This paper presents a semidiscrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot, and Sarti. We propose a new ingredient, namely, working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.

We introduce six families of three-dimensional space-periodic steady solenoidal flows, whose kinetic helicity density is zero at any point. Four families are analytically defined. Flows in four families have zero helicity spectrum. Sample flows from five families are used to demonstrate numerically that neither zero kinetic helicity density nor zero helicity spectrum prohibit generation of large-scale magnetic field by the two most prominent dynamo mechanisms: the magnetic $\alpha$-effect and negative eddy diffusivity. Our computations also attest that such flows often generate small-scale field for sufficiently small magnetic molecular diffusivity. These findings indicate that kinetic helicity and helicity spectrum are not the quantities controlling the dynamo properties of a flow regardless of whether scale separation is present or not.

This article concerns an indirect method to solve state-constrained optimal control problems. The dynamics of the controlled system are given by ordinary differential equations encompassing the effect of a steady flow field in which the moving object is immersed. The proposed method is based on the maximum principle in Gamkrelidze's form. At the core of the approach is a regularity hypothesis which entails the continuity of the measure Lagrange multiplier associated with the state constraint. This property of the multiplier plays a key role in shaping a properly modified shooting method to solve the two-point boundary value problem resulting from the maximum principle. Illustrative applications to time optimal control problems are considered and results of numerical experiments are provided.