A pursuit-evasion differential game with slow pursuers on the edge graph of simplexes. I

We consider the differential game between several pursuing points and one evading point moving along the graph of edges of a simplex when maximal quantities of velocities are given. The normalization of the game in the sense of J. von Neumann including the description of classes of admissible strategies is exposed. In the present part of the paper the qualitative problem for the full graph of three dimensional simplex is solved using the strategy of parallel pursuit for a slower pursuer and some numerical coefficient of a simplex characterizing its proximity to the regular one. Next part will be devoted to higher dimensional cases.

Diagnostic Simplexes for Dissolved-Gas Analysis

A Duval triangle is a diagram used for fault type identification in dissolved-gas analysis of oil-filled high-voltage transformers and other electrical apparatus. The proportional concentrations of three fault gases (such as methane, ethylene, and acetylene) are used as coordinates to plot a point in an equilateral triangle and identify the fault zone in which it is located. Each point in the triangle corresponds to a unique combination of gas proportions. Diagnostic pentagons published by Duval and others seek to emulate the triangles while incorporating five fault gases instead of three. Unfortunately the mapping of five gas proportions to a point inside a two-dimensional pentagon is many-to-one; consequently, dissimilar combinations of gas proportions are mapped to the same point in the pentagon, resulting in mis-diagnosis. One solution is to replace the pentagon with a four-dimensional simplex, a direct generalization of the Duval triangle. In a comparison using cases confirmed by inspection, the simplex outperformed three ratio methods, Duval triangle 1, and two pentagons.

Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

On Some Numerical Integration Formulas on the d-Dimensional Simplex

In this paper, we consider the problem of the approximation of the integral of a function f over a d-dimensional simplex S of $$\mathbb {R}^{d}$$ R d by some quadrature formulas which use only the functional and derivative values of f on the boundary of the simplex S or function data at the vertices of S, at points on its facets and at its center of gravity. The quadrature formulas are computed by integrating over S a polynomial approximant of f which uses functional and derivative values at the vertices of S.

The Pitman–Yor multinomial process for mixture modelling

Summary Discrete nonparametric priors play a central role in a variety of Bayesian procedures, most notably when used to model latent features, such as in clustering, mixtures and curve fitting. They are effective and well-developed tools, though their infinite dimensionality is unsuited to some applications. If one restricts to a finite-dimensional simplex, very little is known beyond the traditional Dirichlet multinomial process, which is mainly motivated by conjugacy. This paper introduces an alternative based on the Pitman–Yor process, which provides greater flexibility while preserving analytical tractability. Urn schemes and posterior characterizations are obtained in closed form, leading to exact sampling methods. In addition, the proposed approach can be used to accurately approximate the infinite-dimensional Pitman–Yor process, yielding improvements over existing truncation-based approaches. An application to convex mixture regression for quantitative risk assessment illustrates the theoretical results and compares our approach with existing methods.

EDSQ Operator on 2DS and Limit Behavior

This paper evaluates the limit behavior for symmetry interactions networks of set points for nonlinear mathematical models. Nonlinear mathematical models are being increasingly applied to most software and engineering machines. That is because the nonlinear mathematical models have proven to be more efficient in processing and producing results. The greatest challenge facing researchers is to build a new nonlinear model that can be applied to different applications. Quadratic stochastic operators (QSO) constitute such a model that has become the focus of interest and is expected to be applicable in many biological and technical applications. In fact, several QSO classes have been investigated based on certain conditions that can also be applied in other applications such as the Extreme Doubly Stochastic Quadratic Operator (EDSQO). This paper studies the behavior limitations of the existing 222 EDSQ operators on two-dimensional simplex (2DS). The created simulation graph shows the limit behavior for each operator. This limit behavior on 2DS can be classified into convergent, periodic, and fixed.

Fast Computation of Polynomial Data Points Over Simplicial Face Values

Polynomial functions [Formula: see text] of degree [Formula: see text] have a form in the Bernstein basis defined over [Formula: see text]-dimensional simplex [Formula: see text]. The Bernstein coefficients exhibit a number of special properties. The function [Formula: see text] can be optimised by the smallest and largest Bernstein coefficients (enclosure bounds) over [Formula: see text]. By a proper choice of barycentric subdivision steps of [Formula: see text], we prove the inclusion property of Bernstein enclosure bounds. To this end, we provide an algorithm that computes the Bernstein coefficients over subsimplices. These coefficients are collected in an [Formula: see text]-dimensional array in the field of computer-aided geometric design. Such a construct is typically classified as a patch. We show that the Bernstein coefficients of [Formula: see text] over the faces of a simplex coincide with the coefficients contained in the patch.