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Tangency properties of sets with finite geometric curvature energies

Sebastian Scholtes (2012)

Fundamenta Mathematicae

We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature ℳ p α ( X ) : = ∫ X ∫ X ∫ X κ p ( x , y , z ) d X α ( x ) d X α ( y ) d X α ( z ) , where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that ℳ p α ( X ) < ∞ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant case p = 3 for...

Teorema de Ramsey aplicado a álgebras de Boole.

F. Benítez Trujillo (1990)

Collectanea Mathematica

Some properties of Boolean algebras are characterized through the topological properties of a certain space of countable sequences of ordinals. For this, it is necessary to prove the Ramsey theorems for an arbitrary infinite cardinal. Also, we define continuous mappings on these spaces from vector measures on the algebra.

The area formula for W 1 , n -mappings

Jan Malý (1994)

Commentationes Mathematicae Universitatis Carolinae

Let f be a mapping in the Sobolev space W 1 , n ( Ω , 𝐑 n ) . Then the change of variables, or area formula holds for f provided removing from counting into the multiplicity function the set where f is not approximately Hölder continuous. This exceptional set has Hausdorff dimension zero.

The Banach–Mazur game and σ-porosity

Miroslav Zelený (1996)

Fundamenta Mathematicae

It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.

The Banach-Saks property and Haar null sets

Eva Matoušková (1998)

Commentationes Mathematicae Universitatis Carolinae

A characterization of Haar null sets in the sense of Christensen is given. Using it, we show that if the dual of a Banach space X has the Banach-Saks property, then closed and convex subsets of X with empty interior are Haar null.

The Beta(p,1) extensions of the random (uniform) Cantor sets

Dinis D. Pestana, Sandra M. Aleixo, J. Leonel Rocha (2009)

Discussiones Mathematicae Probability and Statistics

Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)-Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater...

The box-counting dimension for geometrically finite Kleinian groups

B. Stratmann, Mariusz Urbański (1996)

Fundamenta Mathematicae

We calculate the box-counting dimension of the limit set of a general geometrically finite Kleinian group. Using the 'global measure formula' for the Patterson measure and using an estimate on the horoball counting function we show that the Hausdorff dimension of the limit set is equal to both: the box-counting dimension and packing dimension of the limit set. Thus, by a result of Sullivan, we conclude that for a geometrically finite group these three different types of dimension coincide with the...

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