We consider the mass-subcritical Hartree equation with a homogeneous kernel, in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally wellposed if we simply work in the space of functions whose Fourier transform is integrable. Similar results ar...

In this paper we will present a detailed study of implementing double-precision matrix-matrix multiplication (DGEMM) utilizing the Intel Xeon Phi Coprocessor. We discuss a DGEMM algorithm implementation running "natively" on the coprocessor, minimizing communication with the host CPU. We will run DGEMM across a range of matrix sizes natively as well using Intel Math Kernel Library. Our optimiza...

The present paper introduces a sharp Trudinger type inequality for harmonic functions based on the Cauchy-Riesz kernel function, which includes modified Poisson type kernel in a half plane considered by Xu et al. (Bound. Value Probl. 2013:262, 2013). As applications, we not only obtain Morrey representations of continuous linear maps for harmonic functions in the set of all closed bounded conve...

Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers, and computer scientists in robotics, artificial intelligence data analysis, introducing a myriad of different often confusing notations. The Euclidean 3-space, natural generalization both well-known Gibbs-Heaviside vector Hamilton's quaternions, is used he...

Singular integral operators of two types are considered on a piecewise smooth contour in weighted Lebesgue spaces with generalized Cauchy kernels, related to the parametrix elliptic systems first order plane. The type linear over field complex numbers and represented as usual combination singular operator multiplication by continuous matrix functions. second act space real vector functions and,...

In present paper, a numerical approach for solving Cauchy type singular integral equations is discussed. Lagrange interpolation with Gauss Legendre quadrature nodes and Taylor series expansion are utilized to reduce the computation of integral equations into some algebraic equations. Finally, five examples with exact solution are given to show efficiency and applicability of the method. Also, w...