Displaying similar documents to “A dyadic view of rational convex sets”

A d.c. C 1 function need not be difference of convex C 1 functions

David Pavlica (2005)

Commentationes Mathematicae Universitatis Carolinae

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In [2] a delta convex function on ℝ 2 is constructed which is strictly differentiable at 0 but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class C 1 ( ℝ 2 ) which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.

A Remark on a Paper of Crachiola and Makar-Limanov

Robert Dryło (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line ¹ k , then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.

Ultrafilter-limit points in metric dynamical systems

Salvador García-Ferreira, Manuel Sanchis (2007)

Commentationes Mathematicae Universitatis Carolinae

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Given a free ultrafilter p on ℕ and a space X , we say that x ∈ X is the p -limit point of a sequence ( x n ) n ∈ ℕ in X (in symbols, x = p - lim n → ∞ x n ) if for every neighborhood V of x , { n ∈ ℕ : x n ∈ V } ∈ p . By using p -limit points from a suitable metric space, we characterize the selective ultrafilters on ℕ and the P -points of ℕ * = β ( ℕ ) ∖ ℕ . In this paper, we only consider dynamical systems ( X , f ) , where X is a compact metric space. For a free ultrafilter p on ℕ * , the function f p : X → X is defined by f p ( x ) = p - lim n → ∞ f n ( x ) for each x ∈ X . These functions are not continuous in general....

Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

Sergey A. Antonyan, Natalia Jonard-Pérez (2013)

Fundamenta Mathematicae

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For every n ≥ 2, let cc(ℝⁿ) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝⁿ endowed with the Hausdorff metric topology. Let cb(ℝⁿ) be the subset of cc(ℝⁿ) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝⁿ). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(ℝⁿ). This is applied to show that cb(ℝⁿ)...

About the decision of reachability for register machines

Véronique Cortier (2002)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We study the decidability of the following problem: given p affine functions f 1 , ... , f p over ℕ k and two vectors v 1 , v 2 ∈ ℕ k , is v 2 reachable from v 1 by successive iterations of f 1 , ... , f p (in this given order)? We show that this question is decidable for p = 1 , 2 and undecidable for some fixed p .

Indiscernibles and dimensional compactness

C. Ward Henson, Pavol Zlatoš (1996)

Commentationes Mathematicae Universitatis Carolinae

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This is a contribution to the theory of topological vector spaces within the framework of the alternative set theory. Using indiscernibles we will show that every infinite set u S ⊆ G in a biequivalence vector space 〈 W , M , G 〉 , such that x - y ∉ M for distinct x , y ∈ u , contains an infinite independent subset. Consequently, a class X ⊆ G is dimensionally compact iff the π -equivalence ≐ M is compact on X . This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.