On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.

Teaching as a fractal: from experience to model

The aim of this work is to improve students’ learning by designing a teaching model that seeks to increase student motivation to acquire new knowledge. To design the model, the methodology is based on the study of the students’ opinion on several aspects we think importantly affect the quality of teaching (such as the overcrowded classrooms, time intended for the subject or type of classroom where classes are taught), and on our experience when performing several experimental activities in the classroom (for instance, peer reviews and oral presentations). Besides the feedback from the students, it is essential to rely on the experience and reflections of lecturers who have been teaching the subject several years. This way we could detect several key aspects that, in our opinion, must be considered when designing a teaching proposal: motivation, assessment, progressiveness and autonomy. As a result we have obtained a teaching model based on instructional design as well as on the principles of fractal geometry, in the sense that different levels of abstraction for the various training activities are presented and the activities are self-similar, that is, they are decomposed again and again. At each level, an activity decomposes into a lower level tasks and their corresponding evaluation. With this model the immediate feedback and the student motivation are encouraged. We are convinced that a greater motivation will suppose an increase in the student’s working time and in their performance. Although the study has been done on a subject, the results are fully generalizable to other subjects.

Estudio del comportamiento fractal espacial de la superficie del terreno, considerando factores del relieve climáticos, estructurales y litológicos

El presente estudio parte de la hipótesis de que hay una relación numérica entre la forma de la superficie geomórfica y los factores de modelado del relieve. La metodología utilizada es el análisis multifractal de las curvas de nivel de Modelos Digitales del Terreno de toda España. Para ello hemos utilizado software libre en todos los pasos. Los resultados obtenidos hacen pensar que esta correlación existe, en el factor climático (gradiente de precipitación máxima anual) y estructural (peligrosidad sísmica). El factor litológico no ha dado ajuste algo, probablemente debido a la falta de datos precisos.

Confidentiala "Mode 3" Systems Approach for Knowledge Creation, Diffusion and Use: Towards a 21st Century Fractal Innovation Ecosystem. ACES Working Paper (Report) No. 4, 2007

"Mode 3" allows and emphasizes the co-existence and co-evolution of different knowledge and innovation paradigms: the competitiveness and superiority of a knowledge system is highly determined by its adaptive capacity to combine and integrate different knowledge and innovation modes via co-evolution, co-specialization and coopetition [sic] of knowledge stock and flow dynamics. What results is an emerging fractal knowledge and innovation ecosystem, well-configured for the knowledge economy and society. The intrinsic litmus test of the capacity of such an ecosystem to survive and prosper in the context of continually glocalizing [sic] and intensifying competition represents the ultimate competitiveness benchmark with regards to the robustness and quality of the ecosystem's knowledge and innovation architecture and topology.

Fractal analysis of pyramidal cells in the visual cortex of the galago (Otolemur garnetti): Regional variation in dendritic branching patterns between visual areas

Previously it has been shown that the branching pattern of pyramidal cells varies markedly between different cortical areas in simian primates. These differences are thought to influence the functional complexity of the cells. In particular, there is a progressive increase in the fractal dimension of pyramidal cells with anterior progression through cortical areas in the occipitotemporal (OT) visual stream, including the primary visual area (V1), the second visual area (V2), the dorsolateral area (DL, corresponding to the fourth visual area) and inferotemporal cortex (IT). However, there are as yet no data on the fractal dimension of these neurons in prosimian primates. Here we focused on the nocturnal prosimian galago (Otolemur garnetti). The fractal dimension (D), and aspect ratio (a measure of branching symmetry), was determined for I I I layer III pyramidal cells in V1, V2, DL and IT. We found, as in simian primates, that the fractal dimension of neurons increased with anterior progression from V1 through V2, DL, and IT. Two important conclusions can be drawn from these results: (1) the trend for increasing branching complexity with anterior progression through OT areas was likely to be present in a common primate ancestor, and (2) specialization in neuron structure more likely facilitates object recognition than spectral processing.

Fractal effects on the measurement of the specific surface areas of single-walled carbon nanotubes

The specific surface area (SSA) of single-walled carbon nanotubes (SWNTs) has been measured by different groups. Fujiwara et al. measured the SSA of SWNT bundles by using nitrogen and oxygen as adsorbates, and found that the SSA from O2-adsorption was 6.6% larger than that from N2-adsorption for the same SWNT sample [1]. Also Wei et al. [2] measured the SSA of HiPco SWNTs by using O2, N2 and Ar, and found that, for the same samples, Vm(Ar) > Vm(O2) > Vm(N2), here Vm is the monolayer adsorption capacity at the standard conditions of temperature and pressure (STP). Those research results indicate that, for the same SWNT sample, its measured surface area depends on the employed adsorbate.

Fractal PBG assisted co-planar waveguides

In general, conventional electromagnetic bandgap (PBGs) with uniform distribution show spurious ripples in pass-band and poor stop-band responses. This paper presents a detailed investigation in terms of pass-band and stop-band characteristics of uniplanar transmission line loaded with fractal shape PBGs. (c) 2005 Wiley Periodicals, Inc.

Fractal analysis as a tool for studying specialization in neuronal structure: The study of the evolution of the primate cerebral cortex and human intellect

We review recent findings that, using fractal analysis, have demonstrated systematic regional and species differences in the branching complexity of neocortical pyramidal neurons. In particular, attention is focused on how fractal analysis is being applied to the study of specialization in pyramidal cell structure during the evolution of the primate cerebral cortex. These studies reveal variation in pyramidal cell phenotype that cannot be attributed solely to increasing brain volume. Moreover, the results of these studies suggest that the primate cerebral cortex is composed of neurons of different structural complexity. There is growing evidence to suggest that regional and species differences in neuronal structure influence function at both the cellular and circuit levels. These data challenge the prevailing dogma for cortical uniformity.

Hopping transport on a fractal: ac conductivity of porous silicon

We have measured the frequency dependence of the conductivity and the dielectric constant of various samples of porous Si in the regime 1 Hz-100 kHz at different temperatures. The conductivity data exhibit a strong frequency dependence. When normalized to the dc conductivity, our data obey a universal scaling law, with a well-defined crossover, in which the real part of the conductivity sigma' changes from an sqrt(omega) dependence to being proportional to omega. We explain this in terms of activated hopping in a fractal network. The low-frequency regime is governed by the fractal properties of porous Si, whereas the high-frequency dispersion comes from a broad distribution of activation energies. Calculations using the effective-medium approximation for activated hopping on a percolating lattice give fair agreement with the data.

On Parabolic Subgroups and Hecke Algebras of some Fractal Groups

* The authors thank the “Swiss National Science Foundation” for its support.

Are European equity markets efficient? New evidence from fractal analysis

We report an empirical analysis of long-range dependence in the returns of eight stock market indices, using the Rescaled Range Analysis (RRA) to estimate the Hurst exponent. Monte Carlo and bootstrap simulations are used to construct critical values for the null hypothesis of no long-range dependence. The issue of disentangling short-range and long-range dependence is examined. Pre-filtering by fitting a (short-range) autoregressive model eliminates part of the long-range dependence when the latter is present, while failure to pre-filter leaves open the possibility of conflating short-range and long-range dependence. There is a strong evidence of long-range dependence for the small central European Czech stock market index PX-glob, and a weaker evidence for two smaller western European stock market indices, MSE (Spain) and SWX (Switzerland). There is little or no evidence of long-range dependence for the other five indices, including those with the largest capitalizations among those considered, DJIA (US) and FTSE350 (UK). These results are generally consistent with prior expectations concerning the relative efficiency of the stock markets examined. © 2011 Elsevier Inc.