A numerical method for solving nonlinear Fredholm-Volterra integral equations of general type is presented. This method is based on replacement of unknown function by truncated series of well known Chebyshev expansion of functions. The quadrature formulas which we use to calculate integral terms have been imated by Fast Fourier Transform (FFT). This is a grate advantage of this method which has...

General properties of global covariance matrices representing bipartite Gaussian states can be decomposed into properties of local covariance matrices and their Schur complements. We demonstrate that given a bipartite Gaussian state ρ12 described by a 4 × 4 covariance matrix V, the Schur complement of a local covariance submatrix V1 of it can be interpreted as a new covariance matrix representi...

The fractional calculus has many applications in applied science and engineering. The solution of the differential equation containing fractional derivative is much involved. An effective and easy-to-use method for solving such equations is needed. However not only the analytical solutions exist for a limited number of cases, but also the numerical methods are very complicated and difficult. In...

Due to the importance of quantum entangled states in quantum information and computation [1, 2, 3], much effort has been done recently towards an operational characterization of separable states [4, 5, 6]. The manifestations of mixed-state entanglement can be very subtle [7]. Till now there is no general efficient criterion in judging the separability. The Bell inequalities [8], Peres PPT crite...

Let $a, b, k,in K$ and $u, v in U(K)$. We show for any idempotent $ein K$, $(a 0|b 0)$ is e-clean iff $(a 0|u(vb + ka) 0)$ is e-clean and if $(a 0|b 0)$ is 0-clean, $(ua 0|u(vb + ka) 0)$ is too.

<p style='text-indent:20px;'>Caputo derivative operational matrices of the arbitrary scaled Legendre and Chebyshev wavelets are introduced by deriving directly from these wavelets. The Caputo used in quadratic optimization systems having fractional or integer orders differential equations. Using matrices, a new programming wavelet-based method without doing any integration operation for f...

This paper presents a numerical matrix method based on Bernstein polynomials (BPs) for approximate the solution of a system of m-th order nonlinear Volterra integro-differential equations under initial conditions. The approach is based on operational matrices of BPs. Using the collocation points,this approach reduces the systems of Volterra integro-differential equations associated with the giv...