Probing the arrangement of hyperplanes

AbstractIn this paper we investigate the combinatorial complexity of an algorithm to determine the geometry and the topology related to an arrangement of hyperplanes in multi-dimensional Euclidean space from the “probing” on the arrangement. The “probing” by a flat means the operation from which we can obtain the intersection of the flat and the arrangement. For a finite set H of hyperplanes in Ed, we obtain the worst-case number of fixed direction line probes and that of flat probes to determine a generic line of H and H itself. We also mention the bound for the computational complexity of these algorithms based on the efficient line probing algorithm which uses the dual transform to compute a generic line of H.We also consider the problem to approximate arrangements by extending the point probing model, which have connections with computational learning theory such as learning a network of threshold functions, and introduce the vertical probing model and the level probing model. It is shown that the former is closely related to the finger probing for a polyhedron and that the latter depends on the dual graph of the arrangement.The probing for an arrangement can be used to obtain the solution for a given system of algebraic equations by decomposing the μ-resultant into linear factors. It also has interesting applications in robotics such as a motion planning using an ultrasonic device that can detect the distances to obstacles along a specified direction

Analyzing Loose Contact Oxidation of Diesel Engine Soot and Ag/CeO2 Catalyst Using Nonlinear Regression Analysis

Loose contact (LC) oxidation kinetics of carbon black (CB) model soot and Ag/CeO2 catalyst was deduced from thermogravimetric analysis (TG) experiments. In the LC mode at ≧750 K, CB particles were likely to be combusted also by non-catalyzed oxidation, especially those particles located far from the catalyst surface, as well as Ag/CeO2-catalyed oxidation. Since the non-catalyzed oxidation is not due to catalytic activity, in order to deduce the catalytic activity from TG data, a nonlinear regression analysis method was proposed in this study to extract only the catalyzed oxidation part of the TG data. It was verified that this was successfully done with the equations used by the very good curve fits to the experimental TG data, and the catalytic activity was correctly obtained from LC samples with various degrees of physical contact between the CB and catalyst.

Ring extensions and endomorphism rings of a module

In this paper we consider two conditions of a module related with a ring extension. The one is (T)[?]5R<[?]M[?]･･･[?]M that is, R⊃S is a ring extension and M a right R-module such that ..