A Numerical Method for Computing Double Integrals with Variable Upper Limits

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

The Schott Energy and the Reactive Energy in Electromagnetic Radiation and Mutual Couplings

This is revised version. I have corrected a miss in the derivation and added a figure to show the integration region.

On a method for approximate solution of a mixed boundary value problem for an elliptic equation

A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.

Association betweenembBCodon 306 Mutations, Phenotypic Resistance Profiles and Genotypic Characterization in ClinicalMycobacterium tuberculosisIsolates from Hebei, China

Ethambutol (EMB) is an essential first-line drug for tuberculosis (TB) treatment. Nucleotide substitutions atembBcodon 306 have been proposed as a potential marker for EMB resistance and a predictor for broad drug resistance in clinicalMycobacterium tuberculosis(MTB) isolates. However, discordant findings about the association betweenembB306 mutation and EMB resistance were reported. The Hebei province is located in the Beijing-Tianjin-Hebei integration region; however, little information about the genetic diversity ofembBlocus is available in this area. In this study, we sequenced the region surroundingembB306 codon (codon 207-445) in 62 ethambutol-resistant (EMBr) isolates, 214 ethambutol-susceptible isolates but with resistance to other first-line drugs (EMBs) and 100 pan sensitive isolates. Our data indicated that none of pan sensitive isolates showed mutations atembB306 and 63 drug-resistant isolates harboredembB306 substitutions, with 56.5% (35/62) in EMBrisolates and 13.1% (28/214) in EMBsisolates. Significant association was observed between theembB306 mutation and resistance to INH, RIF, EMB and MDR, and the mutation rates ofembB306 increased with increasing numbers of resistance to first-line drugs. TheembB306 mutation is not the sole causative factor for EMB resistance and the poor sensitivity limits its utility as a marker for DR-TB. However, it may be a potential marker for broad resistance, especially for MDR. The MIRU-VNTR profiles may serve as a potential marker for predicting the possibleembB306 substitutions that may occur in DR-TB isolates under antimicrobial selection pressure.

A resolution of the integration region problem for the supermoduli space integral

The integration region of the supermoduli space integral is defined in the super-Schottky group parametrization. The conditions on the super-period matrix elements are translated to relations on the parameters. An estimate of the superstring amplitude at arbitrary genus is sufficient for an evaluation of the cross-section to all orders in the expansion of the scattering matrix.

Research on Integrals of the Second Category Curved Surface

In this thesis,surface integrals and the relationship between characteristics of the second class and second class double integral surface integrals, analyzes the application of the law of symmetry integral calculation, even if the integration region derived symmetric integrand is an odd function integral value characteristic is not necessarily zero. And pointed out that the feasibility of the method is definitely second class surface integrals.