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Course: MAT 328, Fall 2009
School: Paul Quinn
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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
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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
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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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SUNY Stony Brook - MAT - 311
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
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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
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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
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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
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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
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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
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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.
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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
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MAT 511 Fundamental Concepts of MathProblem Set 7due Thursday, Oct 30Please prove all your answers. Short and elegant proofs are encouraged but not required. Problem 1. The union of n sets B1 , B2 , . . . , Bn can be dened as B1 B2 . . . Bn =
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MAT 511 Fundamental Concepts of MathProblem Set 1due Thursday, Sept 11Please prove all your answers. Short and elegant proofs are encouraged but not required. Problem 1. To win Portias hand, her suitors must nd her portrait hidden in one of the
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SUNY Stony Brook - MAT - 511
MAT 511 Fundamental Concepts of MathProblem Set 8due Thursday, Nov 6Please prove all your answers. Short and elegant proofs are encouraged but not required. This time all the questions are from the textbook: Section 2.2: prove Theorem 2.7, parts
SUNY Stony Brook - MAT - 511
MAT 511 Fundamental Concepts of MathProblem Set 11due Thursday, Dec 4Please prove all your answers. Short and elegant proofs are encouraged but not required. Please do questions 3bcd, 9abd, 10bd (see below for the explanation of projection funct
SUNY Stony Brook - MAT - 511
MAT 511 Fundamental Concepts of MathProblem Set 6due Thursday, Oct 23Please prove all your answers. Short and elegant proofs are encouraged but not required. Problem 1. (a) List all subsets of the set S = {1, , {}. Here stands for the empty set
Rutgers - CS - 111
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SUNY Stony Brook - MAT - 320
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SUNY Stony Brook - MAT - 320
MAT 320 Introduction to AnalysisHomework 12due Friday, December 15 1. (a) Let f : [a, b] R be an integrable function. Consider a sequence b (Pn ) of tagged partitions with |(Pn )| 0. Prove that a g (x) dx = limn S (f, (Pn ). (Here the notatio
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MAT 320 Introduction to AnalysisReview SheetThe nal exam is cumulative and covers everything we have learned in MAT319/MAT320 during the semester. You must know all the denitions and statements of the important theorems, and also understand all pr
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Western Washington - BIOL - 321
Geneticsin the news.Finalcumulative, but overwhelming focus on Sugar Paper, Microarrays and Microarray Paper, lecture and supported readings in Chapters 8 and 9, Chimp Paper, Pheromone Paper, PCR, Northerns, Southerns. 8th, 3:30 - 5:30 Mond