Discrete Approximate Solution of Differential Inclusion with Normal Cone and Prox-Regular Set.

Our aim is to calculate the discrete approximate solution of di⁄erential inclusion with normal cone and prox-regular set, the question is how to calculate this solution? We use the discrete approximation property of a new variant of nonconvex sweeping processes involving normal cone and a nite element method. Knowing that The majority of mathematicians have proved only the existence and uniqueness of the solution for this type of inclusions, like: Mordukhovich, Thibault, Aubin, Messaoud, ...etc.

New Tour on the Subdifferential of Supremum via Finite Sums and Suprema

This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family.

Mouse Models of Achromatopsia in Addressing Temporal “Point of No Return” in Gene-Therapy

Achromatopsia is characterized by amblyopia, photophobia, nystagmus, and color blindness. Previous animal models of achromatopsia have shown promising results using gene augmentation to restore cone function. However, the optimal therapeutic window to elicit recovery remains unknown. Here, we attempted two rounds of gene augmentation to generate recoverable mouse models of achromatopsia including a Cnga3 model with a knock-in stop cassette in intron 5 using Easi-CRISPR (Efficient additions with ssDNA inserts-CRISPR) and targeted embryonic stem (ES) cells. This model demonstrated that only 20% of CNGA3 levels in homozygotes derived from target ES cells remained, as compared to normal CNGA3 levels. Despite the low percentage of remaining protein, the knock-in mouse model continued to generate normal cone phototransduction. Our results showed that a small amount of normal CNGA3 protein is sufficient to form “functional” CNG channels and achieve physiological demand for proper cone phototransduction. Thus, it can be concluded that mutating the Cnga3 locus to disrupt the functional tetrameric CNG channels may ultimately require more potent STOP cassettes to generate a reversible achromatopsia mouse model. Our data also possess implications for future CNGA3-associated achromatopsia clinical trials, whereby restoration of only 20% functional CNGA3 protein may be sufficient to form functional CNG channels and thus rescue cone response.

On inner calmness*, generalized calculus, and derivatives of the normal cone mapping

In this paper, we study continuity and Lipschitzian properties of set-valued mappings, focusing on inner-type conditions. We introduce new notions of inner calmness* and, its relaxation, fuzzy inner calmness*. We show that polyhedral maps enjoy inner calmness* and examine (fuzzy) inner calmness* of a multiplier mapping associated with constraint systems in depth. Then we utilize these notions to develop some new rules of generalized differential calculus, mainly for the primal objects (e.g. tangent cones). In particular, we propose an exact chain rule for graphical derivatives. We apply these results to compute the derivatives of the normal cone mapping, essential e.g. for sensitivity analysis of variational inequalities. Comment: 27 pages

Localizing the Donaldson–Futaki invariant

We use the equivariant localization formula to prove that the Donaldson–Futaki invariant of a compact smooth (Kähler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson–Futaki invariant of the deformation to the normal cone.

Accurate approximated solution to the differential inclusion based on the ordinary differential equation

UDC 517.9 Many problems in applied mathematics can be transformed and described by the differential inclusion involving which is a normal cone to a closed convex set at The Cauchy problem of this inclusion is studied in the paper. Since the change of leads to the change of solving the inclusion becomes extremely complicated. In this paper, we consider an ordinary differential equation containing a control parameter When is large enough, the studied equation gives a solution approximating to a solution of the inclusion above. The theorem about the approximation of these solutions with arbitrary small error (this error can be controlled by increasing ) is proved in this paper.

Modeling of crowds in regions with moving obstacles

<p style='text-indent:20px;'>We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula>-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.</p>