On the Turns of Digital Images

The images built on the basis of rectangular and hexagonal lattices are discussed in the article. For images on a rectangular lattice a formula is proposed, which gives approximate values of the components of a characteristic set of coefficients when turning at an arbitrary angle by the method of the nearest neighbor. The characteristic sets are presented in the form of diagrams, an experimental evaluation of errors is made. It was confirmed a good agreement with the predicted value component of characteristic sets and those which were obtained experimentally. For images built on the basis of a hexagonal lattice was offered a similar formula for the approximation of the components of the characteristic set for rotating at any angle, when this was applied to the modification of the nearest neighbor method for the preservation of coherence, as it was discovered its violation in some cases on a hexagonal lattice. On the basis of four-pixel fragments are built diagrams, which show a good agreement of predicted values and the obtained ones in the experiment. It was defined a system of three-pixel hexagonal fragments to which the theorem is proved on the Eulerian characteristic and were offered analytical expressions, which allow to avoid experimental detection of the characteristic sets of coefficients for all possible reference angles. Their use requires to produce only one such experiment.

Simulation of Transient Liquid Pipeline Flow Using the Deformable Characteristic Grid

This study describes the numerical solution and computational implementation of the isothermal transient flow of several different compressible liquids contained in a single elastic pipeline. The method of characteristics is used to integrate the equations of conservation of mass and momentum along the deforming Eulerian characteristic mesh. Integration along the deforming mesh, which is generated as integration proceeds in time, minimizes the need for interpolation so that problem variables are accurately determined at the computed mesh points. On the other hand, since calculated mesh points are not regularly spaced in the x-t plane, interpolations among the computed points become necessary to find the value of variables at a given fixed instant of time along the length of the pipeline. The computational implementation allows for the simulation of flow across locations where there may be changes in nominal pipe diameter, valves or pumps, as well as tracking the position of fluid interfaces as different batches of fluid move along the pipeline, and results are found to be numerically accurate and stable.

A note on some networks of polygons

Given an infinity of polygons which form the boundary of a finite number of polyhedra, we shall consider the complex K consisting of the polyhedra, and of the faces, edges and vertices of the polygons. We consider only those cases in which the Eulerian Characteristic N of K is finite. Then if the mean number of sides meeting at a vertex is p, and the mean number of sides of a polygon is q, then