Power-Law Exponent for Exponential Growth Network

The question of the power-law exponent of exponential growth networks is studied here.In a discrete case,the degree distribution is defined as the probability distribution of the discrete variable.Based on this,the degree distribution of the pseudofractal scale-free web,an exponential growth network,is obtained.The power-law exponent ln3/ln2 is analyzed according to the maximum likelihood principle.It satisfies consistency and is good for small generations of the network.For many exponential growth networks,their power-law exponent needs to be tested.The work provides a new view on the power-law exponent of an exponential growth network.

Predicted Critical State Based on Invariance of the Lyapunov Exponent in Dual Spaces

Critical states in disordered systems,fascinating and subtle eigenstates,have attracted a lot of research interests.However,the nature of critical states is difficult to describe quantitatively,and in general,it cannot predict a system that hosts the critical state.We propose an explicit criterion whereby the Lyapunov exponent of the critical state should be 0 simultaneously in dual spaces,namely the Lyapunov exponent remains invariant under the Fourier transform.With this criterion,we can exactly predict a one-dimensional quasiperiodic model which is not of self-duality,but hosts a large number of critical states.Then,we perform numerical verification of the theoretical prediction and display the self-similarity of the critical state.Due to computational complexity,calculations are not performed for higher dimensional models.However,since the description of extended and localized states by the Lyapunov exponent is universal and dimensionless,utilizing the Lyapunov exponent of dual spaces to describe critical states should also be universal.Finally,we conjecture that some kind of connection exists between the invariance of the Lyapunov exponent and conformal invariance,which can promote the research of critical phenomena.

Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

In order to quantify the influence of external forcings on the predictability limit using observational data,the author introduced an algorithm of the conditional nonlinear local Lyapunov exponent(CNLLE)method.The effectiveness of this algorithm is validated and compared with the nonlinear local Lyapunov exponent(NLLE)and signal-to-noise ratio methods using a coupled Lorenz model.The results show that the CNLLE method is able to capture the slow error growth constrained by external forcings,therefore,it can quantify the predictability limit induced by the external forcings.On this basis,a preliminary attempt was made to apply this method to measure the influence of ENSO on the predictability limit for both atmospheric and oceanic variable fields.The spatial distribution of the predictability limit induced by ENSO is similar to that arising from the initial conditions calculated by the NLLE method.This similarity supports ENSO as the major predictable signal for weather and climate prediction.In addition,a ratio of predictability limit(RPL)calculated by the CNLLE method to that calculated by the NLLE method was proposed.The RPL larger than 1 indicates that the external forcings can significantly benefit the long-term predictability limit.For instance,ENSO can effectively extend the predictability limit arising from the initial conditions of sea surface temperature over the tropical Indian Ocean by approximately four months,as well as the predictability limit of sea level pressure over the eastern and western Pacific Ocean.Moreover,the impact of ENSO on the geopotential height predictability limit is primarily confined to the troposphere.

The Dual of the Two-Variable Exponent Amalgam Spaces (Lq(),lp())(Ω)

Wiener amalgam spaces are a class of function spaces where the function’s local and global behavior can be easily distinguished. These spaces are ex-tensively used in Harmonic analysis that originated in the work of Wiener. In this paper: we first introduce a two-variable exponent amalgam space (Lq(),lp())(Ω). Secondly, we investigate some basic properties of these spaces, and finally, we study their dual.

Analysis of precipitation characteristics of South and North China based on the power-law tail exponents

Precipitation sequence is a typical nonlinear and chaotic observational series, and studies on precipitation forecasts are restricted to the use of traditional linear statistical methods, especially when analysing the regional characteristics of precipitation. In the context of 20 stations＇ daily precipitation series （from 1956 to 2000） in South China （SC） and North China （NC）, we divide each precipitation series into many self-stationary segments by using the heuristic segmentation algorithm （briefly BG algorithm）. For each station＇s precipitation series, we calculate the exponent of power-law tall （EPT） of the cumulative probability distribution of segments with a length larger than l for precipitation and temperature series. Our results show that the power-law decay of the cumulative probability distribution of stationary segments might be a common attribution for precipitation and other nonstationary time series; the EPT somewhat indicates the precipitation duration and its spatial distribution that might be different from area to area. The EPT in NC is larger than in SC; Meanwhile, EPT might be another effective way to study the abrupt changes in nonlinear and nonstationary time series.

TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS

In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with 0∈?Ωand all the principle curvatures of?Ωat 0 are negative,a∈C1(Ω,R*+),μ>0,0<s<2,1<q<2 and N>2(q+1)/(q-1).By2*:=2N/(N-2)and 2*(s):(2(N-s))/(N-2)we denote the critical Sobolev exponent and Hardy-Sobolev exponent,respectively.

Solutions for Schrodinger-Poisson system involving nonlocal term and critical exponent

In this paper,we consider the following Kirchhoff-Schrodinger-Poisson system:{−(a+b∫_(R^(3))|∇u|^(2))△u+u+ϕu=μQ(x)|u|^(q-2)u+K(x)|u|^(4)u,in R^(3),−△ϕ=u^(2) the nonlinear growth of|u|^(4)u reaches the Sobolev critical exponent.By combining the variational method with the concentration-compactness principle of Lions,we establish the existence of a positive solution and a positive radial solution to this problem under some suitable conditions.The nonlinear term includes the nonlinearity f(u)~|u|^(q-2)u for the well-studied case q∈[4,6),and the less-studied case q∈(2,3),we adopt two different strategies to handle these cases.Our result improves and extends some related works in the literature.

MHD Boundary Layer Flow of a Power-Law Nanofluid Containing Gyrotactic Microorganisms Over an Exponentially Stretching Surface

This study focusses on the numerical investigations of boundary layer flow for magnetohydrodynamic(MHD)and a power-law nanofluid containing gyrotactic microorganisms on an exponentially stretching surface with zero nanoparticle mass flux and convective heating.The nonlinear system of the governing equations is transformed and solved by Runge-Kutta-Fehlberg method.The impacts of the transverse magnetic field,bioconvection parameters,Lewis number,nanofluid parameters,Prandtl number and power-law index on the velocity,temperature,nanoparticle volume fraction,density of motile microorganism profiles is explored.In addition,the impacts of these parameters on local skin-friction coefficient,local Nusselt,local Sherwood numbers and local density number of the motile microorganisms are discussed.The results reveal that the power law index is considered an important factor in this study.Due to neglecting the buoyancy force term,the bioconvection and nanofluid parameters have slight effects on the velocity profiles.The resultant Lorentz force,from increasing the magnetic field parameter,try to decrease the velocity profiles and increase the rescaled density of motile microorganisms,temperature and nanoparticle volume fraction profiles.Physically,an augmentation of power-law index drops the reduced local skin-friction and reduced Sherwood number,while it increases reduced Nusselt number and reduced local density number of motile microorganisms.

General mapping of one-dimensional non-Hermitian mosaic models to non-mosaic counterparts:Mobility edges and Lyapunov exponents

We establish a general mapping from one-dimensional non-Hermitian mosaic models to their non-mosaic counterparts.This mapping can give rise to mobility edges and even Lyapunov exponents in the mosaic models if critical points of localization or Lyapunov exponents of localized states in the corresponding non-mosaic models have already been analytically solved.To demonstrate the validity of this mapping,we apply it to two non-Hermitian localization models:an Aubry-Andre-like model with nonreciprocal hopping and complex quasiperiodic potentials,and the Ganeshan-Pixley-Das Sarma model with nonreciprocal hopping.We successfully obtain the mobility edges and Lyapunov exponents in their mosaic models.This general mapping may catalyze further studies on mobility edges,Lyapunov exponents,and other significant quantities pertaining to localization in non-Hermitian mosaic models.

NEW METHOD TO ESTIMATE SCALING EXPONENTS OF POWER-LAW DEGREE DISTRIBUTION AND HIERARCHICAL CLUSTERING FUNCTION FOR COMPLEX NETWORKS

A new method and corresponding numerical procedure are introduced to estimate scaling exponents of power-law degree distribution and hierarchical clustering function for complex networks. This method can overcome the biased and inaccurate faults of graphical linear fitting methods commonly used in current network research. Furthermore, it is verified to have higher goodness-of-fit than graphical methods by comparing the KS （Kolmogorov-Smirnov） test statistics for 10 CNN （Connecting Nearest-Neighbor） networks.

Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions

A simple stochastic mechanism that produces exact and approximate power-law distributions is presented. The model considers radially symmetric Gaussian, exponential and power-law functions inn= 1, 2, 3 dimensions. Randomly sampling these functions with a radially uniform sampling scheme produces heavy-tailed distributions. For two-dimensional Gaussians and one-dimensional exponential functions, exact power-laws with exponent –1 are obtained. In other cases, densities with an approximate power-law behaviour close to the origin arise. These densities are analyzed using Padé approximants in order to show the approximate power-law behaviour. If the sampled function itself follows a power-law with exponent –α, random sampling leads to densities that also follow an exact power-law, with exponent -n/a – 1. The presented mechanism shows that power-laws can arise in generic situations different from previously considered specialized systems such as multi-particle systems close to phase transitions, dynamical systems at bifurcation points or systems displaying self-organized criticality. Thus, the presented mechanism may serve as an alternative hypothesis in system identification problems.

New seismic attribute:Fractal scaling exponent based on gray detrended fluctuation analysis

Seismic attributes have been widely used in oil and gas exploration and development. However, owing to the complexity of seismic wave propagation in subsurface media, the limitations of the seismic data acquisition system, and noise interference, seismic attributes for seismic data interpretation have uncertainties. Especially, the antinoise ability of seismic attributes directly affects the reliability of seismic interpretations. Gray system theory is used in time series to minimize data randomness and increase data regularity. Detrended fluctuation analysis （DFA） can effectively reduce extrinsic data tendencies. In this study, by combining gray system theory and DFA, we propose a new method called gray detrended fluctuation analysis （GDFA） for calculating the fractal scaling exponent. We consider nonlinear time series generated by the Weierstrass function and add random noise to actual seismic data. Moreover, we discuss the antinoise ability of the fractal scaling exponent based on GDFA. The results suggest that the fractal scaling exponent calculated using the proposed method has good antinoise ability. We apply the proposed method to 3D poststack migration seismic data from southern China and compare fractal scaling exponents calculated using DFA and GDFA. The results suggest that the use of the GDFA-calculated fractal scaling exponent as a seismic attribute can match the known distribution of sedimentary facies.

Exponent及Gauss型遥测数据建模初始化算法

在对某大型航天电子设备的遥测数据建模时,经常遇到含有指数类变化规律的数据,对其进行建模可以为研究设备性能的变化规律、实际与设计的差异等提供手段。针对有指数变化趋势的遥测数据,采用Exponent模型、Gauss模型对遥测数据建模,给出了这种数据模型的表达式,研究了2种模型的参数初始化算法。通过数值实验说明模型参数初始化算法的有效性,为后续利用最优化理论求解模型精确参数提供了良好的初试点。

Construction of Different Kinds of Atomic and Molecular Orbitals Using Complete Orthonormal Sets of ψ^α-ETO in Single Exponent Approximation

Using complete orthonormal sets of ψ^α-exponential type orbitals in single exponent approximation the new approach has been suggested for construction of different kinds of functions which can be useful in the theory of linear combination of atomic orbitals. These functions can be chosen properly according to the nature of the problems under consideration. This is rather important because the choice of the basis set may be play a crucial role in applications to atomic and molecular problems. As an example of application, different atomic orbitals for the ground states of the neutral and the first ten cationic members of the isoelectronic series of He atom are constructed by the solution of Hartree-Fock-Roothaan equations using ψ^1, ψ^0 and ψ^-1 basis sets. The cMculated results are close to the numerical Hartree-Fock values. The total energy, expansion coefficients, orbital exponents and virial ratio for each atom are presented.

CALCULATION OF CRITICAL EXPONENT β(L) OF MAGNETIC THIN FILMS BY USING ISING MODEL

The variational cumulant expansion developed in recent years has been extended to treat the Ising model in statistical physics.In this paper,a detailed calculation of the critical temperature T c (L) and critical exponent β(L) for the magnetic film of L layers are presented by means of the variational cumulant expansion.For L >1,the results of our theoretical calculations are in approximate coincidence with the experimental ones made before,and for the special case of L =1 (2 D),the results of the calculation are identical to the data from other reports.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR SINGULAR ELLIPTIC SYSTEMS WITH CRITICAL SOBOLEV-HARDY AND CONCAVE EXPONENTS

In this paper,we consider a singular elliptic system with both concave non-linearities and critical Sobolev-Hardy growth terms in bounded domains.By means of variational methods,the multiplicity of positive solutions to this problem is obtained.

Determining the Spectrum of the Nonlinear Local Lyapunov Exponents in a Multidimensional Chaotic System

For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent （NLLE） from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.

Multilinear singular integrals and commutators in variable exponent Lebesgue spaces

Boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces are obtained. The vector-valued case is also considered.

Lagrangian-based investigation of multiphase flows by finite-timeLyapunov exponents

Multiphase flows are ubiquitous in our daily life and engineering applications. It is important to investigate the flow structures to predict their dynamical behaviors ef- fectively. Lagrangian coherent structures （LCS） defined by the ridges of the finite-time Lyapunov exponent （FTLE） is utilized in this study to elucidate the multiphase interactions in gaseous jets injected into water and time-dependent turbu- lent cavitation under the framework of Navier-Stokes flow computations. For the gaseous jets injected into water, the highlighted phenomena of the jet transportation can be observed by the LCS method, including expansion, bulge, necking/breaking, and back-attack. Besides, the observation of the LCS reveals that the back-attack phenomenon arises from the fact that the injected gas has difficulties to move toward downstream re- gion after the necking/breaking. For the turbulent cavitating flow, the ridge of the FTLE field can form a LCS to capture the front and boundary of the re-entraint jet when the ad- verse pressure gradient is strong enough. It represents a bar- rier between particles trapped inside the circulation region and those moving downstream. The results indicate that the FFLE field has the potential to identify the structures of mul- tiphase flows, and the LCS can capture the interface/barrier or the vortex/circulation region.

SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN R^N

For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy exponent, h（x） ∈ （D^1,2（R^N））^＊, the dual space of （D^1,2（R^N））, with h（x）≥（≠）0. By Ekeland＇s variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ｜｜h｜｜＊〈CN,sAs^N-s/4-2s（1-μ/μ）^1/2, CN,s=4-2s/N-2（N-2/N＋2-2s）^N＋2-2s/4-2s and As = inf u∈D^1,2（R^N）／{0}∫R^N（｜△↓u｜^2-μu^2/｜x｜^2）dx/（∫R^N｜u｜^2^＊（s）/｜x｜^sdx）^2/2^＊（s）.