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Best approximations and porous sets

Simeon Reich, Alexander J. Zaslavski (2003)

Commentationes Mathematicae Universitatis Carolinae

Let D be a nonempty compact subset of a Banach space X and denote by S ( X ) the family of all nonempty bounded closed convex subsets of X . We endow S ( X ) with the Hausdorff metric and show that there exists a set ℱ ⊂ S ( X ) such that its complement S ( X ) ∖ ℱ is σ -porous and such that for each A ∈ ℱ and each x ˜ ∈ D , the set of solutions of the best approximation problem ∥ x ˜ - z ∥ → min , z ∈ A , is nonempty and compact, and each minimizing sequence has a convergent subsequence.

Best approximations, fixed points and parametric projections

Tiziana Cardinali (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

If f is a continuous seminorm, we prove two f-best approximation theorems for functions Φ not necessarily continuous as a consequence of our version of Glebov's fixed point theorem. Moreover, we obtain another fixed point theorem that improves a recent result of [4]. In the last section, we study continuity-type properties of set valued parametric projections and our results improve recent theorems due to Mabizela [11].

Best N-term approximation in electronic structure calculations. II. Jastrow factors

Heinz-Jürgen Flad, Wolfgang Hackbusch, Reinhold Schneider (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a novel application of best N-term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions ℱ ( 2 ) near electron-electron and electron-nuclear cusps. Based on Nitsche's characterization of best N-term approximation spaces A q α ( H 1 ) , we prove...

Best N-term approximation in electronic structure calculations I. One-electron reduced density matrix

Heinz-Jürgen Flad, Wolfgang Hackbusch, Reinhold Schneider (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

We discuss best N-term approximation spaces for one-electron wavefunctions φ i and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces A q α ( H 1 ) for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted ℓ q spaces of wavelet coefficients to proof that both φ i and ρ are in A q α ( H 1 ) for all α > 0 with α = 1 q - 1 2 . Our proof is based on the...

Best simultaneous L p approximations

Yusuf Karakuş (1998)

Czechoslovak Mathematical Journal

In this paper we study simultaneous approximation of n real-valued functions in L p [ a , b ] and give a generalization of some related results.

Caristi's fixed point theorem and its equivalences in fuzzy metric spaces

Naser Abbasi, Hamid Mottaghi Golshan (2016)

Kybernetika

In this article, we extend Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem to fuzzy metric spaces in the sense of George and Veeramani [A. George , P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems. 64 (1994) 395-399]. Further, a direct simple proof of the equivalences among these theorems is provided.

Currently displaying 61 – 80 of 294