Analytical expressions of the elements of a cross spectral density matrix are derived to describe the partially coherent Lorentz-Gauss beam propagating in uniaxial crystals orthogonal to the x-axis. The intensity and degree of polarization for the partially coherent Lorentz-Gauss beam propagating in uniaxial crystals orthogonal to the x-axis are also presented. The evolution properties of the p...

We study Gauss-Kronrod quadrature formulae for the Jacobi weight function «/"'"'(t) = (l-i)Q(l + t)'3 and its special case a = ß = X^ of the Gegenbauer weight function. We are interested in delineating regions in the (a, /3)-plane, resp. intervals in A, for which the quadrature rule has (a) the interlacing property, i.e., the Gauss nodes and the Kronrod nodes interlace; (b) all nodes contained ...

Recent work by Nesterov and Stich (2016) showed that momentum can be used to accelerate the rate of convergence for block GaussSeidel in the setting where a fixed partitioning of the coordinates is chosen ahead of time. We show that this setting is too restrictive, constructing instances where breaking locality by running non-accelerated Gauss-Seidel with randomly sampled coordinates substantia...

This paper proposes a direct approach for solving optimal control problems. The time domain is divided into multiple subdomains, and a Lagrange interpolating polynomial using the Legendre–Gauss– Lobatto points is used to approximate the states and controls. The state equations are enforced at the Legendre–Gauss–Lobatto nodes in a nonlinear programming implementation by partial Gauss–Lobatto qua...

Any immersion of the circle into the sphere can be described by a Gauss code, which records the order of the self-intersections along the curve. The problem of which Gauss codes can be realized by closed curves is an old one, and has been solved in several ways. The goal of this paper is to generalize the notion of a Gauss code to look at immersed graphs, and to provide an algorithm for decidin...

We address quadrature-based approximations of the bilinear inverse form u>A−1u, where A is a real symmetric positive definite matrix, and analyze properties of the Gauss, Gauss-Radau, and Gauss-Lobatto quadrature. In particular, we establish monotonicity of the bounds given by these quadrature rules, compare the tightness of these bounds, and derive associated convergence rates. To our knowledg...

LetX be a compact, strictly convex C-hypersurface in the (n+1)-dimensional Euclidean space R. The Gauss map ofX maps the hypersurface one-to-one and onto the unit n-sphere S. One may parametrize X by the inverse of the Gauss map. Consequently, the Gauss curvature can be regarded as a function on S. The classical Minkowski problem asks conversely when a positive function K on S is the Gauss curv...

Solving a set of simultaneous linear equations is probably the most important topic in numerical methods. For solving linear equations, iterative methods are preferred over the direct methods specially when the coefficient matrix is sparse. The rate of convergence of iteration method is increased by using Successive Relaxation (SR) technique. But SR technique is very much sensitive to relaxatio...

Abstract. Gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis. Laurie proposed an (n+1)-point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n-point Gauss quadrature rule for all polynomials of degree up to 2n + 1. This rule is referred to as an anti-Gauss ru...

Plants for optimal growth requires absorb water and nutrients absorption from the soil. Magnetic Water downward movement of minerals and makes it easy for plants to absorb nutrients and water. This study was designed to investigate the effect of magnetic salt water on some quantity and quality characteristics of artichoke leaves. The experiment was factorial based on completely randomized desig...