Dissertations and Theses - Mathematics
http://hdl.handle.net/2142/16340
Self improving Orlicz-Poincare inequalities
http://hdl.handle.net/2142/46744
Self improving Orlicz-Poincare inequalities
DeJarnette, Noel
In [20], Keith and Zhong prove that spaces admitting Poincar e inequalities also admit a priori stronger Poincar e inequalities. We use their technique, with slight adjustments, to obtain a similar result in the case of Orlicz-Poincar e inequalities. We give examples in the plane that show all hypotheses are required and develop
the theory of Orlicz-Poincar e inequalities for nondoubling Young functions to show that the ∞-Poincar e inequality does not improve to any Orlicz-Poincar e inequality.
Orlicz Functions
Poincare inequality
maximal function operator
Thu, 16 Jan 2014 18:00:57 GMTLinear and bilinear restriction estimates for the Fourier transform
http://hdl.handle.net/2142/46569
Linear and bilinear restriction estimates for the Fourier transform
Temur, Faruk
This thesis is concerned with the restriction theory of the Fourier transform. We prove two restriction estimates for the Fourier transform. The first is a bilinear estimate for the light cone when the exponents are on a critical line. This extends results proven by Wolff, Tao and Lee-Vargas. The second result is a linear restriction estimate for surfaces with positive Gaussian curvature that improves over estimates proven by Bourgain and Guth, and gives the best known exponents for the well-known restriction conjecture for dimensions that are multiples of three.
Fourier transform
Restriction theory
Wave equation
Linear restriction
Kakeya problem
Schrodinger equation
Thu, 16 Jan 2014 17:54:21 GMTDistribution of some arithmetic sequences
http://hdl.handle.net/2142/45594
Distribution of some arithmetic sequences
Xiao, Jiajie
In this thesis, we first study the correlations of some arithmetic sequences. We prove the existence of the limiting pair correlations of fractions with prime and power denominators, and give the explicit pair correlation density functions. Next, we study the higher level correlations of these fractions, and construct an arithmetic sequence with showing the independence of its different level correlations.
We also study the distribution of angles between common tangents of Ford circles, which is a special case of Apollonian circle packing. We provide the limiting distribution functions of these angles in different situations.
Number theory
analytic number theory.
Thu, 22 Aug 2013 16:48:59 GMTExtremal problems for labelling of graphs and distance in digraphs
http://hdl.handle.net/2142/45527
Extremal problems for labelling of graphs and distance in digraphs
Jahanbekam, Sogol
We study several extremal problems in graph labelling and in weak diameter of digraphs.
In Chapter 2 we apply the Discharging Method to prove the 1,2,3-Conjecture [41] and the
1,2-Conjecture [48] for graphs with maximum average degree less than 8/3. Stronger results on
these conjectures have been proved, but this is the first application of discharging to them,
and the structure theorems and reducibility results are of independent interest. Chapter 2
is based on joint work with D. Cranston and D. West that appears in [17].
In Chapter 3 we focus on digraphs. The weak distance between two vertices x and y in a
digraph G is the length of the shortest directed path from x to y or from y to x. We define
the weak diameter of a digraph to be the maximum directed distance among all pairs of
vertices of the digraph. For a fixed integer D, we determine the minimum number of edges
in a digraph with weak diameter at least D, when D = 2, or when the number of vertices of
the digraph is very large or small with respect to D. Chapter 3 is based on joint work with
Z. Furedi that appears in [26].
In Chapter 4 using Ramsey graphs, we determine the minimum clique size an n-vertex
graph with chromatic number \chi can have if \chi \geq (n+3)/2. For integers n and t, we determine
the maximum number of colors in an edge-coloring of a complete graph Kn that does not
have t edge-disjoint rainbow spanning trees of Kn. For integers t and n, we also determine
the maximum number of colors in an edge-coloring of Kn that does not have any rainbow
spanning subgraph with diameter t. Chapter 4 is based on three papers, the first is joint
work with C. Biro and Z. Furedi [11] and the other two are joint work with D. West [36, 37].
Graph Coloring
Graph Labelling
Ramsey Numbers
AntiRamsey Graph Theory
Weak Diameter in Digraphs
Matching in Graphs
Thu, 22 Aug 2013 16:46:39 GMTUniformly rigid homeomorphisms
http://hdl.handle.net/2142/45507
Uniformly rigid homeomorphisms
Yancey, Kelly
In this dissertation we are interested in the study of dynamical systems that display rigidity and weak mixing. We are particularly interested in the topological analogue of rigidity, called uniform rigidity. A map $T$ defined on a topological space $X$ is called \textit{uniformly rigid} if there exists an increasing sequence of natural numbers $\left( n_m\right)$ such that $\left( T^{n_m}\right)$ converges to the identity uniformly on $X$ and is called \textit{weakly mixing} if there exists a sequence $\left(s_m\right)$ of density one such that $\mu(T^{s_m}A\cap B)$ converges to $\mu(A)\mu(B)$ for every $A,B$ of positive $\mu$-measure (the sequence $\left(s_m\right)$ is called a \textit{mixing sequence}). Uniform rigidity and weak mixing are two properties of a dynamical system that are very different, though not exclusive. Rigidity implies that at certain times the image of an interval is close to the interval, while weak mixing implies that at other times the images of intervals are evenly distributed. Observe that the rigidity times for a weakly mixing map have density zero. This dissertation attempts to better understand the interplay between weak mixing and uniform rigidity.
The underlying theme of this dissertation has two threads: (1) to determine how the topology of a space affects dynamical properties of maps that are defined there and (2) to characterize the structure of uniform rigidity sequences for weakly mixing maps. The work in this dissertation has involved several projects that were designed to provide a better understanding of these maps and their uniform rigidity sequences, thereby yielding information about the dynamical properties that are compatible with certain spaces and information about the structure of those sequences.
weak mixing
topological weak mixing
rigid
uniformly rigid
ergodic theory
topological dynamics
generic
typical homeomorphisms
Thu, 22 Aug 2013 16:42:34 GMTLength functions in flat metrics
http://hdl.handle.net/2142/45475
Length functions in flat metrics
Bankovic, Anja
This dissertation is concerned with equivalence relations on homotopy classes of curves coming
from various spaces of at metrics on a genus g >1 surface. We prove an analog of a result of Randol (building on work of Horowitz) for subfamilies of at metrics coming from q-di erentials. In addition we also describe how these equivalence relations are related to each other.
hyperbolic metric
length functions
Euclidean cone metric
curves on surfaces
Thu, 22 Aug 2013 16:41:19 GMTHyperbolic 3-manifolds of bounded volume and trace field degree
http://hdl.handle.net/2142/45424
Hyperbolic 3-manifolds of bounded volume and trace field degree
Jeon, Bo Gwang
For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings
of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more
cusps, and give the first positive results in this direction. For example, in the 2-cusped case, if a manifold
has linearly independent cusp shapes, we show that the manifold has the desired property. To prove the
results, we use Habegger's proof of the Bounded Height Conjecture in arithmetic geometry.
Hyperbolic 3-Manifolds
Volume
Dehn Filling
Trace Field
Thu, 22 Aug 2013 16:39:45 GMTAnalysis of a 1D approximation of the Boltzmann Equation: the subclass of grossly determined solutions and the asymptotic behavior of the class of general solutions
http://hdl.handle.net/2142/45411
Analysis of a 1D approximation of the Boltzmann Equation: the subclass of grossly determined solutions and the asymptotic behavior of the class of general solutions
Carty, Thomas
In this paper we examine an approximation of the Maxwell-Boltzmann equation for a 1D gas. In the manner of classical gas dynamics, we derive a balance law and use it to determine the grossly
determined solutions, a sub-class of solutions that are functions dependent on the gas's density
field. Then, via spectral decomposition, we derive the class of general solutions and show that they tend asymptotically to the class of grossly determined solutions.
Boltzmann Equation
Grossly Determined Solutions
Spectral Decomposition
Thu, 22 Aug 2013 16:39:23 GMTIdentities involving theta functions and analogues of theta functions
http://hdl.handle.net/2142/45365
Identities involving theta functions and analogues of theta functions
Xu, Ping
My dissertation is mainly about various identities involving theta functions and analogues of theta functions.
In Chapter 1, we give a completely elementary proof of Ramanujan's circular summation formula of theta functions and its generalizations given by S. H. Chan and Z. -G. Liu, and J. M. Zhu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given.
In Chapter 2, we analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many nice explicit examples are given.
In Chapter 3, we study one page in Ramanujan's lost notebook that is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims
are remindful of Gauss sums. In this chapter, we prove all the claims made by Ramanujan about this integral.
Circular summation formula
elementary proof
two-dimensional lattice sums
Poisson equation
theta functions
analogue of theta functions
analogue of Gauss sums
Thu, 22 Aug 2013 16:37:57 GMTGraphical analysis of hard-to-borrow stocks
http://hdl.handle.net/2142/44814
Graphical analysis of hard-to-borrow stocks
Nawaz, Tayyab
We study the graphical analysis for hard to borrow stocks i.e., stock with some constraints such as short selling. The main purpose of this graphical analysis was to introduce some algorithm for calculating the implied dividend yield curve for hard to borrow stocks using the dynamical programming. Implied dividend curve can be used to analyze the options which are hard to borrow. We used the python as our tool for dealing with the financial data taken from the yahoo finance, since it helps in minimizing the memory usage for the data storage. Our main interest was to work with hard to borrowness for at the money options, but due to unavailability for these events we worked with in the money options since they are more crucial for the puts options. Put options have an early exercise for the American options, so considering the put options for in the money options are more interesting to analyze.
hard to borrow stocks
implied dividend yield curve
dynamic programming
American options
Tue, 28 May 2013 19:21:00 GMT