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The Hausdorff Dimension of the Penrose Universe  [PDF]
L. Marek-Crnjac
Physics Research International , 2011, DOI: 10.1155/2011/874302
Abstract: Penrose fractal tiling is one of the simplest generic examples for a noncommutative space. In the present work, we determine the Hausdorff dimension corresponding to a four-dimensional analogue of the so-calledPenrose Universe and show how it could be used in resolving various fundamental problems in high energy physics and cosmology. 1. Introduction As explained in detail in Connes’ [1], Penrose fractal tiling constitutes mathematically a quotient space . Using this fact A. Connes following earlier work due to von Neumann deduced a dimensional function which we generalize to a simple formula function linking both the Menger-Urysohn topological dimension and the corresponding Hausdorff dimension. The present work is subdivided into three main parts. First, we show an explicit application and generalization of the Connes’ dimensional function. Second, we derive the Hausdorff dimension of the Hilbert space which X represents. Finally, we show the relevance of these results in high energy physics and cosmology. 2. The Dimensional Function and the Hilbert Space Let us start from the Connes’ dimensional function for the Penrose universe [1]: Writing and using the Fibonacci sequence, it is easy to see that, starting from the seed and , we obtain the following dimensional hierarchy: By complete induction, one finds We obtain an exceptional Fibonacci sequence : The classical Fibonacci sequence is defined by the recurrence relation where , , and . The first few Fibonacci numbers of the classical Fibonacci sequence are given . The th Fibonacci number is given by the formula which is called the Binet form, named after Jaques Binet, where and are the solutions of the quadratic equation : The Binet form of the th Fibonacci number of the sequence can be expressed similar to the classical Fibonacci sequence: The Fibonacci sequence can be presented as an infinite geometric sequence: The Golden Section principle that connects the adjacent powers of the golden mean is seen from the infinite geometric sequence. The formula for the th Fibonacci number is clearly identical to the bijection formula of E-infinity algebra and rings, namely [2, 3], Here, is the Menger-Urysohn topological dimension which should not be confused with the embedding dimension and is the Hausdorff dimension whose topological dimension is . To see that this extends in a simple fashion to negative dimensions [4], we set and find that the empty set is structured and possesses a finite Hausdorff dimension equal to because Now, we claim that is effectively a random Hilbert space and is four dimensional
Topological diagonalizations and Hausdorff dimension  [PDF]
Tomasz Weiss,Boaz Tsaban
Mathematics , 2002, DOI: 10.1285/i15900932v22n2p83
Abstract: The Hausdorff dimension of a product XxY can be strictly greater than that of Y, even when the Hausdorff dimension of X is zero. But when X is countable, the Hausdorff dimensions of Y and XxY are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than ``being countable'' and stronger than ``having Hausdorff dimension zero''. Fremlin asked whether it is enough for X to have the strongest property in this hierarchy (namely, being a gamma-set) in order to assure that the Hausdorff dimensions of Y and XxY are the same. We give a negative answer: Assuming CH, there exists a gamma-set of reals X and a set of reals Y with Hausdorff dimension zero, such that the Hausdorff dimension of X+Y (a Lipschitz image of XxY) is maximal, that is, 1. However, we show that for the notion of a_strong_ gamma-set the answer is positive. Some related problems remain open.
Hausdorff Dimension and Diophantine Approximation  [PDF]
M. Maurice Dodson,Simon Kristensen
Mathematics , 2003,
Abstract: We begin with a brief treatment of Hausdorff measure and Hausdorff dimension. We then explain some of the principal results in Diophantine approximation and the Hausdorff dimension of related sets, originating in the pioneering work of Vojtech Jarnik. We conclude with some applications of these results to the metrical structure of exceptional sets associated with some famous problems. It is not intended that all the recent developments be covered but they can be found in the references cited.
On the Hausdorff dimension of Newhouse phenomena  [PDF]
Pierre Berger,Jacopo De Simoi
Mathematics , 2014,
Abstract: We show that at the vicinity of a generic dissipative homoclinic unfolding of a surface diffeomorphism, the Hausdorff dimension of the set of parameters for which the diffeomorphism admits infinitely many periodic sinks is at least 1/2.
Hedgehogs of Hausdorff dimension one  [PDF]
Kingshook Biswas
Mathematics , 2009,
Abstract: We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting non-linearisable maps, a common hedgehog of Hausdorff dimension 1, the minimum possible.
David maps and Hausdorff Dimension  [PDF]
S. Zakeri
Mathematics , 2002,
Abstract: David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs {enumerate} [$\bullet$] Given $\alpha$ and $\beta$ in $[0,2]$, there exists a David map $\phi:\CC \to \CC$ and a compact set $\Lambda$ such that $\Hdim \Lambda =\alpha$ and $\Hdim \phi(\Lambda)=\beta$. \vs [$\bullet$] There exists a David map $\phi:\CC \to \CC$ such that the Jordan curve $\Gamma=\phi (\Sen)$ satisfies $\Hdim \Gamma=2$.\vs {enumerate} One should contrast the first statement with the fact that quasiconformal maps preserve sets of Hausdorff dimension 0 and 2. The second statement provides an example of a Jordan curve with Hausdorff dimension 2 which is (quasi)conformally removable.
The Hausdorff Dimension of the Penrose Universe  [PDF]
L. Marek-Crnjac
Physics Research International , 2011, DOI: 10.1155/2011/874302
Abstract: Penrose fractal tiling is one of the simplest generic examples for a noncommutative space. In the present work, we determine the Hausdorff dimension corresponding to a four-dimensional analogue of the so-calledPenrose Universe and show how it could be used in resolving various fundamental problems in high energy physics and cosmology.
On the Hausdorff dimension of the Rauzy gasket  [PDF]
Artur Avila,Pascal Hubert,Alexandra Skripchenko
Mathematics , 2013,
Abstract: In this paper, we prove that the Hausdorff dimension of the Rauzy gasket is less than 2. By this result, we answer a question addressed by Pierre Arnoux. Also, this question is a very particular case of the conjecture stated by S.P. Novikov and A. Ya. Maltsev in 2003.
Hausdorff Dimension of Cantor Series  [PDF]
G. Iommi,B. Skorulski
Mathematics , 2009,
Abstract: In 1996 Y. Kifer obtained a variational formula for the Hausdorff dimension of the set of points for which the frequencies of the digits in the Cantor series expansion is given. In this note we present a slightly different approach to this problem that allow us to solve the variational problem of Kifer's formula.
A Hausdorff dimension for finite sets  [PDF]
Juan M. Alonso
Computer Science , 2015,
Abstract: The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension guarantees that every point of the set has "nearby" neighbors. This property is important for many computer algorithms of great practical value, that obtain solutions by finding nearest neighbors. We also define an analog for finite sets of the classical box-counting dimension, and compute examples. The main result of the paper is a Convergence Theorem. It gives conditions under which, if a sequence of finite sets converges to a compact set (convergence of compact subsets of Euclidean space under the Hausdorff metric), then the finite Hausdorff dimension of the finite sets will converge to the classical Hausdorff dimension of the compact set.
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