CONSTRUCTION OF SECTIONS OF POLYHEDRON METHOD OF TRACES

In this article it is proved that in solving problems on the construction of sections of polyhedra, students not only perform constructions, they also apply axioms, properties of planimetry and stereometry, but also learn algorithmic thinking, the ability to reason logically, make correct arguments and conclusions. It is established that the solution of problems on the construction of sections of polyhedra occupies a special place in the process of forming a spatial representation and in the development of mathematical thinking, both students and schoolchildren. Based on the definition of the trace of the secant plane, the rules for constructing sections of the polyhedron by the traces method are formulated. Problems on construction of sections of polyhedra are developed in the case when:: the section of the prism is given by the trace l , which is located on the plane of the base of the prism and does not have common points with the base of this prism and by point K , belonging to some side rib; the secant plane is defined by the trace l and some point M , belonging to the side rib of the pyramid; the section of the pyramid is determined by points M, N, K two of them are located on different ribs, and the third is the internal point of the face of this pyramid.

H∞Control for a Networked Control Model of Systems with Two Additive Time-Varying Delays

This paper is concerned withH∞control for a networked control model of systems with two additive time-varying delays. A new Lyapunov functional is constructed to make full use of the information of the delays, and for the derivative of the Lyapunov functional a novel technique is employed to compute a tighter upper bound, which is dependent on the two time-varying delays instead of the upper bounds of them. Then the convex polyhedron method is proposed to check the upper bound of the derivative of the Lyapunov functional. The resulting stability criteria have fewer matrix variables but less conservatism than some existing ones. The stability criteria are applied to designing a state feedback controller, which guarantees that the closed-loop system is asymptotically stable with a prescribedH∞disturbance attenuation level. Finally examples are given to show the advantages of the stability criteria and the effectiveness of the proposed control method.

Voronoi polyhedra probing of hydrated OH radical

Voronoi polyhedron method is employed to extract the smallest volume shared by ˙OH radical in liquid water at the biologically important temperature (37 °C). The 3D-visualization and the probability distributions of the metric and topological properties of ˙OH solvation cage are provided.