Analysis of Gegenbauer kernel filtration on the hypersphere

In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated.

Projection-Based Local and Global Lipschitz Moduli of the Optimal Value in Linear Programming

In this paper, we use a geometrical approach to sharpen a lower bound given in [5] for the Lipschitz modulus of the optimal value of (finite) linear programs under tilt perturbations of the objective function. The key geometrical idea comes from orthogonally projecting general balls on linear subspaces. Our new lower bound provides a computable expression for the exact modulus (as far as it only depends on the nominal data) in two important cases: when the feasible set has extreme points and when we deal with the Euclidean norm. In these two cases, we are able to compute or estimate the global Lipschitz modulus of the optimal value function in different perturbations frameworks.

The control of the displacement of a passive particle in a point vortex flow

This work reports numerical explorations in advection of one passive tracer by point vortices living in the unbounded plane. The main objective is to find the energy-optimal displacement of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The direct formulation of the corresponding control problems is presented. The restrictions are due to (i) the ordinary differential equations that govern the displacement of the passive particle around the point vortices, (ii) the available time T to go from the initial position z0 to the final destination zf, and (iii) the maximum absolute value umax that is imposed on the control variables. The latter consist in staircase controls, i.e., the control is written as a finite linear combination of characteristic functions on the real interval. The resulting optimization problems are solved numerically. The numerical results shows the existence nearly/quasi optimal control for the cases of N=1, N=2, N=3, and N=4 vortices.

Asymptotic Expansion for Neural Network Operators of the Kantorovich Type and High Order of Approximation

In this paper, we study the rate of pointwise approximation for the neural network operators of the Kantorovich type. This result is obtained proving a certain asymptotic expansion for the above operators and then by establishing a Voronovskaja type formula. A central role in the above resuts is played by the truncated algebraic moments of the density functions generated by suitable sigmoidal functions. Furthermore, to improve the rate of convergence, we consider finite linear combinations of the above neural network type operators, and also in the latter case, we obtain a Voronovskaja type theorem. Finally, concrete examples of sigmoidal activation functions have been deeply discussed, together with the case of rectified linear unit (ReLu) activation function, very used in connection with deep neural networks.

Single-Chain Magnet Behavior in a Finite Linear Hexanuclear Molecule

The careful monitoring of crystallization conditions of a mixture made of a TbIII building block and a substituted nitronyl nitroxide that typically provides infinite coordination polymers (chains), affords a remarkably...