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  Partner: UNT Libraries
 Department: Department of Mathematics
 Degree Level: Doctoral
Automorphism Groups of Strong Bruhat Orders of Coxeter Groups

Automorphism Groups of Strong Bruhat Orders of Coxeter Groups

Date: August 1986
Creator: Sutherland, David C. (David Craig)
Description: In this dissertation, we describe the automorphism groups for the strong Bruhat orders A_n-1, B_n, and D_n. In particular, the automorphism group of A_n-1 for n ≥ 3 is isomorphic to the dihedral group of order eight, D_4; the automorphism group of B_n for n ≥ 3 is isomorphic to C_2 x C_2 where C_2 is the cyclic group of order two; the automorphism group of D_n for n > 5 and n even is isomorphic to C_2 x C_2 x C_2; and the automorphism group of D_n for n ≥ 5 and n odd is isomorphic to the dihedral group D_4.
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Containment Relations Between Classes of Regular Ideals in a Ring with Few Zero Divisors

Containment Relations Between Classes of Regular Ideals in a Ring with Few Zero Divisors

Date: May 1987
Creator: Race, Denise T. (Denise Tatsch)
Description: This dissertation focuses on the significance of containment relations between the above mentioned classes of ideals. The main problem considered in Chapter II is determining conditions which lead a ring to be a P-ring, D-ring, or AM-ring when every regular ideal is a P-ideal, D-ideal, or AM-ideal, respectively. We also consider containment relations between classes of regular ideals which guarantee that the ring is a quasi-valuation ring. We continue this study into the third chapter; in particular, we look at the conditions in a quasi-valuation ring which lead to a = Jr, sr - f, and a = v. Furthermore we give necessary and sufficient conditions that a ring be a discrete rank one quasi-valuation ring. For example, if R is Noetherian, then ft = J if and only if R is a discrete rank one quasi-valuation ring.
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Invertible Ideals and the Strong Two-Generator Property in Some Polynomial Subrings

Invertible Ideals and the Strong Two-Generator Property in Some Polynomial Subrings

Date: May 1987
Creator: Chapman, Scott T. (Scott Thomas)
Description: Let K be any field and Q be the rationals. Define K^1[X] = {f(X) e K[X]| the coefficient of X in f(X) is zero} and Q^1β[X] = {f(X) e Q[X]| the coefficent of β1(X) in the binomial expansion of f(X) is zero}, where {β1(X)}^∞ i=0 are the well-known binomial polynomials. In this work, I establish the following results: K^1[X] and Q^1β[X] are one-dimensional, Noetherian, non-Prüfer domains with the two-generator property on ideals. Using the unique factorization structure of the overrings K[X] and Q[X], the nonprincipal ideal structures of both rings are characterized, and from this characterization, necessary and sufficient conditions are found for a nonprincipal ideal to be invertible. The nonprincipal invertible ideals are then characterized in terms of the coefficients of the generators, and an explicit formula for the inverse of any proper invertible ideal is found. Finally, the class groups of both rings are shown to be torsion free abelian groups. Let n be any nonnegative integer. Results similar to the above are found in the generalizations of these two rings, K^n[X] and q^nβ[X], where the coefficients on the first n nonconstant basis elements are zero. For the domains K^1[X] and Q^1β[X], the property of strong two-generation is ...
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Radially Symmetric Solutions to a Superlinear Dirichlet Problem in a Ball

Radially Symmetric Solutions to a Superlinear Dirichlet Problem in a Ball

Date: August 1987
Creator: Kurepa, Alexandra
Description: In this paper we consider a radially symmetric nonlinear Dirichlet problem in a ball, where the nonlinearity is "superlinear" and "superlinear with jumping."
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Dually Semimodular Consistent Lattices

Dually Semimodular Consistent Lattices

Date: May 1988
Creator: Gragg, Karen E. (Karen Elizabeth)
Description: A lattice L is said to be dually semimodular if for all elements a and b in L, a ∨ b covers b implies that a covers a ∧ b. L is consistent if for every join-irreducible j and every element x in L, the element x ∨ j is a join-irreducible in the upper interval [x,l]. In this paper, finite dually semimodular consistent lattices are investigated. Examples of these lattices are the lattices of subnormal subgroups of a finite group. In 1954, R. P. Dilworth proved that in a finite modular lattice, the number of elements covering exactly k elements is equal to the number of elements covered by exactly k elements. Here, it is established that if a finite dually semimodular consistent lattice has the same number of join-irreducibles as meet-irreducibles, then it is modular. Hence, a converse of Dilworth's theorem, in the case when k equals 1, is obtained for finite dually semimodular consistent lattices. Several combinatorial results are shown for finite consistent lattices similar to those already established for finite geometric lattices. The reach of an element x in a lattice L is the difference between the rank of x*, the join of x and all ...
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Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension

Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension

Date: May 1988
Creator: Brucks, Karen M. (Karen Marie), 1957-
Description: This dissertation is a study of the dynamics of one-dimensional unimodal maps and is mainly concerned with those maps which are trapezoidal. The trapezoidal function, f_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-e,1]. We study the symbolic dynamics of the kneading sequences and relate them to the analytic dynamics of these maps. Chapter one is an overview of the present theory of Metropolis, Stein, and Stein (MSS). In Chapter two a formula is given that counts the number of MSS sequences of length n. Next, the number of distinct primitive colorings of n beads with two colors, as counted by Gilbert and Riordan, is shown to equal the number of MSS sequences of length n. An algorithm is given that produces a bisection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z)=z^2-2, as counted by P.J. Myrberg, is shown to equal the number of MSS sequences of length n. For an MSS sequence P, let H_ϖ(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one J(P)Σ[0,1] such that the ...
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Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem

Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem

Date: May 1988
Creator: Sumalee Unsurangsie
Description: In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.
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Minimization of a Nonlinear Elasticity Functional Using Steepest Descent

Minimization of a Nonlinear Elasticity Functional Using Steepest Descent

Date: August 1988
Creator: McCabe, Terence W. (Terence William)
Description: The method of steepest descent is used to minimize typical functionals from elasticity.
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Operators on Continuous Function Spaces and Weak Precompactness

Operators on Continuous Function Spaces and Weak Precompactness

Date: August 1988
Creator: Abbott, Catherine Ann
Description: If T:C(H,X)-->Y is a bounded linear operator then there exists a unique weakly regular finitely additive set function m:-->L(X,Y**) so that T(f) = ∫Hfdm. In this paper, bounded linear operators on C(H,X) are studied in terms the measure given by this representation theorem. The first chapter provides a brief history of representation theorems of these classes of operators. In the second chapter the represenation theorem used in the remainder of the paper is presented. If T is a weakly compact operator on C(H,X) with representing measure m, then m(A) is a weakly compact operator for every Borel set A. Furthermore, m is strongly bounded. Analogous statements may be made for many interesting classes of operators. In chapter III, two classes of operators, weakly precompact and QSP, are studied. Examples are provided to show that if T is weakly precompact (QSP) then m(A) need not be weakly precompact (QSP), for every Borel set A. In addition, it will be shown that weakly precompact and GSP operators need not have strongly bounded representing measures. Sufficient conditions are provided which guarantee that a weakly precompact (QSP) operator has weakly precompact (QSP) values. A sufficient condition for a weakly precomact operator to be strongly ...
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Applications of Graph Theory and Topology to Combinatorial Designs

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, ...
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