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The behavior of b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice L g and allowing the source configuration size L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as "faithfulness", because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that b = 2 weighted majority-like RSRG maps acting on the q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule. [ABSTRACT FROM AUTHOR]

The aim of this review paper is to survey the fractal morphology of scale-invariant patterns. We are particularly focusing on the scale and conformal invariance, as well as on the fractal non-uniformity (multifractality), inhomogeneity (lacunarity), and anisotropy (succolarity). We argue that these features can be properly quantified by the following six adimensional numbers: the fractal (e.g., similarity, box-counting, or Assouad) dimension, conformal dimension, degree of multifractal non-uniformity, coefficient of multifractal asymmetry, index of lacunarity, and index of fractal anisotropy. The difference between morphological properties of mathematical and real-world fractals is especially outlined in this review paper. [ABSTRACT FROM AUTHOR]

Two ranges of dielectric permittivity (k) increase in polymer composites upon the modification of BaTiO3 filler with multiwalled carbon nanotubes (MWCNTs) are shown for the first time. The first increase in permittivity is observed at low MWCNT content in the composite (approximately 0.07 vol.%) without a considerable increase in dielectric loss tangent and electrical conductivity. This effect is determined by the intensification of filler–polymer interactions caused by the nanotubes, which introduce Brønsted acidic centers on the modified filler surface and thus promote interactions with the cyanoethyl ester of polyvinyl alcohol (CEPVA) polymer binder. Consequently, the structure of the composites becomes more uniform: the permittivity increase is accompanied by a decrease in the lacunarity (nonuniformity) of the structure and an increase in scale invariance, which characterizes the self-similarity of the composite structure. The permittivity of the composites in the first range follows a modified Lichtenecker equation, including the content of Brønsted acidic centers as a parameter. The second permittivity growth range features a drastic increase in the dielectric loss tangent and conductivity corresponding to the percolation effect with the threshold at 0.3 vol.% of MWCNTs. [ABSTRACT FROM AUTHOR]

This paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in D-dimensional spacetime. These solutions are found by exploiting the dimensional consistency of the radial differential equation for a single massless scalar field, which allows it to transform into an autonomous equation. For massive theories, the radial equation is not exactly solvable, but the massless solutions provide useful approximations to the results for the massive case. The solutions presented here depend on the power of the interaction and on the spatial dimension, both of which may be noninteger. Scalar equations arising in the study of conformal invariance fit into this framework, and classes of new solutions are found. These solutions exhibit two distinct behaviors as D → 2 from above. [ABSTRACT FROM AUTHOR]

For a non-relativistic scale invariant system in two spatial dimensions, the quantum scattering amplitude f() is given as a dispersion relation, with a simple closed form for Im f() as well as the integrated cross-section / Im f(= 0). For fixed 6= 0, the ~ ! 0 classical limit is straightforward to obtain. [ABSTRACT FROM AUTHOR]

In many practical situations, we observe a special Burr probability distribution — e.g., Burr distribution describes the distribution of people by income, it describes the rainfall data, etc. In this paper, we provide a theoretical explanation for the ubiquity of Burr distributions: namely, we show that this distribution naturally follows from scale-invariance. To be more precise, the simplest distribution that can be obtained from scale invariance is the Pareto distribution — a frequent particular case of the general Burr family of distributions. Next simplest are all distributions from the Burr family. We also use the scale-invariance approach to come up with a more general class of distributions that will, hopefully, provide an even more accurate description of the corresponding phenomena. [ABSTRACT FROM AUTHOR]

The modeling of sub-daily extreme rainfall has long constituted a challenge for hydrologists in view of the limited availability of data, both in time and space. In this context, the scale invariance principle, which formally links precipitation intensities across a range of time scales under a rigorous theoretical underpin, comprises a useful and parsimonious tool for deriving probabilistic models for the referred stochastic variables. This paper aims at reviewing the history of application of the scale invariance principle to the marginal distributions of intense precipitation, as well as presenting recent developments in this field. We first address the basic concepts and the mathematical formalism of scale invariant models, discussing distinct scaling regimes across durations and estimation procedures. Next, we present a comprehensive review on at-site and regional stationary scaling models applied in several regions of the world, with focus on their underlying assumptions and main limitations. Finally, we address extensions of the rationale for nonstationary models as a means of accommodating potential effects of climate change in extreme short-duration rainfall. While discussing each of these frameworks, we indicate potential research gaps and modeling developments which could further advance the understanding of scaling behavior of extreme precipitation and improve statistical inference. Hence, this review may constitute a useful guide for practitioners and motivate future research in the modeling of short-duration extreme rainfall. [ABSTRACT FROM AUTHOR]

On the basis of a general action principle, we revisit the scale invariant field equation using the cotensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor λ , which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler's third law is a remarkable scale-invariant property. [ABSTRACT FROM AUTHOR]

Beta‐decay half‐lives for the free neutron, for 6He and 8He, and for 8Li are calculated ab initio from geometrical thermodynamics arguments, independently of any quantum mechanics. Half‐lives for the decay of 8Be to two alphas and for the disintegration of the tetraneutron are also calculated. The calculated values are close to those experimentally observed. [ABSTRACT FROM AUTHOR]

The concept of memory is of central importance for characterizing complex systems and phenomena. Presence of long-term memories indicates how their dynamics can be less sensitive to initial conditions compared to the chaotic cases. On the other hand, it is empirically known that the Feller–Pareto distribution, which decays as the power law i.e. the scale-invariant nature, frequently appears as a statistical law generated by the dynamics of complex systems. However, it is generally not a simple task to determine if a system obeying such a power law possesses a high degree of complexity with a long-term memory. Here, a new method is proposed for characterization of memory. In particular, a scaling relation to be satisfied by any memoryless dynamics generating the Feller–Pareto power-law distribution is presented. Then, the method is applied to the real data of energies released by a series of earthquakes and acceleration of ground motion due to a strong earthquake. It is shown in this way that the sequence of the released energy in seismicity is memoryless in the event time, whereas that of acceleration is memoryful in the sampling time. [ABSTRACT FROM AUTHOR]

The Principle of Indifference ('PI': the simplest non-informative prior in Bayesian probability) has been shown to lead to paradoxes since Bertrand (1889). Von Mises (1928) introduced the 'Wine/Water Paradox' as a resonant example of a 'Bertrand paradox', which has been presented as demonstrating that the PI must be rejected. We now resolve these paradoxes using a Maximum Entropy (MaxEnt) treatment of the PI that also includes information provided by Benford's 'Law of Anomalous Numbers' (1938). We show that the PI should be understood to represent a family of informationally identical MaxEnt solutions, each solution being identified with its own explicitly justified boundary condition. In particular, our solution to the Wine/Water Paradox exploits Benford's Law to construct a non-uniform distribution representing the universal constraint of scale invariance, which is a physical consequence of the Second Law of Thermodynamics. [ABSTRACT FROM AUTHOR]

In this paper, we use the holographic method to compute the entanglement entropy for different regions in the fourth-order scale-invariant theory of gravity. In four dimensions, the action of scale-invariant gravity contains a parameter where the moduli space has some distinguished points, precisely, the space of solutions contains the Log-gravity and for this specific solution, we compute the entanglement entropy. We also use the numerical method to investigate mutual information and tripartite information. Moreover, we make a comment on the sign of these quantities. [ABSTRACT FROM AUTHOR]

We introduce a new class of optimization problems called scale invariant problems that cover interesting problems in machine learning and statistics and show that they are efficiently solved by a general form of power iteration called scale invariant power iteration (SCI-PI). SCI-PI is a special case of the generalized power method (GPM) (Journée et al., 2010) where the constraint set is the unit sphere. In this work, we provide the convergence analysis of SCI-PI for scale invariant problems which yields a better rate than the analysis of GPM. Specifically, we prove that it attains local linear convergence with a generalized rate of power iteration to find an optimal solution for scale invariant problems. Moreover, we discuss some extended settings of scale invariant problems and provide similar convergence results. In numerical experiments, we introduce applications to independent component analysis, Gaussian mixtures, and non-negative matrix factorization with the KL-divergence. Experimental results demonstrate that SCI-PI is competitive to application specific state-of-the-art algorithms and often yield better solutions. [ABSTRACT FROM AUTHOR]

The technique of functional Legendre transforms is used to develop an effective method for calculating the characteristics of critical phenomena in quantum field theory models in the Euclidean space of dimension d. Based on the diagrammatic representation of the second Legendre transform in the theory with a cubic interaction potential, the construction of self-consistent equations is carried out, the solution of which makes it possible to find the dimensions not only of the main fields, but also of the quadratic on the composite operators within the 1 / n -expansion. Application of the proposed methods in the model F has given the opportunity to calculate in the main approximation by 1 / n the anomalous dimensions of both scalar and tensor composite operators quadratic on the fields ϕ. For them, as functions of the spatial dimension d, we obtained explicit analytical expressions in the form of relations of two polynomials with integer coefficients. [ABSTRACT FROM AUTHOR]

A recently developed variational resummation technique incorporating renormalization group properties has been shown to solve the scale dependence problem that plagues the evaluation of physical observables. This method is used in the present work to evaluate thermodynamical quantities within the two‐dimensional nonlinear sigma model, which shares some features with Yang‐Mills theories like asymptotic freedom, trace anomaly, and the nonperturbative generation of a mass gap. Besides the fact that nonperturbative results can be readily generated solely by considering lowest‐order contributions, we also show that its next‐to‐leading correction indicates convergence to the sought‐after scale invariance. [ABSTRACT FROM AUTHOR]

A Bayesian prediction problem for the two-dimensional Wishart model is investigated within the framework of decision theory. The loss function is the Kullback–Leibler divergence. We construct a scale-invariant and permutation-invariant prior distribution that shrinks the correlation coefficient. The prior is the geometric mean of the right invariant prior with respect to permutation of the indices, and is characterized by a uniform distribution for Fisher's |$z$| -transformation of the correlation coefficient. The Bayesian predictive density based on the prior is shown to be minimax. [ABSTRACT FROM AUTHOR]

Knowledge of the loss landscape geometry makes it possible to successfully explain the behavior of neural networks, the dynamics of their training, and the relationship between resulting solutions and hyperparameters, such as the regularization method, neural network architecture, or learning rate schedule. In this paper, the dynamics of learning and the surface of the standard cross-entropy loss function and the currently popular mean squared error (MSE) loss function for scale-invariant networks with normalization are studied. Symmetries are eliminated via the transition to optimization on a sphere. As a result, three learning phases with fundamentally different properties are revealed depending on the learning step on the sphere, namely, convergence phase, phase of chaotic equilibrium, and phase of destabilized learning. These phases are observed for both loss functions, but larger networks and longer learning for the transition to the convergence phase are required in the case of MSE loss. [ABSTRACT FROM AUTHOR]

One of the foremost goals of research in physics is to find the most basic and universal theories that describe our universe. Many theories assume the presence of absolute space and time in which the physical objects are located and physical processes take place. However, it is more fundamental to understand time as relative to the motion of another object, e.g., the number of swings of a pendulum, and the position of an object primarily relative to other objects. This paper aims to explain how using the principle of relationalism (to be introduced below), classical mechanics can be formulated on a most elementary space, which is freed from absolute entities: shape space. In shape space, only the relative orientation and length of subsystems are taken into account. A sufficient requirement for the validity of the principle of relationalism is that by changing a system’s scale variable, all the theory’s parameters that depend on the length, get changed accordingly. In particular, a direct implementation of the principle of relationalism introduces a proper transformation of the coupling constants of the interaction potentials in classical physics. This change leads consequently to a transformation in Planck’s measuring units, which enables us to derive a metric on shape space in a unique way. In order to find out the classical equations of motion on shape space, the method of “symplectic reduction of Hamiltonian systems” is extended to include scale transformations. In particular, we will give the derivation of the reduced Hamiltonian and symplectic form on shape space, and in this way, the reduction of a classical system with respect to the entire similarity group is achieved. [ABSTRACT FROM AUTHOR]

Related to the letters of an alphabet, entropy means the average number of binary digits required for the transmission of one character. Checking tables of statistical data, one finds that, in the first position of the numbers, the digits 1 to 9 occur with different frequencies. Correspondingly, from these probabilities, a value for the Shannon entropy H can be determined as well. Although in many cases, the Newcomb–Benford Law applies, distributions have been found where the 1 in the first position occurs up to more than 40 times as frequently as the 9. In this case, the probability of the occurrence of a particular first digit can be derived from a power function with a negative exponent p > 1. While the entropy of the first digits following an NB distribution amounts to H = 2.88, for other data distributions (diameters of craters on Venus or the weight of fragments of crushed minerals), entropy values of 2.76 and 2.04 bits per digit have been found. [ABSTRACT FROM AUTHOR]

SiamPRN algorithm performs well in visual tracking, but it is easy to drift under occlusion and fast motion scenes because it uses ℓ 1 -smooth loss function to measure the regression location of bounding box. In this paper, we propose a multivariate intersection over union (MIOU) loss in SiamRPN tracking framework. Firstly, MIOU loss includes three geometric factors in regression: the overlap area ratio, the center distance ratio, and the aspect ratio, which can better reflect the coincidence degree of target box and prediction box. Secondly, we improve the definition of aspect ratio loss to avoid gradient explosion, improve the optimization performance of prediction box. Finally, based on SiamPRN tracker, we compared the tracking performance of ℓ 1 -smooth loss, IOU loss, GIOU loss, DIOU loss, and MIOU loss. Experimental results show that the MIOU loss has better target location regression than other loss functions on the OTB2015 and VOT2016 benchmark, especially for the challenges of occlusion, illumination change and fast motion. [ABSTRACT FROM AUTHOR]

To investigate the plastic yielding behaviour of soil, a new plasticity framework using stress Hausdorff derivative was developed in this study. Two hypotheses concerning the effect of scale transform on physical behaviours of soil were proposed, based on which the stress–strain relations in the scaled stress space were provided, and then mapped back to the ordinary stress space for describing the nonlinear strength characteristics of soil. As an alternative approach to the classic plasticity, two case studies on simulating a series of experimental results of sand and clay were made, where it was found that the proposed approach can capture the plastic yielding behaviour of soils. [ABSTRACT FROM AUTHOR]

We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such that symmetries are preserved in the statistical sense. We focus on hidden symmetries, which are broken in the original system but restored in the normalized system. The central motivation and application of this construction is the intermittency phenomenon in turbulence. We show that hidden symmetries yield power law scaling for structure functions, and derive formulas for their exponents in terms of normalized measures. The use of Galilean transformation in the equivalence relation leads to the quasi-Lagrangian description, making the developed theory applicable to the Euler and Navierâ€"Stokes systems. [ABSTRACT FROM AUTHOR]

We address the problem of pedestrian tracking in videos of crowded scene which are captured by first-person viewpoint. The constant motion of camera and pedestrian makes this task challenging. The prime challenges are natural head motion of wearer and target loss and reappearance in a later frame, due to frequent changes in field of view. We propose that the use of first-person vision specific optical flow information and also the modification in the update process along with search region of trackers are useful to identify a lost target in a later frame. This process is termed re-identification in this paper. The specific trackers modified are MEEM and STRUCK. In addition to re-identification we achieve scale invariant tracking (upto 50 % scale variation) and speed up by a factor of 2. We name our tracker as EgoTracker, since it utilizes the information which is specific to egocentric vision. [ABSTRACT FROM AUTHOR]

The length-scale dependence of fatigue crack growth is evaluated for a set of metallic materials, namely titanium Ti-6Al-4V, ductile iron EN-GJS-500-7 and tool steel AISI H13, by performing fatigue crack growth tests on geometrically similar compact C(T) specimens of different sizes. With references to length-scale-invariant variables, notably the crack growth rate normalised by the specimen width, it is demonstrated that fatigue crack growth is not a length-scale-invariant process for the tested conditions. In particular, the length-scale dependence is less significant for larger specimens and at longer crack lengths. The test results also contradict the hypothesis of similitude, i.e., that the growth rate is uniquely related to the stress-intensity-factor range, as smaller specimens manifest a higher growth rate when compared at the same stress-intensity-factor range. The observations are in line with presented fracture-mechanical demonstrations, which show that the Paris–Erdogan law depends on the length scale whenever fatigue crack growth is anticipated to be scale invariant. • Paris–Erdogan law imposes an unexpected length-scale dependency. • Fatigue crack growth is generally length-scale dependent. • For large specimens, long cracks and high stress, the scale dependency is smaller. • The hypothesis of similitude is contradicted by varying the specimen size. [ABSTRACT FROM AUTHOR]

A distinctive fishtail feature in multifractal spectra derived from heart rate variability (HRV) data closely resembles similar features in plots of the Gibbs free energy versus reduced pressure for a van der Waals fluid below the critical temperature, and, in analogy to the nonideal fluid, signals a phase transition from one multifractal state to another. This fishtail feature, often overlooked or dismissed in previous studies, is observed most prevalently in unhealthy patients and suggests a realignment between multifractal variables and their thermodynamic analogs. The van der Waals analogy and subsequent alternate interpretation of multifractal variables and functions lead one to construct a three-dimensional heart health phase diagram reminiscent of a P − V − T diagram for the van der Waals fluid. The diagram which makes use of the Maxwell width as a measure of heart health produces an orderly progression from healthy to unhealthy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate signatures of variation in the behavior of correlated time series by analyzing changes in the topological properties of the corresponding visibility graph. Variations in six different network measures: assortativity, average path length, clustering, transitivity, density, and the average of the mean link length, are explored. We construct visibility graphs from the original and the magnitude and sign of its increment series. Both the horizontal and the natural visibility graphs are studied. Through extensive numerical studies on the time series of fractional Brownian motion (fBm), we first identify network measures that can reflect the changes in correlations in the time series. The efficacy of these markers is examined to identify the transitions in two systems, a two-dimensional (2D) Ising spin system and EEG data with seizures. While all the identified network measures capture the change in the thermal equilibrium correlations for the Ising spin system, they have limited success in the case of the time-dependent fluctuations in the EEG data. We identify some markers relevant to detecting seizures in the EEG data set. [ABSTRACT FROM AUTHOR]

Scale-invariant keypoint detection is a fundamental problem in low-level vision. To accelerate keypoint detectors (e.g. DoG, Harris-Laplace, Hessian-Laplace) that are developed in Gaussian scale-space, various fast detectors (e.g., SURF, CenSurE, and BRISK) have been developed by approximating Gaussian filters with simple box filters. However, there is no principled way to design the shape and scale of the box filters. Additionally, the involved integral image technique makes it difficult to figure out the continuous kernels that correspond to the discrete ones used in these detectors, so there is no guarantee that those good properties such as causality in the original Gaussian space can be inherited. To address these issues, in this paper, we propose a unified B-spline framework for scale-invariant keypoint detection. Owing to an approximate relationship to Gaussian kernels, the B-spline framework provides a mathematical interpretation of existing fast detectors based on integral images. In addition, from B-spline theories, we illustrate the problem in repeated integration, which is the generalized version of the integral image technique. Finally, following the dominant measures for keypoint detection and automatic scale selection, we develop B-spline determinant of Hessian (B-DoH) and B-spline Laplacian-of-Gaussian (B-LoG) as two instantiations within the unified B-spline framework. For efficient computation, we propose to use repeated running-sums to convolve images with B-spline kernels with fixed orders, which avoids the problem of integral images by introducing an extra interpolation kernel. Our B-spline detectors can be designed in a principled way without the heuristic choice of kernel shape and scales and naturally extend the popular SURF and CenSurE detectors with more complex kernels. Extensive experiments on the benchmark dataset demonstrate that the proposed detectors outperform the others in terms of repeatability and efficiency. [ABSTRACT FROM AUTHOR]

We calculate the fourth cluster coefficients of the homogeneous unitary spin 1/2 Fermi gas as functions of the internal-state mass ratio, over intervals constrained by the 3- or 4-body Efimov effect. For this we use our 2016 conjecture (validated for equal masses by Hou and Drut in 2020) in a numerically efficient formulation making the sum over angular momentum converge faster, which is crucial at large mass ratio. The mean cluster coefficient, relevant for equal chemical potentials, is not of constant sign and increases rapidly close to the Efimovian thresholds. We also get the fourth virial coefficients, which we find to be very poor indicators of interaction-induced 4-body correlations. We obtain analytically for all n the cluster coefficients of order n Å1 for an infinity-mass impurity fermion, matching the conjecture for n Æ 3. Finally, in a harmonic potential, we predict a non-monotonic behavior of the 3Å1 cluster coefficient with trapping frequency, near mass ratios where this coefficient vanishes in the homogeneous case. A multilingual version is available in separate files on the open archive HAL at https://hal.archives-ouvertes.fr/hal-03592961. [ABSTRACT FROM AUTHOR]

In this study, heart rate time series have been investigated to assess the presence of scale invariance property of a signal. The electrocardiogram signal is recording from Biopac MP100 system during postural shift from supine to standing. Heart rate time series is extracted from electrocardiogram signal and is analyzed using detrended fluctuation analysis. Since this analysis is based on random walk theory, noise level due to imperfect measurement in recording is reduced and it can systematically eliminate trends of different orders. The essence of this method is to understand the presence of scale invariance in heart rate and to classify the scaling factor changes during postural shift from supine to standing. The obtained scaling factor is invariant with respect to scale sizes, which confirm that heart rate signal is scale invariant. In addition, scaling factor is significantly less for supine position as compared with the standing indicates it is directly proportional to heart rate. Therefore, this method helps in calculating the risk to evaluate the patient status during postural changes to obtain information about heart instability. [ABSTRACT FROM AUTHOR]

Exact flows of an incompressible fluid satisfying the Beltrami equation inside a spherical shell are constructed in the Cartesian coordinates in terms of elementary functions. Two scale-invariant equations defining two infinite series of eigenvalues λn and λ ̃ m ${\tilde {\lambda }}_{m}$ of the operator curl in the shell with the nonpenetration boundary conditions on the boundary spheres are derived. The corresponding eigenfields are presented in explicit form and their symmetries are investigated. Asymptotics of the eigenvalues λn and λ ̃ m ${\tilde {\lambda }}_{m}$ at n, m → ∞ are obtained. [ABSTRACT FROM AUTHOR]

We introduce a novel model of affine gravity, which implements the no-scale scenario. Namely, the Planck mass and Hubble constant emerge dynamically in our model, through the mechanism of spontaneous breaking of scale invariance. This naturally gives rise to inflation, thus introducing a new inflationary mechanism. Moreover, the time direction and nondegenerate metric emerge dynamically as well, which allows considering the usual General Relativity as an effective theory. We show that our model is phenomenologically viable, both from the perspective of the direct tests of gravity and from the standpoint of cosmological evolution. [ABSTRACT FROM AUTHOR]

Background: The output of many healthy physiological systems displays fractal fluctuations with self-similar temporal structures. Altered fractal patterns are associated with pathological conditions. There is evidence that patients with bipolar disorder have altered daily behaviors. Methods: To test whether fractal patterns in motor activity are altered in patients with bipolar disorder, we analyzed 2-week actigraphy data collected from 106 patients with bipolar disorder type I in a euthymic state, 73 unaffected siblings of patients, and 76 controls. To examine the link between fractal patterns and symptoms, we analyzed 180-day actigraphy and mood symptom data that were simultaneously collected from 14 patients. Results: Compared to controls, patients showed excessive regularity in motor activity fluctuations at small time scales (<1.5 h) as quantified by a larger scaling exponent (α1 > 1), indicating a more rigid motor control system. α1 values of siblings were between those of patients and controls. Further examinations revealed that the group differences in α1 were only significant in females. Sex also affected the group differences in fractal patterns at larger time scales (>2 h) as quantified by scaling exponent α2. Specifically, female patients and siblings had a smaller α2 compared to female controls, indicating more random activity fluctuations; while male patients had a larger α2 compared to male controls. Interestingly, a higher weekly depression score was associated with a lower α1 in the subsequent week. Conclusions: Our results show sex- and scale-dependent alterations in fractal activity regulation in patients with bipolar disorder. The mechanisms underlying the alterations are yet to be determined. [ABSTRACT FROM AUTHOR]

Crowd counting is an important vision task, which faces challenges on continuous scale variation within a given scene and huge density shift both within and across images. These challenges are typically addressed using multi-column structures in existing methods. However, such an approach does not provide consistent improvement and transferability due to limited ability in capturing multi-scale features, sensitivity to large density shift, and difficulty in training multi-branch models. To overcome these limitations, a Single-column Scalere-invariant Network (ScSiNet) is presented in this paper, which extracts sophisticated scale-invariant features via the combination of interlayer multi-scale integration and a novel intralayer scale-invariant transformation (SiT). Furthermore, in order to enlarge the diversity of densities, a randomly integrated loss is presented for training our single-branch method. Extensive experiments on public datasets demonstrate that the proposed method outperforms state-of-the-art approaches in counting accuracy and achieves remarkable transferability and scale-invariant property. [ABSTRACT FROM AUTHOR]

• A single IDF model valid for various return periods and durations is developed. • New scaling models of design rainfall curves are presented. • Adjusted rainfall event durations are considered in the context of scale invariance. • Parsimonious intensity–duration–frequency (IDF) structure is obtained. • Good fit to quantile predictions of extreme rainfall data for specified durations. This paper deals with intensity–duration–frequency (IDF) curves, which express the relationship between average rainfall intensity and event duration for various probabilities of non-exceedance (or return periods) for the design/analysis of hydraulic interventions and infrastructures in riverine and urban drainage contexts. New scaling models are proposed to develop a single IDF model valid for all durations, from below 1 h to 24 h. In these models, the scale invariance is applied to rainfall intensity to obtain a parsimonious IDF structure capable of defining a family of IDF curves at various return periods. The main novelty consists of the formulation of simple and multiple scale invariance based on adjusted design event durations. The parameterization is carried out in two phases: in the first phase, the adjustment size is searched for iteratively while the other parameters of the IDF structure are directly obtained in cascade; in the second phase, which enables considering different reliability levels for extreme rainfall data as a function of sample length at different durations, parameter refinement is carried out by means of a local optimization algorithm. The second step accommodates data samples of different extension as a function of available data at different durations. The application to four case studies at various latitudes in Europe, namely Helsinborg (Sweden), Frauenwald (Germany), Pavia and Erice (Italy), proves the IDF structure to fit well the quantile predictions of extreme rainfall data for specified durations below and above one hour. [ABSTRACT FROM AUTHOR]

This paper deals with the violation or retention of symmetries associated with the self-adjoint extension of the Hamiltonian for homogeneous but anisotropic Bianchi I cosmological model. This extension is required to make sure the quantum evolution is unitary. It is found that the scale invariance is lost, but the Noether symmetries are preserved. [ABSTRACT FROM AUTHOR]

Nobel-Prize-winning Black-Scholes equations are actively used to estimate the price of options and other financial instruments. In practice, they provide a good estimate for the price, but the problem is that their original derivation is based on many simplifying statistical assumptions which are, in general, not valid for financial time series. The fact that these equations are effective way beyond their usual assumptions leads to a natural conclusion that there must be an alternative derivation for these equations, a derivation that does not use the usual too-strong assumptions. In this paper, we provide such a derivation in which the only substantial assumption is a natural symmetry: namely, scale-invariance of the corresponding processes. Scale-invariance also allows us to describe possible generalizations of Black-Scholes equations, generalizations that we hope will lead to even more accurate estimates for the corresponding prices. [ABSTRACT FROM AUTHOR]

Starting with a simple two-scalar field system coupled to a modified measure that is independent of the metric, we, first, find a Born–Infeld dynamics sector of the theory for a scalar field and second, show that the initial scale invariance of the action is dynamically broken and leads to a scale charge nonconservation, although there is still a conserved dilatation current. [ABSTRACT FROM AUTHOR]

We report experimental results and test axiomatic models of unstructured bargaining by checking the empirical relevance of the underlying axioms. Our data support strong efficiency, symmetry, independence of irrelevant alternatives and monotonicity, and reject scale invariance. Individual rationality and midpoint domination are violated by a significant fraction of agreements that implement equal division in highly unequal circumstances. Two well-known bargaining solutions that satisfy the confirmed properties explain the observed agreements reasonably well. The most frequent agreements in our sample are the ones suggested by the equal-division solution. In terms of the average Euclidean distance, the theoretical solution that best explains the data is the deal-me-out solution (Sutton, 1986 ; Binmore et al., 1989, 1991). Popular solutions that satisfy scale invariance, individual rationality, and midpoint domination, as the well-known Nash or Kalai-Smorodinsky bargaining solutions, perform poorly in the laboratory. [ABSTRACT FROM AUTHOR]

Scale issues are very meaningful in geography, but nowadays nobody knows how to explain their ubiquitous existence theoretically. Fractality is not an accident for all geographical objects. The aim of this article is to demonstrate to what extent the theory of scale relativity (SR) can be used to solve the problem of geographic scales. With it, we can explain why fractal objects are everywhere. First, we summarize geographic scale position, followed by introducing all tools to understand SR with basic definitions, scale in cartography, how to measure a scale, scales in and from nature, and scale and theoretical geography. Second, we quickly describe the theory of SR. Indeed, it is an elementary geometry around first principles, characterization of scale variables, and scale laws. This article also aims to clarify why geographical objects are non-fractal, in a first calculus, and fractal, in a second calculus with the theory of scale relativity. Third, we will underpin this position through several geographic cases with a karstological example, two urban areas (Montéliard and Avignon), and a hydrographic network and contours of level lines (Gardons). All of them will be carefully analyzed with a fractal analysis. Therefore, we conclude that in this case we are well and truly within the framework of the theory of SR, depending on the results. [ABSTRACT FROM AUTHOR]

Subsurface heterogeneity could influence groundwater flow with implications on structure and productivity of aquifer ecosystems. Here we investigate effects of multifractal heterogeneity on topology of groundwater flow through MODFLOW simulations within a Monte Carlo framework. The results show that heterogeneity leads to focused groundwater advection and the creation of hotspots of bending‐type vortex flow in the fields. We demonstrate for the first time that the vortex structures characterized by Q criterion are 3‐D distributed and greatly deform surrounding pore‐water flow. The structures exhibit scale‐invariant features in multifractal fields and in stationary fields below the correlation scale, indicating that such vortex flow might be widely present with no characteristic scale. Complex spatial patterns of kinetic energy dissipation rate are identified for pore water flowing through heterogeneous porous media and correlate strongly with preferential flows. These findings are important for understanding solute fate and transport in aquifer systems. Plain Language Summary: The findings of this study provide insight into influences of heterogeneity on groundwater flow and solute transport processes in aquifers. Our results reveal that multifractal heterogeneity leads to focused regions of groundwater advection and creates hotspots of bending‐type vortex flow that are 3‐D and greatly deform surrounding pore water. It has strong implications for groundwater contamination and associated implementation of effective bioremediation strategies. This is because solutes/contaminants would be spun around these structures and therefore have less interaction with the soil matrix in the inner part of the vortices. We also demonstrate that the structures of the vortex flow are scale invariant in multifractal K fields and below the correlation scale of stationary K fields. It indicates that these vortex flow structures may be widely present in the field with no characteristic scale. Substantial energy dissipation along the high‐velocity zones also has strong implications for understanding activity, diversity, and the distribution of subterranean microorganisms. Key Points: Deformation and rotation of pore‐water flow occur near high vorticity zones, formed as blobs and correlated strongly to preferential flowsVorticity‐dominated blobs exhibit scale invariance in both multifractal K fields and up to the correlation scale in variogram K fieldsHeterogeneity dissipates substantial kinetic energy within large pore‐water velocity zones [ABSTRACT FROM AUTHOR]

In this paper, we propose two strategies of reducing the amount of data needed for binary tomographic reconstructions. We study how the direction dependency changes by reducing the resolution of an image and we point out how to specify the most informative angles for the original image using its downscaled version. We also show how to predict the final acceptable resolution. Applications of the proposed strategies are also mentioned. [ABSTRACT FROM AUTHOR]

We consider an online supervised learning problem, in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of the instances. Our goal is to design online algorithms which also enjoy this property, i.e. are scale-invariant. We start with the case of coordinate-wise invariance, in which the individual coordinates (features) can be arbitrarily rescaled. We give an algorithm, which achieves essentially optimal regret bound in this setup, expressed by means of a coordinate-wise scale-invariant norm of the comparator. We then study general invariance with respect to arbitrary linear transformations. We first give a negative result, showing that no algorithm can achieve a meaningful bound in terms of scale-invariant norm of the comparator in the worst case. Next, we compliment this result with a positive one, providing an algorithm which "almost" achieves the desired bound, incurring only a logarithmic overhead in terms of the relative size of the instances. [ABSTRACT FROM AUTHOR]

We employed high-density microelectrode arrays to investigate spontaneous firing patterns of neurons in brain circuits of the primary somatosensory cortex (S1) in mice. We recorded from over 150 neurons for 10 min in each of eight different experiments, identified their location in S1, sorted their action potentials (spikes), and computed their power spectra and inter-spike interval (ISI) statistics. Of all persistently active neurons, 92% fired with a single dominant frequency — regularly firing neurons (RNs) — from 1 to 8 Hz while 8% fired in burst with two dominant frequencies — bursting neurons (BNs) — corresponding to the inter-burst (2–6 Hz) and intra-burst intervals (20–160 Hz). RNs were predominantly located in layers 2/3 and 5/6 while BNs localized to layers 4 and 5. Across neurons, the standard deviation of ISI was a power law of its mean, a property known as fluctuation scaling, with a power law exponent of 1 for RNs and 1.25 for BNs. The power law implies that firing and bursting patterns are scale invariant: the firing pattern of a given RN or BN resembles that of another RN or BN, respectively, after a time contraction or dilation. An explanation for this scale invariance is discussed in the context of previous computational studies as well as its potential role in information processing. [ABSTRACT FROM AUTHOR]

Fractal geometry is fascinating due to its ability to describe the complex geometries naturally occurring in real world. The fractal or Hausdorff dimension d is used to describe the laws of scale invariance, which are exemplified by functions satisfying a mathematical law of the form f(sx) = sdf(x), being s the scale factor. Closely related is the problem of determining the fixed point of physical systems. Mathematically, the fixed point corresponds to a point in the domain of a function which is mapped to itself. In dynamical systems, it typically corresponds to the value to which the response of the system converge and stabilize. The present contribution aims to present the general concepts of scale invariance and fixed or critical points, employing such aspects to the problem of complex impedance association, both in series and in ladder configuration. The complex impedance, and transfer function of transmission lines and flow graphs related to the convergence the iterative equations to a fixed points are discussed in more details. [ABSTRACT FROM AUTHOR]

Reservoir computing (RC) is a relatively new machine-learning framework that uses an abstract neural network model, called reservoir. The reservoir forms a complex system with high dimensionality, nonlinearity, and intrinsic memory effect due to recurrent connections among individual neurons. RC manifests a best-in-class performance in processing information generated by complex dynamical systems, yet little is known about its microscopic/macroscopic dynamics underlying the computational capability. Here, we characterize the neuronal and network dynamics of liquid state machines (LSMs) using numerical simulations and Modified National Institute of Standards and Technology (MNIST) database classification tasks. The computational performance of LSMs largely depends on a dynamic range of neuronal avalanches whereby the avalanche patterns are determined by the neuron and network models. A larger dynamic range leads to higher performance—the MNIST classification accuracy is highest when the avalanche sizes follow a slowly decaying power-law distribution with an exponent of ∼1.5, followed by the power-law statistics with a larger exponent and the mixture of power-law/log-normal distributions. Network-theoretic analysis suggests that the formation of large functional clusters and the promotion of dynamic transitions between large and small clusters may contribute to the scale-invariant nature. This study provides new insight into our understanding of the computational principles of RC concerning the actions of the individual neurons and the system-level collective behavior. • Scale-invariant avalanches promote the reservoir's computational capability. • The scale-free nature depends on the properties of neurons and networks. • Network clustering mediates the neuronal and network dynamics. • Self-organized criticality may underlie the reservoir dynamics. [ABSTRACT FROM AUTHOR]

Mine subsidence is a regional geological hazard in China. To evaluate whether a power law describes the frequency–size statistics of mine subsidence, as for earthquakes and floods, we studied the frequency–size statistics of mine subsidence at three metal mines in China (Jinchuan Nickel Mine, Sanshandao Gold Mine, and Jingerquan Nickel Mine). Data sets for these mines consisted of 1088, 345, and 101 Global Positioning System (GPS) monitoring points, covering monitoring periods of 14.5, 4, and 3.5 years, respectively. Although these mines had different geological and hydrological settings, mining methods, and stress fields, their noncumulative frequency–size distributions for subsidence and uplift events can be described using power laws. The subsidence power-law exponent for these three mines ranged from 1.20 to 1.67, 1.49 to 1.94, and 1.01 to 1.17, with mean values of 1.46, 1.76, and 1.09, respectively. The power-law scaling for each mine was valid over the range from 2 to 455 mm, 2 to 566 mm, and 2 to 277 mm, respectively; scaling was positively correlated with the power-law exponent. The frequency–size statistics for subsidence events having different time scales showed an identical power-law dependence. The power-law behavior of uplift events was similar to subsidence events. This power-law behavior, its underlying mechanisms, factors influencing the power-law exponent, and the threshold between normal and extreme subsidence events are discussed herein. We conclude that the power-law distribution of mine subsidence events reflects the scale invariance of the subsidence system. This has important practical applications for subsidence hazard assessment and subsidence event prediction. [ABSTRACT FROM AUTHOR]

We embed in a generalized Borel procedure the notion of renormalization and renormalons. While there are several efforts in literature to have a semi-classical understanding of the renormalons, here we argue that this is not the fundamental issue and show how to deal with the problem. We find that the effective Lagrangians describing the effects of renormalons are non-local in space but local in time. The quark–antiquark potential in QCD with an infinite number of fermions is also analyzed. The connection between the analyzable functions, the Callan–Symanzyk equation and the renormalons, provides an insight of a non-perturbative renormalization from the standard perturbative renormalization approach. [ABSTRACT FROM AUTHOR]

Based on the extreme value theory, self–affinity, and scale invariance, we studied the temporal and spatial relationship and the variation of water level and established a model of Gumbel–Pareto distribution for designed flood calculation. The model includes the previous extreme value models, the over–threshold data, and the fractal features shared by previous extreme value models. The model was simplified into a logarithmic normal distribution and a Pareto distribution for specific parameter values, and was used to calculate the designed flood values for the Shanghai Wusong Station in 100– and 1 000–year return periods. The calculated results show that the value of the designed flood height calculated in the Gumbel–Pareto distribution is between those in the Gumbel and Pearson–III distributions. The designed flood values in the 100– and 1 000–year return periods of the model were 0.03% and 0.11% lower, respectively, than the Gumbel distribution and 0.06% and 1.54% higher, respectively, than the Pearson–III distribution. Compared to the traditional model based solely on extreme probability, the Gumbel–Pareto distribution model could better describe the probabilistic characteristics of extreme marine elements and better use the data. [ABSTRACT FROM AUTHOR]

In the early twentieth century, a handful of French geographers and historians famously suggested that mainland France comprised two agrarian systems: enclosed field systems with scattered settlements in the central and western France and openfield systems with grouped settlements in eastern France. This division between grouped and scattered settlements can still be found on the outskirts of urban areas. The objective of this paper is to determine whether the shape of urban areas varies with the type of built patterns in their periphery. To this end, we identify and characterise the local and global deviations from scale invariance of built patterns in mainland France. For this, we propose a new method—Geographically Weighted Fractal Analysis—that can characterise built patterns at a fine spatial resolution without making any a priori distinction between urban patterns and suburban or rural patterns. By applying GWFA to the spatial distribution of buildings throughout mainland France we identify six geographically consistent types of built patterns that are distinctive in the way buildings are either concentrated or dispersed across scales. The relationship between the local built textures and the global shape of twenty metropolitan areas is then analysed statistically. It is found that the proportion of dispersed (or concentrated) outer suburban built patterns in metropolitan areas is closely related to the distance threshold that marks the morphological limit of their urban areas. [ABSTRACT FROM AUTHOR]

SAR (species area relationship) is a classic ecological theory that has been extensively investigated and applied in the studies of global biogeography and biodiversity conservation in macro-ecology. It has also found important applications in microbial ecology in recent years thanks to the breakthroughs in metagenomic sequencing technology. Nevertheless, SAR has a serious limitation for practical applications—ignoring the species abundance and treating all species as equally abundant. This study aims to explore the biogeography discoveries of human microbiome over 18 sites of 5 major microbiome habitats, establish the baseline DAR (diversity-area scaling relationship) parameters, and perform comparisons with the classic SAR. The extension from SAR to DAR by adopting the Hill numbers as diversity measures not only overcomes the previously mentioned flaw of SAR but also allows for obtaining a series of important findings on the human microbiome biodiversity and biogeography. Specifically, two types of DAR models were built, the traditional power law (PL) and power law with exponential cutoff (PLEC), using comprehensive datasets from the HMP (human microbiome project). Furthermore, the biogeography "maps" for 18 human microbiome sites using their DAR profiles for assessing and predicting the diversity scaling across individuals, PDO profiles (pair-wise diversity overlap) for measuring diversity overlap (similarity), and MAD profile (for predicting the maximal accrual diversity in a population) were sketched out. The baseline biogeography maps for the healthy human microbiome diversity can offer guidelines for conserving human microbiome diversity and investigating the health implications of the human microbiome diversity and heterogeneity. [ABSTRACT FROM AUTHOR]