Wyniki wyszukiwania - Biblioteka Nauki

1

Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets

100%

Pallaschke D. , Urbański R.

Schedae Informaticae

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2012

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tom 21

PL

The quasidifferential calculus developed by V.F. Demyanov and A.M. Rubinov provides a complete analogon to the classical calculus of differentiation for a wide class of nonsmooth functions. Although this looks at the first glance as a generalized subgradient calculus for pairs of subdifferentials it turns out that, after a more detailed analysis, the quasidifferential calculus is a kind of Fréchet-differentiation whose gradients are elements of a suitable Minkowski–Rådström–Hörmander space. One aim of the paper is to point out this fact. The main results in this direction are Theorem 1 and Theorem 5. Since the elements of the Minkowski–Rådström–Hörmander space are not uniquely determined, we focus our attention in the second part of the paper to smallest possible representations of quasidifferentials, i.e. to minimal representations. Here the main results are two necessary minimality criteria, which are stated in Theorem 9 and Theorem 11.

2

Point derivations for Lipschitz functions andClarke's generalized derivative

100%

Demyanov V. F. , Pallaschke D.

Applicationes Mathematicae

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1996-1997

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tom 24

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nr 4

465-474

EN

Clarke's generalized derivative $f^0(x,v)$ is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed $x\in X$ and fixed $v\in E$ the function $f^0(x,v)$ is continuous and sublinear in $f\in Lip(X,d)$. It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz's product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.

3

Some remarks on the space of differences of sublinear functions

100%

Bartels S. G. , Pallaschke D.

Applicationes Mathematicae

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1993-1995

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tom 22

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nr 3

419-426

EN

Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X,‖·‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = ℝ^n this construction yields a norm such that D(ℝ^n) becomes a Banach space. Furthermore, we show that for a real Banach space with a smooth dual every sublinear Lipschitzian function can be expressed by the Fenchel conjugate of the farthest point mapping to its subdifferential at the origin. This leads to a simple family of sublinear functions which contains an exhaustive family of upper convex approximations for any quasidifferentiable function.

4

Spaces of Lipschitz functions on metric spaces

100%

Pallaschke D. , Pumplün D.

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tom 35

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nr 1

5-23

EN

In this paper the universal properties of spaces of Lipschitz functions, defined over metric spaces, are investigated.

5

Algebraic Separation and Shadowing of Arbitrary Sets

80%

Grzybowski J. , Pallaschke D. , Urbański R.

Commentationes Mathematicae

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2013

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tom 53

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nr 2

EN

In this paper we consider a generalization of the separation technique proposed in [10,4,7] for the separation of finitely many compact convex sets $A_i,~i \in I $ by another compact convex set $S$ in a locally convex vector space to arbitrary sets in real vector spaces. Then we investigate the notation of shadowing set which is a generalization of the notion of separating set and construct separating sets by means of a generalized Demyanov-difference in locally convex vector spaces.

6

Minimal pairs of bounded closed convex sets as minimal representations of elements of the Minkowski-Rådström-Hörmander spaces

80%

Grzybowski J. , Pallaschke D. , Urbański R.

Banach Center Publications

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2009

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tom 84

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nr 1

31-55

EN

The theory of minimal pairs of bounded closed convex sets was treated extensively in the book authored by D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, Fractional Arithmetic with Convex Sets. In the present paper we summarize the known results, generalize some of them and add new ones.

7

Support functions and subdifferentials

80%

Grzybowski J. , Pallaschke D. , Urbański R.

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tom 56

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nr 1

EN

In this paper we study Minkowski duality, i.e. the correspondence between sublinear functions and closed convex sets in the context of dual pairs of vector spaces.