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Decision-making for long memory data in technical-economic design, fractals and decision area bubbles

Václav Beran (2003)

Applications of Mathematics

Economic and management theories are very often based in their applications on the perception of homogeneity of the application space. The purpose of this article is to query such a conviction and indicate new possible directions of discipline development. The article deals with symbiosis of process and his steering model as a process of management. It is possible that in relative near future it will be necessary to accept approaches and changes in interpretations of decision-making. Applications...

Differential and integral calculus for a Schauder basis on a fractal set (I) (Schauder basis 80 years after)

Julian Ławrynowicz, Tatsuro Ogata, Osamu Suzuki (2009)

Banach Center Publications

In this paper we introduce a concept of Schauder basis on a self-similar fractal set and develop differential and integral calculus for them. We give the following results: (1) We introduce a Schauder/Haar basis on a self-similar fractal set (Theorems I and I'). (2) We obtain a wavelet expansion for the L²-space with respect to the Hausdorff measure on a self-similar fractal set (Theorems II and II'). (3) We introduce a product structure and derivation on a self-similar fractal set (Theorem III)....

Diffusion and propagation problems in some ramified domains with a fractal boundary

Yves Achdou, Christophe Sabot, Nicoletta Tchou (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of ℝ 2 with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous...

Digit sets of integral self-affine tiles with prime determinant

Jian-Lin Li (2006)

Studia Mathematica

Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system ϕ d ( x ) = M - 1 ( x + d ) d ∈ D has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that D = M γ D ₀ , where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results of Kenyon,...

Dimension de Hausdorff de certains fractals aléatoires

Fathi Ben Nasr (1992)

Journal de théorie des nombres de Bordeaux

On construit des ensembles de Cantor aléatoires par partages successifs de rectangles, en partant d’un carré, (le nombre de divisions de la longueur peut être différent de celui de la largeur). La construction est stationnaire : elle fait intervenir des variables aléatoires indépendantes et équidistribuées. Sur ces ensembles il existe une mesure naturelle, μ , aléatoire elle aussi. Des résultats concernant les boréliens portant μ et leur dimension de Hausdorff ont déjà été obtenus par J. Peyrière...

Dimension of a measure

Pertti Mattila, Manuel Morán, José-Manuel Rey (2000)

Studia Mathematica

We propose a framework to define dimensions of Borel measures in a metric space by formulating a set of natural properties for a measure-dimension mapping, namely monotonicity, bi-Lipschitz invariance, (σ-)stability, etc. We study the behaviour of most popular definitions of measure dimensions in regard to our list, with special attention to the standard correlation dimensions and their modified versions.

Dimension of the harmonic measure of non-homogeneous Cantor sets

Athanasios Batakis (2006)

Annales de l’institut Fourier

We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets is a continuous function of the parameters determining these sets. This results extends a previous one of the author and do not use ergotic theoretic tools, not applicables to our case.

Dimensions of non-differentiability points of Cantor functions

Yuanyuan Yao, Yunxiu Zhang, Wenxia Li (2009)

Studia Mathematica

For a probability vector (p₀,p₁) there exists a corresponding self-similar Borel probability measure μ supported on the Cantor set C (with the strong separation property) in ℝ generated by a contractive similitude h i ( x ) = a i x + b i , i = 0,1. Let S denote the set of points of C at which the probability distribution function F(x) of μ has no derivative, finite or infinite. The Hausdorff and packing dimensions of S have been found by several authors for the case that p i > a i , i = 0,1. However, when p₀ < a₀ (or equivalently...

Dirichlet forms on quotients of shift spaces

Manfred Denker, Atsushi Imai, Susanne Koch (2007)

Colloquium Mathematicae

We define thin equivalence relations ∼ on shift spaces ∞ and derive Dirichlet forms on the quotient space Σ = ∞ / ∼ in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.

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