and elegant proofs are encouraged but not required. Problem 1. Suppose X is a set with an equivalence relation , X/ is the set of equival