Counting Dimension Research Articles

On the dimensions of attractors of random self-similar graph directed iterated function systems

In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, V -variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of 1 -variable and \infty -variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separation condition we give an expression for the almost sure Hausdorff and Assouad dimension.

Mixed labyrinth fractals

Labyrinth fractals are self-similar fractals that were introduced and studied in recent work [2,3]. In the present paper we define and study more general objects, called mixed labyrinth fractals, that are in general not self-similar and are constructed by using sequences of labyrinth patterns. We show that mixed labyrinth fractals are dendrites and study properties of the paths in the graphs associated to prefractals, and of arcs in the fractal, e.g., the path length, and the box counting dimension and length of arcs. We also consider more general objects related to mixed labyrinth fractals, formulate two conjectures about arc lengths, and establish connections to recent results on generalised Sierpiński carpets.

Numerical Simulation on Propagation of a Ubiquitiform Crack in Rock Materials

In this paper, quasi-static crack propagation in the rock materials is modelled using the extended element method (X-FEM). In the proposed model, the heterogeneity of material properties was characterized by the Weibull distribution. The complexity of the ubiquitiformal crack can then be determined by the box counting dimension. A ubiquitiformal fracture surface which is resulted from the heterogeneity of the material properties is obtained, which is more consistent with experimental result than the fracture surface obtained in the homogenous materials. Moreover, the calculated numerical result of the complexity is found to be in good agreement with previous experimental data.

Mechanical and fatigue properties of pearlitic ductile iron castings characterized by long solidification times

Abstract In this paper, the fatigue behaviour of heavy section pearlitic ductile iron castings has been investigated. The inoculation treatment has been changed for each casting in order to investigate its influence on the mechanical and fatigue properties of the materials. Tensile tests and axial fatigue tests under nominal ratio R = 0.01 have been performed on specimens taken from the core of casting components characterized by long solidification times. Scanning Electron Microscopy has been used to investigate the fracture surface of the broken samples in order to identify crack initiation points and fracture mechanisms. Metallographic analyses have been carried out to measure nodule count and nodules dimensions and to identify matrices structures. It has been found that fatigue behaviour is strongly influenced by defects, such as microshrinkages or degenerated graphite particles near to specimens' surface. It has been also found that inoculation process influences the microstructure and the fatigue resistance of heavy section pearlitic ductile iron castings.

Mono-fractal analysis of CuXO films: Texture and phase distribution

Several studies on thermally-grown copper oxide films dealing with their crystalline properties and main phases as functions of the growth temperature have been published; notwithstanding, few research works have proposed a description of the surface morphology of the films using mono-fractal analysis. In this work, copper oxide films were grown by direct thermal oxidation in atmospheric air at different temperatures and times to obtain experimental proof of the phases present in the films by XRD, FTIR and Raman. The textural properties of the copper oxide films were analyzed from atomic force microscopy (AFM) images. The box counting and information fractal dimensions were used to perform the mono-fractal analysis to establish a correlation between the textural properties and the phases of the CuxO films.

Prevalence and clinical correlates of sensory phenomena in obsessive compulsive disorder

IntroductionA substantial number of patients suffering from obsessive compulsive disorder (OCD) report a subjective distressing experience prior to the repetitive behavior, known as sensory phenomena(SP).ObjectivesNeed to systematically evaluate SP and the clinical correlates in OCD.AimsAssess prevalence of SP and clinical correlates in OCD.MethodsSubjects (n = 71) fulfilling the criteria for DSM IV-TR OCD were recruited consecutively from a specialty OCD clinic in Southern India and were assessed using the Yale brown obsessive and compulsive scale (YBOCS), dimensional Yale-Brown obsessive compulsive scale (D-YBOCS) and the University of São Paulo Sensory Phenomena Scale (USP-SPS).ResultsThe prevalence of the SP was found to be 50.7%. Prevalence of SP is significantly greater in the patients with early age of onset (P = 0.47). In subtypes of SP, Tactile was 12.7%, “just right” for look was 26.8%, “just right” for sound was 9.9%, “just right” for feeling was 16.9%, feeling of incompleteness leading to repetitive behavior was 22.5%, “energy release” sensation leading to repetitive behavior was 4.2% and “urge only” leading to repetitive behavior was 11.3%. SP was found to have significant correlation with symmetry/ordering/arranging/counting dimension (P = 0.003). Significant positive correlation existed between SP severity and the severity of the compulsions (P = 0.02).ConclusionConsidering its high prevalence in OCD, it might be useful to incorporate SP assessment during the routine clinical assessment of OCD. It might warrant a place in the phenomenological and nosological description of OCD. Additionally, the neurobiological correlates of SP need to be explored.Disclosure of interestThe authors have not supplied their declaration of competing interest.

Minimal partition coverings and generalized dimensions of a complex network

Computing the generalized dimensions Dq of a complex network requires covering the network by a minimal number of “boxes” of size s. We show that the current definition of Dq is ambiguous, since there are in general multiple minimal coverings of size s. We resolve the ambiguity by first computing, for each s, the minimal covering that is summarized by the lexicographically minimal vector x(s). We show that x(s) is unique and easily obtained from any box counting method. The x(s) vectors can then be used to unambiguously compute Dq. Moreover, x(s) is related to the partition function, and the first component of x(s) can be used to compute D∞ without any partition function evaluations. We compare the box counting dimension and D∞ for three networks.

The Fractal Characteristics of Drainage Networks and Erosion Evolution Stages of Ten Kongduis in the Upper Reaches of the Yellow River, China

Abstract: The fractal characteristics of drainage in the ten kongduis of the upper Yellow River were obtained using the box counting dimension, and the evolution stages of the watershed topography were defined by different ranges of the fractal dimensions of river networks (Dg). The results show that the fractal scaleless range of the Maobula River is 20–370 m based on a combination of artificial judgment, correlation coefficient test and fitting error. Other kongduis show good fractal characteristics in this fractal scaleless range as well. The box counting dimension can be used as a quantitative index of watershed topography fractal characteristics. The fractal dimension of stream networks is independent of the threshold contributing area used for extracting the drainage networks from the DEM. The values of Dg in the upper ten kongduis are in the range of 1.08–1.14. Both the runoff yield and the sediment yield are positively and linearly related with Dg. The positive relation between the sediment yield ...

Novel fractal characteristic of atomic force microscopy images

Fractal dimension (DF) is one of the important parameters in the description of object's properties in different fields including biology and medicine. The present paper is focused on the application of the fractal dimension (the box counting dimension) in the analysis of the properties of cell surface on the base of its images obtained by atomic force microscopy (AFM). Fractal dimension of digital 3D AFM images depends on interpoint distances determined by the scanning step in the XY-plane and Z-scale factor t. We have studied the dependence of DF of AFM images on the Z-scale factor (DF=φ(t)) with purpose to reveal the features of the dependence and its usefulness in the analysis of the maps of surface properties. Using the model digital surfaces such as the plane, sinusoidal surfaces and “hilly” surface, we revealed that the sizes and spatial frequency of surface structural elements determined the basic features of the dependence (the parameters of peaks on the curve DF=φ(t)) and the element of chance in the localization of the structural elements on the surface had no significant influence on the dependence. Our findings demonstrate that the dependence of the fractal dimension on the Z-scale factor characterizes the structure of the AFM images more comprehensively than the roughness Ra and fractal dimension DF evaluated at a certain t. The dependence DF=φ(t) can be considered as a novel characteristic of AFM images. On analyzing the AFM images (lateral force maps) of glutaraldehyde-fixed adhered human fibroblasts and A549 human lung epithelial cells we found the significant difference in the dependences DF=φ(t) for different cell types that could be related to the difference of structural and mechanical surface properties of the studied cells.

Fatigue strength improvement of heavy-section pearlitic ductile iron castings by in-mould inoculation treatment

Casting defects like microshrinkage porosity or degenerate graphite play a major role in fatigue behaviour of ductile cast iron. This phenomenon is even more evident in heavy-section ductile iron castings where defects are more frequent and difficult to control. This work is aimed to investigate the fatigue behaviour of heavy-section pearlitic ductile iron castings obtained with different in-mould inoculation treatments. It has been found that the inoculation process modifies the microstructure of the alloy, dimension and morphology of defects. The higher the nodule count, the lower the microshrinkage cavities dimensions.The fracture surface of broken samples has been investigated by means of Scanning Electron Microscopy in order to identify crack initiation points and fracture mechanisms. Metallographic analyses have been carried out to measure nodule count and nodules dimensions and to identify the matrix microstructure. Statistics of extreme values and Murakami approach have been used to analyse the influence of defects on the fatigue behaviour of the materials analysed. It was found that a synergic effect between degenerate graphite particles and microshrinkage porosities reduces the fatigue strength of pearlitic ductile iron castings.

Maximal entropy coverings and the information dimension of a complex network

Computing the information dimension dI of a complex network G requires covering G by a minimal collection of “boxes” of size s to obtain a set of probabilities, computing the entropy H(s), and quantifying how H(s) scales with log⁡s. We show that to determine whether dI≤dB holds for G, where dB is the box counting dimension, it is not sufficient to determine a minimal covering for each s. We introduce the new notion of a maximal entropy minimal covering of G, and a corresponding new definition of dI. The use of maximal entropy minimal coverings in many cases enhances the ability to compute dI.

Random affine code tree fractals: Hausdorff and affinity dimensions and pressure

AbstractWe prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems are equal to the zero of the pressure. In particular, we do not presume the validity of the Falconer–Sloan condition or any other additional assumptions which have been essential in all the previously known results.

Minimal box size for fractal dimension estimation

We extend Kenkel’s model for determining the minimal allowable box size s* to be used in computing the box counting dimension of a self-similar geometric fractal. This minimal size s* is defined in terms of a specified parameter e which is the deviation of a computed slope from the box counting dimension. We derive an exact implicit equation for s* for any e. We solve the equation using binary search, compare our results to Kenkel’s, and illustrate how s* varies with e. A listing of the Python code for the binary search is provided. We also derive a closed form estimate for s* having the same functional form as Kenkel’s empirically obtained expression.

Marstrand-type Theorems for the Counting and Mass Dimensions in ℤ

The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are $$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$ where ⌊A⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ $\mathbb{R}^d$ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings: $$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$ where $${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha \ \bigg| \ 1 \leq \|C_i\| \leq r \|C\| \biggr\}$$ in which the infimum is taken over cubic coverings {Ci} of A ∩ C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ $\mathbb{R}^d$ under orthogonal projections with range of dimension k have counting dimension at least min(k, D(A)); if we assume D(A) = D(A), then the mass dimension of A under the typical orthogonal projection is equal to min(k, D(A)). This work extends recent work of Y. Lima and C. G. Moreira.

Exploring the topological sources of robustness against invasion in biological and technological networks.

For a network, the accomplishment of its functions despite perturbations is called robustness. Although this property has been extensively studied, in most cases, the network is modified by removing nodes. In our approach, it is no longer perturbed by site percolation, but evolves after site invasion. The process transforming resident/healthy nodes into invader/mutant/diseased nodes is described by the Moran model. We explore the sources of robustness (or its counterpart, the propensity to spread favourable innovations) of the US high-voltage power grid network, the Internet2 academic network, and the C. elegans connectome. We compare them to three modular and non-modular benchmark networks, and samples of one thousand random networks with the same degree distribution. It is found that, contrary to what happens with networks of small order, fixation probability and robustness are poorly correlated with most of standard statistics, but they depend strongly on the degree distribution. While community detection techniques are able to detect the existence of a central core in Internet2, they are not effective in detecting hierarchical structures whose topological complexity arises from the repetition of a few rules. Box counting dimension and Rent’s rule are applied to show a subtle trade-off between topological and wiring complexity.

Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians

We consider the spectrum of discrete Schrödinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.

Sets which are not tube null and intersection properties of random measures

We show that in R d there are purely unrectifiable sets of Hausdorff (and even box counting) dimension d − 1 which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same authors and by Carbery. Our method extends also to ‘convex tube null sets’, establishing a contrast with a theorem of Alberti, Csörnyei and Preiss on Lipschitz-null sets. The sets we construct are random, and the proofs depend on intersection properties of certain random fractal measures with curves.

Experiment on Water Absorbing and Surface Pore Property of Concrete

In the purpose of finding the relationship between the pore property and water absorbing property of concrete of different strength, scanning electron microscope (SEM for shot) is used to observe the pore in the concrete as well as computer software is used to gain the box counting dimension, water absorbing experiment is also done to the concrete to figure out the water absorbing curve. It's found that the water absorbing property has nothing to do with the strength of concrete; C30 concrete has the lowest box counting dimension and poorest ability absorbing water among the concrete C20, C30 and C40.

Determination of reservoir properties through the use of computed X-ray microtomography – eolian sandstone examples

A possibility of examination of the rock pore structure by means of the X-ray computed microtomography was presented. Parameters characterizing the pore structure, i.e. porosity and coefficient of homogeneity of pore structure by local porosity examination and box counting dimension and mean chord length and normalized Euler number and coordination number were determined through pore structure image analyses. Complementary methods as pycnometry, mercury injection porosimetry and NMR were used to determine porosity and other factors to make the comparison and determine mutual relationships between petrophysical properties obtained from various sources.The study covered Rotliegend sandstones of eolian origin. Reservoir properties laboratory investigations were focused in three areas located in the marginal part of the eastern erg and middle part of the Polish Lowland Permian Basin. Rotliegend sediments in the study showed changeability in petrography and reservoir properties so the data were grouped into three parts according to geological regions. Computed X-ray microtomography gave differentiation of the investigated areas with respect to their pore structure and porosity development. This differentiation was confirmed by means of other applied laboratory methods.

The gender similarities hypothesis is untestable as formulated.

Comments on the original article by Zell et al. (see record 2015-00137-002) regarding gender similarities and differences using a metasynthesis approach. The authors concluded that the average difference between males and females across psychological domains is relatively small (d = 0.21), with the majority of effects being either small or very small, thereby supporting Hyde's (2005) gender similarities hypothesis that males and females are similar on most psychological dimensions. Unfortunately, their study suffers from the same flaw as nearly all studies testing that hypothesis, namely that it does not provide a metric for either the psychological importance or relevance of the dimensions considered nor does it supply an individuation rule for counting dimensions.