Decreased long-range temporal correlations in the resting-state functional magnetic resonance imaging blood-oxygen-level-dependent signal reflect motor sequence learning up to 2 weeks following training

Decreased long-range temporal correlations (LRTC) in brain signals can be used to measure cognitive effort during task execution. Here, we examined how learning a motor sequence affects long-range temporal memory within resting-state functional magnetic resonance imaging signal. Using the Hurst exponent (HE), we estimated voxel-wise LRTC and assessed changes over 5 consecutive days of training, followed by a retention scan 12 days later. The experimental group learned a complex visuomotor sequence while a complementary control group performed tightly matched movements. An interaction analysis revealed that HE decreases were specific to the complex sequence and occurred in well-known motor sequence learning associated regions including left supplementary motor area, left premotor cortex, left M1, left pars opercularis, bilateral thalamus, and right striatum. Five regions exhibited moderate to strong negative correlations with overall behavioral performance improvements. Following learning, HE values returned to pretraining levels in some regions, whereas in others, they remained decreased even 2 weeks after training. Our study presents new evidence of HE's possible relevance for functional plasticity during the resting-state and suggests that a cortical subset of sequence-specific regions may continue to represent a functional signature of learning reflected in decreased long-range temporal dependence after a period of inactivity.

Self-similarity and vanishing diffusion in fluvial landscapes

Complex topographies exhibit universal properties when fluvial erosion dominates landscape evolution over other geomorphological processes. Similarly, we show that the solutions of a minimalist landscape evolution model display invariant behavior as the impact of soil diffusion diminishes compared to fluvial erosion at the landscape scale, yielding complete self-similarity with respect to a dimensionless channelization index. Approaching its zero limit, soil diffusion becomes confined to a region of vanishing area and large concavity or convexity, corresponding to the locus of the ridge and valley network. We demonstrate these results using one dimensional analytical solutions and two dimensional numerical simulations, supported by real-world topographic observations. Our findings on the landscape self-similarity and the localized diffusion resemble the self-similarity of turbulent flows and the role of viscous dissipation. Topographic singularities in the vanishing diffusion limit are suggestive of shock waves and singularities observed in nonlinear complex systems.

Seeing through noise in power laws

Despite widespread claims of power laws across the natural and social sciences, evidence in data is often equivocal. Modern data and statistical methods reject even classic power laws such as Pareto's law of wealth and the Gutenberg-Richter law for earthquake magnitudes. We show that the maximum-likelihood estimators and Kolmogorov-Smirnov (K-S) statistics in widespread use are unexpectedly sensitive to ubiquitous errors in data such as measurement noise, quantization noise, heaping and censorship of small values. This sensitivity causes spurious rejection of power laws and biases parameter estimates even in arbitrarily large samples, which explains inconsistencies between theory and data. We show that logarithmic binning by powers of λ > 1 attenuates these errors in a manner analogous to noise averaging in normal statistics and that λ thereby tunes a trade-off between accuracy and precision in estimation. Binning also removes potentially misleading within-scale information while preserving information about the shape of a distribution over powers of λ, and we show that some amount of binning can improve sensitivity and specificity of K-S tests without any cost, while more extreme binning tunes a trade-off between sensitivity and specificity. We therefore advocate logarithmic binning as a simple essential step in power-law inference.

Golden ratio and self-similarity in swimming: breast-stroke and the back-stroke

INTRODUCTION

Dynamics-on-graph concepts and generalized finite-length Fibonacci sequences have been used to characterize, from a temporal point of view, both human walking & running at a comfortable speed and front-crawl & butterfly swimming strokes at a middle/long distance pace. Such sequences, in which the golden ratio plays a crucial role to describe self-similar patterns, have been found to be subtly experimentally exhibited by healthy (but not pathological) walking subjects and elite swimmers, in terms of durations of gait/stroke-subphases with a clear physical meaning. Corresponding quantitative indices have been able to unveil the resulting hidden time-harmonic and self-similar structures.

RESULTS

In this study, we meaningfully extend such latest findings to the remaining two swimming strokes, namely, the breast-stroke and the back-stroke: breast-stroke, just like butterfly swimming, is highly technical and involves the complex coordination of the arm and leg actions, while back-stroke is definitely similar to front-crawl swimming. An experimental validation with reference to international-level swimmers is included.

Engineering Themes in Plant Forms and Functions

Living structures constantly interact with the biotic and abiotic environment by sensing and responding via specialized functional parts. In other words, biological bodies embody highly functional machines and actuators. What are the signatures of engineering mechanisms in biology? In this review, we connect the dots in the literature to seek engineering principles in plant structures. We identify three thematic motifs-bilayer actuator, slender-bodied functional surface, and self-similarity-and provide an overview of their structure-function relationships. Unlike human-engineered machines and actuators, biological counterparts may appear suboptimal in design, loosely complying with physical theories or engineering principles. We postulate what factors may influence the evolution of functional morphology and anatomy to dissect and comprehend better the why behind the biological forms.

Sudden bio-mathematical self-similarity and the uniqueness of human mass societies: from T-patterns and T-strings to T-societies

With the explosive growth of human knowledge especially in the twenteeth century with even greater facilitation of access to knowledge, the world of even relatively recent great thinkers becomes daunting as seen from a modern viewpoint. Recently, humans ignored the existence of the complex intracellular world of cell organs, giant information molecules such as DNA, societies of specialized worker molecules (proteins), and generally the surprising nanoscale world visible to humanity since only a few decades ago. Moreover, computational power and video technology were inaccessible to all scientists from, for example, Aristotle to Freud, so new views and ideas seem to be expected about phenomena at all scales including nano and human. Some have arrived very recently. Thus urgently needed knowledge about the biology of animal and human behavior received the first Nobel Prize as late as 1973, in Physiology and Medicine, shared by Karl von Frisch, Konrad Lorenz, and Niko Tinbergen. Lorenz's Nobel lecture was entitled "Analogy as a Source of Knowledge" which did not mention self-analogy (self-similarity) as none of the species studied were part of others and knowledge of the nanoscale phenomena at the heart of this article had barely become available. The views and empirical findings presented in this article depend on such recent intracellular nanoscale insights and the development of a set of mathematical patterns, called T-system, of which only two are considered, the self-similar (i.e., parts having a structure similar to the whole) T-pattern and the derived T-string, a T-patterned material string (here, polymer or text). Specially developed algorithms implemented in the THEME^TM software for T-pattern detection and analysis (TPA) allowed the detection of interaction T-patterns in humans, animals, and brain neuronal networks, showing self-similarity between animal interaction patterns and neuronal interaction patterns in their brains. TPA of DNA and text also showed unique self-similarity between modern human literate mass societies and the protein societies of their body cells, both with Giant Extra-Individual Purely Informational T-strings (GEIPIT; genomes or textomes) defining the behavioral potentials of their specialized citizens. This kind of society is here called T-society and only exists in humans and proteins, while the self-similarity between them only exists in human T-societies.

Similarity to the self influences memory for social targets

The construct of the self is important in the domain of memory research. Recent work has shown that person memory is influenced by similarity of social targets to the self. The current experiments investigate self-similarity as defined by traits and political ideology to better understand how memory for social targets is organised. Across three experiments, participants formed positive or negative impressions based on each target's picture, a trait-implying behavior (Experiments 1 & 2), and/or political ideology (conservative/liberal label in Experiment 2; political-ideological belief statements in Experiment 3) followed by a memory test. Results showed a self-similarity effect dependent on valence in Experiment 1, but not in Experiments 2 or 3 when participants processed ideological information associated with targets. These results suggest that self-similarity has an effect on memory for social targets, but that ideological information disrupts self-focused processing of others, suggesting that ideological information also has a powerful influence on what people remember about others (i.e. social targets).

Kernel Estimation Using Total Variation Guided GAN for Image Super-Resolution

Various super-resolution (SR) kernels in the degradation model deteriorate the performance of the SR algorithms, showing unpleasant artifacts in the output images. Hence, SR kernel estimation has been studied to improve the SR performance in several ways for more than a decade. In particular, a conventional research named KernelGAN has recently been proposed. To estimate the SR kernel from a single image, KernelGAN introduces generative adversarial networks(GANs) that utilize the recurrence of similar structures across scales. Subsequently, an enhanced version of KernelGAN, named E-KernelGAN, was proposed to consider image sharpness and edge thickness. Although it is stable compared to the earlier method, it still encounters challenges in estimating sizable and anisotropic kernels because the structural information of an input image is not sufficiently considered. In this paper, we propose a kernel estimation algorithm called Total Variation Guided KernelGAN (TVG-KernelGAN), which efficiently enables networks to focus on the structural information of an input image. The experimental results show that the proposed algorithm accurately and stably estimates kernels, particularly sizable and anisotropic kernels, both qualitatively and quantitatively. In addition, we compared the results of the non-blind SR methods, using SR kernel estimation techniques. The results indicate that the performance of the SR algorithms was improved using our proposed method.

Far-from-equilibrium universality in the two-dimensional Heisenberg model

We characterize the universal far-from-equilibrium dynamics of the two-dimensional quantum Heisenberg magnet isolated from its environment. For a broad range of initial conditions, we find a long-lived universal prethermal regime characterized by self-similar behavior of spin-spin correlations. We analytically derive the spatial-temporal scaling exponents and find excellent agreement with numerics using phase space methods. The scaling exponents are insensitive to the choice of initial conditions, which include coherent and incoherent spin states with values of total magnetization and energy in a wide range. Compared to previously studied self-similar dynamics in nonequilibrium O(n) field theories and Bose gases, we find qualitatively distinct scaling behavior originating from the presence of spin modes that remain gapless at long times and are protected by the global SU(2) symmetry. Our predictions, which suggest a distinct nonequilibrium universality class from Bose gases and O(n) theories, are readily testable in ultracold atoms simulators of Heisenberg magnets.

Emergent entanglement structures and self-similarity in quantum spin chains

We introduce an experimentally accessible network representation for many-body quantum states based on entanglement between all pairs of its constituents. We illustrate the power of this representation by applying it to a paradigmatic spin chain model, the XX model, and showing that it brings to light new phenomena. The analysis of these entanglement networks reveals that the gradual establishment of quasi-long range order is accompanied by a symmetry regarding single-spin concurrence distributions, as well as by instabilities in the network topology. Moreover, we identify the existence of emergent entanglement structures, spatially localized communities enforced by the global symmetry of the system that can be revealed by model-agnostic community detection algorithms. The network representation further unveils the existence of structural classes and a cyclic self-similarity in the state, which we conjecture to be intimately linked to the community structure. Our results demonstrate that the use of tools and concepts from complex network theory enables the discovery, understanding and description of new physical phenomena even in models studied for decades. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Self-similar hierarchy of coherent tubular vortices in turbulence

Energy transfers from larger to smaller scales in turbulence. This energy cascade is a process of the creation of smaller-scale coherent vortices by larger ones. In our recent study (Yoneda, Goto and Tsuruhashi 2022 Nonlinearity 35, 1380-1401), we reformulated the energy cascade in terms of this stretching process and derived the [Formula: see text] law of the energy spectrum under physically reasonable assumptions. In the present study, we provide a quantitative verification of these assumptions by using direct numerical simulations. We decompose developed turbulence in a periodic cube into scales by using the band-pass filter and identify the axes of coherent tubular vortices by the low-pressure method. Even when the turbulent kinetic energy and its dissipation rate temporally fluctuate about their temporal means, the total length of the vortices at each scale varies little with time. This result is consistent with our assumption of the temporal stationarity on the vorticity decomposition. The present numerical analysis also shows that the hierarchy of vortex axes is self-similar in a wide range of scales, i.e. in the inertial range and a lower part of the dissipation range and that the volume fraction occupied by the tubular vortices at each scale is independent of the scale. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Self-similarity in turbulence and its applications

First, we discuss the non-Gaussian type of self-similar solutions to the Navier-Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne et al. (1996 Commun. Partial. Differ. Equ. 21, 179-193). In order to shed some light on it, we study self-similar solutions to the one-dimensional Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution, we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer's function [Formula: see text]. For the two-dimensional Navier-Stokes equations, on top of the celebrated Burgers vortex, we derive yet another solution to the associated Fokker-Planck equation. This can be regarded as a 'conjugate' to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the three-dimensional (3D) Navier-Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier-Stokes equations, we consider what we could tell about more general solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Network Traffic Prediction Incorporating Prior Knowledge for an Intelligent Network

Network traffic prediction is an important tool for the management and control of IoT, and timely and accurate traffic prediction models play a crucial role in improving the IoT service quality. The degree of burstiness in intelligent network traffic is high, which creates problems for prediction. To address the problem faced by traditional statistical models, which cannot effectively extract traffic features when dealing with inadequate sample data, in addition to the poor interpretability of deep models, this paper proposes a prediction model (fusion prior knowledge network) that incorporates prior knowledge into the neural network training process. The model takes the self-similarity of network traffic as a priori knowledge, incorporates it into the gating mechanism of the long short-term memory neural network, and combines a one-dimensional convolutional neural network with an attention mechanism to extract the temporal features of the traffic sequence. The experiments show that the model can better recover the characteristics of the original data. Compared with the traditional prediction model, the proposed model can better describe the trend of network traffic. In addition, the model produces an interpretable prediction result with an absolute correction factor of 76.4%, which is at least 10% better than the traditional statistical model.

Generalized Finite-Length Fibonacci Sequences in Healthy and Pathological Human Walking: Comprehensively Assessing Recursivity, Asymmetry, Consistency, Self-Similarity, and Variability of Gaits

Healthy and pathological human walking are here interpreted, from a temporal point of view, by means of dynamics-on-graph concepts and generalized finite-length Fibonacci sequences. Such sequences, in their most general definition, concern two sets of eight specific time intervals for the newly defined composite gait cycle, which involves two specific couples of overlapping (left and right) gait cycles. The role of the golden ratio, whose occurrence has been experimentally found in the recent literature, is accordingly characterized, without resorting to complex tools from linear algebra. Gait recursivity, self-similarity, and asymmetry (including double support sub-phase consistency) are comprehensively captured. A new gait index, named Φ-bonacci gait number, and a new related experimental conjecture—concerning the position of the foot relative to the tibia—are concurrently proposed. Experimental results on healthy or pathological gaits support the theoretical derivations.

Supervised Learning of Neural Networks for Active Queue Management in the Internet

The paper examines the AQM mechanism based on neural networks. The active queue management allows packets to be dropped from the router’s queue before the buffer is full. The aim of the work is to use machine learning to create a model that copies the behavior of the AQM PIα mechanism. We create training samples taking into account the self-similarity of network traffic. The model uses fractional Gaussian noise as a source. The quantitative analysis is based on simulation. During the tests, we analyzed the length of the queue, the number of rejected packets and waiting times in the queues. The proposed mechanism shows the usefulness of the Active Queue Management mechanism based on Neural Networks.

Scaling up real networks by geometric branching growth

Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or species. Here, we provide empirical evidence for self-similar growth of network structure in the evolution of real systems-the journal-citation network and the world trade web-and present the geometric branching growth model, which predicts this evolution and explains the symmetries observed. The model produces multiscale unfolding of a network in a sequence of scaled-up replicas preserving network features, including clustering and community structure, at all scales. Practical applications in real instances include the tuning of network size for best response to external influence and finite-size scaling to assess critical behavior under random link failures.

A Framework to Assess the Citation Performance of Complex Innovation Systems

Scientometric indicators are useful to evaluate the relevance of scientific research, to prepare rankings, and to evaluate and inform research policies. That is why the choice of appropriate indicators is a matter of primary concern. This article aims to introduce a framework to decide the appropriate type of indicator for assessing the citation-based performance of complex innovation systems. The framework is two-fold: First, it brings the methodology to decide when the use of standard average based indicators is granted, and when scale-invariant indicators are mandatory. Second, it provides the procedures to build scale-invariant indicators to assess the relative impact of complex innovation systems. The framework is validated empirically through the evaluation of the relative impact of the Chilean science system in 2017. The result suggests that the Chilean science system has characteristics of a complex innovation system such as the distribution of citations fits to a power law with an exponential cutoff −2.77±0.09 and a power-law correlation between the size of the system and its impact 1.29±0.11. Furthermore, the framework shows to be efficient to compare fields of vastly different sizes.

Depressurization-Induced Nucleation in the "Polylactide-Carbon Dioxide" System: Self-Similarity of the Bubble Embryos Expansion

Self-similar expansion of bubble embryos in a plasticized polymer under quasi-isothermal depressurization is examined using the experimental data on expansion rates of embryos in the CO₂-plasticized d,l-polylactide and modeling the results. The CO₂ initial pressure varied from 5 to 14 MPa, and the depressurization rate was 5 × 10⁻³ MPa/s. The constant temperature in experiments was in a range from 310 to 338 K. The initial rate of embryos expansion varied from ≈0.1 to ≈10 µm/s, with a decrease in the current external pressure. While modeling, a non-linear behavior of CO₂ isotherms near the critical point was taken into account. The modeled data agree satisfactorily with the experimental results. The effect of a remarkable increase in the expansion rate at a decreasing external pressure is interpreted in terms of competing effects, including a decrease in the internal pressure, an increase in the polymer viscosity, and an increase in the embryo radius at the time of embryo formation. The vanishing probability of finding the steadily expanding embryos for external pressures around the CO₂ critical pressure is interpreted in terms of a joint influence of the quasi-adiabatic cooling and high compressibility of CO₂ in the embryos.

Fractality and Variability in Canonical and Non-Canonical English Fiction and in Non-Fictional Texts

This study investigates global properties of three categories of English text: canonical fiction, non-canonical fiction, and non-fictional texts. The central hypothesis of the study is that there are systematic differences with respect to structural design features between canonical and non-canonical fiction, and between fictional and non-fictional texts. To investigate these differences, we compiled a corpus containing texts of the three categories of interest, the Jena Corpus of Expository and Fictional Prose (JEFP Corpus). Two aspects of global structure are investigated, variability and self-similar (fractal) patterns, which reflect long-range correlations along texts. We use four types of basic observations, (i) the frequency of POS-tags per sentence, (ii) sentence length, (iii) lexical diversity, and (iv) the distribution of topic probabilities in segments of texts. These basic observations are grouped into two more general categories, (a) the lower-level properties (i) and (ii), which are observed at the level of the sentence (reflecting linguistic decoding), and (b) the higher-level properties (iii) and (iv), which are observed at the textual level (reflecting comprehension/integration). The observations for each property are transformed into series, which are analyzed in terms of variance and subjected to Multi-Fractal Detrended Fluctuation Analysis (MFDFA), giving rise to three statistics: (i) the degree of fractality ( H ), (ii) the degree of multifractality ( D ), i.e., the width of the fractal spectrum, and (iii) the degree of asymmetry ( A ) of the fractal spectrum. The statistics thus obtained are compared individually across text categories and jointly fed into a classification model (Support Vector Machine). Our results show that there are in fact differences between the three text categories of interest. In general, lower-level text properties are better discriminators than higher-level text properties. Canonical fictional texts differ from non-canonical ones primarily in terms of variability in lower-level text properties. Fractality seems to be a universal feature of text, slightly more pronounced in non-fictional than in fictional texts. On the basis of our results obtained on the basis of corpus data we point out some avenues for future research leading toward a more comprehensive analysis of textual aesthetics, e.g., using experimental methodologies.

A universal model describing the structure and functions of living systems

Can Life be explained based on the fundamental Laws of Nature? This question is central in Science since its answer could unify Physics and Biology and open new routes for Medicine. The present study introduces a clear and well-documented hypothesis addressing the unified description of all living systems. The proposed universal model is based on two established characteristics of Life. First, the concept of Functional Self-similarity (FSS) is introduced. As shown by several authors, all living systems can be classified in a multi-level hierarchy of increasing complexity. Systems in all hierarchical levels are characterized by a small set of the same attributes defining Life. This observation implies the existence of an elementary living system (i.e., a quantum of Life) having all the necessary functionalities of living systems. Secondly, the non-equilibrium nature of living systems implies that they should be able to process information since such a function is required for reducing entropy. Therefore, all living systems necessarily perform logical operations similar to electronic circuits. This conclusion, which is based on the requirement to overcome the constraints of the Second Law of Thermodynamics, indicates a close correspondence between living systems and information processing machines, namely computers. Consequently, important theoretical principles and concepts regarding computer design may also apply in the study of living systems. The above considerations lead to the Hypothesis of a Universal Architecture (UAH).

The Fractal Geometry of the Nymphalid Groundplan: Self-Similar Configuration of Color Pattern Symmetry Systems in Butterfly Wings

Simple Summary

Highly diverse color patterns of butterfly wings can be explained as modifications of an archetypical color pattern of nymphalid butterflies called the nymphalid groundplan. The nymphalid groundplan contains three major symmetry systems and a discal symmetry system, but their relationships have been elusive. Here, the morphological and spatial relationships among these symmetry systems were studied based on cross-species color-pattern comparisons of the hindwings in nymphalid butterflies. It was shown that all symmetry systems can be expressed as various structures, suggesting the equivalence (homology) of these systems in developmental potential. In some cases, the discal symmetry system is circularly surrounded by the central symmetry system, which may then be surrounded by the border and basal symmetry systems, indicating a unified supersymmetry system covering the entire wing. These results suggest that butterfly color patterns are hierarchically constructed; one system is nested within another system, which is a self-similar relationship that achieves the fractal geometry. This self-similarity is likely mediated by the serial induction of organizers during development, and a possible mechanism is proposed for symmetry breaking of the system morphology, which contributes to the diversity of butterfly wing color patterns.

Abstract

The nymphalid groundplan is an archetypical color pattern of nymphalid butterflies involving three major symmetry systems and a discal symmetry system, which share the basic morphogenesis unit. Here, the morphological and spatial relationships among these symmetry systems were studied based on cross-species comparisons of nymphalid hindwings. Based on findings in Neope and Symbrenthia, all three major symmetry systems can be expressed as bands, spots, or eyespot-like structures, suggesting equivalence (homology) of these systems in developmental potential. The discal symmetry system can also be expressed as various structures. The discal symmetry system is circularly surrounded by the central symmetry system, which may then be surrounded by the border and basal symmetry systems, based mainly on findings in Agrias, indicating a unified supersymmetry system covering the entire wing. The border symmetry system can occupy the central part of the wing when the central symmetry system is compromised, as seen in Callicore. These results suggest that butterfly color patterns are hierarchically constructed in a self-similar fashion, as the fractal geometry of the nymphalid groundplan. This self-similarity is likely mediated by the serial induction of organizers, and symmetry breaking of the system morphology may be generated by the collision of opposing signals during development.

Long-Range Dependent Traffic Classification with Convolutional Neural Networks Based on Hurst Exponent Analysis

The paper examines the ability of neural networks to classify Internet traffic data in terms of self-similarity expressed by the Hurst exponent. Fractional Gaussian noise is used for the generation of synthetic data for modeling the genuine ones. It is presented that the trained model is capable of classifying the synthetic data obtained from the Pareto distribution and the real traffic data. We present the results of training for different optimizers of the cost function and a different number of convolutional layers in the neural network.

Graph fractal dimension and the structure of fractal networks

Fractals are geometric objects that are self-similar at different scales and whose geometric dimensions differ from so-called fractal dimensions. Fractals describe complex continuous structures in nature. Although indications of self-similarity and fractality of complex networks has been previously observed, it is challenging to adapt the machinery from the theory of fractality of continuous objects to discrete objects such as networks. In this article, we identify and study fractal networks using the innate methods of graph theory and combinatorics. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colourings and graph descriptive complexity, and analyse the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework in evolutionary biology and virology by analysing networks of viral strains sampled at different stages of evolution inside their hosts. Our methodology revealed gradual self-organization of intra-host viral populations over the course of infection and their adaptation to the host environment. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

Multi-Frequency Resonance Behaviour of a Si FractalNEMS Resonator

Novel Si-based nanosize mechanical resonator has been top-down fabricated. The shape of the resonating body has been numerically derived and consists of seven star-polygons that form a fractal structure. The actual resonator is defined by focused ion-beam implantation on a SOI wafer where its 18 vertices are clamped to nanopillars. The structure is suspended over a 10 μm trench and has width of 12 μm. Its thickness of 0.040 μm is defined by the fabrication process and prescribes Young’s modulus of 76 GPa which is significantly lower than the value of the bulk material. The resonator is excited by the bottom Si-layer and the interferometric characterisation confirms broadband frequency response with quality factors of over 800 for several peaks between 2 MHz and 16 MHz. COMSOL FEM software has been used to vary material properties and residual stress in order to fit the eigenfrequencies of the model with the resonance peaks detected experimentally. Further use of the model shows how the symmetry of the device affects the frequency spectrum. Also, by using the FEM model, the possibility for an electrical read out of the device was tested. The experimental measurements and simulations proved that the device can resonate at many different excitation frequencies allowing multiple operational bands. The size, and the power needed for actuation are comparable with the ones of single beam resonator while the fractal structure allows much larger area for functionalisation.

Day and Night Changes of Cardiovascular Complexity: A Multi-Fractal Multi-Scale Analysis

Recently, a multifractal-multiscale approach to detrended fluctuation analysis (DFA) was proposed to evaluate the cardiovascular fractal dynamics providing a surface of self-similarity coefficients α(q,τ), function of the scale τ, and moment order q. We hypothesize that this versatile DFA approach may reflect the cardiocirculatory adaptations in complexity and nonlinearity occurring during the day/night cycle. Our aim is, therefore, to quantify how α(q, τ) surfaces of cardiovascular series differ between daytime and night-time. We estimated α(q,τ) with -5 ≤ q ≤ 5 and 8 ≤ τ ≤ 2048 s for heart rate and blood pressure beat-to-beat series over periods of few hours during daytime wake and night-time sleep in 14 healthy participants. From the α(q,τ) surfaces, we estimated short-term (<16 s) and long-term (from 16 to 512 s) multifractal coefficients. Generating phase-shuffled surrogate series, we evaluated short-term and long-term indices of nonlinearity for each q. We found a long-term night/day modulation of α(q,τ) between 128 and 256 s affecting heart rate and blood pressure similarly, and multifractal short-term modulations at q < 0 for the heart rate and at q > 0 for the blood pressure. Consistent nonlinearity appeared at the shorter scales at night excluding q = 2. Long-term circadian modulations of the heart rate DFA were previously associated with the cardiac vulnerability period and our results may improve the risk stratification indicating the more relevant α(q,τ) area reflecting this rhythm. Furthermore, nonlinear components in the nocturnal α(q,τ) at q ≠ 2 suggest that DFA may effectively integrate the linear spectral information with complexity-domain information, possibly improving the monitoring of cardiac interventions and protocols of rehabilitation medicine.

Bubble pinch-off in turbulence

Although bubble pinch-off is an archetype of a dynamical system evolving toward a singularity, it has always been described in idealized theoretical and experimental conditions. Here, we consider bubble pinch-off in a turbulent flow representative of natural conditions in the presence of strong and random perturbations, combining laboratory experiments, numerical simulations, and theoretical modeling. We show that the turbulence sets the initial conditions for pinch-off, namely the initial bubble shape and flow field, but after the pinch-off starts, the turbulent time at the neck scale becomes much slower than the pinching dynamics: The turbulence freezes. We show that the average neck size, [Formula: see text], can be described by [Formula: see text], where [Formula: see text] is the pinch-off or singularity time and [Formula: see text], in close agreement with the axisymmetric theory with no initial flow. While frozen, the turbulence can influence the pinch-off through the initial conditions. Neck shape oscillations described by a quasi-2-dimensional (quasi-2D) linear perturbation model are observed as are persistent eccentricities of the neck, which are related to the complex flow field induced by the deformed bubble shape. When turbulent stresses are less able to be counteracted by surface tension, a 3-dimensional (3D) kink-like structure develops in the neck, causing [Formula: see text] to escape its self-similar decrease. We identify the geometric controlling parameter that governs the appearance of these kink-like interfacial structures, which drive the collapse out of the self-similar route, governing both the likelihood of escaping the self-similar process and the time and length scale at which it occurs.

Tree crowns grow into self-similar shapes controlled by gravity and light sensing

Plants have developed different tropisms: in particular, they reorient the growth of their branches towards the light (phototropism) or upwards (gravitropism). How these tropisms affect the shape of a tree crown remains unanswered. We address this question by developing a propagating front model of tree growth. Being length-free, this model leads to self-similar solutions after a long period of time, which are independent of the initial conditions. Varying the intensities of each tropism, different self-similar shapes emerge, including singular ones. Interestingly, these shapes bear similarities to existing tree species. It is concluded that the core of specific crown shapes in trees relies on the balance between tropisms.

Modeling and Optimization of Time-Of-Use Electricity Pricing Systems

Time-of-Use (TOU) pricing is an important strategy for electricity providers to manage supply and hence making the grid more efficient and for consumers to manage their costs. In this paper, we discuss a general stochastic modeling framework for consumer's power demand based on which the TOU contract characteristics can be selected, so as to minimize the mean electricity price paid by the customer. We exploit the characteristics of power demand observed in real grids to propose to model it during homogeneous peak periods as a constant level with fluctuations described by a scaled fractional Brownian motion. We analyze the exceedance process over pre-specified thresholds and use this information for formulating an optimization problem to determine the key features of the TOU contract. Due to the analytical intractability of certain expressions with the exception of short-range dependence fluctuations, the solution of the posited optimization problem requires using techniques such as Monte Carlo simulation and numerical search. The methodology for two pricing schemes is illustrated using real data.

Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets

We study the self-similarity and dissipation scalings of a turbulent planar jet and the theoretically implied mean flow scalings. Unlike turbulent wakes where such studies have already been carried out (Dairay et al. 2015 J. Fluid Mech. 781, 166-198. (doi:10.1017/jfm.2015.493); Obligado et al. 2016 Phys. Rev. Fluids 1, 044409. (doi:10.1103/PhysRevFluids.1.044409)), this is a boundary-free turbulent shear flow where the local Reynolds number increases with distance from inlet. The Townsend-George theory revised by (Dairay et al. 2015 J. Fluid Mech. 781, 166-198. (doi:10.1017/jfm.2015.493)) is applied to turbulent planar jets. Only a few profiles need to be self-similar in this theory. The self-similarity of mean flow, turbulence dissipation, turbulent kinetic energy and Reynolds stress profiles is supported by our experimental results from 18 to at least 54 nozzle sizes, the furthermost location investigated in this work. Furthermore, the non-equilibrium dissipation scaling found in turbulent wakes, decaying grid-generated turbulence, various instances of periodic turbulence and turbulent boundary layers (Dairay et al. 2015 J. Fluid Mech. 781, 166-198. (doi:10.1017/jfm.2015.493); Vassilicos 2015 Annu. Rev. Fluid Mech. 95, 114. (doi:10.1146/annurev-fluid-010814-014637); Goto & Vassilicos 2015 Phys. Lett. A 3790, 1144-1148. (doi:10.1016/j.physleta.2015.02.025); Nedic et al. 2017 Phys. Rev. Fluids 2, 032601. (doi:10.1103/PhysRevFluids.2.032601)) is also observed in the present turbulent planar jet and in the turbulent planar jet of (Antonia et al. 1980 Phys. Fluids 23, 863055. (doi:10.1063/1.863055)). Given these observations, the theory implies new mean flow and jet width scalings which are found to be consistent with our data and the data of (Antonia et al. 1980 Phys. Fluids 23, 863055. (doi:10.1063/1.863055)). In particular, it implies a hitherto unknown entrainment behaviour: the ratio of characteristic cross-stream to centreline streamwise mean flow velocities decays as the -1/3 power of streamwise distance in the region, where the non-equilibrium dissipation scaling holds.

Trait Mindfulness Is Associated With the Self-Similarity of Heart Rate Variability

Previous studies have linked trait mindfulness with better self-regulation and adaptation. Heart rate variability (HRV) is a good physiological indicator of the capacity for self-regulation and adaptation. The present study explored the relationship between trait mindfulness and HRV from the viewpoint of crosstalking between different HRV parameter pairs, which would reflect the dynamic interactions between each pair of HRV parameters in different processes. We measured the trait mindfulness of seventy-four undergraduate students and recorded nine HRV parameters during the following four consecutive experimental phases: (1) calming phase, (2) mental arithmetic task phase, (3) recovery phase, and (4) mindfulness practice phase. The relationship between trait mindfulness and HRV was explored at the following three levels: (1) the absolute level, i.e., HRV parameters in four different states, (2) the difference-change level, i.e., differences in HRV parameters between different states, and (3) the crosstalking level, i.e., self-similarity of crosstalking HRV parameter pairs. The results supported the following hypothesis: trait mindfulness, as measured by the Mindful Attention Awareness Scale (MAAS), was significantly and positively correlated with the self-similarity of crosstalking HRV parameter pairs but was not significantly correlated with the HRV parameters at the difference-change and absolute levels. These findings indicate that as trait mindfulness increases, the ability to maintain ANS function homeostasis improves. HIGHLIGHTS -Trait mindfulness is associated with better self-regulation and adaptation.-Heart rate variability (HRV) is a good physiological indicator of the capacity for self-regulation and adaptation.-Trait mindfulness is significantly correlated with self-similarity of crosstalking HRV parameter pairs but not with the HRV parameters at the difference-change or absolute levels.

Assessment of Driver's Drowsiness Based on Fractal Dimensional Analysis of Sitting and Back Pressure Measurements

The most effective way of preventing motor vehicle accidents caused by drowsy driving is through a better understanding of drowsiness itself. Prior research on the detection of symptoms of drowsy driving has offered insights on providing drivers with advance warning of an elevated risk of crash. The present study measured back and sitting pressures during a simulated driving task under both high and low arousal conditions. Fluctuation of time series of center of pressure (COP) movement of back and sitting pressure was observed to possess a fractal property. The fractal dimensions were calculated to compare the high and low arousal conditions. The results showed that under low arousal (the drowsiness state) the fractal dimension was significantly lower than what was calculated with high arousal. Accumulated drowsiness thus contributed to the loss of self-similarity and unpredictability of time series of back and sitting pressure measurement. Drowsiness further reduces the complexity of the posture control system as viewed from back and sitting pressure. Thus, fractal dimension is a necessary and sufficient condition of a decreased arousal level. It further is a necessary condition for detecting the interval or point in time with high risk of crash.

Oscillating path between self-similarities in liquid pinch-off

Many differential equations involved in natural sciences show singular behaviors; i.e., quantities in the model diverge as the solution goes to zero. Nonetheless, the evolution of the singularity can be captured with self-similar solutions, several of which may exist for a given system. How to characterize the transition from one self-similar regime to another remains an open question. By studying the classic example of the pinch-off of a viscous liquid thread, we show experimentally that the geometry of the system and external perturbations play an essential role in the transition from a symmetric to an asymmetric solution. Moreover, this transient regime undergoes unexpected log-scale oscillations that delay dramatically the onset of the final self-similar solution. This result sheds light on the strong impact external constraints can have on predictions established to explain the formation of satellite droplets or on the rheological tests applied on a fluid, for example.

Symmetric coalescence of two hydraulic fractures

The formation of a fracture network is a key process for many geophysical and industrial practices from energy resource recovery to induced seismic management. We focus on the initial stage of a fracture network formation using experiments on the symmetric coalescence of two equal coplanar, fluid-driven, penny-shaped fractures in a brittle elastic medium. Initially, the fractures propagate independently of each other. The fractures then begin to interact and coalesce, forming a bridge between them. Within an intermediate period after the initial contact, most of the fracture growth is localized along this bridge, perpendicular to the line connecting the injection sources. Using light attenuation and particle image velocimetry to measure both the fracture aperture and velocity field, we characterize the growth of this bridge. We model this behavior using a geometric volume conservation argument dependent on the symmetry of the interaction, with a 2D approximation for the bridge. We also verify experimentally the scaling for the bridge growth and the shape of the thickness profile along the bridge. The influence of elasticity and toughness of the solid, injection rate of the fluid, and initial location of the fractures are captured by our scaling.

Fractal Structure and Non-Extensive Statistics

The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non-extensive statistics have been formulated over the last several years, in particular a fractal structure in thermodynamic functions was recently proposed as a possible origin for non-extensive statistics in physical systems. In the present work, we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such a system. It is shown that a system with the fractal structure described here presents temperature fluctuation following an Euler Gamma Function, in accordance with previous works that provided evidence of the connections between those fluctuations and Tsallis statistics. Finally, the scale invariance of the fractal thermodynamical system is discussed in terms of the Callan–Symanzik equation.

Assessing and Interpreting the Within-Body Biogeography of Human Microbiome Diversity

A human body hosts a relatively independent microbiome including five major regional biomes (i.e., airway, oral, gut, skin, and urogenital). Each of them may possess different regional characteristics with important implications to our health and diseases (i.e., so-termed microbiome associated diseases). Nevertheless, these regional microbiomes are connected with each other through diffusions and migrations. Here, we investigate the within-body (intra-individual) distribution feature of microbiome diversity via diversity area relationship (DAR) modeling, which, to the best of our knowledge, has not been systematically studied previously. We utilized the Hill numbers for measuring alpha and beta-diversities and built 1,200 within-body DAR models with to date the most comprehensive human microbiome datasets of 18 sites from the human microbiome project (HMP) cohort. We established the intra-DAR profile (z-q pattern: the diversity scaling parameter z of the power law (PL) at diversity order q = 0–3), intra-PDO (pair-wise diversity overlap) profile (g-q), and intra-MAD (maximal accrual diversity) profile (D_max-q) for the within-body biogeography of the human microbiome. These profiles constitute the “maps” of the within-body biogeography, and offer important insights on the within-body distribution of the human microbiome. Furthermore, we investigated the heterogeneity among individuals in their biogeography parameters and found that there is not an “average Joe” that can represent majority of individuals in a cohort or population. For example, we found that most individuals in the HMP cohort have relatively lower maximal accrual diversity (MAD) or in the “long tail” of the so-termed power law distribution. In the meantime, there are a small number of individuals in the cohort who possess disproportionally higher MAD values. These findings may have important implications for personalized medicine of the human microbiome associated diseases in practice, besides their theoretical significance in microbiome research such as establishing the baseline for the conservation of human microbiome.

Branch Thinning and the Large-Scale, Self-Similar Structure of Trees

Branch formation in trees has an inherent tendency toward exponential growth, but exponential growth in the number of branches cannot continue indefinitely. It has been suggested that trees balance this tendency toward expansion by also losing branches grown in previous growth cycles. Here, we present a model for branch formation and branch loss during ontogeny that builds on the phenomenological assumption of a branch carrying capacity. The model allows us to derive approximate analytical expressions for the number of tips on a branch, the distribution of growth modules within a branch, and the rate and size distribution of tree wood litter produced. Although limited availability of data makes empirical corroboration challenging, we show that our model can fit field observations of red maple (Acer rubrum) and note that the age distribution of discarded branches predicted by our model is qualitatively similar to an empirically observed distribution of dead and abscised branches of balsam poplar (Populus balsamifera). By showing how a simple phenomenological assumption-that the number of branches a tree can maintain is limited-leads directly to predictions on branching structure and the rate and size distribution of branch loss, these results potentially enable more explicit modeling of woody tissues in ecosystems worldwide, with implications for the buildup of flammable fuel, nutrient cycling, and understanding of plant growth.

The Fractality of Polar and Reed-Muller Codes

The generator matrices of polar codes and Reed-Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed-Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those rows, in the limit as the blocklength tends to infinity. We compute the Lebesgue measure and the Hausdorff dimension of these sets. We furthermore show that these sets are finely structured and self-similar in a well-defined sense, i.e., they have properties that are common to fractals.

Statistical Image Properties in Large Subsets of Traditional Art, Bad Art, and Abstract Art

Several statistical image properties have been associated with large subsets of traditional visual artworks. Here, we investigate some of these properties in three categories of art that differ in artistic claim and prestige: (1) Traditional art of different cultural origin from established museums and art collections (oil paintings and graphic art of Western provenance, Islamic book illustration and Chinese paintings), (2) Bad Art from two museums that collect contemporary artworks of lesser importance (© Museum Of Bad Art [MOBA], Somerville, and Official Bad Art Museum of Art [OBAMA], Seattle), and (3) twentieth century abstract art of Western provenance from two prestigious museums (Tate Gallery and Kunstsammlung Nordrhein-Westfalen). We measured the following four statistical image properties: the fractal dimension (a measure relating to subjective complexity); self-similarity (a measure of how much the sections of an image resemble the image as a whole), 1st-order entropy of edge orientations (a measure of how uniformly different orientations are represented in an image); and 2nd-order entropy of edge orientations (a measure of how independent edge orientations are across an image). As shown previously, traditional artworks of different styles share similar values for these measures. The values for Bad Art and twentieth century abstract art show a considerable overlap with those of traditional art, but we also identified numerous examples of Bad Art and abstract art that deviate from traditional art. By measuring statistical image properties, we quantify such differences in image composition for the first time.

A Statistical Description of Plant Shoot Architecture

Plant architectures can be characterized statistically by their spatial density function, which specifies the probability of finding a branch at each location in the territory occupied by a plant. Using high-precision 3D scanning, we analyzed 557 plant shoot architectures, representing three species, grown across three to five environmental conditions, and through 20-30 developmental time points. We found two elegant properties in the spatial density functions of these architectures: all functions could be nearly modified in one direction without affecting the density in orthogonal directions (called "separability"), and all functions shared the same underlying shape, aside from stretching and compression (called "self-similarity"). Surprisingly, despite their striking visual diversity, we discovered that all architectures could be described as variations on a single underlying function: a Gaussian density function truncated at roughly two SDs. We also observed systematic variation in the spatial density functions across species, growth conditions, and time, which suggests functional specialization despite following the same general design form.

Using CNN Features to Better Understand What Makes Visual Artworks Special

One of the goal of computational aesthetics is to understand what is special about visual artworks. By analyzing image statistics, contemporary methods in computer vision enable researchers to identify properties that distinguish artworks from other (non-art) types of images. Such knowledge will eventually allow inferences with regard to the possible neural mechanisms that underlie aesthetic perception in the human visual system. In the present study, we define measures that capture variances of features of a well-established Convolutional Neural Network (CNN), which was trained on millions of images to recognize objects. Using an image dataset that represents traditional Western, Islamic and Chinese art, as well as various types of non-art images, we show that we need only two variance measures to distinguish between the artworks and non-art images with a high classification accuracy of 93.0%. Results for the first variance measure imply that, in the artworks, the subregions of an image tend to be filled with pictorial elements, to which many diverse CNN features respond (richness of feature responses). Results for the second measure imply that this diversity is tied to a relatively large variability of the responses of individual CNN feature across the subregions of an image. We hypothesize that this combination of richness and variability of CNN feature responses is one of properties that makes traditional visual artworks special. We discuss the possible neural underpinnings of this perceptual quality of artworks and propose to study the same quality also in other types of aesthetic stimuli, such as music and literature.

Statistical evidence of anasymptotic geometric structure to the momentum transporting motions in turbulent boundary layers

The turbulence contribution to the mean flow is reflected by the motions producing the Reynolds shear stress (〈-uv〉) and its gradient. Recent analyses of the mean dynamical equation, along with data, evidence that these motions asymptotically exhibit self-similar geometric properties. This study discerns additional properties associated with the uv signal, with an emphasis on the magnitudes and length scales of its negative contributions. The signals analysed derive from high-resolution multi-wire hot-wire sensor data acquired in flat-plate turbulent boundary layers. Space-filling properties of the present signals are shown to reinforce previous observations, while the skewness of uv suggests a connection between the size and magnitude of the negative excursions on the inertial domain. Here, the size and length scales of the negative uv motions are shown to increase with distance from the wall, whereas their occurrences decrease. A joint analysis of the signal magnitudes and their corresponding lengths reveals that the length scales that contribute most to 〈-uv〉 are distinctly larger than the average geometric size of the negative uv motions. Co-spectra of the streamwise and wall-normal velocities, however, are shown to exhibit invariance across the inertial region when their wavelengths are normalized by the width distribution, W(y), of the scaling layer hierarchy, which renders the mean momentum equation invariant on the inertial domain.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.

On the shape of giant soap bubbles

We study the effect of gravity on giant soap bubbles and show that it becomes dominant above the critical size [Formula: see text], where [Formula: see text] is the mean thickness of the soap film and [Formula: see text] is the capillary length ([Formula: see text] stands for vapor-liquid surface tension, and [Formula: see text] stands for the liquid density). We first show experimentally that large soap bubbles do not retain a spherical shape but flatten when increasing their size. A theoretical model is then developed to account for this effect, predicting the shape based on mechanical equilibrium. In stark contrast to liquid drops, we show that there is no mechanical limit of the height of giant bubble shapes. In practice, the physicochemical constraints imposed by surfactant molecules limit the access to this large asymptotic domain. However, by an exact analogy, it is shown how the giant bubble shapes can be realized by large inflatable structures.

Statistical Image Properties in Works from the Prinzhorn Collection of Artists with Schizophrenia

The Prinzhorn Collection preserves and exhibits thousands of visual artworks by patients who were diagnosed to suffer from mental disease. From this collection, we analyzed 1,256 images by 14 artists who were diagnosed with dementia praecox or schizophrenia. Six objective statistical properties that have been used previously to characterize visually aesthetic images were calculated. These properties reflect features of formal image composition, such as the complexity and distribution of oriented luminance gradients and edges, as well as Fourier spectral properties. Results for the artists with schizophrenia were compared to artworks from three public art collections of paintings and drawings that include highly acclaimed artworks as well as artworks of lesser artistic claim (control artworks). Many of the patients' works did not differ from these control images. However, the artworks of 6 of the 14 artists with schizophrenia possess image properties that deviate from the range of values obtained for the control artworks. For example, the artworks of four of the patients are characterized by a relative dominance of specific edge orientations in their images (low first-order entropy of edge orientations). Three patients created artworks with a relatively high ratio of fine detail to coarse structure (high slope of the Fourier spectrum). In conclusion, the present exploratory study opens novel perspectives for the objective scientific investigation of visual artworks that were created by persons who suffer from schizophrenia.

On the organization of human T-cell receptor loci: log-periodic distribution of T-cell receptor gene segments

The human T-cell repertoire is complex and is generated by the rearrangement of variable (V), diversity (D) and joining (J) segments on the T-cell receptor (TCR) loci. The T-cell repertoire demonstrates self-similarity in terms clonal frequencies when defined by V, D and J gene segment usage; therefore to determine whether the structural ordering of these gene segments on the TCR loci contributes to the observed clonal frequencies, the TCR loci were examined for self-similarity and periodicity in terms of gene segment organization. Logarithmic transformation of numeric sequence order demonstrated that the V and J gene segments for both T-cell receptor α (TRA) and β (TRB) loci are arranged in a self-similar manner when the spacing between the adjacent segments was considered as a function of the size of the neighbouring gene segment, with an average fractal dimension of approximately 1.5. Accounting for the gene segments occurring on helical DNA molecules with a logarithmic distribution, sine and cosine functions of the log-transformed angular coordinates of the start and stop nucleotides of successive TCR gene segments showed an ordered progression from the 5' to the 3' end of the locus, supporting a log-periodic organization. T-cell clonal frequency estimates, based on V and J segment usage, from normal stem cell donors were plotted against the V and J segment on TRB locus and demonstrated a periodic distribution. We hypothesize that this quasi-periodic variation in gene-segment representation in the T-cell clonal repertoire may be influenced by the location of the gene segments on the periodic-logarithmically scaled TCR loci. Interactions between the two strands of DNA in the double helix may influence the probability of gene segment usage by means of either constructive or destructive interference resulting from the superposition of the two helices.

The self-similarity theory of high pressure torsion

By analyzing the problem of high pressure torsion (HPT) in the rigid plastic formulation, we show that the power hardening law of plastically deformed materials leads to self-similarity of HPT, admitting a simple mathematical description of the process. The analysis shows that the main parameters of HPT are proportional to β ^q , with β being the angle of the anvil rotation. The meaning of the parameter q is: q = 0 for velocity and strain rate, q = 1 for shear strain and von Mises strain, q = n for stress, pressure and torque (n is the exponent of a power hardening law). We conclude that if the hardening law is a power law in a rotation interval β, self-similar regimes can emerge in HPT if the friction with the lateral wall of the die is not too high. In these intervals a simple mathematical description can be applied based on self-similarity. Outside these ranges, the plasticity problem still has to be solved for each value of β. The results obtained have important practical implications for the proper design and analysis of HPT experiments.

The dynamic generalization of the Eshelby inclusion problem and its static limit

The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.

Box-Counting Dimension Revisited: Presenting an Efficient Method of Minimizing Quantization Error and an Assessment of the Self-Similarity of Structural Root Systems

Fractal dimension (FD), estimated by box-counting, is a metric used to characterize plant anatomical complexity or space-filling characteristic for a variety of purposes. The vast majority of published studies fail to evaluate the assumption of statistical self-similarity, which underpins the validity of the procedure. The box-counting procedure is also subject to error arising from arbitrary grid placement, known as quantization error (QE), which is strictly positive and varies as a function of scale, making it problematic for the procedure's slope estimation step. Previous studies either ignore QE or employ inefficient brute-force grid translations to reduce it. The goals of this study were to characterize the effect of QE due to translation and rotation on FD estimates, to provide an efficient method of reducing QE, and to evaluate the assumption of statistical self-similarity of coarse root datasets typical of those used in recent trait studies. Coarse root systems of 36 shrubs were digitized in 3D and subjected to box-counts. A pattern search algorithm was used to minimize QE by optimizing grid placement and its efficiency was compared to the brute force method. The degree of statistical self-similarity was evaluated using linear regression residuals and local slope estimates. QE, due to both grid position and orientation, was a significant source of error in FD estimates, but pattern search provided an efficient means of minimizing it. Pattern search had higher initial computational cost but converged on lower error values more efficiently than the commonly employed brute force method. Our representations of coarse root system digitizations did not exhibit details over a sufficient range of scales to be considered statistically self-similar and informatively approximated as fractals, suggesting a lack of sufficient ramification of the coarse root systems for reiteration to be thought of as a dominant force in their development. FD estimates did not characterize the scaling of our digitizations well: the scaling exponent was a function of scale. Our findings serve as a caution against applying FD under the assumption of statistical self-similarity without rigorously evaluating it first.

Ants determine their next move at rest: motor planning and causality in complex systems

To find useful work to do for their colony, individual eusocial animals have to move, somehow staying attentive to relevant social information. Recent research on individual Temnothorax albipennis ants moving inside their colony's nest found a power-law relationship between a movement's duration and its average speed; and a universal speed profile for movements showing that they mostly fluctuate around a constant average speed. From this predictability it was inferred that movement durations are somehow determined before the movement itself. Here, we find similar results in lone T. albipennis ants exploring a large arena outside the nest, both when the arena is clean and when it contains chemical information left by previous nest-mates. This implies that these movement characteristics originate from the same individual neural and/or physiological mechanism(s), operating without immediate regard to social influences. However, the presence of pheromones and/or other cues was found to affect the inter-event speed correlations. Hence we suggest that ants' motor planning results in intermittent response to the social environment: movement duration is adjusted in response to social information only between movements, not during them. This environmentally flexible, intermittently responsive movement behaviour points towards a spatially allocated division of labour in this species. It also prompts more general questions on collective animal movement and the role of intermittent causation from higher to lower organizational levels in the stability of complex systems.