This paper is concerned with finding approximate solution for the singular integral equations. Relating the singular integrals to Cauchy principal-value integrals, we expand the kernel and the density function of singular integral equation by the sum of the chebyshev polynomials of the first, second, third and fourth kinds. Some numerical examples are presented to illustrate the accuracy and ef...

We formulate a unique continuation principle for the inhomogeneous Cauchy-Riemann equations near a boundary point z0 of a smooth domain in complex euclidean space. The principle implies that the Bergman projection of a function supported away from z0 cannot vanish to infinite order at z0 unless it vanishes identically. We prove that the principle holds in planar domains and in domains where the...

If H denotes the classical Hilbert transform and Hu(x) = v(x), then the functions u(x) and v(x) are the values on the real axis of a pair of conjugate functions, harmonic in the upper half-plane. This note gives a generalization of the above concepts in which the Laplace equation Deltau = 0 is replaced by the Yukawa equation Deltau = mu(2)u and in which the Cauchy-Riemann equations have a corre...

This paper contributes a tutorial level discussion of some interesting properties of the recent Cauchy–Schwarz (CS) divergence measure between probability density functions. This measure brings together elements from several different machine learning fields, namely information theory, graph theory and Mercer kernel and spectral theory. These connections are revealed when estimating the CS dive...

An analytical approach to the construction of a singular integral equation (SIE) with Cauchy kernel is proposed, which makes it possible clarify stress-strain state multi-wedge system loaded radial cracks. This based on use theory residues and clarification solutions periodicity characteristic equation. It illustrated example SIE for two-wedge interfacial crack under conditions anti-plane defor...

A new method for performing a nonlinear form of manifold-oriented stochastic neighbor projection method is proposed. By the use of kernel functions, one can operate in the feature space without ever computing the coordinates of the data in that space, but rather by simply computing the inner products between the images of all pairs of data in the feature space. The proposed method is termed as ...

We consider the problem of local (quasi-)geoid modelling from terrestrial gravity anomalies. Whereas this problem is uniquely solvable (up to spherical harmonic degree one) if gravity anomalies are globally available, the problem is non-unique if gravity anomalies are only available within a local area, which is the typical situation in local/regional gravity field modelling. We derive a mathem...

In this paper, the Schwarz boundary value problem (BVP) for the inhomogeneous Cauchy-Riemann equation in a triangle is investigated explicitly. Firstly, by the technique of parquetingreflection and the Cauchy-Pompeiu representation formula a modified Cauchy-Schwarz representation formula is obtained. Then, the solution of the Schwarz BVP is explicitly solved. In particular, the boundary behavio...

This study examined a Cauchy problem for multi-dimensional Laplace equation with mixed boundary. is severely ill-posed in the sense of Hadamard. To solve this problem, mollification approach suggested based on bilateral exponential kernel and new approach. The stable error estimates are obtained under priori posteriori rule, which numerical findings much influenced by unknown information. An es...