We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n)T where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We conjecture that along with operators previously constructed by Grünbaum, Haine, Horozov, and Iliev they exhaust all bispectral regular (i.e. a(n) 6= 0, c(n) 6= 0,∀n ∈ Z) ...

Nonlinear observers based on the well-known concept of minimum energy estimation are discussed. The approach relies an output injection operator determined by a Hamilton–Jacobi–Bellman equation whose solution is subsequently approximated neural network. A suitable optimization problem allowing to learn network parameters proposed and numerically investigated for linear nonlinear oscillators.

We consider a Riemannian manifold, (M, g), of dimension n with boundary ∂M . We analyze the inverse problem, originally formulated by Dix [4], of reconstructing g from boundary measurements associated with the single scattering of seismic waves on this manifold. The measurements determine the shape operator on the boundary. We develop an explicit reconstruction procedure involving the solution ...

In this paper a non-relativistic particle moving on a hypersurface in a curved space and the multidimensional rotator are investigated using the Hamilton-Jacobi formalism. The equivalence with the Dirac Hamiltonian formalism is demonstrated in both Cartesian and curvilinear coordinates. The energy spectrum of the multi-dimensional rotator is equal to that of a pure Laplace-Beltrami operator wit...

A new complex Jacobi iterative technique adapted for the solution of three-dimensional (3D) non-paraxial beam propagation is presented. The beam propagation equation for analysis of optical propagation in waveguide structures is based on a novel modified Padé(1,1) approximant operator we recently proposed. The effectiveness of our new approach is demonstrated in comparison with the traditional ...