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A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema
In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function π which is the uniform limit of a sequence of sawtooth functions πβ : [0, 1] β [0, 1] with increasingly sharp teeth. Let π = [0, 1] x [0, 1] and πΉ(π) be the Hausdorff metric space determined by π. We define contraction maps π€β , π€β , π€β on π. These maps define a contraction map π€ on πΉ(π) via π€(π΄) = π€β(π΄) β π€β(π΄) β π€β(π΄). The iteration under π€ of the diagonal in π defines a sequence of graphs of continuous functions πβ. Since π€ is a contraction map in the compact metric space πΉ(π), π€ has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function π. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in πΆ[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set β¦
Date:
December 1993
Creator:
Huggins, Mark C. (Mark Christopher)
System:
The UNT Digital Library
The Continuous Wavelet Transform and the Wave Front Set
In this paper I formulate an explicit wavelet transform that, applied to any distribution in S^1(R^2), yields a function on phase space whose high-frequency singularities coincide precisely with the wave front set of the distribution. This characterizes the wave front set of a distribution in terms of the singularities of its wavelet transform with respect to a suitably chosen basic wavelet.
Date:
December 1993
Creator:
Navarro, Jaime
System:
The UNT Digital Library
Weak and Norm Convergence of Sequences in Banach Spaces
We study weak convergence of sequences in Banach spaces. In particular, we compare the notions of weak and norm convergence. Although these modes of convergence usually differ, we show that in βΒΉ they coincide. We then show a theorem of Rosenthal's which states that if {πβ} is a bounded sequence in a Banach space, then {πβ} has a subsequence {π'β} satisfying one of the following two mutually exclusive alternatives; (i) {π'β} is weakly Cauchy, or (ii) {π'β} is equivalent to the unit vector basis of βΒΉ.
Date:
December 1993
Creator:
Hymel, Arthur J. (Arthur Joseph)
System:
The UNT Digital Library
Applications of Rapidly Mixing Markov Chains to Problems in Graph Theory
In this dissertation the results of Jerrum and Sinclair on the conductance of Markov chains are used to prove that almost all generalized Steinhaus graphs are rapidly mixing and an algorithm for the uniform generation of 2 - (4k + 1,4,1) cyclic Mendelsohn designs is developed.
Date:
August 1993
Creator:
Simmons, Dayton C. (Dayton Cooper)
System:
The UNT Digital Library
Property (H*) and Differentiability in Banach Spaces
A continuous convex function on an open interval of the real line is differentiable everywhere except on a countable subset of its domain. There has been interest in the problem of characterizing those Banach spaces where the continuous functions exhibit similar differentiability properties. In this paper we show that if a Banach space E has property (H*) and B_Eβ’ is weak* sequentially compact, then E is an Asplund space. In the case where the space is weakly compactly generated, it is shown that property (H*) is equivalent for the space to admit an equivalent Frechet differentiable norm. Moreover, we define the SH* spaces, show that every SH* space is an Asplund space, and show that every weakly sequentially complete SH* space is reflexive. Also, we study the relation between property (H*) and the asymptotic norming property (ANP). By a slight modification of the ANP we define the ANP*, and show that if the dual of a Banach spaces has the ANP*-I then the space admits an equivalent FrΓ©chet differentiability norm, and that the ANP*-II is equivalent to the space having property (H*) and the closed unit ball of the dual is weak* sequentially compact. Also, we show that in the β¦
Date:
August 1993
Creator:
Obeid, Ossama A.
System:
The UNT Digital Library
Ο-regular Rings
The dissertation focuses on the structure of Ο-regular (regular) rings.
Date:
May 1993
Creator:
Badawi, Ayman R.
System:
The UNT Digital Library
Universal Branched Coverings
In this paper, the study of k-fold branched coverings for which the branch set is a stratified set is considered. First of all, the existence of universal k-fold branched coverings over CW-complexes with stratified branch set is proved using Brown's Representability Theorem. Next, an explicit construction of universal k-fold branched coverings over manifolds is given. Finally, some homotopy and homology groups are computed for some specific examples of Universal k-fold branched coverings.
Date:
May 1993
Creator:
Tejada, DΓ©bora
System:
The UNT Digital Library