Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics

Summary The members of the power divergence family of statistics all have an asymptotically equivalent χ² distribution (Cressie and Read [1]). An asymptotic expansion for the distribution function is derived which shows that the speed of convergence to this asymptotic limit is dependent on λ. Known results for Pearson'sX ² statistic and the log-likelihood ratio statistic then appear as special cases in a continuum rather than as separate (unrelated) expansions.     

A Priori Bounds for Periodic Solutions of a Kind of Second Order Neutral Functional Differential Equation with Multiple Deviating Arguments

Thc main aim of this paper is to use the continuation theorem of coincidence degree theory for studying the existence of periodic solutions to a kind of neutral functional differential equation as follows：（x（t）-^n∑i=1cix（t-ri））″=f（x（t））x′＋g（x（t-τ））＋p（t）.In order to do so, we analyze the structure of the linear difference operator A ： C2π→C2π, [Ax]（t） =x（t）-∑^ni=1cix（t-ri）to determine some flmdamental properties first, which we are going to use throughout this paper. Meanwhile, we also prove some new inequalities which are useful for estimating a priori bounds of periodie solutions.     

Divergence points with fast growth orders of the partial quotients in continued fractions

This paper is concerned with the divergence points with fast growth orders of the partial quotients in continued fractions. Let S be a nonempty interval. We are interested in the size of the set of divergence points

Equivalent definition of some weighted Hardy spaces

We present an equivalent definition of functions analytic in the half-plane ℂ₊ = {z: Re z > 0} for which

Inequalities for absolute maxima of a polynomial and its derivative

Suppose z ₁, z ₂, ... z _n are complex numbers with absolute values more than 1 and Arg z _j Arg z _k for j k where Arg w stands for the argument of the complex number w in [0,2). In this note we show that

Onextensions of the Gale-Berlekamp switching problem and constants ofl q —spaces

For positive integersn, m and realp≥1, let Upper and lower bounds for this quantity are derived, extending results of Brown and Spencer forB ₁(n,n), corresponding to the Gale-Berlekamp switching problem. For a Minkowski spaceM of dimensionm, define a quantity investigated by Dvoretzky and Rogers.     

On the zeros of a class of generalised Dirichlet series-XIV

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A.Let 0<θ<1/2and let {a _n }be a sequence of complex numbers satisfying the inequality for N = 1,2,3,…,also for n = 1,2,3,…let α _n be real and |α_n| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T ₀(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a _n to and |a_N| ≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)^-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided for every ε > 0. Dedicated to the memory of Professor K G Ramanathan     

On boundary values of solutions of the dirichlet problem for second order elliptic equation

The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for the general elliptic equation of the second order

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Almost sure convergence of sample range

Let X ₁, X ₂, ... be i.i.d. random variables. The sample range is R _n = max {X _i , 1 ≤ i ≤ n} − min {X _i , 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α _k ), (β _k ) then we have

Regolarità hölderiana delle derivate dei minimi di funzionali con parte principale quadratica

Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali dei funzionali sotto opportune ipotesi suA _ij ^αβ e sug.

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Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA _ij ^αβ andg.

     

Some convergence theorems on a supercritical Galton-Watson process

Summary LetX=(X _n; n≧0,X ₀=1) be a supercritical Galton-Watson process. The limiting distribution of ) where is the m.l.e. of the offspring mean, is derived. As an application of this result, some limit theorems leading ultimately to a parameter free result of statistical interest, are also established.     

Periodicity in a Nonlinear Predator-prey System with State Dependent Delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear state dependent delays predator-prey system where a_i(t),c_j(t),d_i(t) are continuous positive periodic functions with periodic ω>0, b_1(t),b_2(t) are continuous periodic functions with periodic ωand ∫_0~ωbi(t)dt>0. T_i,σ_j, p_i (i=1,2,…,n, j=1, 2,…,m) are continuous and ω-periodic with respect to their first arguments, respectively, α_i, β_j,γ_i(i=1,2,…,n, j=1,2, …, m) are positive constants.     

The generalized Roper-Suffridge extension operator in Banach spaces (III)

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) on the domain $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } defined by

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Integral estimates of the solutions to the helmholtz equation in unbounded domains

The following boundary value problem is studied:

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

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such that an↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞ Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that ∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0 if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.     

Composition operators on the Lipschitz space in polydiscs

Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ