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.

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.

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.

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 Hβ,b generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from Lp1(Rn) to Lp2 (Rn) if and only if b is a C(M)O(Rn) function, where 1/p1 - 1/p2 = β/n, 1 ＜ p1 ＜∞, 0 ≤β＜ n. Furthemore,the characterization of Hβ,b on the homogenous Herz space (K)qα,p(Rn) was obtained.

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.

A NOTE ON "THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS"

In this article, we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations…… admits a unique global generalized solution in ……and a unique global classical solution in…… the sufficient conditions of the blow up of the solution in finite time are given, and also two examples are given.

m-degree Center-connecting Line of Finite Points Set and Its Properties

In this paper, we first give the concept of m-degree center-connecting line in n-dimensional Euclidean space Enand investigate its several properties, then we obtain the length of m-degree center-connecting line formula in finite points set. As its application,we extend the Leibniz formula and length of medians formula in n-dimensional simplex to polytope.

Discreteness Criterion for Subgroups in PU（1,n;C）

In this paper,the authors introduce the discreteness criterion for n-dimensional subgroups in PU(1, n; C).These generalize the well-known discreteness criterion first established by T.Jorgensen.

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.

High Dimensional Cluster Analysis Using Path Lengths

A hierarchical scheme for clustering data is presented which applies to spaces with a high number of dimensions (). The data set is first reduced to a smaller set of partitions (multi-dimensional bins). Multiple clustering techniques are used, including spectral clustering;however, new techniques are also introduced based on the path length between partitions that are connected to one another. A Line-of-Sight algorithm is also developed for clustering. A test bank of 12 data sets with varying properties is used to expose the strengths and weaknesses of each technique. Finally, a robust clustering technique is discussed based on reaching a consensus among the multiple approaches, overcoming the weaknesses found individually.

一类多线性Hardy型算子在变指数Herz-Morrey空间的有界性（英文）

In this paper, we will obtain that the boundedness of multilinear n-dimensional fractional Hardy operators of variable order β(x) on variable exponent Herz-Morrey spaces.

Preliminary Exploration on Properties of the Equidistant Conjugate Points of Higher Dimensional Simplex

In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the property of the equidistant conjugate points of a triangle is generalized to high dimensional simplex.

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.

Operator Equations Inducing Some Generalizations of Slant Hankel Operators

An extension of slant Hankel operator,namely,the kth-orderλ-slant Hankel operator on the Lebesgue space L^(2)(T^(n)),where T is the unit circle and n≥1,a natural number,is described in terms of the solution of a system of operator equations,which is subsequently expressed in terms of the product of a slant Hankel operator and a unitary operator.The study is further lifted in Calkin algebra in terms of essentially kth-orderλ-slant Hankel operators on L^(2)(T^(n)).