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Displaying 41 – 54 of 54

On the range of the derivative of a smooth function and applications.

Robert Deville (2006)

RACSAM

We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X,Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F'(X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear: we discuss the existence of a mapping F...

On the range of the derivative of a smooth mapping between Banach spaces.

Deville, Robert (2005)

Abstract and Applied Analysis

On the Size of Balls Covered by Analytic Transformations.

Lawrence A. Harris (1977)

Monatshefte für Mathematik

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space

Daniel Azagra, Mar Jiménez-Sevilla (2002)

Bulletin de la Société Mathématique de France

We study the size of the sets of gradients of bump functions on the Hilbert space $ℓ_{2}$ , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in $ℓ_{2}$ can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space $ℓ_{2}$ can be uniformly approximated by $C^{1}$ smooth Lipschitz functions $ψ$ so that the cones generated by the ranges of its derivatives $ψ^{'} (ℓ_{2})$ have empty interior. This implies that there are $C^{1}$ smooth Lipschitz bumps...

On the stability of mappings and an answer to a problem of TH.M. Rassias

P. Găvruţa, M. Hossu, D. Popescu, C. Căprău (1995)

Annales mathématiques Blaise Pascal

On the Structure of the Spaces ... .

Barbro Grevholm (1970)

Mathematica Scandinavica

On the structure of universal differentiability sets

Michael Dymond (2017)

Commentationes Mathematicae Universitatis Carolinae

A subset of $ℝ^{d}$ is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function $f : ℝ^{d} \to ℝ$ . We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.

On the Subdifferential of the Supremum of Two Convex Functions.

Arne Brondsted (1972)

Mathematica Scandinavica

On the tangency of multifunctions

A. Meimaridou (1985)

Annales Polonici Mathematici

On Uniform Differentiability

S. Rolewicz (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

We introduce the notion of uniform Fréchet differentiability of mappings between Banach spaces, and we give some sufficient conditions for this property to hold.

On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions

S. Rolewicz (1999)

Studia Mathematica

Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction $Γ : X \to 2^{Φ}$ to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.

One counterexample concerning the Fréchet differentiability of convex functions on closed sets

Eva Matoušková (1993)

Acta Universitatis Carolinae. Mathematica et Physica

One-parameter submonoids in locally complete differentiable monoids

Mitch Anderson (1980)

Colloquium Mathematicae

Optimal integrability of the Jacobian of orientation preserving maps

Andrea Cianchi (1999)

Bollettino dell'Unione Matematica Italiana

Dato un qualsiasi spazio invariante per riordinamenti $X (Ω)$ su un insieme aperto $Ω \subset R^{n}$ , si determina il più piccolo spazio invariante per riordinamenti $Y (Ω)$ con la proprietà che se $u : Ω \to R^{n}$ è una applicazione che mantiene l'orientamento e ${|D u|}^{n} \in X (Ω)$ , allora $det D u$ appartiene localmente a $Y (Ω)$ .

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