THE COMPACT COMPOSITION OPERATOR ON THE μ-BERGMAN SPACE IN THE UNIT BALL

nAbstract Let p 0 and μ be a normal function on [0, 1), ν(r) =(1-r~2)~(1+(n/p)) μ(r) for r ∈ [0, 1). In this article, the bounded or compact weighted composition operator T_φ,ψfrom the μ-Bergman space A~p(μ) to the normal weight Bloch type space β_ν in the unit ball is characterized. The briefly sufficient and necessary condition that the composition operator C_φis compact from A~p(μ) to β_ν is given. At the same time, the authors give the briefly sufficient and necessary condition that C_φis compact on β_μ for a 1.     

Hausdorff dimensions of some irregular sets associated with β-expansions

The Hausdorff dimensions of some refined irregular sets associated with β-expansions are determined for any β 1. More precisely, Hausdorff dimensions of the sets {x ∈ [0, 1) :lim inf(n→∞) S_n(x, β)/n= α_1, lim sup (n→∞) S_n(x, β)/n= α_2}, α_1, α_2≥0 are obtained completely, where S_n(x, β) =sum ε_k(x, β) from k=1 to n denotes the sum of the first n digits of the β-expansion of x. As an application, we present another concise proof of that the set of points x ∈ [0, 1) satisfying lim_(n→∞) S_n(x,β)/n does not exist is of full Hausdorff dimension.     

Bagley-Torvik型分数阶微分方程和包含非局部积分边值问题

In this article,we consider the Bagley-Torvik type fractional differential equation ~cD~(ν1) l(t)-a~cD~(ν2) l(t) = g(t, l(t)) and differential inclusion ~cD~(ν1) l(t)-a~cD~(ν2) l(t) ∈ G(t, l(t)),t ∈(0, 1) subjecting to l(0) = l_0,and■ ds where 1 ν_1 ≤ 2, 1 ≤ν_2 ν_1,0 ω≤ 1, χ = ν_1-ν_2 0, a, λ′are given constants. By using Leray-Schauder degree theory and fixed point theorems, we prove the existence of solutions. Our results extend the existence theorems for the classical Bagley-Torvik equation and some related models.     

Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems with continuous potentials

In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials:{-ε~2?u +(1 + δP(x))u = μ1 u~3+ βuv~2 in ?,-ε~2?v +(1 + δQ(x))v = μ2 v~3+ βu~2 v in ?,u 0, v 0 in ?,(?u)/(?v)=(?ν)/(?ν)=0on ??,(A_ε)where ? is a smooth bounded domain in R~N for N = 2, 3, δ, ε, μ_1 and μ_2 are positive parameters, β∈ R,P(x) and Q(x) are two smooth potentials defined on ?, the closure of ?. Due to Liapunov-Schmidt reduction method, we prove that(A_ε) has at least O(1/(ε| ln ε|)~N) synchronized and O(1/(ε| ln ε|)~(2 N)) segregated vector solutions for ε and δ small enough and some β∈ R. Moreover, for each m ∈(0, N) there exist synchronized and segregated vector solutions for(A_ε) with energies in the order of ε~(N-m). Our results extend the result of Lin et al.(2007) from the Lin-Ni-Takagi problem to the nonlinear Schr¨odinger elliptic systems with continuous potentials.     

Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear N-Coupled Schr?dinger System

In this paper, we study the following N-coupled nonlinear Schr?dinger system ■where n ≤ 3, N ≥ 3, μ_j 0, β_(i,j)= β_(j,i) 0 are constants and βj,j= μj, j = 1, ···, N.There have been intensive studies for the system on existence/non-existence and classification of ground state solutions when N = 2. However fewer results about the classification of ground state solution are available for N ≥ 3. In this paper, we first give a complete classification result on ground state solutions with Morse indices 1,2 or 3 for three-coupled Schr?dinger system. Then we generalize our results to Ncoupled Schr?dinger system for ground state solutions with Morse indices 1 and N.We show that any positive ground state solutions with Morse index 1 or Morse index N must be the form of(d_1w, d_2w, ···, d_Nw) under suitable conditions, where w is the unique positive ground state solution of certain equation. Finally, we generalize our results to fractional N-coupled Schr?dinger system.     

ON THE PROPERTIES OFε(≥0) OPTIMAL POLICIES IN DISCOUNTED UNBOUNDED RETURN MODEL

This paper investigates the properties of ε(≥0) optimal policies in the model of [2].It is shownthat,if π~*=(π_0~*,π_1~*,…,π_n~*,π_(n+1)~*,…)is a β-discounted optimal policy,then(π_0~*,π_1~*,…,π_n~*)~∞ for alln≥0 is also a β-discounted optimal policy.Under some condition we prove that stochastic stationarypolicy π_n~(*∞)corresponding to the decision rule π_n~* is also optimal for the same discounting factor β.Wehave also shown that for each β-optimal stochastic stationary policy π_0~(*∞),π_0~(*∞) can be decomposed intoseveral decision rules to which the corresponding stationary policies are also β-optimal separately;and conversely,a proper convex combination of these decision rules is identified with the former π_0~*.We have further proved that for any (ε,β)-optimal policy,say π~*=(π_0~*,π_1~*,…,π_n~*,π_(n+1)~*,…),(π_0~*,π_1~*,…,π_(n-1)~*)∞ is ((1-β~n)~(-1)ε,β)optimal for n>0.At the end of this paper we mention that the resultsabout convex combinations and de     

Commuting dual Toeplitz operators on the harmonic Bergman space

We completely characterize commuting dual Toeplitz operators with bounded harmonic symbols on the harmonic Bergman space of the unit disk. We show that for harmonic ■ and ψ, S■Sψ= SψS■ on(L2h)⊥if and only if ■ and ψ satisfy one of the following conditions:(1) Both ■ and ψ are analytic on D.(2) Both ■ and ψ are anti-analytic on D.(3) There exist complex constants α and β, not both 0, such that ■ = αψ + β.Furthermore, we give the necessary and sufficient conditions for S■Sψ= S■ψ.     

Composition Cesàro Operator on the Normal Weight Zygmund Space in High Dimensions

Let n>1 and B be the unit ball in n dimensions complex space Cⁿ.Suppose thatφis a holomorphic self-map of B andψ∈H(B)withψ(0)=0.A kind of integral operator,composition Cesàro operator,is defined by T_φψ(f)(z)=∫¹0f[φ(tz)]Rψ(tz)dt/t,f∈(B)z∈B.In this paper,the authors characterize the conditions that the composition Cesàro operator T_φ,ψis bounded or compact on the normal weight Zygmund space Z_μ(B).At the same time,the sufficient and necessary conditions for all cases are given.     

The largest eigenvalue distribution of the laguerre unitary ensemble

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in(0, t), that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, α_n(t) and β_n(t), as well as three auxiliary quantities, denoted by r_n(t), R_n(t), and σ_n(t). We establish the second order differential equations for both β_n(t) and r_n(t). By investigating the soft edge scaling limit when α = O(n) as n →∞ or α is finite, we derive a P_Ⅱ, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials P_n(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n,asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy's equation for ρ(t) = Ξ′(t) with Ξ(t) satisfying the σ-form of P_Ⅳ or P_Ⅴ.     

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

For partial linear model Y=X~τβ_0 _(g0)(T) εwith unknown β_0∈R~d and an unknown smooth function go, this paper considers the Huber-Dutter estimators of β_0, scale σfor the errors and the function go respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β_0 and σare asymptotically normal with convergence rate n~((-1)/2) and the B-spline Huber-Dutter estimator of go achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β_0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.