Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Vita. Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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Distance geometry problems (DGPs) are concerned with the construction of structures given partial information about distances between vertices. I present a generalisation which I call the angle and distance geometry problem (ADGP), in which partial angle information may be given as well. The work is primarily concerned with the algebraic and theoretical aspects of this problem, although it contains some information on practical applications. The embedding space is typically real three dimensional space for applications such as computer aided design and molecular chemistry, although other embedding spaces are possible. I show that both DGP and ADGP are NP-hard, but that in some sense the ADGP is more expressive than the DGP. To combat the problems of NP-hardness I present some graph theoretic heuristics which may be applied to both DGP and ADGP, and so reduce the time required by general purpose algorithms for their solution. I discuss the general purpose algorithms Cylindrical Algebraic Decomposition and Gröbner bases and their application to this field. In addition, I present an O(n) parallel algorithm for computing convex hulls in three dimensions, using O(n²) processors connected in a mesh-like topology with no shared memory.

The classical Steiner Problem may be stated: Given n points [formula omitted] in the Euclidean plane, to construct the shortest tree(s) (i.e. undirected, connected, circuit free graph(s)) whose vertices include [formula omitted]. The problem is generalised by considering sets in a metric space rather than points in E² and also by minimising a more general graph function than length, thus yielding a large class of network minimisation problems which have a wide variety of practical applications, The thesis is concerned with the following aspects of these problems. 1. Existence and uniqueness or multiplicity of solutions. 2. The structure of solutions and demonstration that minimising trees of various problems share common properties. 3. Solvability of problems by Euclidean constructions or by other geometrical methods. / Science, Faculty of / Mathematics, Department of / Graduate

Ng, Kwong Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 123-126). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.iii / Contents --- p.v / Introduction --- p.vii / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Basic Notations and Conventions --- p.1 / Chapter 1.2 --- Fundamental Results in Banach Space Theory and Variational Analysis --- p.4 / Chapter 1.3 --- Set-Valued Mappings --- p.6 / Chapter 1.4 --- Enlargements and Projections --- p.8 / Chapter 1.5 --- Subdifferentials --- p.11 / Chapter 1.6 --- Sets of Normals --- p.18 / Chapter 1.7 --- Coderivatives --- p.24 / Chapter 2 --- The Generalized Distance Function - Basic Estimates --- p.27 / Chapter 2.1 --- Elementary Properties of the Generalized Distance Function --- p.27 / Chapter 2.2 --- Frechet-Like Subdifferentials of the Generalized Distance Function --- p.32 / Chapter 2.3 --- Limiting and Singular Subdifferentials of the Generalized Distance - Function --- p.44 / Chapter 3 --- The Generalized Distance Function - Estimates via Intermediate Points --- p.73 / Chapter 3.1 --- Frechet-Like and Limiting Subdifferentials of the Generalized Dis- tance Function via Intermediate Points --- p.74 / Chapter 3.2 --- Frechet and Proximal Subdifferentials of the Generalized Distance Function via Intermediate Points --- p.90 / Chapter 4 --- The Marginal Function --- p.95 / Chapter 4.1 --- Singular Subdifferentials of the Marginal Function --- p.95 / Chapter 4.2 --- Singular Subdifferentials of the Generalized Marginal Function . . --- p.102 / Chapter 5 --- The Perturbed Distance Function --- p.107 / Chapter 5.1 --- Elementary Properties of the Perturbed Distance Function --- p.107 / Chapter 5.2 --- The Convex Case - Subdifferentials of the Perturbed Distance Function --- p.111 / Chapter 5.3 --- The Nonconvex Case - Frechet-Like and Proximal Subdifferentials of the Perturbed Distance Function --- p.113 / Bibliography --- p.123

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Let $A \subseteq \Z_n$ with $|A| = k$ for some $k \in \Z^+$. We consider the metric space $(\Z_n,\delta)$ in which $\delta$ is the distance metric on $\Z_n$ defined as follows: for every $x,y \in \Z_n$, $\delta(x,y) = |x-y|_n$ where $|z|_n = \min(z,n-z)$ for $z \in \{0,\ldots,n-1\}$. We say that $A$ is \emph{Erd\H{o}s-deep} if, for every $i \in \{1,2,\dots,k-1\}$, there is a positive number $d_i$ satisfying $$|\{\{x,y\} \subseteq A: \delta(x,y)=d_i\}|=i.$$ Erd\H{o}s-deep sets in $\Z_n$ have been previously classified as translates of: $\{0,1,2,4\}$ when $n = 6$; and, modular arithmetic progressions $\{0,g,2g,\cdots,(k-1)g\} \subseteq \Z_n$ for some generator $g$ and size $k$. Erd\H{o}s-deep sets have primarily been considered in metric spaces $(\Z_n,\delta)$ and $(\R^d,\norm{\cdot})$ for $d = 2$, but some exploration for $d > 2$ has been done as well. We introduce the notion of an \emph{Erd\H{o}s-deep family}. Let $\mathcal{F}=\{A_1,A_2,\dots,A_s\}$, where $A_1,\ldots, A_s \subseteq \Z_n$. Then we say $\mathcal{F}$ is Erd\H{o}s-deep if for some $k \in \Z^+$, for every $i \in \{1,2,\dots,k-1\}$ there is exactly one positive number $d_i$ satisfying $$\sum_{j=1}^s |\{\{x,y\} \subseteq A_j: \delta(x,y)=d_i\}|=i,$$ and no such $d_i$ for any $i \ge k$. We provide a complete existence theorem for Erd\H{o}s-deep pairs of arithmetic progressions $A_1,A_2 \subseteq \Z_n$ and also give a conjectured classification for Erd\H{o}s-deep families of three arithmetic progressions. Using an identity on triangular numbers, we show a general construction for larger families whose size $s$ is the square of an integer. This construction suggests the existence of Erd\H{o}s-deep families often relies on such number-theoretic identities. We define an extremal case of the Erd\H{o}s-deep family in $(\Z_n,\delta)$ in which both the distances and multiplicities are in $\{1,\ldots,k-1\}$; such families are called Winograd families. We conjecture that Winograd families of arithmetic progressions do not exist in the metric space $(\Z,|\cdot|)$. Erd\H{o}s-deep sets in $(\Z_n,\delta)$ correspond to a class of interesting musical rhythms. We conclude this work with a variety of musical demonstrations and original compositions using Erd\H{o}s-deep rhythm families as a creative constraint in composing multi-voiced rhythms. / Graduate