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.

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.

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.

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.

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.

Parameterized Littlewood-Paley Operators and Their Commutators on Lebesgue Spaces with Variable Exponent

In this paper, by applying the technique of the sharp maximal function and the equivalent representation of the norm in the Lebesgue spaces with variable exponent, the boundedness of the parameterized Litflewood-Paley operators, including the parameterized Lusin area integrals and the parameterized Littlewood-Paley gλ^＊- functions, is established on the Lebesgue spaces with variable exponent. Furthermore, the boundedness of their commutators generated respectively by BMO functions and Lipschitz functions are also obtained.

Boundedness for Commutators of Calder o′n-Zygmund Operator on Morrey Spaces with Variable Exponent

Our aim in the present paper is to prove the boundedness of commutators on Morrey spaces with variable exponent. In order to obtain the result, we clarify a relation between variable exponent and BMO norms.

Boundedness for Hardy Type Operators on Herz Spaces with Variable Exponents

In this paper, we will prove the boundedness of Hardy type operators Hβ（x） and Hβ^＊（x） of variable order β（x） on Herz spaces Kp（·）^α（·）q and Kp（·）^α（·）q′,where α（·） and p（·）are both variable.

The regulation of ferrocene-based catalysts on heat transfer in highpressure combustion of ammonium perchlorate/hydroxyl-terminated polybutadiene/aluminum composite propellants

The regulation of the burning rate pressure exponent for the ammonium perchlorate/hydroxylterminated polybutadiene/aluminum(AP/HTPB/Al)composite propellants under high pressures is a crucial step for its application in high-pressure solid rocket motors.In this work,the combustion characteristics of AP/HTPB/Al composite propellants containing ferrocene-based catalysts were investigated,including the burning rate,thermal behavior,the local heat transfer,and temperature profile in the range of 7-28 MPa.The results showed that the exponent breaks were still observed in the propellants after the addition of positive catalysts(Ce-Fc-MOF),the burning rate inhibitor((Ferrocenylmethyl)trimethylammonium bromide,Fc Br)and the mixture of Fc Br/catocene(GFP).However,the characteristic pressure has increased,and the exponent decreased from 1.14 to 0.66,0.55,and 0.48 when the addition of Ce-FcMOF,Fc Br and Fc Br/GFP in the propellants.In addition,the temperature in the first decomposition stage was increased by 7.50℃ and 11.40℃ for the AP/Fc Br mixture and the AP/Fc Br/GFP mixture,respectively,compared to the pure AP.On the other hand,the temperature in the second decomposition stage decreased by 48.30℃ and 81.70℃ for AP/Fc Br and AP/Fc Br/GFP mixtures,respectively.It was also found that Fc Br might generate ammonia to cover the AP surface.In this case,a reaction between the methyl in Fc Br and perchloric acid caused more ammonia to appear at the AP surface,resulting in the suppression of ammonia desorption.In addition,the coarse AP particles on the quenched surface were of a concave shape relative to the binder matrix under low and high pressures when the catalysts were added.In the process,the decline at the AP/HTPB interface was only exhibited in the propellant with the addition of Ce-Fc-MOF.The ratio of the gas-phase temperature gradient of the propellants containing catalysts was reduced significantly below and above the characteristic pressure,rather than 3.6 times of the difference in the blank propellant.Overall,the obtained results demonstrated that the pressure exponent could be effectively regulated and controlled by adjusting the propellant local heat and mass transfer under high and low pressures.

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.

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.

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）.

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.

NONTRIVIAL SOLUTION FOR A CLASS OF SEMILINEAR BIHARMONIC EQUATION INVOLVING CRITICAL EXPONENTS

In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.

New prediction of chaotic time series based on local Lyapunov exponent

A new method of predicting chaotic time series is presented based on a local Lyapunov exponent, by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in state space. After recon- structing state space from one-dimensional chaotic time series, neighboring multiple-state vectors of the predicting point are selected to deduce the prediction formula by using the definition of the locaI Lyapunov exponent. Numerical simulations are carded out to test its effectiveness and verify its higher precision over two older methods. The effects of the number of referential state vectors and added noise on forecasting accuracy are also studied numerically.

EXISTENCE OF MULTIPLE SOLUTIONS FOR SINGULAR QUASILINEAR ELLIPTIC SYSTEM WITH CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX TERMS

The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means of variational methods, that under certain conditions, the system has at least two positive solutions.

SOLUTIONS FOR THE QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS

In this article, we study the quasilinear elliptic problem involving critical Hardy Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign-changing solutions to the problem.

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.

Estimating the largest Lyapunov exponent in a multibody system with dry friction by using chaos synchronization

Using the properties of chaos synchronization, the method for estimating the largest Lyapunov exponent in a multibody system with dry friction is presented in this paper. The Lagrange equations with multipliers of the systems are given in matrix form, which is adequate for numerical calculation. The approach for calculating the generalized velocity and acceleration of the slider is given to determine slipping or sticking of the slider in the systems. For slip-slip and stick-slip multibody systems, their largest Lyapunov exponents are calculated to characterize their dynamics.

Multilinear Calder′on Zygmund operators on variableexponent Morrey spaces over domains

The boundedness of multilinear singular integrals of Calder′on-Zygmund type onproduct of variable exponent Lebesgue spaces over both bounded and unbounded domains areobtained. Further more, the boundedness for this type multilinear operators on product ofvariable exponent Morrey spaces over domains is shown in the paper.