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1

On some Lr-biharmonic Euclidean hypersurfaces

Mohammadpouri A., Pashaie F.

Journal of Mathematics and Applications

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2016

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Vol. 39

91--104

EN

In decade eighty, Bang-Yen Chen introduced the concept of biharmonic hypersurface in the Euclidean space. An isometrically immersed hypersurface x : Mn → En+1 is said to be biharmonic if ∆2x = 0, where ∆ is the Laplace operator. We study the Lr-biharmonic hypersurfaces as a generalization of biharmonic ones, where Lr is the linearized operator of the (r + 1)th mean curvature of the hypersurface and in special case we have L0 = ∆. We prove that Lr-biharmonic hypersurface of Lr-finite type and also Lr-biharmonic hypersurface with at most two distinct principal curvatures in Euclidean spaces are r-minimal.

2

Correcting theorem 1 from "When lines go bad in hyperspace"

Schlicker S., Bay C., Lembcke A.

Demonstratio Mathematica

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2009

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Vol. 42, nr 2

237-240

EN

This is in regards to the paper "When Lines go bad in hyperspace" by Christopher Bay, Amber Lembcke, and Steven Schlicker which appears in Demonstratio Mathematica, No. 3, Volume 38 (2005), p. 689-701. It has recently been brought to our attention that Theorem 1 from this paper is not correct. Please note that the main conclusions of the paper do not depend at all on this theorem. However, as the authors we feel it is our responsibility to bring this erroneous theorem to your attention. As stated in the paper, Theorem 1 intends to demonstrate that there can be infinitely many elements at a given location between two sets A and B.

3

The spaces of closed convex sets in Euclidean spaces with the fell topology

Sakai K., Yang Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 2

139-143

EN

Let ConvF(Rn) be the space of all non-empty closed convex sets in Euclidean space Rn endowed with the Fell topology. We prove that ConvF(lRn) ≈ Rn x Q for every n > 1 whereas ConvF(R) ≈ R x I.

4

When lines go bad in hyperspace

Bay C., Lembcke A., Schlicker S.

Demonstratio Mathematica

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2005

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Vol. 38, nr 3

689--701

EN

Let H(Rn) denote the hyperspace of all non-empty compact subsets of Rn. The Hausdorff metric h provides a way to measure distances between two elements of H(Rn) and generates the complete metric space (H(Rn),h)). In this paper, we examine geometric properties of lines in H(Rn), as determined by the Hausdorff metric, and compare and contrast the properties of these lines with Euclidean lines in Rn. Several surprising properties of these objects will be highlighted.

5

Hausdorff convergence of domains and their boundaries for shape optimal design

Holzleitner L.

Control and Cybernetics

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2001

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Vol. 30, no 1

23-44

EN

A question, which arises frequently in shape optimal design, is the convergence of domains. If the objective function is defined by using the solution of a PDE with boundary conditions, then also the convergence of the boundary is of importance. In this paper a criterion for a set of domains is defined, such that from [Omega_n] --> Omega follows [Gamma_n] --> Gamma if one is restricting to this set of domains. Moreover it is proved that this criterion is sharp, meaning that if [Omega_n] --> Omega implies [Gamma_n] --> Gamma holds for any sequence of this set, then this criterion has to be fulfilled. A similar criterion for the convergence of the Lebesgue measure of the boundaries my(Gamma_n) --> my(Gamma) is given.

6

On non-injectivity of Borel functions selecting points from compact sets. 2

Balcerzak M., Peredko J., Pol R.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 2

103-105

EN

The note is a supplement to (1]. We refine a result from [1] on the non-injectivity of Borel selections on the hyperspaces and we discuss the relations of the results in [1] with some results obtained by Lecomte (4].

7

Some metric properties of hyperspaces

Bogdewicz A.

Demonstratio Mathematica

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2000

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Vol. 33, nr 1

135-149