Noncommutative (deformed, quantum) spaces are deformations of the usual commutative space-time. They depend on parameters, such that for certain values of parameters they become the usual space-time. The symmetry acting on them is given in terms of a deformed quantum group symmetry. In this work we discuss two special examples, the θ-deformed space and the κ-deformed space. In the case of the θ...

Motivated by the work of Colin de Verdière and Saint-Raymond on spectral theory for zeroth-order pseudodifferential operators tori, we consider viscosity limits in which operators, P, are replaced P + iν Δ, ν > 0. By adapting Helffer–Sjöstrand scattering resonances, show that, a complex neighbourhood continuous spectrum, eigenvalues Δ have as goes to In simplified setting this justifies claims ...

We propose and analyze a randomized zeroth-order optimization method based on approximating the exact gradient by finite differences computed in set of orthogonal random directions that changes with each iteration. A number previously proposed methods are recovered as special cases including spherical smoothing, coordinate descent, well discretized descent. Our main contribution is proving conv...

In this paper, we obtain the sharp uniqueness for an inverse $ x $-source problem a one-dimensional time-fractional diffusion equation with zeroth-order term by minimum possible lateral Cauchy data. The key ingredient is unique continuation which holds weak solutions.

In this paper, we study, in the semiclassical sense, global approximate controllability small time of quantum density and momentum 1-D cubic Schrödinger equation with two controls between states positive densities. We first control asymptotic expansions zeroth order physical observables via Agrachev–Sarychev method. Then conclude proof through techniques approximation nonlinear equation.

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self tim...