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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.

Beyond Lebesgue and Baire II: Bitopology and measure-category duality

N. H. Bingham, A. J. Ostaszewski (2010)

Colloquium Mathematicae

We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density topologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on infinitary combinatorics due to Kestelman and to Borwein and Ditor. We hence give a unified proof of the measure and category cases of the Uniform Convergence Theorem for slowly varying functions. We also extend results on very slowly varying functions...

BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups

Sean Li (2015)

Analysis and Geometry in Metric Spaces

Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that BZcan be decomposed into a controlled number of pieces, the restriction of f on each of which is quantitatively biLipschitz. This extends a result of [14], which proved the same result, but with the restriction that G has an appropriate discretization. We provide an example of a Carnot group not admitting such...

Bi-Lipschitz embeddings of hyperspaces of compact sets

Jeremy T. Tyson (2005)

Fundamenta Mathematicae

We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in ℓ². If X is a countable compact metric space containing at most n nonisolated points, there is a Lipschitz embedding of K(X) in ℝ n + 1 ; in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. By way of contrast, the hyperspace K([0,1]) of the...

Blow-up of regular submanifolds in Heisenberg groups and applications

Valentino Magnani (2006)

Open Mathematics

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence...

Borel extensions of Baire measures in ZFC

Menachem Kojman, Henryk Michalewski (2011)

Fundamenta Mathematicae

We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

Borel partitions of unity and lower Carathéodory multifunctions

S. Srivastava (1995)

Fundamenta Mathematicae

We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in A ( ℰ ⊗ ℬ ( X ) ) into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential equations...

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

For a linear operator T in a Banach space let σ p ( T ) denote the point spectrum of T, let σ p , n ( T ) for finite n > 0 be the set of all λ ∈ σ p ( T ) such that dim ker(T - λ) = n and let σ p , ∞ ( T ) be the set of all λ ∈ σ p ( T ) for which ker(T - λ) is infinite-dimensional. It is shown that σ p ( T ) is ℱ σ , σ p , ∞ ( T ) is ℱ σ δ and for each finite n the set σ p , n ( T ) is the intersection of an ℱ σ set and a δ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more detailed decomposition...

Currently displaying 21 – 40 of 51