Korovkin-type approximation of set-valued functions with convex graphs

Different Korovkin type results have been obtained in cones of set-valued continuous functions. Here we show that if we consider the subcone of set-valued continuous functions having a convex graph, then we can consider a Korovkin system which contains only affine functions. In this way we give a non trivial example where the number of functions to be used in a Korovkin system can be reduced

Modelling opinion formation by means of kinetic equations

In this chapter, we review some mechanisms of opinion dynamics that can be modelled by kinetic equations. Beside the sociological phenomenon of compromise, naturally linked to collisional operators of Boltzmann kind, many other aspects, already mentioned in the sociophysical literature or no, can enter in this framework. While describing some contributions appeared in the literature, we enlighten some mathematical tools of kinetic theory that can be useful in the context of sociophysics

Gastrokine-1, an anti-amyloidogenic protein secreted by the stomach, regulates diet-induced obesity

Obesity and its sequelae have a major impact on human health. The stomach contributes to obesity in ways that extend beyond its role in digestion, including through effects on the microbiome. Gastrokine-1 (GKN1) is an anti-amyloidogenic protein abundantly and specifically secreted into the stomach lumen. We examined whether GKN1 plays a role in the development of obesity and regulation of the gut microbiome. Gkn1−/− mice were resistant to diet-induced obesity and hepatic steatosis (high fat diet (HFD) fat mass (g) = 10.4 ± 3.0 (WT) versus 2.9 ± 2.3 (Gkn1−/−) p < 0.005; HFD liver mass (g) = 1.3 ± 0.11 (WT) versus 1.1 ± 0.07 (Gkn1−/−) p < 0.05). Gkn1−/− mice also exhibited increased expression of the lipid-regulating hormone ANGPTL4 in the small bowel. The microbiome of Gkn1−/− mice exhibited reduced populations of microbes implicated in obesity, namely Firmicutes of the class Erysipelotrichia. Altered metabolism consistent with use of fat as an energy source was evident in Gkn1−/− mice during the sleep period. GKN1 may contribute to the effects of the stomach on the microbiome and obesity. Inhibition of GKN1 may be a means to prevent obesity

On the Korovkin-type approximation of set-valued continuous functions

This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous setvalued functions and in spaces of vector-valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions

A Korovkin-type theorem for set-valued Hausdorff continuous functions

In this paper, we give a generalization of a Korovkin-type theorem for set-valued Hausdorff continuous functions (cf. [1] and [2]), by means of upper and lower envelopes.</span

Approximation of set-valued continuous functions in Fréchet spaces, II

We consider set-valued functions defined on a cornpact Hausdorff topological space and wìth compact convex values in a Fréchet space. We give some conditions which ensure that a set of set-valued functions is a Korovkin set with respect to monotone operators

Korovkin-type approximation in spaces of vector-valued and set-valued functions

We consider some Korovkin-type approximation results for sequences of linear continuous operators in spaces of vector-valued and set-valued continuous functions without assuming the existence of the limit operator. Even in spaces of real continuous functions, where similar results have already been established, we replace the positivity assumption with a weaker condition. We also give some quantitative estimate of the convergence and some applications where previous results cannot be applied