This paper is devoted to the analysis of a special kind of regularity of multifunction which we call metric pseudo-(sub)regularity, when the multifunction is not metrically (sub)regular at a given point but is metrically (sub)regular at certain points in a neighborhood with moduli possibly tending to infinity with a certain order. By using advanced techniques of generalized differentiation we d...

In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert s...

In computable mathematics, there are known definitions of computable numbers, computable metric spaces, computable compact sets, and computable functions. A traditional definition of a computable function, however, covers only continuous functions. In many applications (e.g., in phase transitions), physical phenomena are described by discontinuous or multi-valued functions (a.k.a. constraints)....

In a complete Riemannian manifold (M,g) if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of (M,g). In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds (M,g) admitting a real valued function u such that the hessian of u has atmost two eigenvalues −u and − 2 , under some mild hypothesis on (M,g)...

We investigate analyticity of joint spectra of Am-valued holomorphic mappings, where A denotes a complex Banach algebra. We show also that if K is an analytic set-valued function whose values are compact subsets of Cn and d is the transfinite diameter in Cn, then the upper-semicontinuous regularization of logd(X) is plurisubharmonic. Moreover, we give higher dimensional extensions of Aupetit's ...

In this paper, the chaotic behavior of a set-valued mapping F : X → 2 , where X is a compact space, is investigated. The existence of the generalized shadowing property in the hyperspace 2 is proved. Based on the generalized shadowing property of the set-valued mappings F and the assumption of the existence of an unstable chain recurrent point of the mapping F , it is shown that the Bernoulli s...

Integer-valued polynomials on subsets of discrete valuation domains are well studied. We undertake here a systematical study of integer-valued polynomials on subsets S of valued fields and of several connected notions: the polynomial closure of S, the Bhargava’s factorial ideals of S and the v-orderings of S. A sequence of numbers is naturally associated to the subset S and a good description c...