The solution of themulti-dimensional Schrödinger equation in the generalized spherical coordinates is constructed in the Lagrange-meshmethod. Laguerre and Jacobimeshes are used to constructmatrix elements for the generalizedHamiltonian. Thematrix elements are functions of quadrature abscissas and involve few free parameters for each angular dimension. A ring-shaped non-central separable potenti...

We introduce a discrete-time fractional calculus of variations on the time scale hZ, h > 0. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of ...

In 1798 J.-L. Lagrange published an extensive book on the solution of numerical equations. Lagrange had developed a general systematic algorithm for detecting, isolating, and approximating all real and complex roots of a polynomial equation with real coeecients, with arbitrary precision. In contrast to Newton's Method, Lagrange's algorithm is guaranteed to converge. We discuss some lesser known...

As is often the case, some results get rediscovered over time. In particular, some rather striking results of Lagrange are often recreated. For instance, in [6], a result pertaining to the Pell equation for a prime discriminant was recast in the light of nonabelian cohomology groups. Yet, in [1], the authors acknowledged the fact that the result “has been discovered before,” and provided an ele...

Equation (1.1)ρ is called the mean field equation because it often arises in the context of statistical mechanics of point vortices in the mean field limits. Recently, there has been interest in (1.1)ρ because it also arises from the Chern-Simons-Higgs model vortex theory when some parameter tends to zero. (For these recent developments, we refer the readers to [5], [2], [3], [10], [11], [13], ...

The need of power and equipment that can serve to distribute the electric energy becomes primordial things this period almost domains’ specialists work in, one is microgrid system gain more places in not just industrial area but domestic area, however, rural or urban. This be worked synchronization with main grid alone provides its own needs alimentation characteristics via distributed generato...

The development of the GPS (Global Positioning System) signal simulator involving to a number of key technologies, in which the generation of navigation message has important significance. Based on analysis of the structure of GPS navigation data, the paper researches the production of telemetry word and handover word, parity check code, time parameters and star clock. Using disturbing force eq...

Setting. Let (X, g) be a given Riemannian manifold and let ∇ be the corresponding LeviCivita connection. Let G be a compact Lie group with a bi-invariant Riemannian metric κ. Let π : Q → X be a principal bundle with structure group G acting on the left, let A be a principal connection on Q, and let B be the curvature of A. 46 4. Wong’s Equations and Coordinate Formulas Now define the Lagrangian...