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Gâteaux derivative

About: Gâteaux derivative is a research topic. Over the lifetime, 219 publications have been published within this topic receiving 3550 citations.


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Journal ArticleDOI
TL;DR: This work considers the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals and revisits this problem using the notion of a shape derivative and shows that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals.
Abstract: We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the otherby solving Poisson's orHelmholtz's equation with well-chosen boundar y conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding Euler-Lagrange equations. We revisit this problem using the notion of a shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gateaux derivatives and Euler-Lagrange equations. Finally we apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results.

288 citations

Journal ArticleDOI
TL;DR: In this paper, the authors show that a continuous convex function of one real variable is not differentiable, except perhaps at a countable subset of its interval of continuity, except for a subset of the variables in the norm topology.
Abstract: A continuous convex function of one real variable is differentiable, except perhaps at a countable subset of its interval of continuity. The present paper deals with generalizat ions of this e lementary s ta tement to convex functions which are defined on some Banach space E, and continuous in the norm topology, with \"differentiable\" replaced either by \"Frdchet differentiable\" or \"Gateaux differentiable\". Since for E = L ~ ( 0 , 1 ) the very norm funct ion/ (x) = Ilxll for x in E, which is convex and continuous on all of E, is nowhere even G~teaux differentiable (Mazur [13]), this amounts to a classification of the category of all Banach spaces depending upon whether certain differentiability s ta tements hold. Therefore we say tha t a Banach space is a strong di//erentiability space (SDS) if the following theorem holds for it.

274 citations

Book
01 Jan 1997
TL;DR: In this article, a characterisation of WCG Spaces and of Eberlein Compacta is presented, and two concrete classes of Banach Spaces that lie in. Fragmentability.
Abstract: Canonical Examples of Weak Asplund Spaces. Properties of Gateaux Differentiability Spaces and Weak Asplund Spaces. Stegall's Classes. Two More Concrete Classes of Banach Spaces that Lie in . Fragmentability. "Long Sequences" of Linear Projections. Vasak Spaces and Gul'ko Compacta. A Characterization of WCG Spaces and of Eberlein Compacta. Main Open Questions and Problems. References. Index.

223 citations

Journal ArticleDOI
10 Jan 2017-Calcolo
TL;DR: In this paper, a general conformable fractional derivative (GCFD) is proposed to describe the physical world, which is generalized from the concept of CFD proposed by Khalil.
Abstract: Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term $$t^{1-\alpha }$$ in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD.

216 citations

Journal ArticleDOI
TL;DR: In this paper, the authors compared various definitions of directional derivatives in topological vector spaces and pointed out that in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent.
Abstract: Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of Gâteaux, Frechet, and Hadamard are singled out from the general framework of σ-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent. The chain rule for directional derivatives of a composite mapping is discussed.

209 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20219
20207
20192
20184
20176
20168