Strong laws of large numbers for weighted sums of $$\tilde \rho $$ -mixing random variables

Strong laws are established for linear statistics that are weighted sums of a -mixing random sample. The results obtained generalize the results of Baxter et al. [SLLN for weighted independent indentically distributed random variables, J. Theoret. Probab. 17 (2004), 165–181] to -mixing random variables. This paper is supported by Key discipline of Zhejiang Province (Key discipline of Statistics of Zhejiang Gongshang University) and National Natural Science Foundation of China.     

The Commutators of Fractional Integrals on Besov Spaces

Abstract   In this paper we consider the commutators of fractional integrals

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Periodic Solutions for a Class of Forced Liénard-type Equations

Abstract   By applying the topological degree theory, we establish some sufficient conditions for the existence on T-periodic solutions for the Liénard-type equation

Positive Solutions for Semipositone <Emphasis Type="Italic">m</Emphasis>-point Boundary-value Problems

Abstract   Let ξ _i ∈ (0, 1) with 0 < ξ₁ < ξ₂ < ··· < ξ _m−2 < 1, a _i , b _i ∈ [0,∞) with and . We consider the m-point boundary-value problem

Existence of Positive Solutions for a Class of Sigular Boundary Value Problem of Fourth Order

Some results concerning existence of positive solutions for the singular boundary value problems u^(4)(t)=f(t,u(t)) t∈（0,1) u(0)=u(1)=0 u‘(0)=u‘(1)=0 have been given, where f(t, x) may be singular at t = 0, 1.     

Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract  Let Ω be the unit ball centered at the origin in . We study the following problem

Structure theorems for generalized random processes

Generalized random processes are classified by various types of continuity. Representation theorems of a generalized random process on {M _p } on a set with arbitrary large probability, as well as representations of a correlation operator of a generalized random process on {M _p } and L ^r (R), r > 1, are given. Especially, Gaussian generalized random processes are proven to be representable as a sum of derivatives of classical Gaussian processes with appropriate growth rate at infinity. Examples show the essence of all the proposed assumptions. In order to emphasize the differences in the concept of generalized random processes defined by various conditions of continuity, the stochastic differential equation y′(ω; t) = f(ω; t) is considered, where y is a generalized random process having a point value at t = 0 in the sense of Lojasiewicz. This paper was supported by the project Functional analysis, ODEs and PDEs with singularities, No. 144016, financed by the Ministry of Science, Republic of Serbia.     

Inverse-Definiteness of the Fourth-Order Symmetric Differential Operator （Ⅰ）

Abstract   We give a linear symmetric differential operator L defined by

Invariant Tori and Boundedness in Asymmetric Oscillations

In this paper,we are concerned with the boundedness of all the solutions of the equation x″ ax^ -bx- Ф(x)=p(t),where p(t) is a smooth 2π-periodic function,a and b are positive constants,and the perturbation Ф(x) is bounded.     

Blow-up Solutions for Mixed Nonlinear Schrodinger Equations

This paper is concerned with the initial boundary-value problem for the following nonlinear evolution equation: Фt=iαФxx βФ^2Ф^-x γ|Ф|^2Фx ig(|Ф|^2)Ф Under certain conditions on the initial data and the function g(s),we study the existence and nonexistence of global solution for this equation.The blow-up solution and the blow-up time are also investigated.     

Linear Volterra Integral Equations as the Limit of Discrete Systems

We consider the multidimensional abstract linear integral equation of Volterra type x（t） （*）∫Rtα（s)x(s)ds=f(t),t∈R,（1）as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Bauach space-valued defined on a compact interval R of R^n Rt is a subinterval of R depending on t ∈ R and (*) f denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems, The Henstock integral setting enables us to consider highly oscillating functions.     

On the size of the Jacobians of curves over finite fields

Abdract  Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field of q elements, we obtain the asymptotic formula q ^g+o(g) for the size of set of the -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.      

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems

Abstract   The singular second-order m-point boundary value problem

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.     

Cesàro α-Integrability and Laws of Large Numbers I

For a sequence of integrable random variables, we introduce a new set of conditions called Cesàro -Integrability and Strong Cesàro -Integrability and show that, for <1/2, these conditions that are strictly weaker than Cesàro Uniform Integrability and Strong Cesàro Uniform Integrability respectively, are sufficient for WLLN and SLLN to hold for a sequence of pairwise independent random variables. For some special kinds of dependent sequences of random variables also, Cesàro -integrability for appropriate is shown to be sufficient for WLLN to hold.     

Precise rate in the law of iterated logarithm for <Emphasis Type="Italic">ρ</Emphasis>-mixing sequence

Let {X, X _n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim _n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.      

Eden-Staudacher and Beisert-Eden-Staudacher equations in the $$\mathcal{N} = 4$$ supersymmetric gauge theory

We investigate the Eden-Staudacher and Beisert-Eden-Staudacher equations for the anomalous dimension of twist-2 operators at a large spin s in the supersymmetric gauge theory. We reduce these equations to a set of linear algebraic equations and calculate their kernels analytically. We demonstrate that in the perturbation theory, the anomalous dimension is a sum of products of the Euler functions ζ(k) having the maximum transcendentality property. We also show that at a large coupling, the “singular” solution of the Beisert-Eden-Staudacher equation reproduces the anomalous dimension constants predicted from the string side of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 117–129, April, 2008.