Optimization is the science of making a best choice in the face of conflicting requirements. Any convex optimization problem has geometric interpretation. If a given optimization problem can be transformed to a convex equivalent, then this interpretive benefit is acquired. That is a powerful attraction: the ability to visualize geometry of an optimization problem. Conversely, recent advances in geometry hold convex optimization within their proofs' core. This book is about convex optimization, convex geometry (with particular attention to distance geometry), geometrical problems, and problems that can be transformed into geometrical problems. Euclidean distance geometry is, fundamentally, a determination of point conformation from interpoint distance information; e.g., given only distance information, determine whether there corresponds a realizable configuration of points; a list of points in some dimension that attains the given interpoint distances. large black & white paperback

Convex Optimization & Euclidean Distance Geometry

Theory of cones

The notion of contra-continuity was introduced and investigated by Dontchev (3). In this paper, we introduce and investigate a new generalization of contra-continuity called contra-precontinuity.

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On Contra-Precontinuous Functions

We will continue the study of p-closed spaces. This class of spaces is strictly placed between the class of strongly compact spaces and the class of quasi-H-closed spaces. We will provide new characterizations of p-closed spaces and investigate their relationships with some other classes of topological spaces.

ON p-CLOSED SPACES

In 1990, Ganster and Reilly proved that a function is continuous if and only if it is precontinuous and LC-continuous. In this paper we extend their decomposition of continuity in terms of ideals. We show that a function $f \colon (X,\tau,{\cal I}) \to (Y,\sigma)$ is continuous if and only if it is pre-I-continuous and I-LC-continuous. We also provide a decomposition of I-continuity.

Idealization of Ganster-Reilly decomposition theorems

A space is said to be resolvable if it has two disjoint dense subsets. It is shown thatX is a Baire space with no resolvable open subsets iff every real function defined onX has a dense set of points of continuity. Thus almost resolvable spaces, as defined by Bolstein, are shown to be characterized as the union of a first category set and a closed resolvable set.

A characterization of almost resolvable spaces

Self-dual cones in euclidean spaces

An $s$-set in Euclidean space is a set of finite, non-zero, Hausdorff $s$-dimensional measure. Call an $s$-set straight if its $s$-measure agrees with its Method I $s$-outer measure. Examples are given where there is a continuous, one-to-one function $f$ on $\mathbb{R}^n$ which is measure preserving on $ E$ so that $f(E)$ is straight (such an $f$ will be called a straightening of $E$). It is shown that any $s$-set can be written as a countable union of sets for which there are straightenings.

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Measure preserving continuous straightening of fractional dimensional sets

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The Length of the Graph of a One to One Function from [0,1] to [0,1]

(1981). On Functions Whose Inverse Is Their Reciprocal. Mathematics Magazine: Vol. 54, No. 4, pp. 185-189.

James Foran

Papers

A characterization of almost resolvable spaces

Self-dual cones in euclidean spaces

Measure preserving continuous straightening of fractional dimensional sets

The Length of the Graph of a One to One Function from [0,1] to [0,1]

On Functions Whose Inverse Is Their Reciprocal