## You limited your search to:

**Partner:**UNT Libraries

**Department:**Department of Mathematics

**Degree Level:**Doctoral

### Applications of Graph Theory and Topology to Combinatorial Designs

**Date:**December 1988

**Creator:**Somporn Sutinuntopas

**Description:**This dissertation is concerned with the existence and the isomorphism of designs. The first part studies the existence of designs. Chapter I shows how to obtain a design from a difference family. Chapters II to IV study the existence of an affine 3-(p^m,4,λ) design where the v-set is the Galois field GF(p^m). Associated to each prime p, this paper constructs a graph. If the graph has a 1-factor, then a difference family and hence an affine design exists. The question arises of how to determine when the graph has a 1-factor. It is not hard to see that the graph is connected and of even order. Tutte's theorem shows that if the graph is 2-connected and regular of degree three, then the graph has a 1-factor. By using the concept of quadratic reciprocity, this paper shows that if p Ξ 53 or 77 (mod 120), the graph is almost regular of degree three, i.e., every vertex has degree three, except two vertices each have degree tow. Adding an extra edge joining the two vertices with degree tow gives a regular graph of degree three. Also, Tutte proved that if A is an edge of the graph satisfying the above conditions, ...

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc331968/

### Hausdorff, Packing and Capacity Dimensions

**Date:**August 1989

**Creator:**Spear, Donald W.

**Description:**In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation. A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that ...

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc330990/

### The Maximum Size of Combinatorial Geometries Excluding Wheels and Whirls as Minors

**Date:**August 1989

**Creator:**Hipp, James W. (James William), 1956-

**Description:**We show that the maximum size of a geometry of rank n excluding the (q + 2)-point line, the 3-wheel W_3, and the 3-whirl W^3 as minor is (n - 1)q + 1, and geometries of maximum size are parallel connections of (q + 1)-point lines. We show that the maximum size of a geometry of rank n excluding the 5-point line, the 4-wheel W_4, and the 4-whirl W^4 as minors is 6n - 5, for n ≥ 3. Examples of geometries having rank n and size 6n - 5 include parallel connections of the geometries V_19 and PG(2,3).

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc330849/

### Polynomial Isomorphisms of Cayley Objects Over a Finite Field

**Date:**December 1989

**Creator:**Park, Hong Goo

**Description:**In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc331144/

### Aspects of Universality in Function Iteration

**Date:**December 1991

**Creator:**Taylor, John (John Allen)

**Description:**This work deals with some aspects of universal topological and metric dynamic behavior of iterated maps of the interval.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc278799/

### Characterizations of Some Combinatorial Geometries

**Date:**August 1992

**Creator:**Yoon, Young-jin

**Description:**We give several characterizations of partition lattices and projective geometries. Most of these characterizations use characteristic polynomials. A geometry is non—splitting if it cannot be expressed as the union of two of its proper flats. A geometry G is upper homogeneous if for all k, k = 1, 2, ... , r(G), and for every pair x, y of flats of rank k, the contraction G/x is isomorphic to the contraction G/y. Given a signed graph, we define a corresponding signed—graphic geometry. We give a characterization of supersolvable signed graphs. Finally, we give the following characterization of non—splitting supersolvable signed-graphic geometries : If a non-splitting supersolvable ternary geometry does not contain the Reid geometry as a subgeometry, then it is signed—graphic.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc277894/

### π-regular Rings

**Date:**May 1993

**Creator:**Badawi, Ayman R.

**Description:**The dissertation focuses on the structure of π-regular (regular) rings.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc279388/

### Universal Branched Coverings

**Date:**May 1993

**Creator:**Tejada, Débora

**Description:**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.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc279340/

### Applications of Rapidly Mixing Markov Chains to Problems in Graph Theory

**Date:**August 1993

**Creator:**Simmons, Dayton C. (Dayton Cooper)

**Description:**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.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc277740/

### Property (H*) and Differentiability in Banach Spaces

**Date:**August 1993

**Creator:**Obeid, Ossama A.

**Description:**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_E• 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 Fréchet 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 ...

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc277852/