Subset Of N-space Research Articles

A representation of convex semilinear sets

If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of translates of closed halfspaces, where a closed halfspace is the set of all points obeying a homogeneous weak linear inequality with coefficients from F. Andradas, Rubio, and Velez have shown that closed (open) convex semilinear sets are finite intersections of translates of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This paper represents arbitrary convex semilinear sets in a manner analogous to that of Andradas, Rubio, and Velez.

Visualization and data mining of high-dimensional data

Visualization provides insight through images and can be considered as a collection of application specific mappings: ProblemDomain→VisualRange. For the visualization of multivariate problems a multidimensional system of parallel coordinates (abbreviated as ∥-coords) is constructed which induces a one-to-one mapping between subsets of N-space and subsets of 2-space. The result is a rigorous methodology for doing and seeing N-dimensional geometry. Starting with an the overview of the mathematical foundations, it is seen that the display of high-dimensional datasets and search for multivariate relations among the variables is transformed into a 2-D pattern recognition problem. This is the basis for the application to Visual Data Mining which is illustrated with real dataset of Very Large Scale Integration (VLSI—“chip”) production. Then a recent geometric classifier is presented and applied to three real datasets. The results compared to those of 23 other classifiers have the least error. The algorithm has quadratic computational complexity in the size and number of parameters, provides comprehensible and explicit rules, does dimensionality selection—where the minimal set of original variables required to state the rule is found—and orders these variables so as to optimize the clarity of separation between the designated set and its complement. Finally, a simple visual economic model of a real country is constructed and analyzed in order to illustrate the special strength of ∥-coords in modeling multivariate relations by means of hypersurfaces.

Uniform local strong uniqueness on finite subsets

LetF be an approximating function with parameter setP, a subset ofn-space, on an intervalI satisfying the hypotheses of Meinardus and Schwedt (the local Haar condition and propertyZ) which result in an alternating theory. Consider uniform approximation to functionf onX, a finite subset ofI. A sufficient condition is given, involving best parameters being in a closed set, on which the degree isn that a family of functionsf have a uniform (parameterwise) local strong uniqueness constant > 0. The necessity of the condition in this and related problems, in particular ordinary rational approximation on an interval, is examined.