A New Polyconvolution Operator with the Weight Function Related to the Hartley Integral Transforms $$H_2,H_1$$ and Applications

In this paper the integral transformation of a random process is analyzed with a sufficiently General formulation of the problem. Similar problems were studied by the author earlier in relation to random functions in the electric power industry. In principle, the problem is to analyze the changes in the correlation function under the influence of the process of the integral filter with the weight function, the “inverse problem” is considered - the output signal is assumed to be known, the parameters of the input signal to be determined. The idealization used in this case consists in the assumption of the stationary of the initial process, as well as the difference structure of the weight function. The paper continues the mathematical description of the action of the linear integral operator on a random function described in the book [6], §7.

The paper proposes a matrix implementation of the collocation method for constructing a solution to Volterra integral equations of the second kind using systems of orthogonal Chebyshev polynomials of the first kind and Legendre polynomials. The integrand in the equations considered in this work is represented as a partial sum of a series for these polynomials. The roots of the Chebyshev and Legendre polynomials are chosen as collocation points. Using matrix and integral transformations, properties of finite sums of products of these polynomials and weight functions at the zeros of the corresponding polynomials with degree equal to the number of nodes, integral equations are reduced to systems of linear algebraic equations for unknown values of the sought functions at these points. As a result, solutions to Volterra integral equations of the second kind are found by polynomial interpolations of the obtained function values at collocation points using inverse matrices, the elements of which are written on the basis of orthogonal relations for these polynomials. In the presented work, the elements of integral matrices are also given in explicit form. Error estimates for the constructed solutions with respect to the infinite norm are obtained. The results of computational experiments are presented, which demonstrate the effectiveness of the collocation method used.

With the aid of the Laplace transform, the canonical expression of the second-order many-body perturbation correction to an electronic energy is converted into the sum of two 13-dimensional integrals, the 12-dimensional parts of which are evaluated by Monte Carlo integration. Weight functions are identified that are analytically normalizable, are finite and non-negative everywhere, and share the same singularities as the integrands. They thus generate appropriate distributions of four-electron walkers via the Metropolis algorithm, yielding correlation energies of small molecules within a few mE(h) of the correct values after 10(8) Monte Carlo steps. This algorithm does away with the integral transformation as the hotspot of the usual algorithms, has a far superior size dependence of cost, does not suffer from the sign problem of some quantum Monte Carlo methods, and potentially easily parallelizable and extensible to other more complex electron-correlation theories.

Registration is a classical problem in the application of remote sensing images. The existing methods prefer to fit the relationship between target and source images with the same model on the whole. In fact, the geometrical relationship between two images is not always consistent, especially for the wide-field-viewed images of GaoFen-1 (GF-1) launched by the China Aerospace Science and Technology Corporation (CASC) in April 2013. Generally, The existing methods didn't take the local deformation into consideration. Towards this end, we solve the problem with three stages in this paper. Firstly, the coarse registration obtains the integral perspective transformation model of images. Secondly, the fine registration partitions image into many blocks and improves the relationship of every block with the inverse distance weighting (IDW) function. Finally, the coordinate transformation and resampling are the final step. Compared to other methods, the experiments demonstrate that the proposed algorithm is capable of generating satisfied results which are robust against deformation at local area.

The polyconvolution with the weight functionγof three functionsf,g, andhfor the integral transforms Fourier sine(Fs), Fourier cosine(Fc), and Kontorovich-Lebedev(Kiy), which is denoted by∗γ(f,g,h)(x), has been constructed. This polyconvolution satisfies the following factorization propertyFc(∗γ(f,g,h))(y)=sin y(Fsf)(y)⋅(Fcg)(y)⋅(Kiyh)(y), for ally>0. The relation of this polyconvolution to the Fourier convolution and the Fourier cosine convolution has been obtained. Also, the relations between the polyconvolution product and others convolution product have been established. In application, we consider a class of integral equations with Toeplitz plus Hankel kernel whose solution in closed form can be obtained with the help of the new polyconvolution. An application on solving systems of integral equations is also obtained.

With the increasing research in the field of contact mechanics, different types of contact models have been investigated by many researchers by employing various complex material models. To ascertain the orthotropy effect and modeling parameters on a receding contact model, the double frictional receding contact problem for an orthotropic bilayer loaded by a cylindrical punch is taken into account in this study. Assuming plane strain sliding conditions, the governing equations are found analytically using Fourier integral transformation technique. Then, the resulting singular integral equations are solved numerically using an iterative method. The weight function describing the asymptotic behavior of the stresses are investigated in detail and powers of the stress singularities are provided. To control the trustworthiness and correctness of the analytical formulation and to compare the resulting stress distributions and contact boundaries, a numerically efficient finite element method was employed using augmented Lagrange contact algorithm. The aim of this paper is to investigate the orthotropy effect, modeling parameters and coefficients of friction on the surface and interface stresses, surface and interface contact boundaries, powers of stress singularities, weight function and to provide highly parametric benchmark results for tribological community in designing wear resistant systems.

Characteristics of space- and surface-wave fields produced by an electromagnetic source in a multilayered structure are explored. Using the integral transformation technique it is shown that the space and surface-wave modes are orthogonal along the longitudinal direction with respect to an appropriate weighting function. It is demonstrated that these properties, together with reciprocity, can be utilized to determine the amplitudes of various surface-wave modes produced by an arbitrarily shaped source. Numerical results for the space- and surface-wave power for a circular patch antenna are presented. The study may find application for millimeter-wave printed antennas where the surface waves will play an important role in determining the radiation and impedance characteristics.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The half-range orthogonality property for neutron transport eigenfunctions leads to the definition of two weight functions. Each of these yields a set of orthonormal polynomials and related nodes and quadrature coefficients. The polynomials of different set, same order, are here shown to be linked by singular integral transformations. As a consequence, they obey almost identical recursion formulae. The transformations also generate cross-set relations involving nodes and quadrature coefficients of different set and equal order. Calculation procedures are described and examples of numerical tables for nodes and quadrature coefficients presented. Appendices deal with the reduction to the fullrange case and the limit behaviour of nonasymptotic quadrature sums.

The purpose of this article is to analyse the influence of a the generalized thermal transport on the free convection flow of a viscous nanofluid inside a cylindrical tube. The generalized Atangana-Baleanu time-fractional differential operator is used to generalized the heat transport equation. The effect of transverse magnetic field is also considered. The close form solutions are found by use of integral transformations, more precisely Laplace and finite Hankel transformations. The graphs depict the influence of fractional and other physical characteristics. Tables illustrate the effect of the volume fraction parameter. It has found that the heat flux given by the fractional equation influence heat transfer because the values of the temperature gradient are attenuated by the weight function in the heat flux expression. The temperature in the nanofluid has a different evolution concerning the fractional parameter.

A new family of bivariate densities with specified marginal densities is described. The bivariate dependence structure takes the form of a density weighting function constructed as follows. The space of continuous variates X and Y is mapped into a unit square through probability integral transformations. A polygonal partition of the unit square supports a bivariate covariance characteristic, which is structured as a functional of a univariate regression characteristic. This structure (a) exhibits all geometric features of positive (or negative) dependence implied by Fréchet bounds, which it can attain, and (b) prescribes positive (or negative) mutual regression dependence between X and Y. To obtain a parametric family of bivariate densities, the regression characteristic is given a power form. The resultant two-parameter model has specified marginals and, within certain constraints, allows one to separately control the shape of the bivariate density and the degree of association between X and Y; their Spearman's correlation coefficient p can attain any value: −1 < p < 1.

The main difficulty in using modified Clenshaw-Curtis integration for computing Singular and oscillatory integrals is the computation of the modified moments. In this paper we give recurrence formulae for computing modified moments for a number of important weight functions.