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1 
Convex metrics and manifoldsRolfsen, Dale. January 1967 (has links)
Thesis (Ph. D.)University of WisconsinMadison, 1967. / Vita. Typescript. eContent providerneutral record in process. Description based on print version record. Includes bibliographical references.

2 
COMPLETE PROBABILISTIC METRIC SPACES AND RANDOM VARIABLE GENERATED SPACESSherwood, Howard, 1938 January 1966 (has links)
No description available.

3 
PROBABILISTIC METRICS AND PROBABILITY MEASURES ON METRICSStevens, Robert Ray, 1935 January 1965 (has links)
No description available.

4 
PRODUCTS AND QUOTIENTS OF PROBABILISTIC METRIC SPACESEgbert, Russell James, 1937 January 1966 (has links)
No description available.

5 
Generalizations of metric spacesBaxley, John Virgil 08 1900 (has links)
No description available.

6 
Angle and distance geometry problemsKay, Andrew January 1991 (has links)
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 NPhard, but that in some sense the ADGP is more expressive than the DGP. To combat the problems of NPhardness 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(n<sup>2</sup>) processors connected in a meshlike topology with no shared memory.

7 
On the steiner problemCockayne, Ernest January 1967 (has links)
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

8 
Subdifferentials of distance functions in Banach spaces.January 2010 (has links)
Ng, Kwong Wing. / Thesis (M.Phil.)Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 123126). / 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  SetValued 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  FrechetLike 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  FrechetLike 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  FrechetLike and Proximal Subdifferentials of the Perturbed Distance Function  p.113 / Bibliography  p.123

9 
TOPOLOGIES FOR PROBABILISTIC METRIC SPACESFritsche, Richard Thomas, 1936 January 1967 (has links)
No description available.

10 
ErdősDeep Families of Arithmetic ProgressionsGaede, Tao 30 August 2022 (has links)
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 $x,y \in \Z_n$, $\delta(x,y) = xy_n$ where $z_n = \min(z,nz)$ for $z \in \{0,\ldots,n1\}$. We say that $A$ is \emph{Erd\H{o}sdeep} if, for every $i \in \{1,2,\dots,k1\}$, there is a positive number $d_i$ satisfying
$$\{\{x,y\} \subseteq A: \delta(x,y)=d_i\}=i.$$
Erd\H{o}sdeep sets in $\Z_n$ have been previously classified as translates of: $\{0,1,2,4\}$ when $n = 6$; and, modular arithmetic progressions $\{0,g,2g,\cdots,(k1)g\} \subseteq \Z_n$ for some generator $g$ and size $k$.
Erd\H{o}sdeep 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}sdeep 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}sdeep if for some $k \in \Z^+$, for every $i \in \{1,2,\dots,k1\}$ 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}sdeep pairs of arithmetic progressions $A_1,A_2 \subseteq \Z_n$ and also give a conjectured classification for Erd\H{o}sdeep 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}sdeep families often relies on such numbertheoretic identities.
We define an extremal case of the Erd\H{o}sdeep family in $(\Z_n,\delta)$ in which both the distances and multiplicities are in $\{1,\ldots,k1\}$; 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}sdeep 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}sdeep rhythm families as a creative constraint in composing multivoiced rhythms. / Graduate

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