Allard, William K. - Department of Mathematics, Duke University

• The central limit theorem. To prove the central limit theorem we make use of the Fourier transform which is one of the most useful
• We fix a first order language and we let be its set of symbols. (We may have called this the alphabet previously.)
• The Betweenness Axioms. For each pair of distinct points a and b there is a subset
• Integrating the Gaussian. For each real x we let
• Expectation Suppose E is an experiment the set of possible outcomes of which is the sample space S. Let E be an
• Random walk, gambler's ruin and other good stuff. Suppose 0 < p < 1 and let q = 1 -p. Let
• Homotopies and the Poincare Lemma. Let I = [0, 1] and let T be the vector field on R Rn
• 1. Uniform convergence. Suppose X is a set and (Y, ) is a metric space. We let
• Congruence. The Congruence Axiom.(Cong) There is a group
• Test Two Mathematics 131.01 Spring 2005 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• 1. Binary operations. Suppose X is a set
• 1. n-tuples. N = {0, 1, 2, . . . , n, . . .}
• 1. Sets, relations and functions. 1.1 Set theory. We assume the reader is familiar with elementary set theory as it is used in mathematics
• 1. Interpretations. 1.1. Review of terms and statements.
• 1. Arclength reparameterization. Suppose I is an interval and
• Integration of a scalar function over a submanifold. Suppose n is a positive integer, V is an n-dimensional inner product space, 0 m < n and M Mm(V ).
• 1. The Riemann and Lebesgue integrals. 1 2. The theory of the Lebesgue integral. 7
• Relations and functions. A relation is a set of ordered pairs.
• Hermitian inner products. Suppose V is vector space over C and
• Topics for Term Projects 1. Report on the proof that it is impossible to trisect an arbitrary angle using only a compass and
• Bayes' Rule. Suppose B1, . . . , Bn are disjoint events each of which with positive probability and whose union is the entire sample space. Then for any event A and any j {1, . . . , n} we have
• Definitions to memorize. 1. Sets relations and functions.
• 1. Lines and planes in Rn Definition 1.1. We say a subset L of Rn
• 1. Initial segments, well ordering and the axiom of choice. 1.1. Initial segments. We suppose throughout this subsection that < linearly
• Stokes' Theorem. Let n be a positive integer, let V be an open subset of Rn
• More on expectation and random variables. Definition. Suppose (S, E, P) is a probability space and X : S R is a random variable. Suppose
• The Parallel Projection Theorem. We assume the Parallel Postulate. Let j j be a length function for segments.
• Congruence. The Congruence Axiom.(Cong) There is a group
• Hilbert's axioms for (two dimensional) neutral geometry. We spell these out below. It will take a while. There will be several groups of axioms: the incidence
• Computing expectation by conditioning. Let (S, E, P) be a probability space, let F E be such that
• Angle sums and more. Among other things, we will prove the following Theorems.
• Test Two Answer Key Mathematics 114.01 Spring 2005 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• Fourier series. Preliminary material on inner products.
• 1. Differentiation of vector valued functions of a real variable. Definition 1.1. Suppose A R, E is a normed vector space,
• 1. The real numbers. 1.1. Ordered rings.
• Planetary motion. This material comes from an article by Robert Osserman in the American Mathematical Monthly of
• 1. Integrals in polar coordinates. be defined by
• Inscribing triangles to compute area. Proposition. Suppose k, l L(Rm
• Fractional linear transformations. Definition. We let GL(2, C) be the set of invertible 2 2 matrices
• 1. The general transformation formula. Theorem 1.1. Suppose
• Probem 7(a) on page 116. A paraphrase of the problem. An urn contains a finite nonempty set B of black balls and a finite
• Independent families of random variables. Definition. Suppose X is a family of random variables. We say X is independent if
• Test Two Mathematics 135.01 Fall 2007 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• 1. Power series. Definition 1.1. Suppose c is a sequence in C. (c will be a coefficient sequence.)
• Some xed point theorems. Suppose X is nonempty set and < is a complete linear ordering of X . Given x; y 2 X we write
• The multinomial distribution. Let n and r be positive integers and let
• TOTAL VARIATION REGULARIZATION FOR IMAGE DENOISING; III. EXAMPLES.
• Definition. Suppose V1, . . . , Vm and W are vector spaces. We say a function : V1 Vm W
• Problem 25 on page 55. A pair of fair dice are rolled until a sum of five or seven appears. Find the probability that a five appears
• Some worked problems from the Chapter Six homework. p. 292 n. 20. In the first case X and Y are independent because the joint density factors; in fact,
• n-coordinates. Let S be a set. We say y is an m-variable on S if y : S Rm
• 1. Vectors. In what follows n will always be a positive integer.
• Test One Mathematics 135.01 Fall 2007 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• It's dark and you're trying to open the door... We begin with a very useful formula.
• 1. Expressibility and representability. Definition 1.1. (Unique existence.) Suppose x is a variable and A is a state-
• Mathematics 282. Fall 2006. Elliptic Partial Differential Equations.
• 1. Sets, relations and functions. 1.1. Set theory. We assume the reader is familiar with elementary set theory as
• Curves in Rn 1. Limits, continuity and differentiation.
• Test Two Mathematics 135.02 Fall 2009 Answer Key.
• Let X be a normed vector space. Definition. Suppose v X and C X. We say C is a cone with vertex v if
• 1. Holder's Inequality and Minkowski's inequality. We fix p, q [1, ] such that
• Integration of a differential form over an oriented submanifold. Let m and n be positive integers, let V be an open subset of V and let M Mm(V ) and let o be an
• Linear transformations and Gaussian random vectors. We fix a positive integer n.
• 1. The natural numbers and arithmetic. Theorem 1.1. Suppose, for each i = 1, 2, Xi is a nonempty set,
• Hilbert's axioms for (two dimensional) neutral geometry. We spell these out below. It will take a while. There will be several groups of axioms: the incidence
• 1. Equivalence is preserved under substitution. We fix a first order logic F and we let S be its set of statements.
• Random vectors. Fix a positive integer n. We say a subset R of Rn
• Mathematics 103.07 Test One October 5, 2006 I have neither given nor received aid in the completion of this test.
• 1. A class of good context free grammars. G = (T, N, s, P)
• 1. Introduction to the theory of infinite sets. Theorem 1.1. N is infinite.
• 1. Integration. Definition 1.1. We say f : R2
• The Poisson process. I1, I2, . . . , Im, . . .
• 1. The Fourier transform on R. is integrable; this means, by definition, f Leb1. For each R we set
• ) = {f L(R2 Suppose s F(R2
• 1. Recursive functions. For each n N we let
• Vector fields and divergence. Let U be an open subset of Rn
• 1. The Theorems of Fubini and Tonelli. Suppose n, m N+
• Differentiation and Tangency. Definition. Suppose X and Y are normed vector spaces, A is a subset of X and f : A Y .
• 1. Back to formal theories. I'm going to modify some notions involving formal theories. Suppose
• The differential. Let n be a positive integer.
• 1. The integers and the rational numbers. 1.1. The integers. Let
• Inner products. Let V be a vector space.
• The change of variables formula for multiple integrals. Let n be a positive integer.
• On the Performance of a Distributed Object Oriented Adaptive Mesh Refinement Code
• Mathematics 103.07 Test Two November 16, 2006 I have neither given nor received aid in the completion of this test.
• Differentiation with respect to a coordinate. A numerical function is a function whose domain and range are subsets of R, the set of real numbers.
• A take home final exam problem. Introduction. We have already considered the following. Letf P be such that
• Fractional linear transformations. De nition. We let GL(2; C) be the set of invertible 2 2 matrices
• Betweenness preserving permutations. be the identity map of the set of points.
• The reconstruction of surfaces in R 3 by reflection. by William K. Allard
• Linear Algebra Definition. A vector space (over R) is an ordered quadruple
• Differential Forms. Suppose U is an open subset of Rn
• Differential geometry upstairs William K. Allard
• 1. Sets, relations and functions. 1.1. Set theory. We assume the reader is familiar with elementary set theory as
• 1. Topological spaces Definition 1.1. We say a family of sets T is a topology if
• 1. Metric spaces. Definition 1.1. Let X be a set. We say is a metric on X if
• 1. An extremely useful abstract closure principle. Suppose X is a vector space over R and
• 1. Summation. Let X be a set.
• 1. More on the exponential function. Proposition 1.1. We have
• 1. Algebras of sets and the integration of elementary functions. Let X be a set.
• 1. Fourier series. Definition 1.1. Given a real number P, we say a complex valued function f on R
• Final Problem Set. 1. Getting caught up.
• Definition. Whenever m and n are positive integers we let n = R{1,...,m}{1,...,n}
• The Inverse and Implicit Function Theorems. Proposition. Suppose X and Y are normed vector spaces and L is a linear isomorphism from X onto Y .
• Two examples illustrating the Inverse Function Theorem. Example One. Let L(x) = x for x R, a = 0,let
• The Implicit Function Theorem. Suppose (1) X, Y and Z are Banach spaces;
• Alternating and symmetric multilinear functions. Suppose V and W are normed vector spaces.
• How elementary linear maps change areas. Fix an integer n 2. Let
• The critical set goes to a set of measure zero. Proposition. Suppose S is an r-dimensional linear subspace of Rn
• Submanifolds. Let n be a positive integer.
• 1. Trees; context free grammars. 1.1. Trees.
• 1. Axioms and rules of inference for propositional logic. Suppose T = (L, A, R) is a formal theory. Whenever H is a finite subset of L
• Definition 1.1. Suppose A is an alphabet, a A, s (A) and n = |s|. We let
• 1. Model existence theorem. We fix a first order logic F such that
• 1. The tough one. We introduce the following notation. Suppose A is a statement and
• 1. More on the function F. Z = {(z, A) R Sym(Rn
• The Inverse and Implicit Function Theorems. Proposition. Suppose X and Y are normed vector spaces and L is a linear isomorphism from X onto Y .
• Test One Mathematics 135.02 Fall 2009 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• Test One Mathematics 135.01 Fall 2007 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• Test Two Mathematics 135.01 Fall 2007 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• Probability spaces. We are going to give a mathematical definition of probability space. We will first make some remarks
• p. 113, n. 60. There is a 50-50 chance the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia. If the queen has had three princes without the disease,
• Problem 74, page 113. In successive rolls of a pair of fair dice, what is the probability of getting 2 sevens before 6 even numbers?
• Random variables. Definition. We say
• Some worked problems from Chapter Five. Page 228, n. 2. A system consisting of one original unit plus a spare can function for a random amount
• Convolutions and Fourier Transforms. Definition. Suppose
• Some basic facts about integration. The case R2
• Conditional distributions. The continuous case. Suppose Y is a continous random vector.
• Linear transformations and Gaussian random vectors. Remember, n-vectors are the same as n 1 matrices.
• Math 103.02; Fall 2010; Test Two I have neither given nor received aid in the completion of this test.
• Math 103.02; Fall 2010; Test Three I have neither given nor received aid in the completion of this test.
• Math 103.02 Quiz Ten Due Monday, November 29.
• Minima and maxima. We fix a positive integer n.
• 1. More on differentiability, differentials and linear approximation. Let m and n be positive integers.
• Green's Theorem. r : [a, b] [0, 1] R2
• The flow of a vector field. Suppose F = Pi + Qj is a vector field in the plane1
• Ordinary differential equations, initial conditions and integral equations. We fix a positive integer n. The first time you read this I suggest you take n = 1.
• Problem 41 on page 55. Suppose 0 < T ; m and v are continuous [0, T); m is positive and nondecreasing on (0, T); v is
• Variation of Constants? Let us recall how we solved
• A nifty example. Suppose (1) A, B, C, D, E, F are positive
• Definition. For z C we define ); cosh z =
• 1. Sets, relations and functions. 1.1 Set theory. We assume the reader is familiar with elementary set theory as it is used in mathematics
• Some fixed point theorems. Suppose X is nonempty set and < is a complete linear ordering of X. Given x, y X we write
• Betweenness preserving permutations. be the identity map of the set of points.
• Angle sums and more. Among other things, we will prove the following Theorems.
• Ruler and compass constructions. 1. Definition. Suppose S C. We let
• Topics for Term Projects 1. Report on the proof that it is impossible to trisect an arbitrary angle using only a compass and
• 1. Preliminaries. 1.1. Tangents and normals.
• Math 103.02; Fall 2010; Test One I have neither given nor received aid in the completion of this test.
• 1. Godel's Theorem. 1.1. -consistency.
• 1. Arithmetic modulo 2. b = {0, 1}.
• Critical points and stability. Suppose J is an open interval in R and
• The Betweenness Axioms. For each pair of distinct points a and b there is a subset
• Definition. (See p.70) Suppose (S, A, P) is a probability space and X is a random variable. We define FX (x) = P(X x) for x R.
• Whenever X is a set we let be the number of members of X if X is finite and we let it be if X is infinite. Here are some basic counting
• Orientation. Let n be a positive integer, let m be a positive integer not exceeding n and let V be an n-dimensional
• Test One Mathematics 131.01 Spring 2005 TO GET FULL CREDIT YOU MUST SHOW ALL WORK!
• SIAM J. MATH. ANAL. c XXXX Society for Industrial and Applied Mathematics Vol. 0, No. 0, pp. 000000
• Ruler and compass constructions. 1. De nition. Suppose S C. We let
• Partitions of Unity. Theorem. Suppose a Rn
• The Brouwer Fixed Point Theorem. Fix a positive integer n and let Dn
• 1. Differentiability, differentials, linear approximation and all that stuff.