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The development of an efficient image-based computer identification system for plants or other organisms is an important ambitious goal, which is still far from realization. This paper presents three new methods potentially usable for such a system: fractal-based measures of complexity of leaf outline, a heuristic algorithm for automatic detection of leaf parts — the blade and the petiole, and a hierarchical perceptron — a kind of neural network classifier. The next few sets of automatically extractable features of leaf blades, encompassed those presented and/or traditionally used, are compared in the task of plant identification using the simplest known "nearest neighbor" identification algorithm, and more realistic neural network classifiers, especially the hierarchical. We show on two real data sets that the presented techniques are really usable for automatic identification, and are worthy of further investigation.

The electronic packaging and systems are very important topics as the limitation of miniaturization approaches in semiconductor industry. Regarding the optimal materials microstructure for these applications, we studied different alloys such as Sn-3.0Ag-0.5Cu (wt.%)/organic solderability preservative (SAC305/OSP) Cu and SAC305–0.05Ni/OSP Cu solder joints. We implemented the fractal dimension characterization and microstructure morphology reconstruction. This is the first time that we applied fractals on such alloys. The morphology reconstruction is important for predicting and designing the optimal microstructure for the advanced desirable properties these alloys. These analyzed parameters are important for the hand-held devices and systems especially for the exploitation. The fractal reconstruction was applied on the prepared microstructures with five different magnifications. The results confirmed successful application of fractals in this area of materials science considering the grains and shapes reconstructions.

Physicist Clint Sprott demonstrated a relationship between aesthetic judgments of fractal images and their fractal dimensions [1993]. Scott Draves, aka Spot, a computer scientist and artist, has created a space of images called fractal flames, based on attractors of two-dimensional iterated function systems. A large community of users run software that automatically downloads animated fractal flames, known as "sheep", and displays them as their screen-saver. The users may vote electronically for the sheep they like while the screen-saver is running. In this report we proceed from Sprott to Spot. The data show an inverted U-shaped curve in the relationship between aesthetic judgments of flames and their fractal dimension, confirming and clarifying earlier reports.

Tanpura is a multi-stringed accompanying drone instrument extensively used in classical music in India. The instrument is plucked by finger. We know that jitter, shimmer and complexity perturbations (CP) are found also in tanpura signals. The source of origin of these perturbations was reported to be related to some sort of nonlinearity associated with the strings and their mode of attachment.

The objective of the present study is to see in what way fractal-dimensional analysis may be helpful to relate the apparent nonlinearity and also if there is any relation of these dimensions with different part of the signals like attack time, quasi-steady state and the decay.

In the paper fractal dimension, D₀ and generalized dimensions, D₂ to D₁₀ and their dynamic behavior over time are studied for 15 tanpuras having four strings each, tuned to Pa (5th note), Sa (tonic), Sa (tonic) and Sa′ (lower octave tonic). The obeying of power law indicates nonlinearity in the production source system.

The presence of multifractality is studied through an examination of relationship between q and D_q and the functional relationship between D_qs.

Scanning electron microscopy (SEM) is of great importance for studying fractal permeability. In this work, we presented a new technique, by applying the high-order upwind compact difference schemes to solve the hyperbolic conservation laws, to enhance textural differences for accurate segmentation of the SEM images. From the enhanced SEM images, the channels and pores can be obtained by using the two-stage image segmentation. Combining with the box counting method, the key parameters for evaluation of the fractal permeability such as the tortuosity fractal dimension, the pore area fractal dimension and the maximum pore area can be derived from the segmented images. Application of the technique to the SEM images of a red sandstone from south China shows remarkable enhancement of edge details, allowing the more accurate segmentation of the SEM images. Rather than the original image algorithm, the fractal permeability derived from this new approach is closer to the experimental value, especially when the magnification falls in the range of 500–600. The results evidence that our enhanced images approach may provide stronger constraints on evaluations of permeability of sandstones.

In recent years, because complex networks can be used to model real-world complex systems, such as the Internet, urban infrastructure networks, and gene interaction networks, such research has been widely applied in engineering, social sciences, and life sciences and has caused widespread concern. Fractal dimension, as a concept concerning the filling ability and complexity of an object space, has great significance for the study of the robustness of complex networks. This paper studies the relationship between fractal dimension and the robustness of different types of complex networks from the perspective of network structure and network scale. We find that fractal dimension is strongly correlated with robustness under certain conditions and can be used as an important index to evaluate the robustness of complex networks.

Visual sense has an important role in shaping human understanding of the natural world. Nevertheless, it is not clear how the complexity of visual stimuli influences the complexity of information processing in the brain. In this study, we hypothesized that changes in the fractal pattern of electroencephalogram (EEG) signals directly follow the changes in the fractal dimension of animations. Therefore, 12 types of 2D fractal animations were presented to a group of healthy students (15 males, $25.93\pm 1.79$ years old, 3 left-handed) while their brain signals were recorded using a 32-channel amplifier. Regression analysis between the fractal dimension of EEG signals and the fractal dimension of animations indicated that the complexity of fractal animations is directly sensed by changes in the fractal dimension of EEG signals at the centro-parietal and parietal regions. It may indicate that when the complexity of visual stimuli increases, the mechanism of information processing in the brain also enhances its complexity to better attend to and comprehend the stimuli.

In this paper, we have explored the local structure and fractal characteristics of fractal functions with certain fractal dimensions. The conclusion that points with inconsistent oscillation amplitudes with the upper Box dimension of the corresponding fractal functions have been proved to be nowhere dense. This will play an important supporting role in exploring the fractal dimension estimation of the combination of fractal functions.

The C/SiC composite is a promising material for ablation-resistant thermal protection in near-space hypersonic environments. The formation of an SiO₂ oxide layer through passive oxidation on the surface of the composite is a significant factor influencing its performance. It is essential to accurately predict the thickness of the SiO₂ oxide layer and the recession and mass loss of the C/SiC composite during passive oxidation. The SiO₂ oxide layer is a typical porous media exhibiting self-similarity and thus fractal theory can be applied to establish the relation between the oxygen flow rate and microstructural parameters of the oxide layer. The Weierstrass–Mandelbrot (WM) function is employed to simulate the rough interfaces between the SiO₂ oxide layer and the C/SiC composite to evaluate the influence of the fractal dimensions of the oxide layer on the performance of thermal protection of the C/SiC composite. The results show that the C/SiC composite exhibits improved thermal protection performance when accompanied by a lower tortuosity fractal dimension and a higher pore area fractal dimension of the oxide layer. Conversely, the composite demonstrates enhanced ablation resistance with a higher tortuosity fractal dimension and a lower pore area fractal dimension of the oxide layer. The predictions of the calculation model show good agreement with the experimental data and demonstrate the critical influence of microstructural parameters of the oxide layer on passive oxidation of the composite, providing practical implications for designing materials with desired thermal protection or ablation resistance properties.

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

The research object of this paper is the mixed $(\kappa ,s)$-Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed $(\kappa ,s)$-Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under $\sigma =({\sigma }_1,{\sigma }_2)$ order of the mixed integral is $3-min\{\frac{{\sigma }_1}{\kappa },\frac{{\sigma }_2}{\kappa }\}$ where $\kappa >0$.

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a complexity function bounded by a given function $f$ with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the word entropy$E_W(f)$ associated to a given function $f$ and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by $f$ in terms of its word entropy. We present a combinatorial proof of the fact that $E_W(f)$ is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by $f$ and we give several examples showing that even under strong conditions on $f$, the word entropy $E_W(f)$ can be strictly smaller than the limiting lower exponential growth rate of $f$.

Coherent upper conditional expectations are introduced through fractal outer measures to consider conditioning events that have zero probability with respect to the initial probability, as fractal sets. The model can be applied when the probability is concentrated on sets with zero Lebesgue measure and the density does not exist. It can be applied to provide a probabilistic representation of quantum states in cases where the density probability is absent. However, when the density probability exists, the two probabilistic representations align. This new approach can have intriguing data analytics applications, ranging from addressing probabilistic challenges in quantum state representation in quantum physics and managing extreme events in financial markets to capturing rare occurrences with significant implications in biomedical research.

Fractional calculus has recently attracted considerable attention. In particular, various fractional differential equations are used to model nonlinear wave theory that arises in many different areas of physics such as Josephson junction theory, field theory, theory of lattices, etc. Thus one may expect fractional calculus, in particular fractional differential equations, plays an important role in quantum field theories which are expected to satisfy fractional generalization of Klein–Gordon and Dirac equations. Until now, in high-energy physics and quantum field theories the derivative operator has only been used in integer steps. In this paper, we want to extend the idea of differentiation to arbitrary non-integers steps. We will address multi-dimensional fractional action-like problems of the calculus of variations where fractional field theories and fractional differential Dirac operators are constructed.

This paper studies the statistical characteristics of a unique long-term high-resolution precipitable water vapor (PWV) data set at Darwin, Australia, from 12 March 2002 to 28 February 2011. To understand the convective precipitation processes for climate model development, the U.S. Department of Energy’s Atmospheric Radiation Measurement (ARM) program made high-frequency radar observations of PWV at the Darwin ARM site and released the best estimates from the radar data retrievals for this time period. Based on the best estimates, we produced a PWV data set on a uniform 20-s time grid. The gridded data were sufficient to show the fractal behavior of precipitable water with Hausdorff dimension equal to 1.9. Fourier power spectral analysis revealed modulation instability due to two sideband frequencies near the diurnal cycle, which manifests as nonlinearity of an atmospheric system. The statistics of PWV extreme values and daily rainfall data show that Darwin’s PWV has El Nino Southern Oscillation (ENSO) signatures and has potential to be a predictor for weather forecasting. The right skewness of the PWV data was identified, which implies an important property of tropical atmosphere: ample capacity to hold water vapor. The statistical characteristics of this long-term high-resolution PWV data will facilitate the development and validation of climate models, particularly stochastic models.

A practical method is proposed to evaluate the fractal dimensions of the self-affine data whose power spectra follow a power law. It is on the ground of the logarithmic divergence of the moment of the power spectrum. From a crossover point of the logarithmic behaviour, a critical exponent of the moment can be obtained and related to the fractal dimension. A practical application involves an evaluation of the fractal dimensions of such self-affine data as vowels. As a result of the analyses of the critical behaviour of their moments, those dimensions are found to range over 1.60∼1.71.