Effective Hausdorff Dimension in General Metric Spaces

We introduce the concept of effective dimension for a wide class of metric spaces whose metric is not necessarily based on a measure. Effective dimension was defined by Lutz (Inf. Comput., 187(1), 49–79, 2003) for Cantor space and has also been extended to Euclidean space. Lutz effectivization uses gambling, in particular the concept of gale and supergale, our extension of Hausdorff dimension to other metric spaces is also based on a supergale characterization of dimension, which in practice avoids an extra quantifier present in the classical definition of dimension that is based on Hausdorff measure and therefore allows effectivization for small time-bounds. We present here the concept of constructive dimension and its characterization in terms of Kolmogorov complexity, for which we extend the concept of Kolmogorov complexity to any metric space defining the Kolmogorov complexity of a point at a certain precision. Further research directions are indicated

An excursion to the Kolmogorov random strings

AbstractWe study the sets of resource-bounded Kolmogorov random strings:Rt={x|Ct(n)(x)⩾|x|} fort(n)=2nk. We show that the class of sets that Turing reduce toRthas measure 0 inEXPwith respect to the resource-bounded measure introduced by Lutz. From this we conclude thatRtis not Turing-complete forEXP. This contrasts with the resource-unbounded setting. ThereRis Turing-complete forco-RE. We show that the class of sets to whichRtbounded truth-table reduces, hasp2-measure 0 (therefore, measure 0 inEXP). This answers an open question of Lutz, giving a natural example of a language that is not weakly complete forEXPand that reduces to a measure 0 class inEXP. It follows that the sets that are ⩽pbbt-hard forEXPhavep2-measure 0

Dimension Spectra of Lines

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim(a, b) is equal to the effective packing dimension Dim(a, b), then sp(L) contains a unit interval. We also show that, if the dimension dim(a, b) is at least one, then sp(L) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite

Universal fluctuations in subdiffusive transport

Subdiffusive transport in tilted washboard potentials is studied within the fractional Fokker-Planck equation approach, using the associated continuous time random walk (CTRW) framework. The scaled subvelocity is shown to obey a universal law, assuming the form of a stationary Levy-stable distribution. The latter is defined by the index of subdiffusion alpha and the mean subvelocity only, but interestingly depends neither on the bias strength nor on the specific form of the potential. These scaled, universal subvelocity fluctuations emerge due to the weak ergodicity breaking and are vanishing in the limit of normal diffusion. The results of the analytical heuristic theory are corroborated by Monte Carlo simulations of the underlying CTRW

Análisis filogenético molecular: Diseño e implementación de algoritmos escalables y fiables y verificación automática de propiedades de una filogenia

La filogenética es la ciencia que estudia las relaciones entre organismos basándose en suproximidad evolutiva. La forma más visual y conveniente de representar estas relaciones es através de los árboles filogenéticos. El análisis filogenético es un proceso formado por distintasetapas cuya finalidad es poder reconstruir dichos árboles. Estas etapas pueden incluir: estudiode modelos evolutivos (modelos matemáticos que intentan explicar de la forma más fielposible la evolución real de los organismos), análisis estadístico, alineamiento de secuencias,...Actualmente el coste computacional limita de forma práctica tanto la realización de filogeniasextensivas (tratando miles y decenas de miles de secuencias) como la aplicación de modelosevolutivos más generales, interesantes y explicativos que el modelo uniforme (usado entamaños de problema reducidos). Por otro lado, los procesos de secuenciación de cadenasbiológicas no están exentos de errores, los cuales pueden aparecer en cualquier lugar de lasecuencia. Hasta la fecha, todo proceso de verificación requiere la actuación manual de unexperto en el campo, lo que es un proceso muy costoso. El trabajo aborda tres aspectosespecíficos: i) el desarrollo e implementación de un sistema de inferencia filogenética queconcentra varios métodos sobre análisis de secuencias y estudio de filogenias que no se habíanunido hasta el momento; ii) el desarrollo e implementación de una aplicación para la detecciónautomática de errores en las cadenas obtenidas en los procesos de secuenciación; iii) elestudio teórico de nuevos algoritmos para la caracterización de problemas entre aquellos quese consideran como no resolubles en la actualidad

New insights on the seismogenic potential of the Eastern Betic Shear Zone (SE Iberia): Quaternary activity and paleoseismicity of the SW segment of the Carrascoy Fault Zone

The Carrascoy Fault (CAF) is one of the main active faults that form part of the Eastern Betic Shear Zone, a 450 km fault system that accommodates most of the convergence between the Eurasian (Iberia) and Nubian plates in the Betic Cordillera, south Spain. Although the CAF represents a major earthquake threat to the nearby City of Murcia, studies on its Quaternary tectonics and seismogenic potential are scarce to date. We present evidence that supports the division of the CAF into two overlapping segments with contrasting tectonic structure, Quaternary activity, and landform control: a SW segment, characterized by a broad fold-and-thrust zone similar to the forebergs defined in the Gobi-Altai region, and a NE segment, characterized by a sharp mountain front controlled by strike-slip tectonics. We attribute the differentiation into these two segments to the stresses associated with topography, which in turn is a consequence of the shortening component, at the middle Pleistocene, after circa 217.4 ka. For the SW segment we infer the occurrence of 9 to 11, Mw 6.7 paleoearthquakes in the last 30.2 kyr, and a slip rate of 0.37 ± 0.08 m/kyr. We date the occurrence of the last surface rupture event after 2750 B.P., and we estimate an average recurrence period of major events of 3.3 ± 0.7 kyrThis work was supported by SISMOGEN (IGME, 2279) and FASEGEO (CGL2009-09726) research projects and a technical assistance of the Civil Protection Service of Murci

Constructive Dimension and Turing Degrees

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim_H(S) and constructive packing dimension dim_P(S) is Turing equivalent to a sequence R with dim_H(R) <= (dim_H(S) / dim_P(S)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0, then dim_P(R) >= 1 - epsilon. The reduction thus serves as a *randomness extractor* that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim_H(S) / dim_P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim_H(S) = dim_P(S)) such that dim_H(S) > 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.Comment: The version of this paper appearing in Theory of Computing Systems, 45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to insufficient care with the choice of delta. This version modifies that proof to fix the error

Impact of the coronavirus pandemic on maxillofacial trauma:a retrospective study in southern Spain

The coronavirus pandemic has impacted health systems worldwide, with Spain being one of the most affected countries. However, little is known about the extent to which the effects of staying home, social distancing, and quarantine measures have influenced the epidemiology of patients with maxillofacial trauma. The aim of this study was to analyze the impact of the coronavirus pandemic on the incidence, demographic patterns, and characteristics of maxillofacial fractures in the largest hospital in southern Spain. Data from patients who underwent surgery for maxillofacial fractures during the first year of the pandemic between 16 March 2020 and 14 March 2021 (pandemic group) were retrospectively compared with a control group during the equivalent period of the previous year (pre-pandemic group). The incidence was compared by weeks and by lockdown periods of the population. Demographic information, aetioloy, fracture characteristics, treatment performed, and days of preoperative stay were evaluated. Descriptive and bivariate statistics were calculated (p<0.05). During the first year of the pandemic, there was a 35.2% reduction in maxillofacial fractures (n=59) compared to the pre-pandemic year (n=91, p=0.040). A significant drop was detected during the total home lockdown period of the population (p=0.028). In the pandemic group, there was a reduction in fractures due to interpersonal aggressions, an increase in panfacial fractures, a significant increase in other non-facial injuries associated with polytrauma (p=0.037), a higher number of open reduction procedures with internal fixation, and a significantly longer mean preoperative stay (p=0.016). The first pandemic year was associated with a decline in the frequency of maxillofacial trauma and a change in the pattern and characteristics of fractures. Inter-annual epidemiological knowledge of maxillofacial fractures may be useful for more efficient planning of resource allocation and surgical practice strategy during future coronavirus outbreaks and population lockdowns