Hypothesis Of Scale Invariance Research Articles

Note on dark energy and cosmic transit in a scale-invariance cosmology

In general, the laws of physics are not invariant under a change of scale. To find out whether the “scale-invariance hypothesis” corresponds to nature or not, a careful examination to its implications is required. As a consequence, the scale-invariance cosmological models need to be carefully checked with many tests in order to confirm or disconfirm them. In this paper, three different toy models have been introduced in the framework of a scale-invariance cosmology to examine dark energy and cosmic transit. Although cosmic transit exists in the three models, the pressure stays always negative during cosmic evolution. In addition, there is always a singularity in the evolution of the equation of state parameter which is not suitable for a complete investigation of dark energy evolution. The undesirable features of the parameters have been discussed, and a comparison with other cosmological contexts has been done.

Modified multifractal large deviation spectrum based on CID for financial market system

We modify the basic roughness grain exponent, only available for application of one single series, to complexity-invariant distance (CID) for studying multifractal features between two time series. CID is taken into consideration as a new roughness grain exponent with the large deviation spectrum to detect the similarity and correlation between different stock markets in this work.The new method is firstly applied to artificial series in order to test the scale invariance hypothesis with both the Legendre spectrum and large deviation spectrum. Results show that the large deviation spectrum is better in the detection of multifractal analysis between two time series as it exhibit plentiful scaling structures. Then we investigate the scaling behavior between financial time series so as to test whether there is scale invariance of stock markets in China and the US. Besides, from the α values responding to the maximal fα in the 5 pair series and the widths of the spectrum, a new way of measurement of the similarity and correlation has been found. To overcome the limitation of q, we then compute the surface area of the large deviation spectrum under each scale and further detect the evolution of the slope by a scaling/non-scaling criterion which helps in the measurement of scaling behavior. Moreover, the proposed method can give us more information between different markets and distinguish them from different angles.

Sensitivity Issues in Finite-Difference Large-Eddy Simulations of the Atmospheric Boundary Layer with Dynamic Subgrid-Scale Models

The neutral atmospheric boundary layer (ABL) is simulated by finite-difference large-eddy simulations (LES) with various dynamic subgrid-scale (SGS) models. The goal is to understand the sensitivity of the results to several aspects of the simulation set-up: SGS model, numerical scheme for the convective term, resolution, and filter type. Three dynamic SGS models are tested: two scale-invariant models and the Lagrangian-averaged scale-dependent (LASD) model. The results show that the LASD model has the best performance in capturing the law-of-the-wall, because the scale invariance hypothesis is violated in finite-difference LES. Two forms of the convective term are tested, the skew-symmetric and the divergence forms. The choice of the convective term is more important when the LASD model is used and the skew-symmetric scheme leads to better simulations in general. However, at fine resolutions both in space and time, the sensitivity to the convective scheme is reduced. Increasing the resolution improves the performance in general, but does not better capture the law of the wall. The box and Gaussian filters are tested and it is found that, combined with the LASD model, the Gaussian filter is not sufficient to dissipate the small numerical noises, which in turn affects the large-scale motions. In conclusion, to get the most benefits of the LASD model within the finite-difference framework, the simulations need to be set up properly by choosing the right combination of numerical scheme, resolution, and filter type.

GRS 1915+105 AS A GALACTIC ANALOG OF A FANAROFF-RILEY II QUASAR

We study the long term time averaged kinetic luminosity, $\bar{Q}$, of the major flares of the Galactic microquasar GRS 1915+105 and the relationship to the intrinsic X-ray (bolometric) luminosity, $L_{\mathrm{bol}}$, and scale it to that of a complete sample of SDSS/FIRST FR II quasars. If the scale invariance hypothesis for black holes (BHs) holds then we show that the expected distribution in the $\bar{Q}$ - $L_{\mathrm{bol}}$ scatter plane of GRS 1915+105 is consistent with FR II quasars for distances D = 10.7 - 11 kpc. We compare the specific values of kinetic luminosity and $L_{\mathrm{bol}}$ during flares of GRS 1915+105 to that predicted by several 3-D MHD simulations of BH accretion flows with relativistic ejections. If FR II quasars are a scaled up version of GRS 1915+105, the data are consistent with numerical models when they contain an ergospheric disk jet and the BH spin is $a/M= 0.99$ or $a/M=0.998$ (we estimate $a/M>0.984$). In the framework of scale invariance of BHs, our results may imply that FR II quasars also hold rapidly rotating BHs.

Extrapolating the Fractal Characteristics of an Image Using Scale-Invariant Multiple-Point Statistics

The resolution of measurement devices can be insufficient for certain purposes. We propose to stochastically simulate spatial features at scales smaller than the measurement resolution. This is accomplished using multiple-point geostatistical simulation (direct sampling in the present case) to interpolate values at the target scale. These structures are inferred using hypothesis of scale invariance and stationarity on the spatial patterns found at the coarse scale. The proposed multiple-point super-resolution mapping method is able to deal with “both continuous and categorical variables,” and can be extended to multivariate problems. The advantages and limitations of the approach are illustrated with examples from satellite imaging.

A scale-invariant model for snow slab avalanches

On the basis of the evidence of fractality reported in the literature as regards snow materialand snow avalanches, the paper introduces a statistical model for the natural release of slabavalanches characterized by the differing cohesion of the snow cover at different scales.Under the hypothesis of scale invariance of shear strength, this model can beused to estimate the stability of snow slopes of different types, and to examinewhether defects can propagate from the microscale (cohesion and friction) to themacroscale of the weak plane (big avalanche) or whether they eventually stop at acertain scale (small event). This may shed light on the apparently inexplicabledifferences in behaviour of snow covers which look macroscopically similar, under thesame mechanical loads, and which may eventually trigger an avalanche or not.

Effects of grasping force magnitude on the coordination of digit forces in multi-finger prehension

The study addresses three main questions: (1) Does the magnitude of the grasping force affect the prehension synergies, i.e., conjoint changes of finger forces and moments? (2) Do individual finger forces scale with the total grasping forces ('scale-invariance hypothesis')? (3) How specification of the grasping force magnitude affects the inverse optimization of digit forces. Subjects (n = 7) grasped with minimal force an instrumented handle and maintained it at rest in the air. Then, the subjects doubled the initial grasping force. The forces and moments exerted by individual digits were recorded with six-component sensors. External torques that the subjects should resist (9 in total) varied among the trials from 0 to 0.46 Nm both in clockwise and counterclockwise directions. After the force doubling, the moments of the normal forces (M (n)) increased in the pronation effort tasks (PR-tasks) and decreased in the supination effort tasks (SU-tasks). The changes in the moments of the tangential forces (M (t)) were opposite to the M (n) changes; the moments increased in the SU-tasks and decreased in the PR-tasks. The opposite effects of force doubling on the M (t)s in the SU-tasks and PR-tasks were a consequence of the unidirectional changes of the thumb tangential forces: in all the tasks the contribution of the thumb tangential force to the total tangential force increased after the grasping force doubling (and the total contribution of the four fingers decreased). The decrease of the virtual finger (VF) tangential force was mainly due to the decrease of the index finger force (VF is an imagined finger that exerts the same force and moment as all the fingers together). In the non-zero torque tasks the individual finger forces did not scale proportionally with the grasping force, the sharing percentage of the individual finger forces in the VF normal force changed with the grasping force increase. The root mean square differences between the actual finger sharing percentages in the VF force and the sharing percentages predicted from optimization procedures in which different cost functions were used were in all cases smaller after the doubling than before the doubling. Hence the answers to the three questions formulated above are: (1) the alteration of the grasping force magnitude induces complex coordinated changes of all digit forces and moments; (2) the scale invariance hypothesis is confirmed only for the zero-torque tasks and rejected for the non-zero tasks, and (3) the specification of the grasping force magnitude at the level of twice the initial grasping force-which essentially restricts the control task to the object tilt prevention-improves the accuracy of the employed optimization procedures.

Zero Constant Formula for First-Passage Observables in Bounded Domains

In this Letter, we develop an analytical approach which provides an explicit determination of mean first-passage times (MFPTs) for random walks in bounded domains for a wide class of transport processes. In particular, we derive for the first time explicit expressions of MFPTs for emblematic models of transport in complex media, such as diffusion on deterministic and random fractals. This approach relies on a scale-invariance hypothesis and a large volume expansion of the MFPT, which actually proves to be very accurate even for small system sizes as shown by numerical simulations. This explicit determination of MFPTs can be straightforwardly generalized to further useful first-passage observables such as occupation times and splitting probabilities.

Scaling of tropical rainfall as observed by TRMM precipitation radar

We used a three-year (1998–2000) dataset of TRMM Precipitation Radar observations to investigate the scaling properties of spatial rainfall fields. This dataset allows consideration of spatial scales ranging from about 4.3 km to 138 km and short temporal scales corresponding to the sensor overpasses. The focus is on the marginal spatial moment scaling, which allows estimation of the scaling parameters from a single scene of data. Here we present a global perspective of the scaling properties of tropical rainfall in terms of its spatial variability, atmospheric forcing, predictability, and applicability. Our results reveal the following: 1) the scaling parameters exhibit strong variability associated with land/ocean contrast and mean precipitation at the synoptic scale; 2) there exists a one-to-one relationship between the scaling parameters and the large-scale spatial average rain rate of a universal functional form; 3) the majority of the scenes are consistent with the hypothesis of scale invariance at the moment orders of 0 and 2; 4) relatively there are more scale-invariant rain scenes over land than over ocean; and 5) for the scenes that are non-scale-invariant, deviation from scale-invariance mainly arises from the increasingly intermittent behavior of rainfall as spatial scale decreases. These results have important implications for the development and calibration of downscaling procedures designed to reproduce rainfall properties at different spatial scales and lead to a better understanding of the nature of tropical rainfall at various spatial resolutions.

Constraints on the role of synchrotron X-rays from jets of accreting black holes

We present an extension of the scale-invariance formalism by Heinz & Sunyaev which includes the effects of radiative cooling on the X-ray synchrotron emission from core-dominated jets and outflows. We derive the scaling relations between the radio luminosity L I , the synchrotron X-ray luminosity L x , the mass M, and the accretion rate m of an accreting black hole. We argue that the inclusion of synchrotron losses in the scaling relations is required by the data used in Merloni et al. to define the 'fundamental plane' correlation between L r , L x and M in active galactic nuclei and Galactic X-ray binaries. We then fit the new scaling relations to the 'fundamental plane' correlation. This allows us to derive statistical constraints on the contribution from jet synchrotron emission to the X-rays from black holes, or, alternatively, on the particle acceleration scheme at work in these jets. We find that, in order for jet synchrotron X-rays to be consistent with the observed L r -L x -M correlation, the global particle spectrum must be both steep, f(y) α γ -(3.4+0.6-0.5) , and unbroken down to energies well below the radio synchrotron regime. Such a steep particle distribution would imply steeper X-ray spectra than observed in most sources contributing to the correlation with available X-ray spectral index measurements. We suggest that, unless some assumptions of the scale-invariance hypothesis are broken, another emission mechanism (likely either disc radiation and/or inverse Compton scattering in the jet) contributes significantly to the X-rays of a sizeable fraction of the sources.

Refining earthquake clustering models

Assuming that earthquakes are the realization of a stochastic point process and that the magnitude distribution of all earthquakes is described by the Gutenberg‐Richter law with a constant b value, we model the occurrence rate density of earthquakes in space and time by means of an epidemic model. The occurrence rate density is computed by the sum of two terms, one representing the independent, or spontaneous activity, and the other representing the activity induced by previous earthquakes. While the first term depends only on space, the second one is factored into three terms that include the magnitude, time, and location, respectively, of the past earthquakes. In this paper we use the modified Omori law for the time term, focusing our investigation on the magnitude and space terms. We formulate two different hypotheses for each of them, and we find the respective maximum likelihood parameters on the basis of the catalog of instrumental seismicity recorded in Italy from 1987 to 2000. The comparison of the respective likelihood computed for the seismicity recorded in 2001 provides a way for choosing the best model. The confidence level of our choice is then assessed by means of a Monte Carlo simulation on the various hypotheses. Our study shows that an inverse power density function is more reliable than a normal density function for the space distribution and that the hypothesis of scale invariance of aftershock productivity with respect to magnitude can be rejected with high confidence level. The final model is suitable for computing earthquake occurrence probability in real circumstances.

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2. The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.

Science of complexity: Phenomenological basis and possibility of application to problems of chemical engineering

The basic principles of flicker-noise spectroscopy, a new method of analysis of chaotic time and space series, are presented. This method can be considered the phenomenological basis of the currently developing science of complexity, which deals with open nonlinear dissipative systems of various essences-real natural objects. The method is based on the postulate that irregularities in measured dynamic variables are of crucial informational significance in accepting a new hypothesis of scale invariance. In this approach, the power spectra and the structure functions of different orders are governed by irregularities of various types, namely, dynamic spikes and jumps of measured variables. For processes under investigation, the expressions for both the power spectra and the structure functions prove to be identical (invariant) at each of the spatiotemporal levels of the system. The introduced phenomenological parameters completely and unambiguously characterize the state of the evolving system, thus serving as the “passport data” of the system. The possibility of applying the proposed methodology to solving topical problems of chemical engineering is discussed.

Finite Size Scaling in Ecology

Finite size scaling techniques are applied to ecology and are shown to predict relationships between power-law exponents as well as a connection between scaling phenomena in physics and ecology. The framework presented here ought to be useful for testing the hypothesis of scale invariance in the spatial distribution of species across a landscape.

Scale invariance and the gell-mann-low function of the conductance of a layer with an exponentially broad resistance spectrum

A study is made of a layer having an exponentially broad spectrum of local resistances, one of whose dimensions is smaller than the self-averaging dimension. An investigation is made of the hypothesis of scale invariance and the Gell-Mann-Low function for finite scaling in systems with an exponentially broad spread of resistances. A comparative analysis is made of the scale behavior of these systems and the case of strong localization.

Role of coherent structures in the stochastic‐dynamic variability of precipitation

Using time‐frequency‐scale elements obtained from wavelet packets as a basis, we describe a broad framework of analysis which can be used to reveal the essential dynamics, identified as coherent structures, of precipitation. We show that the matching pursuits algorithm with nearly symmetric orthogonal wavelets provides an optimal representation of the inner structure of rainfall time series and can describe features that range from scales of isolated singularity to synoptically forced large‐scale features. We describe the analysis of time series of several storms and show that there exist distinct scales of variation identifiable with rain cell and synoptic‐scale activity, which is in contradistinction to the scale invariance hypothesis.

An analysis of the rainfall time structure by box counting—some practical implications

The scale-invariant behavior of the rainfall time structure was investigated by applying the box counting method to rainfall time series. Two years of minute observations, 90 years of daily observations and 170 years of monthly observations were analyzed and the results were interpreted and related to physical properties of the rainfall process. This paper discusses the question of whether an hypothesis of temporal scale invariance is valid for rainfall and the possibilities of using it in practical hydrology.

Invariant features of contrast detection: an explanation in terms of self-similar detector arrays.

Human-detection performance for extended spatial contrasts appears to be subjected to several invariants. Such invariant features put general constraints on possible mechanistic interpretations. We show that the hypothesis of scale invariance is sufficient to determine the distribution of elementary sample apertures (receptive field centers) with respect to aperture size. The hypothesis that the elementary sample apertures are virtually identical except for size, which is augmented with general functional forms for pooling of elementary responses and coding properties for the elementary samples, suffices to derive the phenomenological facts. Because of the division of labor among units of different size, the concept of line-spread function loses a physiologically clear interpretation.