Lipschitz Estimates for Commutators of N-dimensional Fractional Hardy Operators

In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q 〈 ∞ and 1/P - 1/q = （α＋β）/n. Furthermore, the boundedness of Hβ,b on the homogenous Herz space Kq^α,p（R^n） is obtained.

General calculation formulas and recurrence relations of radial matrix elements for relativistic n-dimensional hydrogen atom of spin S=0

In this paper, the general calculation formulas of radial matrix elements for relativistic n-dimensional hydrogen atom of spin S=0 are obtained, and the recurrence relation of different power order radial matrix elements are also derived.

On Multiplicative Generators of n-Dimensional Overlap Functions

In this paper, inspired by the multiplicative generators of overlap functions, we mainly propose the concepts of multiplicative generator pairs of n-dimensional overlap functions, in order to extend the dimensionality of overlap functions from 2 to n. We present the condition under which the pair (g, h) can multiplicatively generate an n-dimensional overlap function Og,h. we focus on the homogeneity and idempotency property on multiplicatively generated n-dimensional overlap functions.

Solutions of Laplace Equation in n-Dimensional Spaces

The solutions of the Laplace equation in n-dimensional space are studied. The angular eigenfunctions have the form of associated Jacob/polynomials. The radial solution of the Helmholtz equation is derived.

Relation Between Dimension and Angular Momentum for Radially Symmetric Potential in N-Dimensional Space

It is proved that when solving SchrSdinger equations for radially symmetric potentials the effect of higher dimensions on the radial wave function is equivalent to the effect of higher angular momenta in lower-dimensional cases. This result is applied to giving solutions for several radially symmetric potentials in N dimensions.

L_p(1≤p≤∞) APPROXIMATION BY AN n-DIMENSIONAL SUBSPACE ON A SET OF n+1 POINTS

Some new characterizations and immediate explicit expressions of best L(1≤p≤∞) approximation and their deviations by an n-dimensional subspace on a set of n+1 points are given.

Uncertainty Relations for Some Central Potentials in N-Dimensional Space

We study the uncertainty relation for three quantum systems in the N-dimensional space by using the virial theorem (VT). It is shown that this relation depends on the energy spectrum of the system as well as on the space dimension N. It is pointed out that the form of lower bound of the inequality, which is governed by the ground state, depends on the system and on the space dimension N. A comparison between our result for the lower bound and recent results, based on information-theoretic approach, is pointed out. We examine and analyze these derived uncertainties for different angular momenta with a special attention made for the large N limit.

CANONICAL REPRESENTATIONS AND DEGREE OF FREEDOM FORMULAE OF ORTHOGONAL TENSORS IN N-DIMENSIONAL EUCLIDEAN SPACE

In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part ofn \2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.

Explicit expressions and recurrence formulas of radial average value for N-dimensional hydrogen atom

In this paper, two recurrence formulas for radial average values of N-dimensional hydrogen atom are derived. Explicit expressions for <n rJ N-2 |r s|n rJ N-2 > are given for 3≥s≥-6. These results can be applied to discuss average value of centrifugal potential energy and other physical quantities. The relevant results of the usual hydrogen atom are contained in more general conclusion of this paper as special cases.

Sharp Bound for the Generalized m-Linear n-Dimensional Hardy-Littlewood-Polya Operator

In this paper,we calculate the sharp bound for the generalized m-linear n-dimensional Hardy-Littlewood-Polya operator on power weighted central and non-central homogeneous Morrey spaces.As an application,the sharp bound for the Hardy-Littlewood-Polya operator on power weighted central and noncentral homo-geneous Morrey spaces is obtained.Finally,we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral homogeneous Morrey spaces,which generalizes the previous results.

Characterization for commutators of n-dimensional fractional Hardy operators

In this paper,it was proved that the commutator H_(β,b)generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L^(p1)(R^n)to L^(p2)(R^n)if and only if b is a CMO(R^n)function,where 1/p1-1/p2=β/n,1<p1<∞,0≤β<n.Furthermore, the characterization of H_(β,b)on the homogenous Herz space K_q^(α,p)(R^n)was obtained.

Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.

Commutators of n-dimensional rough Hardy operators

In this paper, we study central BMO estimates for commutators of n-dimensional rough Hardy operators. Furthermore, λ-central BMO estimates for commutators on central Morrey spaces are discussed.

A new approach to fuzzy dynamic fault tree analysis using the weakest n-dimensional t-norm arithmetic

Dynamic fault tree analysis is widely used for the reliability analysis of the complex system with dynamic failure characteristics. In many circumstances, the exact value of system reliability is difficult to obtain due to absent or insufficient data for failure probabilities or failure rates of components. The traditional fuzzy operation arithmetic based on extension principle or interval theory may lead to fuzzy accumulations. Moreover, the existing fuzzy dynamic fault tree analysis methods are restricted to the case that all system components follow exponential time-to-failure distributions. To overcome these problems, a new fuzzy dynamic fault tree analysis approach based on the weakest n-dimensional t-norm arithmetic and developed sequential binary decision diagrams method is proposed to evaluate system fuzzy reliability. Compared with the existing approach,the proposed method can effectively reduce fuzzy cumulative and be applicable to any time-tofailure distribution type for system components. Finally, a case study is presented to illustrate the application and advantages of the proposed approach.

Four kinds of raising and lowering operators of n-dimensional hydrogen atom and isotropic harmonic oscillator

The factorization of the radial Schrodinger equation of n-dimensional (n≥2) hydrogen atoms and isotropic harmonic oscillators was investigated and four kinds of raising and lowering operators were derived.The relation between n -dimensional (n≥2) and one-dimensional hydrogen atoms and harmonic oscillators was discussed.

Asymptotic Study to the N-Dimensional Radial Schrdinger Equation for the Quark-Antiquark System

Here an asymptotic study to the N-dimensional radial Schrdinger equation for the quark-antiquark interaction potential employing asymptotic iteration method via an ansatz to the wavefunction is carried out. The complete energy spectra of the consigned system is obtained by computing and adding energy eigenvalues for ground state, for large " r" and for small " r". From this analysis, the mass spectra of heavy quarkonia is derived in three dimensions. Our analytical and numerical results are in good correspondence with other experimental and theoretical studies.

Irregular but Nonfractal Drums and n-Dimensional Weyl Conjecture

In this paper, we study the asymptotics of the spectrum of the Dirichlet （or Neumann） Laplacian in a bounded open set ΩRn（n≥1） with irregular but nonfractal boundary Ω. We give a partial resolution of the Weyl conjecture, i.e. for the counting function Ni（λ）（i=0: Dirichlet; i=i: Neumann）, we have got a precise estimate of the remainder term Ψi（λ）=φ（λ）-Ni（λ） for large λ, where φ（λ） is the Weyl term. This implies that for the irregular but nonfractal drum Ω, not only the volume |Ω|n is spectral invariant but also the area of boundary |Ω|n-1 might be spectral invariant as well.

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u_0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u_0∈ L1(Rn) ∩ L^2(Rn) and the external force f ∈ L^1(Rn× R+) ∩ L^1(R+, L^2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t > 0, where the dimension n ≥ 2, C > 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.

PROPERTIES OF SOLUTIONS OF n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.

三维氮磷共掺杂石墨烯的制备及电催化氧还原性能研究

氧还原反应(ORR)是燃料电池和金属-空气电池的阴极反应,其缓慢的动力学是目前掣肘燃料电池和金属空气电池发展的主要因素。为了使反应顺利进行,常常需要使用金属(包括贵金属和过渡金属)基催化剂,但这些催化剂存在价格高、稳定性差等缺陷。非金属原子掺杂的碳基催化剂近年来被认为是一种具有前景的ORR电催化剂。在此,使用单一前驱体磷酸三聚氰胺和氧化石墨烯经水热然后退火制备了三维氮磷共掺杂石墨烯材料(3DNPG)。通过扫描电镜(SEM)、透射电镜(TEM)、X射线衍射(XRD)、X射线光电子能谱(XPS)等手段对3DNPG的结构与组成进行了表征。结果表明:3DNPG具有独特的三维网状结构、较大的比表面积(514.38m 2/g),特别是还具有非常高的N(15.53%)和P(6.61%)掺杂量。用作碱性溶液里ORR的催化剂时,其起始电位为0.994V,半波电位为0.817V,极限电流为5.34mA/cm^(2),在48h后电流密度保持率仍维持在99%,因此是一种兼具优异活性和循环稳定性的ORR催化剂,在燃料电池、金属-空气电池等领域具有非常重要的应用价值。