The stable graph: the metric space scaling limit of a critical random
graph with i.i.d. **power**-**law** degrees

**power**-

**law**tail behaviour with exponent $\alpha+1$, where $\alpha \in (1,2)$. The limiting components are constructed from random $\mathbb{R}$-trees encoded by the excursions above its running infimum of a process whose

**law**is locally absolutely continuous with respect to that of a spectrally positive $\alpha$-stable L\'evy process. These spanning $\mathbb{R}$-trees are measure-changed $\alpha$-stable trees. In each such $\mathbb{R}$-tree, we make a random number of vertex-identifications, whose locations are determined by an auxiliary Poisson process. This generalises results which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erd\H{o}s--R\'enyi random graph) and where the role of the $\alpha$-stable L\'evy process is played by a Brownian motion.

10/10 relevant

arXiv

Alteration of **Power** **Law** Scaling of Spontaneous Brain Activity in Schizophrenia

**power**

**law**scaling analysis may serve as a novel functional brain imaging marker for evaluating patients with mental illness. Expand abstract.

**Power**

**law**scaling is a well-validated method in physics that has been used to describe the complex nature of a system across different time scales. In this research, we investigated the change of

**power**-

**law**characteristics in a large-scale resting-state fMRI data of schizophrenia (N = 200) and healthy participants (N = 200) derived from Taiwan Aging and Mental Illness cohort. Fourier transform was used to determine the

**power**spectral density (PSD) of resting-state fMRI signal. We estimated the

**power**

**law**scaling of PSD of resting-state fMRI signal by determining the slope of the regression line fitting to the log-log plot of PSD. The

**power**

**law**scaling represents the dynamical properties of resting-state fMRI signal ranging from noisy oscillation (e.g., white noise) to complex fluctuations (e.g., slope approaches -1). Linear regression model was used to assess the statistical difference in

**power**

**law**scaling between schizophrenia and healthy participants. The significant differences in

**power**

**law**scaling were found in six brain regions. Schizophrenia patients has significantly more positive

**power**

**law**scaling (i.e., frequency components become more homogenous) at four brain regions: left precuneus, left medial dorsal nucleus, right inferior frontal gyrus, and right middle temporal gyrus, compared with healthy participants. Additionally, schizophrenia exhibited less positive

**power**

**law**scaling (i.e., frequency components are more dominant at lower frequency range) in bilateral putamen. Significant correlations of

**power**

**law**scaling with the severity of psychosis were found in these identified brain areas in schizophrenia. These findings suggest that schizophrenia has abnormal brain signal complexity toward random patterns, which is linked to psychotic symptoms. The

**power**

**law**scaling analysis may serve as a novel functional brain imaging marker for evaluating patients with mental illness.

10/10 relevant

bioRxiv

Curvature of magnetic field lines in compressible magnetized turbulence: Statistics, magnetization predictions, gradient curvature, modes and self-gravitating media

**power**-

**law**relations to the magnetization. Moreover, the

**power**-

**law**tail of the curvature probability distribution function is also proportional to the Alfvenic Mach number. We also explore whether the curvature method could be used in the field-tracing Velocity Gradient Technique. In particular, we observe that there is a relation between the mean and standard deviation of the curvature probed by velocity gradients to $M_A$. Finally we discuss how curvature is contributed by different MHD modes in interstellar turbulence, and suggests that the eigenvectors of MHD modes could be possibly represented by the natural Fernet-Serrat frame of the magnetic field lines. We discuss possible theoretical and observational applications of the curvature technique, including the extended understanding on a special length scale that characterize the importance of magnetic field curvature in driving MHD turbulence, and how it could be potentially used to study self-gravitating system.

5/10 relevant

arXiv

A non linear Ohm's **law** -a phenomenological approach

**power**

**law**for conduction resistivity Expand abstract.

**Law**the medium constitutive Ohm's

**Law**for an electrical flow with a square

**power**

**law**for conduction resistivity

4/10 relevant

engrXiv

Categorizing SHR and WKY rats by chi2 algorithm and decision tree

**power**

**law**can differentiate human behavior. Expand abstract.

**power**-

**law**distribution. It has been proposed that distinct and yet robust exponents of the

**power**

**law**can differentiate human behavior. Using the mentally disordered SHR and WKY rats as samples, we employ chi2 algorithm and Decision Tree to classify different states of mental disorder by analyzing different traits in degree of connectivity.

5/10 relevant

bioRxiv

Statistical mass function of prestellar cores from the density distribution of their natal clouds

**power**-

**law**with slope $q$. The variety of MCs is divided in ensembles according to the PDF slope and each ensemble is represented by a single spherical cloud. The cores are considered as elements of self-similar structure typical for fractal clouds and are modeled by spherical objects populating each cloud shell. Our model assumes relations between size, mass and density of the statistical cores. Out of them a core mass-density relationship $\rho\propto m^x$ is derived where $x=1/(1+q)$. We found that $q$ determines the existence or non-existence of a threshold density for core collapse. The derived general CMF is a

**power**

**law**of slope $-1$ while the CMF of gravitationally unstable cores has a slope $(-1 + x/2)$, comparable with the slopes of the high-mass part of the stellar initial mass function and of observational CMFs.

5/10 relevant

arXiv

Scaling **Laws** for Neural Language Models

**power**-

**law**with model size, dataset size, and the amount of compute used for training, with some trends spanning more than seven orders of magnitude. Expand abstract.

**laws**for language model performance on the cross-entropy loss. The loss scales as a

**power**-

**law**with model size, dataset size, and the amount of compute used for training, with some trends spanning more than seven orders of magnitude. Other architectural details such as network width or depth have minimal effects within a wide range. Simple equations govern the dependence of overfitting on model/dataset size and the dependence of training speed on model size. These relationships allow us to determine the optimal allocation of a fixed compute budget. Larger models are significantly more sample-efficient, such that optimally compute-efficient training involves training very large models on a relatively modest amount of data and stopping significantly before convergence.

5/10 relevant

arXiv

NuSTAR and Parkes observations of the transitional millisecond pulsar
binary XSS J12270-4859 in the rotation-**powered** state

**power**-

**law**with photon index Gamma = 1.17+/-0.08 giving a 3-79keV luminosity of 7.6(-0.8;+3.8)x10**32 erg/s, for a distance of 1.37(-0.15;+0.69)kpc. Expand abstract.

**powered**state, complemented with a 2.5yr-long radio monitoring at Parkes telescope and archival XMM-Newton and Swift X-ray and optical data. The radio pulsar is mainly detected at 1.4GHz displaying eclipses over about 40% of the 6.91h orbital cycle. We derive a new updated radio ephemeris to study the 3-79keV light curve that displays a significant orbital modulation with fractional amplitude of 28+/-3%, a structured maximum centred at the inferior conjunction of the pulsar and no cycle-to-cycle or low-high-flaring mode variabilities. The average X-ray spectrum, extending up to about 70keV without a spectral break, is well described by a simple

**power**-

**law**with photon index Gamma = 1.17+/-0.08 giving a 3-79keV luminosity of 7.6(-0.8;+3.8)x10**32 erg/s, for a distance of 1.37(-0.15;+0.69)kpc. Energy resolved orbital light curves reveal that the modulation is not energy dependent from 3keV to 25keV and is undetected with an upper limit of about 10% above 25keV. Comparison with previous X-ray XMM-Newton observations in common energy ranges confirms that the modulation amplitudes vary on timescales of a few months, indicative of a non-stationary contribution of the intrabinary shock formed by the colliding winds of the pulsar and the companion. A more detailed inspection of energy resolved modulations than previously reported gives hints of a mild softening at superior conjunction of the pulsar below 3keV, likely due to the contribution of the thermal emission from the neutron star. The intrabinary shock emission, if extending into the MeV range, would be energetically capable alone to irradiate the donor star.

4/10 relevant

arXiv

The `Red Supergiant Problem': the upper luminosity boundary of type-II supernova progenitors

**power**-

**law**with an upper and lower cutoff, and find an upper luminosity limit of $\log(L_{\rm hi}/L_\odot) = 5.20^{+0.17}_{-0.11}$ (68\% confidence), though this increases to $\sim$5.3 if one fixes the

**power**-

**law**slope to be that expected from theoretical arguments. Again, the results point to the significance of the RSG Problem being within $\sim 2 \sigma$. Under the assumption that all progenitors are the result of single-star evolution, this corresponds to an upper mass limit for the parent distribution of $M_{\rm hi} = 19.2{\rm M_\odot}$, $\pm1.3 {\rm M_\odot (systematic)}$, $^{+4.5}_{-2.3} {\rm M_\odot}$ (random) (68\% confidence limits).

5/10 relevant

arXiv

On the Independence of Fundamental Decompositions of **Power**-**Law** Kinetic
Systems

**power**-

**law**kinetic systems of Hernandez et al. Expand abstract.

**power**-

**law**kinetic systems of Hernandez et al. In addition to our previous work, we provide important properties of the independence (i.e., the network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and the incidence-independence (i.e., the image of the network's incidence map is the direct sum of the incidence maps' images of the subnetworks) of these decompositions. Feinberg established the essential relationship between independent decompositions and the set of positive equilibria of a network, which we call the Feinberg Decomposition Theorem (FDT). Moreover, Farinas et al. recently documented its version for incidence-independence. Fundamental decomposition divides the network into subnetworks of deficiency either 0 or 1 only. Hence, available results for lower deficiency networks, such as the Deficiency Zero Theorem (DZT), can be used. These justify the study of independent fundamental decompositions. A MATLAB program which (i) computes the subnetworks of a CRN under the fundamental decomposition and (ii) is useful for determining whether the decomposition is independent and incidence-independent is also created. Finally, we provide the following solution for determining multistationarity of CRNs with the following steps: (1) the use of the program, (2) the application of available results for CRNs with deficiency 0 or 1 (e.g., DZT), and (3) the use of FDT. We illustrate the solution by showing that the generalization of a subnetwork of Schmitz's carbon cycle model by Hernandez et al., endowed with mass action kinetics, has no capacity for multistationarity.

10/10 relevant

arXiv