Hirota bilinear identity for Cauchy-Baker-Akhieser (CBA) kernel is introduced as a basic tool to construct integrable hierarchies containing lattice and q-difference times. Determinant formula for the action of meromorphic function on CBA kernel is derived. This formula gives opportunity to construct generic solutions for integrable lattice equations.

The Cauchy problem is studied for an homogeneous quantum kinetic equation describing the Compton effect. Since the collision kernel commonly used in physics is highly singular, numerical simulations are performed for related collision kernels to get a preliminary insight into the behavior of the solutions. Some of the numerical results are then given a theoretical explanation. Global existence ...

Given a curve Γ⊂C with specified regularity, we investigate boundedness and positivity for certain three-point symmetrization of Cauchy-like kernel KΓ whose definition is dictated by the geometry complex function theory domains bounded Γ. Our results show that S[ReKΓ] S[ImKΓ] (namely, symmetrizations real imaginary parts KΓ) behave very differently from their counterparts Cauchy previously stud...

The method of stable random projections is useful for efficiently approximating the lα distance (0 < α ≤ 2) in high dimension and it is naturally suitable for data streams. In this paper, we propose to use only the signs of the projected data and we analyze the probability of collision (i.e., when the two signs differ). Interestingly, when α = 1 (i.e., Cauchy random projections), we show that t...

We study the total positivity of the kernel 1/(x+2 cos(πα)xy+y). The case of infinite order is characterized by an application of Schoenberg’s theorem. We then give necessary conditions for the cases of any given finite order with the help of Chebyshev polynomials of the second kind. Sufficient conditions for the finite order cases are also obtained, thanks to Propp’s formula for the Izergin-Ko...

For one class of the singular integro-differential equations with Cauchy kernel on an interval, a Galerkin method is justified. The convergence is proved and the error estimation is given.

The importance of the support vector machine and its applicability to a wide range of problems is well known. The strength of the support vector machine lies in its kernel. In our recent paper, we have shown how the Laplacian kernel overcomes some of the drawbacks of the Gaussian kernel. However this was not a total remedy for the shortcomings of the Gaussian kernel. In this paper, we design a ...

The importance of the support vector machine and its applicability to a wide range of problems is well known. The strength of the support vector machine lies in its kernel. In our recent paper, we have shown how the Laplacian kernel overcomes some of the drawbacks of the Gaussian kernel. However this was not a total remedy for the shortcomings of the Gaussian kernel. In this paper, we design a ...

The importance of the support vector machine and its applicability to a wide range of problems is well known. The strength of the support vector machine lies in its kernel. In our recent paper, we have shown how the Laplacian kernel overcomes some of the drawbacks of the Gaussian kernel. However this was not a total remedy for the shortcomings of the Gaussian kernel. In this paper, we design a ...