For an infinitesimal symplectic action of a Lie algebra g on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and Lie algebra g. We obtain its second crossed product in case g = R and show an infinitesimal version for a theorem type of Takesaki duality.

An infinitesimal bending of the curve at E is considered and the infinitesimal bending field is determined and discussed. CurveBend, tool for graphical presentation of non rigid curves is presented. Influence of infinitesimal bending field on curves is discussed and visualized by the tool.

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis ("SIA"), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis ("CA") without resort to the method of limits. Formally, however, unlike Robinsonian "nonstandard analysis", SIA conflicts with CA, deriving, e.g., "not every quantity is either...

although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.here, after criticizi...

For an infinitesimal symplectic action of a Lie algebra g on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and Lie algebra g. We obtain its second crossed product in case g = R and show an infinitesimal version for a theorem type of Takesaki duality.

The two-dimensional cubic nonlinear Schrödinger equation (NLS) can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. In this paper, we establish that every one-dimensional traveling wave solution of NLS with linear phase is unstable with respect to some infinitesimal perturbation with two-dimensional structure. If the coefficients ...

In the first part of this work [4] we have studied the approximation problem of Ε/(Χχ) by Ef(X%), where (Xt) is the solution of a stochastic differential equation, (X?) is defined by the Euler discretization scheme with step ^, and /(·) is a given function, only supposed measurable and bounded; we have proven that the error can be expanded in terms of powers of £, under a nondegeneracy conditio...

The method of one parameter, point symmetric, approximate Lie group invariants is applied to the problem of determining solutions of systems of pure one-dimensional, diffusion equations. The equations are taken to be non-linear in the dependent variables but otherwise homogeneous. Moreover, the matrix of diffusion coefficients is taken to differ from a constant matrix by a linear perturbation w...