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### Analysis2003Aug

Course: MATH 600, Spring 2011
School: University of North...
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University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
Preliminary Exam Jan 20071. Let X be a metric space and let Aj be subsets of X , j = 1, 2, . . . . For each of the following statements, prove it or give a counterexample (the means limit points):(i) (A1 A2 ) A1 A2(ii)j =1Aj Ajj =12. Prove that th
University of North Carolina School of the Arts - MATH - 600
Analysis Preliminary ExamAugust, 20081. Let f : R2 R be given by the formulax2 yx2 + y 2f (x, y ) =0if (x, y ) = (0, 0)if (x, y ) = (0, 0).(a) Show that f is continuous at (0, 0).(b) Prove that the rst order partial derivatives of f at (0, 0) ex
University of North Carolina School of the Arts - MATH - 600
Analysis Preliminary ExamAugust 24, 20091. If F1 and F2 are closed subsets of R1 and dist(F1 , F2 ) = 0 then F1 F2 = . Prove orgive a counterexample.2. Newtons method for nding zeroes of a function f : R1 R1 is based on the recursionformulaf (xn )x
University of North Carolina School of the Arts - MATH - 600
Analysis preliminary examJan. 8, 20091. Let C be the standard Cantor set on the interval [0, 1] and let A = C c be its complement on the real line. Identify the set of all limit points A of A, explaining your answer.2. (a) Provenk=k=1n(n + 1)2(b)
University of North Carolina School of the Arts - MATH - 600
August 2010 Preliminary Exam in Analysis1. Suppose that f : R R is a function such that f (f (x) = x for all x R. Prove thatthere exists an irrational number t such that f (t) is also irrational.2. Find three subsets A, B, C of the real line R such tha
University of North Carolina School of the Arts - MATH - 600
January 2010 Preliminary Exam in Analysis.1. Let X be a connected metric space. Given two points p, q X and a numberprove that there exist an integer n&gt; 0,0 and points a0 , a1 , . . . , an X such that a0 = p,an = q , andd(aj , aj 1 ) &lt;for all j = 1
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
Topics for the Ph.D. Preliminary ExaminationANALYSISFundamentals of Analysis (MAT 601-602)The basic material is contained in Rudin's &quot;Principles of Mathematical Analysis&quot; 3rdEdition, Chapters 1 through 9. Important topics are:1. Properties of real an
University of North Carolina School of the Arts - MATH - 601
University of North Carolina School of the Arts - MATH - 601
University of North Carolina School of the Arts - MATH - 601
Qualifying ExaminationJanuary 10, 2008Algebra Part Please do all ve questions. Problem #5 is worth twice as much as each of the othersWe will always assume that rings have an identity element and that modulesare unitary left modules.1. Let I be an
University of North Carolina School of the Arts - MATH - 601
Algebra P art o f Qualifying Examination, A ugust 25, 2009Instructions: Do all questions, justify your answers with the necessary proofs.All rings are associative (not necessarily commutative) with identity a nd all modulesare left unitary modules. We
University of North Carolina School of the Arts - MATH - 601
Q ualifying Exam - January 2009Algebra PartInstructions: C omplete as m any q uestions a s possible. Answers should be justified withthe necessary proofs. All rings are a ssumed to be n oncom mutative unless statedotherwise. All rings have an identity
University of North Carolina School of the Arts - MATH - 601
Algebra Part of Qualifying Examination, August 23, 2010Instructions: Do all questions, justify your answers with the necessary proofs. Allrings are associative (not necessarily commutative) with identity, and all modules are leftunitary modules. We den
University of North Carolina School of the Arts - MATH - 601
(Topics for qualifying exam in algebra) 731 SYLLABUSI. Set-up (over rings with unity, including noncommutative)Modules and Homomorphism TheoremsDirect Sums and Products, Free Modules(including Universal Mapping Properties)Projective and Injective Mod
University of North Carolina School of the Arts - MATH - 601
August 2008Qualifying ExaminationAlgebra PartThere are only 6 questions. Do them all.1. Let A be a nite abelian group. Prove that A is not a projective Z-moduleand also that it is not an injective Z-module.2. Prove that Q/Z Z Q/Z = 03. Let I be an
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Qualifying Exam in Combinatorics12 January, 20071. Suppose that G is a graph with no vertex of valence &lt; 5, exactly 13vertices of valence 5, no vertex of valence 7, and possibly an assortmentof vertices of other valences.(a) What is the least number
University of North Carolina School of the Arts - MATH - 680
Qualifying Exam in Combinatorics19 August, 20081. Let be a 3-connected planar graph with planar dual . Prove ordisprove:If is a Cayley graph and is is vertex-transitive, then is a Cayleygraph.If you believe the statement to be true, give a proof, or
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Combinatorics Qualifying ExamPractice QuestionsJuly 21, 20081. (a) Dene (v , k , )-design. Derive some identities involving these three parameters.(b) How are these designs related to nite pro jective planes? Give a picture of the simplest nitepro je
University of North Carolina School of the Arts - MATH - 680
Topics for Qualifying Exam in CombinatoricsI. EnumerationA. Selections with and without repetitions (combinations &amp;permutations)B. Partitions1. Stirling numbers of the first and second kindC. Principle of Inclusion-Exclusion1. Surjections2. Derang
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
QUALIFYING EXAM COMPLEX ANALYSISThursday, January 8, 2009Show ALL your work. Write all your solutions in clear, logical steps. Good luck!Your Name:Problem Score Max120220330430Total100Problem 1. Let f = f (z ) be analytic in the unit disk,
University of North Carolina School of the Arts - MATH - 680
Qualifying Exam, Complex Analysis, August 20101. Let n &gt; 0 be an integer. How many solutions does the equation 3z n = ez have in theopen unit disk? Justify your answer in full detail.2. Let f (z ) =n0an z n be holomorphic in the unit disk U such that
University of North Carolina School of the Arts - MATH - 680
Complex Part1. Show that the function f (z ) = 1/z has no a holomorphic antiderivative on cfw_&lt; 1 &lt; |z | &lt; 2.2. Suppose that f is an entire function and f 2 is a holomorphicpolynomial. Show that f is also a holomorphic polynomial.3. Suppose that a fun
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Topics for Qualifying Exam in Complex AnalysisIComplex Plane and Elementary Function.a) Complex Numbersb) Polar Representationc) Stereographic Projectiond) The Square and Square Root Functionse) The Exponential Functionf) The Logarithm Functiong)
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Qualifying ExamProbabilityAugust, 20101. Let Xi , i 1 be IID and set N = inf cfw_n 1 : Xn &gt; X1 (inf = ). Prove thatP (N &gt; n) 1/n for n 1, and use this to show EN = .2. Let Xi , i 1 be independent withP (Xi = i) = P (Xi = i) =12iandP (Xi = 0) =
University of North Carolina School of the Arts - MATH - 680
PhD Qualifying Exam, January 2010ProbabilityYou may use the following facts.(i) Y is Poisson with parameter &gt; 0 if P (Y = k ) = e k /k ! for k = 0, 1, . . . , in which case E (Y ) = andV ar(Y ) = .(ii) Y is exponential with parameter &gt; 0 if Y has den
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Analysis Qualifying ExamAugust 2010You must justify your answers in full detail, andexplicitly check all the assumptions of any theorem you use.1. Assume that f, f1 , f2 , L1 (R) (Lebesgue measure), and that as n (i) fn f pointwise on Rand (ii) fn1
University of North Carolina School of the Arts - MATH - 680
Real analysis qualifying examJan. 13, 20101. (a) Let f be a continuous map of a metric space X into a metric space Y .True or False. If false either give a counterexample, or make the statement true byeither adding a hypothesis or modifying the conclu
University of North Carolina School of the Arts - MATH - 680
Topics for Qualifying Exam in Analysisr-algebrasMeasures, outer measures, Borel measuresMeasurable FunctionsLebergee integration in abstract measure spaces and in R , Lebesgue measureRiesz representation theorem for positive Borel measures and for co
University of North Carolina School of the Arts - MATH - 684
MAT 682Numerical AnalysisAugust, 2008NAME:1. Suppose A be an n n invertible matrix. Let A = U V be the singularvalue decomposition of A, where = diagcfw_1 , 2 , . . . , n and 1 2 n &gt; 0.(a) Show that 2 (A) = 1 /n , where 2 (A) A(b) Show that(n 1)
University of North Carolina School of the Arts - MATH - 684
MAT 683Numerical AnalysisAugust, 2008NAME:1. For which values of s [0, 1], will there exist a unique quadratic polynomial p thatsatises the following conditionsp(0) = p0 ,p(1) = p1 ,p (s) = p2 ?2. Consider the equation3x + g (x) = 0(1)with g :
University of North Carolina School of the Arts - MATH - 684
MAT 683Numerical AnalysisJanuary, 2008NAME:1. Find a polynomial p(x) of degree 2 that satisesp(x0 ) = a,p (x0 ) = b,p (x1 ) = c,where a, b, c are given constants and x0 , x1 are two dierent points.2. Let f (0), f (h) and f (2h) be the values of a
University of North Carolina School of the Arts - MATH - 684
1MAT 683Numerical AnalysisQualifying Exam Syracuse UniversityAugust 24, Fall 2010Ex 1. Let x0 &lt; x1 be two distinct real numbers. Letx1 x0 . Find a polynomial p of degree 3 such thatbe such that 0 &lt;&lt;p ( x0 ) = p ( x0 + ) = 1p ( x1 ) = p ( x1 + )
University of North Carolina School of the Arts - MATH - 684
MAT 683Numerical AnalysisJanuary, 2010NAME:1. (a) Let S be a linear spline function that interpolates f at a sequence of nodes 0 =1x0 &lt; x1 &lt; &lt; xn = 1. Find an expression of 0 S (x) dx in terms of xi and f (xi ),i = 0, 1, . . . , n.(b) Can a and b
University of North Carolina School of the Arts - MATH - 684
MAT 684Numerical AnalysisAugust, 2010NAME:1. Let f : [0, 1] R be a given continuous function. Consider the following boundary valueproblem with Dirichlet boundary conditionsu (x) + u(x) = f (x),u(0) = 0,0&lt;x&lt;1u(1) = 0.(a) Derive the variational e
University of North Carolina School of the Arts - MATH - 684
1MAT 684Qualifying ExamSyracuse UniversityJanuary, Spring 2010Ex 1. Consider the boundary value problem od determining u(t), 0 &lt; t &lt; 1 which satisesu (t) = 6u (t) tu(t) + u2 (t)for0&lt;t&lt;1u(0) = 20u(1) = 10(a) Formulate a nite dierence method to c
University of North Carolina School of the Arts - MATH - 684
University of North Carolina School of the Arts - MATH - 661
University of North Carolina School of the Arts - MATH - 661
University of North Carolina School of the Arts - MATH - 661
University of North Carolina School of the Arts - MATH - 661
Topology Qualifying ExamAugust 20, 2007Do all of the following problems.1. (20 points) Let X be a set.(a) Dene what it means to say that the collection B of subsets of X is a basis fora topology on X .(b) Suppose now that X is endowed with a topolog