# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

2 Pages

### Analysis2009aug

Course: MATH 600, Spring 2011
School: University of North...
Rating:

Word Count: 345

#### Document Preview

Preliminary Analysis Exam August 24, 2009 1. If F1 and F2 are closed subsets of R1 and dist(F1 , F2 ) = 0 then F1 F2 = . Prove or give a counterexample. 2. Newtons method for nding zeroes of a function f : R1 R1 is based on the recursion formula f (xn ) xn+1 = xn , n 1. f (xn ) Show that if f C 1 , f (a) = 0 and f (a) = 0, then there exists a &gt; 0 such that if |x1 a| &lt; then xn a....

Register Now

#### Unformatted Document Excerpt

Coursehero

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Preliminary Analysis Exam August 24, 2009 1. If F1 and F2 are closed subsets of R1 and dist(F1 , F2 ) = 0 then F1 F2 = . Prove or give a counterexample. 2. Newtons method for nding zeroes of a function f : R1 R1 is based on the recursion formula f (xn ) xn+1 = xn , n 1. f (xn ) Show that if f C 1 , f (a) = 0 and f (a) = 0, then there exists a > 0 such that if |x1 a| < then xn a. (Suggestion: use the Mean Value Theorem.) 3. Let f : [0, ) [0, ) and for h > 0 and k 1 set Mk (h) = sup f (x), mk (h) = (k1)hx<kh Let inf (k1)hx<kh f (x). U ( h) = Mk (h)h, L(h) = k=1 mk (h)h. k=1 We say f is directly Riemann integrable if U (h) < for all h > 0 and lim(U (h) L(h)) = 0. h0 Recall f is improperly Riemann integrable on [0, ) if f is Riemann integrable on [0, a] for every a > 0, and a lim a f (t) dt < Show . 0 (a) that if f is continuous and nonincreasing, then f is directly Riemann integrable whenever f is improperly Riemann integrable on [0, ). (b) Give an example of a continuous function f which is improperly Riemann integrable on [0, ) but not directly Riemann integrable. 4. Suppose f : [0, ) [0, ) is such that for any sequence an of nonnegative terms we have an < = n=1 f (an ) < n=1 Prove that lim sup x0+ f (x) < x 5. Let f be a continuous real valued function dened on the unit square and for each 0 x 1 let fx be the function on the unit interval dened by fx (y ) = f (x, y ). Prove that for any sequence xn in [0,1] there is a subsequence nk such that fxnk converges uniformly on [0,1]. 6. If c is a real parameter prove that x7 + x + c = 0 has a unique real root and that this root is a dierentiable function of c.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

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
University of North Carolina School of the Arts - MATH - 661
University of North Carolina School of the Arts - MATH - 661
August 2008Qualifying ExaminationTopologyProblems 1-4 consist of true or false statements. Each statement is to be proved ordisproved with brief but complete reasoning. Provide definitions of all underlined,italicized words and phrases. On page 2 fin
University of North Carolina School of the Arts - MATH - 661
T OPOLOGY QUALIFYING EXAMAUGUST 2009There are six problems. Begin y our a nswer to any p roblem on a n ew p age in y our bluebook(s). Make a space between answers to separate p arts of a question to facilitategrading. A ll a nswers must be j ustified
University of North Carolina School of the Arts - MATH - 661
Topology Qualifying ExamJanuary 9, 2009Do all of the following problems each of which is worth 20 points.1. Let X be a topological space.(a) Dene what it means to say that X is compact.(b) State the Tube Lemma.(c) Prove the Tube Lemma.2. Let X and
University of North Carolina School of the Arts - MATH - 661
Topology Qualifying ExamAugust 23, 2010Do all of the following problems.1. Let X be a topological space.(a) (5 points) Dene what it means to say that X is regular.(b) (10 points) Give a complete proof that a product of two regular spaces isregular.
University of North Carolina School of the Arts - MATH - 661
Topology Qualifying ExamTopics CoveredMAT 661Point Set TopologyTopological Spaces: Topologies, neighborhoods, basis, closure operations, continuity,product and quotient topologies.Topological Properties: Separation axioms, compactness, local compact
University of North Carolina School of the Arts - MATH - 651
Statistics Qualifying Exam for MAT 651/652August, 20091. Let X1 , X2 , . . . , X10 be a random sample of size 10 from a continuous uniform distribution on(0, 2). Let W = X(10) X(1) where X(10) is the largest order statistic and X(1) the smallestorder
University of North Carolina School of the Arts - MATH - 651
Probability and Statistics I (MAT651)1. Axiomatic Foundations, The Calculus of Probabilities, Counting, EnumeratingOutcomes. Conditional Probability and Independence2. Random Variables, Distribution Functions, Density and Mass Functions3. Distribution