On exact solution of some extremal problems on classes of functions defined by majorant of integral modulus of continuity

The error of the interval quadrature rectangle formula on the class of periodic functions with convex majorant of the integral modulus of continuity of even derivative are obtained.

Partial Enclosure Range Searching

A new type of range searching problem, called the partial enclosure range searching problem, is introduced in this paper. Given a set of geometric objects [Formula: see text] and a query region [Formula: see text], our goal is to identify those objects in [Formula: see text] which intersect the query region [Formula: see text] by at least a fixed proportion of their original size. Two variations of this problem are studied. In the first variation, the objects in [Formula: see text] are axis-parallel line segments and the goal is to count the total number of members of [Formula: see text] so that their intersection with [Formula: see text] is at least a given proportion of their size. Here, [Formula: see text] can be an axis-parallel rectangle or a parallelogram of arbitrary orientation. In the second variation, [Formula: see text] is a polygon and [Formula: see text] is an axis-parallel rectangle. The problem is to report the area of the intersection between the polygon [Formula: see text] and a query rectangle [Formula: see text].

Difference Schemes for Three Kinds of Nonlinear Flow Model in Low Permeability Porous Media

This paper discusses three kinds of typical one-dimensional nonlinear equations coming from low permeability reservoir seepage models with different boundary conditions. The several finite difference methods including forward difference method and second central order difference quotient method are used for the respective discrete process of three models. With these difference methods, the discrete schemes of models are obtained. Then the corresponding nonlinear discrete equations are deduced. While dealing with the boundary condition, the mid-rectangle formula is used. Finally, integrated discrete equations of three nonlinear equations are formed. The results should be meaningful for the numerical simulation of non-Darcy flow model of the low-permeability oil wells.