Fractal dimension for Gaussian colored processes

The exact analytical expression for the Hausdorff dimension of free processes driven by Gaussian noise in n-dimensional space is obtained. The fractal dimension solely depends on the time behavior of the arbitrary correlation function of the noise, ranging from DX=1 for Orstein-Uhlenbeck input noise to any real number greater than 1 for fractional Brownian motions.

Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics

We prove that the stable holonomies of a proper codimension 1 attractor Λ, for a Cr diffeomorphism f of a surface, are not C1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.

Hausdorff dimension versus smoothness

There is a one-to-one correspondence between C1+H Cantor exchange systems that are C1+H fixed points of renormalization and C1+H diffeomorphisms f on surfaces with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, there is no such C1+α Cantor exchange system with bounded geometry that is a C1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set. The proof of the last result uses that the stable holonomies of a codimension 1 hyperbolic attractor Λ are not C1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ.

Degradación de efluentes de las industrias del cuero y textil mediante procesos fotocatalíticos heterogéneos empleando la luz solar. Degradation of effluents from the leather and textile industries through heterogeneous photocatalytic processes using sunlight.

El creciente desarrollo de la industria del cuero y textil en nuestro país, y específicamente en la provincia de Córdoba, ha hecho resurgir en los ultimos años una problemática aún no resuelta que es la elevada contaminación de los recursos hídricos. En ambas industrias, la operación de teñido involucra principalmente colorantes de tipo azoico los cuales son "no biodegradables" y se fragmentan liberando aminas aromáticas cancerígenas. Para abordar esta problemática, la fotocatálisis heterogénea aparece como una nueva tecnología que permitiría la completa mineralización de estos colorantes. A través de radiación y un fotocatalizador sólido adecuado se pueden generan radicales libres eficientes para la oxidación de materia orgánica (colorantes) en medio acuoso. En este sentido, se proponen tamices moleculares mesoporosos modificados con metales de transición (MT) como fotocatalizadores potencialmente aptos para la degradación de estos contaminantes. El propósito principal de este proyecto es el diseño, síntesis, caracterización y evaluación de materiales mesoporosos que presenten actividad fotocatalítica ya sea mediante la modificación de su estructura con diversos metales fotosensibles y/o empleándolos como soporte de óxido de titanio. Se pretende evaluar estos materiales en la degradación de colorantes intentando desplazar su fotosensibilidad hacia la radiación visible para desarrollar nuevas tecnologías con menor impacto ambiental y mayor aprovechamiento de la energía solar. Para ello se sintetizarán materiales del tipo MCM-41 modificados con distintos MT tales como Fe, Cr, Co, Ni y Zn mediante incorporación directa del ión metálico o impregnación. Al mismo tiempo, tanto estos últimos materiales como el MCM-41 silíceo serán empleados como soporte de TiO2. Sus propiedades fisicoquímicas se caracterizarán mediante distintas técnicas instrumentales y su actividad fotocatalítica se evaluará en la degradación de colorantes azoicos bajo radiación visible. Se seleccionará el catalizador más eficiente y se estudiarán los diversos factores que afectan el proceso de fotodegradación. Así mismo, el análisis de la concentración del colorante y los productos presentes en el medio en función del tiempo de reacción permitirá inferir sobre la cinética de la decoloración y postular posibles mecanismos de fotodegradación. Con esta propuesta se espera contribuír al desarrollo de un sector industrial importante en nuestra provincia como es el de las industrias del cuero y textil, mediante la generación de nuevas tecnologías que empleen la energía solar para la degradación de sus efluentes (colorantes). En este sentido, se espera desarrollar nuevos materiales optimizados para lograr la mayor eficiencia fotocatalítica. Esto conduciría entonces hacia la remediación de un problema ambiental de alto impacto tanto para nuestra provincia y nuestro país como para la población mundial, como es la contaminación de los recursos hídricos. Finalmente, con este proyecto se contribuirá a la formación de dos doctorandos y un maestrando, cuyos temas de tesis están vinculados con nuestro objeto de estudio. The increasing development of the textile and leather industries in our country, and specifically in Córdoba, has revived an unresolved problem that is the high contamination of water resources. In both industries, the dyeing involves mainly type azoic dyes which are not biodegradable and break releasing carcinogenic aromatic amines. Heterogeneous photocatalysis appears as a new technology that would allow the complete mineralization of these pollutants. Through radiation and a suitable solid it is possible to generate free radicals for efficient oxidation of organic matter (dyes) in aqueous medium. In this respect, mesoporous molecular sieves modified with transition metals are proposed as potential photocatalysts. The main purpose of this project is the synthesis of mesoporous materials having photocatalytic activity for the degradation of dyes. We will try to move their photosensitivity to visible radiation to develop new technologies with lower environmental impact and greater use of solar energy. Materials MCM-41 modified with metals (Fe, Cr, Co, Ni and Zn) will be synthesized by direct incorporation or impregnation. These materials and the siliceous MCM-41 will be then employed as support of TiO2. The materials will be evaluated in the photocatalytic degradation of azoic dyes under visible radiation. The influence of different factors on the photodegradation proccess will be studied. Kinetic studies will be carried out and a possible reaction way will be proposed. Thus, this work will contribute to the advancement of an important industrial sector and the remediation of an environmental problem with high impact for our province and our country. Moreover, this proyect will contribute to the development of two doctoral tesis and one magister tesis which are vinculated with our study subject.

Influence of reactant absorption and dissolution in heterogeneous precipitation processes

In this research work, the aim was to investigate the volumetric mass transfer coefficient [kLa] of oxygen in stirred tank in the presence of solid particle experimentally. The kLa correlations as a function of propeller rotation speed and flow rate of gas feed were studied. The O2 and CO2 absorption in water and in solid-liquid suspensions and heterogeneous precipitation of MgCO3 were thoroughly examined. The absorption experiments of oxygen were conducted in various systems like pure water and in aqueous suspensions of quartz and calcium carbonate particles. Secondly, the precipitation kinetics of magnesium carbonate was also investigated. The experiments were performed to study the reactive crystallization with magnesium hydroxide slurry and carbon dioxide gas by varying the feed rates of carbon dioxide and rotation speeds of mixer. The results of absorption and precipitation are evaluated by titration, total carbon (TC analysis), and ionic chromatrography (IC). For calcium carbonate, the particle concentration was varied from 17.4 g to 2382 g with two size fractions: 5 µm and 45-63 µm sieves. The kLa and P/V values of 17.4 g CaCO3 with particle size of 5µm and 45-63 µm were 0.016 s-1 and 2400 W/m3. At 69.9 g concentration of CaCO3, the achieved kLa is 0.014 s-1 with particle size of 5 µm and 0.017 s-1 with particle size of 45 to 63 µm. Further increase in concentration of calcium carbonate, i.e. 870g and 2382g , does not affect volumetric mass transfer coeffienct of oxygen. It could be concluded from absorption results that maximum value of kLa is 0.016 s-1. Also particle size and concentration does affect the transfer rate to some extend. For precipitation experiments, the constant concentration of Mg(OH)2 was 100 g and the rotation speed varied from 560 to 750 rpm, whereas the used feed rates of CO2 were 1 and 9 L/min. At 560 rpm and feed rate of CO2 is 1 L/min, the maximum value of Mg ion and TC were 0.25 mol/litre and 0.12 mol/litre with the residence time of 40 min. When flow rate of CO2 increased to 9 L/min with same 560 rpm, the achieved value of Mg and TC were 0.3 mol/litre and 0.12 mol/L with shorter residence time of 30 min. It is concluded that feed rate of CO2 is dominant in precipitation experiments and it has a key role in dissociation and reaction of magnesium hydroxide in precipitation of magnesium carbonate.

On universal and periodic β-expansions, and the Hausdorff dimension of the set of all expansions

We study the topology of a set naturally arising from the study of β-expansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of β-expansions are equal and explicitly calculable.

Sparse block-Jacobi matrices with arbitrarily accurate Hausdorff dimension

We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i.e. nonrandom, block-Jacobi matrices may be determined with any degree of precision, improving a result of Zlatos [Andrej Zlatos,. Sparse potentials with fractional Hausdorff dimension, J. Funct. Anal. 207 (2004) 216-252]. (C) 2010 Elsevier Inc. All rights reserved.

Testing the Markov property with ultra high frequency financial data

This paper develops a framework to test whether discrete-valued irregularly-spaced financial transactions data follow a subordinated Markov process. For that purpose, we consider a specific optional sampling in which a continuous-time Markov process is observed only when it crosses some discrete level. This framework is convenient for it accommodates not only the irregular spacing of transactions data, but also price discreteness. Further, it turns out that, under such an observation rule, the current price duration is independent of previous price durations given the current price realization. A simple nonparametric test then follows by examining whether this conditional independence property holds. Finally, we investigate whether or not bid-ask spreads follow Markov processes using transactions data from the New York Stock Exchange. The motivation lies on the fact that asymmetric information models of market microstructures predict that the Markov property does not hold for the bid-ask spread. The results are mixed in the sense that the Markov assumption is rejected for three out of the five stocks we have analyzed.

Hausdorff dimension of non-hyperbolic repellers. I: Maps with holes

This is the first paper in a two-part series devoted to studying the Hausdorff dimension of invariant sets of non-uniformly hyperbolic, non-conformal maps. Here we consider a general abstract model, that we call piecewise smooth maps with holes. We show that the Hausdorff dimension of the repeller is strictly less than the dimension of the ambient manifold. Our approach also provides information on escape rates and dynamical dimension of the repeller.

Hausdorff dimension for non-hyperbolic repellers II: Da diffeomorphisms

We study non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomorphisms with unstable dimension 2 through a Hopf bifurcation. Using some recent abstract results about non-uniformly expanding maps with holes, by ourselves and by Dysman, we show that the Hausdorff dimension and the limit capacity (box dimension) of the repeller are strictly less than the dimension of the ambient manifold.

On the Mellin Transforms of Dirac’S Delta Function, The Hausdorff Dimension Function, and The Theorem by Mellin

Mathematics Subject Classification: 44A05, 46F12, 28A78

Chemodiversity and molecular plasticity: recognition processes as explored by property spaces.

In the last few years, a need to account for molecular flexibility in drug-design methodologies has emerged, even if the dynamic behavior of molecular properties is seldom made explicit. For a flexible molecule, it is indeed possible to compute different values for a given conformation-dependent property and the ensemble of such values defines a property space that can be used to describe its molecular variability; a most representative case is the lipophilicity space. In this review, a number of applications of lipophilicity space and other property spaces are presented, showing that this concept can be fruitfully exploited: to investigate the constraints exerted by media of different levels of structural organization, to examine processes of molecular recognition and binding at an atomic level, to derive informative descriptors to be included in quantitative structure--activity relationships and to analyze protein simulations extracting the relevant information. Much molecular information is neglected in the descriptors used by medicinal chemists, while the concept of property space can fill this gap by accounting for the often-disregarded dynamic behavior of both small ligands and biomacromolecules. Property space also introduces some innovative concepts such as molecular sensitivity and plasticity, which appear best suited to explore the ability of a molecule to adapt itself to the environment variously modulating its property and conformational profiles. Globally, such concepts can enhance our understanding of biological phenomena providing fruitful descriptors in drug-design and pharmaceutical sciences.

Can the fractal dimension be applied for the early diagnosis of non-proliferative diabetic retinopathy?

The fractal dimension has been employed as a useful parameter in the diagnosis of retinal disease. Avakian et al. (Curr Eye Res 2002; 24: 274-280), comparing the vascular pattern of normal patients with mild to moderate non-proliferative diabetic retinopathy (NPDR), found a significant difference between them only in the macular region. This significant difference in the box-counting fractal dimension of the macular region between normal and mild NPDR patients has been proposed as a method of precocious diagnosis of NPDR. The aim of the present study was to determine if fractal dimensions can really be used as a parameter for the early diagnosis of NPDR. Box-counting and information fractal dimensions were used to parameterize the vascular pattern of the human retina. The two methods were applied to the whole retina and to nine anatomical regions of the retina in 5 individuals with mild NPDR and in 28 diabetic but opthalmically normal individuals (controls), with age between 31 and 86 years. All images of retina were obtained from the Digital Retinal Images for Vessel Extraction (DRIVE) database. The results showed that the fractal dimension parameter was not sensitive enough to be of use for an early diagnosis of NPDR.