First digit law from Laplace transform

The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevalence of Benford's law. The slight variation from the Benford term leads to deviations from Benford's law for distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completely monotonic distributions can satisfy Benford's law within a small bound. Our study suggests that the origin of Benford's law is from the way that we write numbers, thus should be taken as a basic mathematical knowledge.     

Validity of the permeability Carman-Kozeny equation: A volume averaging approach

A volume averaging approach is used to estimate the porous media permeability. Contrary to traditional methods that rely on solving the Navier-Stokes equations for laminar flow, this approach has the advantage that it does not require the specification of some physical conditions and parameters (pressure drop and viscosity). Numerical results on synthetic models of porous media showed that (i) the local porous medium configuration has an important effect on the permeability value, and (ii) the Carman-Kozeny equation cannot describe the permeability behavior as a function of porosity and characteristic lengths. In turn, our results indicate that simple empirical equations, commonly used in practice, are unable to describe the permeability functionalities over a broad range of porous media configurations.     

An extended integrable fractional-order KP soliton hierarchy

In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.     

Extension of Planck's law of thermal radiation to systems with a steady heat flux

Planck's law of thermal radiation is limited to equilibrium systems that have a definite temperature and do not carry any heat flux. Here we extend it to steady‐state systems with a constant heat flux. The obtained formulas explicitly describe the spectrum of thermal radiation in every direction and provide a sound basis for the self‐consistent analysis of radiative heat transport across interfaces, gaps, layered and other important structures.     

Fluctuation scaling in complex systems: Taylor's law and beyond1

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form ‘fluctuations ≈ constant × average^α’, where the exponent α is predominantly in the range [1/2, 1]. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names Taylor's law or fluctuation scaling. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling.     

On the effective viscosity for the Darcy-Brinkman equation

Up-scaling of the Stokes equations with non-slip boundary condition describing the flow in a porous medium, leads to the Darcy-Brinkman equation

?_βμ_βv_D,β=-K_β·(∇P_m,β-ρ_βg)+K_β·μ_β∇²v_D,β.     

Giving birth to the refined unambiguous statement of Snell's law in ray optics

This paper reports on the ambiguity present in the long-running Snell's law of refraction of light, which is based on the traditional ambiguous definitions of the angles of incidence and refraction in ray optics. To get rid of the said ambiguity, the refined unambiguous definitions of angles of incidence and refraction reported earlier by the author have been employed to give birth to the refined unambiguous statement of Snell's law of refraction. The most interesting physical insights that resulted from the refined unambiguous statement of Snell's law are: (i) in case of refraction of light of a particular colour while passing from one optical medium to another, the ratio of the sine of the angle of incidence and the sine of the angle of refraction is approximately equal to the refractive index of the second optical medium with respect to the first optical medium and (ii) the refractive index of an optical medium ‘b’ with respect to another optical medium ‘a’ is approximately equal to the reciprocal of the refractive index of the optical medium ‘a’ with respect to the optical medium ‘b’, and vice versa. These results are entirely novel and different from those existing in the traditional literature of ray optics.     

Nonlinear constitutive law for ferroelectric/ferroelastic material and its finite element realization

The hysteresis phenomena of ferroelectric/ferroelastic material in polarization procedure are investigated. Some assumptions are presented based on the published experimental data. The electrical yielding criterion, mechanical yielding criterion and isotropic hardening model are established. The flow theory in incremental forms in polarization procedure is presented. The nonlinear constitutive law for electrical-mechanical coupling is proposed phenomenologically. Finally, the nonlinear constitutive law expressed in a form of matrices and vectors, which is immediately associated with finite element analysis, is formulated. In the example problem of a rectangular specimen subjected to a uniaxial electric field, the procedure from virgin state to fully polarized state is simulated. Afterward, a uniaxial compressive loading is applied to depolarizing the specimen. Results are in agreement with the experimental data.     

A power law distribution in patients’ lengths of stay in hospital

The distribution of patients’ lengths of stay in English hospitals is measured by using routinely collected data from 11 years. It is found to be well approximated by a power law distribution spanning over more than three decades. To explain this observation, a theoretical resource allocation model is presented. It is based on iterative long-term scheduling of hospital beds, and its main assumption is that future beds are allocated preferentially. This represents a situation where different parts of the health care system compete for resources, with bargaining powers proportional to current resource levels.     

A microscopic view of the Fourier law

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.     

The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.     

Local structure-preserving methods for the generalized Rosenau-RLW-KdV equation with power law nonlinearity

Local structure-preserving algorithms including multi-symplectic, local energy-and momentum-preserving schemes are proposed for the generalized Rosenau–RLW–Kd V equation based on the multi-symplectic Hamiltonian formula of the equation. Each of the present algorithms holds a discrete conservation law in any time–space region. For the original problem subjected to appropriate boundary conditions, these algorithms will be globally conservative. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results show that the proposed algorithms have satisfactory performance in providing an accurate solution and preserving the discrete invariants.     

A physically based connection between fractional calculus and fractal geometry

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.     

The law of the leading digits and the world religions

Benford’s law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger ones. We investigate here numerical data on the country-wise adherent distribution of seven major world religions i.e. Christianity, Islam, Buddhism, Hinduism, Sikhism, Judaism and Baha’ism to see if the proportion of the leading digits occurring in the distribution conforms to Benford’s law. We find that the adherent data of all the religions, except Christianity, excellently does conform to Benford’s law. Furthermore, unlike the adherent data on Christianity, the significant digit distribution of the three major Christian denominations i.e. Catholicism, Protestantism and Orthodoxy obeys the law. Thus in spite of their complexity general laws can be established for the evolution of religious groups.     

Steady state ensembles of thermal radiation in a layered media with a constant heat flux

This paper describes steady‐state ensembles of thermally excited electromagnetic radiation in nano‐scale layered media with a constant non‐vanishing heat flux across the layers. It is shown that Planck's law of thermal radiation, the principle of equivalence, and the laws of wave propagation in layered media, imply that in order for the ensemble of thermally excited electromagnetic fields to exist in a medium consisting of a stack of layers between two half‐space, the net heat flux across the layers must exceed a certain threshold that is determined by the temperatures of the half spaces and by the reflective properties of the entire structure. The obtained results provide a way for estimating the radiative heat transfer coefficient of nano‐scale layered structures.     

Rayleigh-sommerfeld diffraction theory and Lambert’s law

The Rayleigh-Sommerfeld diffraction theory is used to derive expressions for (1) the spectral irradiance on the surface of a hemisphere covering the aperture and (2) the spectral radiant intensity. For a uniform, noncoherent source-aperture, both calculations predict a cosθ angular variation, as is known to be the case of Lambertian sources. A cosine-fourth dependence of the spectral irradiance on a plane parallel to the aperture plane is also indicated.     

Stochastic perturbation of non-Kerr law optical solitons

The dynamics of optical solitons with non-Kerr law non-linearities, in presence of stochastic perturbation terms, is studied in this paper. The Langevin equations are derived and it is proved that the solitons travel through a fiber with a fixed mean velocity. The non-linearities that are considered here are the power law, parabolic law and the dual-power law types.     

Fourier's law based on microscopic dynamics

While Fourier's law is empirically confirmed for many substances and over an extremely wide range of thermodynamic parameters, a convincing microscopic derivation still poses difficulties. With current machines, the solution to Newton's equations of motion can be obtained with high precision and for a reasonably large number of particles. For simplified model systems, one thereby arrives at a deeper understanding of the microscopic basis for Fourier's law. We report on recent, and not so recent, advances.     

验证马吕斯定律的实验方法

利用光电池接收入射的偏振光,通过A／D转换芯片（ICL7170）,将电压值显示在LED上,验证了马吕斯定律。     

Transmission properties of random point scatterers for waves with two-band dispersion law

The scattering problem is considered for the one-dimensional Dirac equation whose potential is a system of randomly distributed point scatterers. Types of such scatterers are described. The probability densities for the transmission and transformation disbalance coefficients are determined in the high-energy region and various averaged characteristic are calculated.