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A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.

On $C^{1}$ -stability and $C^{1}$ -determinacy

A. A. Du Plessis, C. T. C. Wall (1989)

Publications Mathématiques de l'IHÉS

On Cohomology of Singularities of C°° Mapping.

Er Jian Xiao (1983)

Mathematische Annalen

On collective compactness of derivatives

Jiří Durdil (1976)

Commentationes Mathematicae Universitatis Carolinae

On conability of singlevalued mappings

Le Van Hot (1982)

Časopis pro pěstování matematiky

On conjugacy of diffeomorphisms infinitely tangent to the identity. II.

H. Kawabe (1985)

Inventiones mathematicae

On differential equations and inclusions with mean derivatives on a compact manifold

S.V. Azarina, Yu.E. Gliklikh (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We introduce and investigate a new sort of stochastic differential inclusions on manifolds, given in terms of mean derivatives of a stochastic process, introduced by Nelson for the needs of the so called stochastic mechanics. This class of stochastic inclusions is ideologically the closest one to ordinary differential inclusions. For inclusions with forward mean derivatives on manifolds we prove some results on the existence of solutions.

On differential inclusions of velocity hodograph type with Carathéodory conditions on Riemannian manifolds

Yuri E. Gliklikh, Andrei V. Obukhovski (2004)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We investigate velocity hodograph inclusions for the case of right-hand sides satisfying upper Carathéodory conditions. As an application we obtain an existence theorem for a boundary value problem for second-order differential inclusions on complete Riemannian manifolds with Carathéodory right-hand sides.

On Finite Ck Left Determinacy.

C.T.C. Wall (1982/1983)

Inventiones mathematicae

On Finite Determinacy of Formal Vector Fields.

Fumio Ichikawa (1982/1983)

Inventiones mathematicae

On generalized “ham sandwich” theorems

Marek Golasiński (2006)

Archivum Mathematicum

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_{1}, ..., A_{m} \subseteq ℝ^{n}$ be subsets with finite Lebesgue measure. Then, for any sequence $f_{0}, ..., f_{m}$ of $ℝ$ -linearly independent polynomials in the polynomial ring $ℝ [X_{1}, ..., X_{n}]$ there are real numbers $λ_{0}, ..., λ_{m}$ , not all zero, such that the real affine variety ${x \in ℝ^{n}; λ_{0} f_{0} (x) + \dots + λ_{m} f_{m} (x) = 0}$ simultaneously bisects each of subsets $A_{k}$ , $k = 1, ..., m$ . Then some its applications are studied.

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