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Algebra2007Jan
Course: MATH 601, Spring 2011School: University of North...
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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 < 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 &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 > 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_< 1 < |z | < 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 > X1 (inf = ). Prove thatP (N > 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 > 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 > 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 > 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 < x1 be two distinct real numbers. Letx1 x0 . Find a polynomial p of degree 3 such thatbe such that 0 <<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 < x1 < < 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<x<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 < t < 1 which satisesu (t) = 6u (t) tu(t) + u2 (t)for0<t<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
Kennesaw - POLS - 1000
Cynthia OlaoluwaPols 1000Gunning4/26/11Federal Election Campaign Contribution Assignment1. Re-elections rates for the US House and US Senate are high.2. No, there isnt any difference between House members and Senators?3. The House incumbent has ove
Kennesaw - POLS - 1000
Cynthia OlaoluwaPols 1000Gunning4/26/11Federal Election Campaign Contribution Assignment1. Re-elections rates for the US House and US Senate are high.2. There is a difference between House members and Senators, but not really much of one.The only
Kennesaw - POLS - 1000
The Legislative Branch1. Representation: giving voice to viewpoints in society.2. Lawmaking: finding agreement on new laws and the budget.There is a tradeoff between these two goals.Who should represent us?Descriptive Representation: A legislature sh
Kennesaw - POLS - 1000
Executive OrganizationPyramid shaped the command sturctuePresident at the top (POTUS)Cabinet SecretariesDeputy SecretariesUnder SecretariesCareer Civil ServantsExecutive Office of the PresidentChief of staffPress SecretaryNational Security Counc
Kennesaw - POLS - 1000
4/19/11Presidential ApprovalBush Trend with Rally PointsBureaucracyObservation: Any large organization has some form of bureaucracy.Why do we need bureaucracies?What do they do for us?Why do we hate bureaucracy? ( They treat us alike).Why are they
Kennesaw - POLS - 1000
Judicial BranchOrganization of Federal Judiciary1. Supreme court of the USA (1)2. Federal Appeals Courts (11)3. Federal District Courts (95)Supreme Court (9 justices)Appeals (3 judge panels)Districts (1 judge presides)Difference between LevelsDis
Kennesaw - POLS - 1000
How a bill becomes a law.Committee/SubcommitteeHearingsMarkupReportingFloor ActionsDebate (House: time limits, Senate: unlimited debate-Filibuster/Cloture 60 votes)Amendments (House: Open/Closed Rules, Senate: very open to amendments)Passage Vote
Kennesaw - BIO - 1000
D. The lungs supply oxygen to the circulatory system (blood), which in turn, supplies oxygen to the restof the body. For example the lungs provide the oxygen that is needed to metabolize glucose. It alsosupplies air to the brain, which keeps the brain f