On the positive nonoscillatory solutions of the difference equation x_(n+1)= α +(x_(n-K)/x_(n-m))~p

The aim of this paper is to show that the following difference equation:xn+1=α+(xn-k/xn-m)p,n=0,1,2…,where α＞-1,p＞0,k,m ∈ N are fixed, 0≤m＜k,x-k,x-k+1,…,x-m,…,x-1,x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α+1.It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.     

Bergman kernel function on Hua Construction of the second type

In this paper, we give an explicit formula of the Bergman kernel function on Hua Construction of the second type when the parameters 1/p1,…,1/(pr-1) are positive integers and 1/pr is an arbitrary positive real number.     

L p bounds for singular integrals with rough kernels on product domains

This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface.Some rather weaker size conditions,which imply the Lp-boundedness of such operators for certain fixed p(1 p ∞),are given.     

THE GENERALIZED SMASH PRODUCT AND COPRODUCT

91. PreliminariesThroughout the paper all spaces are over a fixed ground field K. If C is a coalgebra,then we always denote the comultiplication and counit by b and E respectively. Set b(c) Z of @ cd for c E C. If M is a right (left) C -comodule, then we use p for the structure mapof M, and set p(m) = ac(o) @ m(1) (p(m) ~ ac(--1) @ m(o)) for m E M. Let H be abialgebra. Denote left H-module category by HM and left H-comodule category by HM.If M and N are in HM, then M @ N is also a l…     

8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Let F = Q（√-p1p2） be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol （p1/p2） = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold： （1） The quartic residue symbols （p1/p2）4 = （p2/p1）4 = 1; （2） Either both p1 and p2 are represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=ε（2x^2-p1y^2）,x,y∈Z,ε∈{±1},where h＋（2p1） is the narrow class number of Q（√2p1）,Moreover, we also generalize these results.     

On the positive nonoscillatory solutions of the difference equation Xn＋1=α＋（xn-k/xn-m）^p

The aim of this paper is to show that the following difference equation：Xn＋1=α＋（xn-k/xn-m）^p, n=0,1,2,…,
where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k＋1,…,x-m,…,X-1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α＋1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.     

随机规划中若干凸性命题(Ⅱ)

Peter kall discussed the convexity statement if a_(i1)(ω),…,a_(in)(ω),b_i(ω) are independent random variables with known normal distributions.In theorem 1, we discuss the convexity statement if a_(ik)(ω)～N(m_k,σ_k) (k=1,…,N;1≤N≤n) and others are fixed. Theorem 1 If a_(ik)(ω)～N(m_k,σ_k)(k=1,…,N;1≤N≤n) and a_(ik)(ω)(k=1,…,N are independent and others are fixed, then (i) Suppose then X_i(a_i) is conyex.     

ON PROPERTIES OF MEROMORPHIC SOLUTIONS FOR COMPLEX DIFFERENCE EQUATION OF MALMQUIST TYPE

In this paper, we present the properties on zeros, fixed points, poles, Borel exceptional value of finite order transcendental meromorphic solutions of complex difference equation of Malmquist typewhere n（∈ N） 〉 2, P（f（z）） and Q（f（z）） are relatively prime polynomials in f（z） with rational coefficients a8 （s = 0, 1,…,p） and bt （t = 0, 1,… ,q） such that aoapbq 7≠ O, and also consider the existence and the forms on rational solutions of this type of difference equations. Some examples are also listed to show that the assumptions of theorems, in certain senses, are the best possible.     

Global blow-up for a heat system with localized sources and absorptions

In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic systemut=Δu vp(x0, t)-aur, x∈Ω, t＞0,vt=Δv uq(x0,t)-bvs, x∈Ω, t＞0subject to homogeneous Dirichlet conditions and nonnegative initial data, where x0 ∈Ω is a fixed point, p, q, r, s ≥ 1 and a, b ＞ 0 are constants. In the situation when nonnegative solution (u, v) of the above problem blows up in finite time, it is showed that the blow-up is global and this differs from the local sources case. Moreover, for the special case r = s = 1,are obtained uniformly on compact subsets of Ω, where T* is the blow-up time.     

The Subgroups of Finite Metacyclic Groups

In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples(k, l, β) ∈ N~3 and all the subgroups of K. The results are applied to compute some subgroups of K as well as to study the structure and the number of p-subgroups of K, where p is a fixed prime number. In addition, the author gets a factorization of K, and then studies the metacyclic p-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic p-group when p ≥ 3. A "reciprocity" relation on enumeration of subgroups of a metacyclic group is also given.     

A KERNEL-TYPE ESTIMATOR OF A QUANTILE FUNCTION UNDER RANDOMLY TRUNCATED DATA

A kernel-type estimator of the quantile function Q(p) = inf {t : F(t)≥p}, 0≤p≤1, is proposed based on the kernel smoother when the data are subjected to random truncation. The Bahadur-type representations of the kernel smooth estimator are established, and from Bahadur representations the authors can show that this estimator is strongly consistent, asymptotically normal, and weakly convergent.     

关于正整数p~αq~β的正约数的个数

命题若一个正整数可以写成为pαqβrθ(p,q,r为互不相等的质数,α,β,θ为不小于1的自然数)．则它的正约数的个数N=(α 1)×(β 1)×(θ 1)．分析任取a∈{1,p,p2,…,pα},b∈{1, q,q2,…,qβ},c∈{1,r,r2,…,rθ}．则abc必为正整数pαqβrθ的正约数,这里的a有α 1种取法,b有β 1种取法,c有θ 1种取法．反之     

用Walsh-Fourier级数的(c,β)平均来逼近函数的问题

设p≥2是固定的整数.x∈[0,1]的p进表示是x=(0.x_1x_2…x_n…),其中x_k∈{0,1,…,p-1},k∈N={1,2,…}。並且约定对p进有理点取有限表示。对任意非负整数k≥0,写k=sum from j=0 to n (k_jp~j),k_j∈{0,1,…,p-1}。设,则p进的Walsh函数定义为。     

INVOLUTIONS FIXING ^mUi=1 CPi（1） ×HPi（n）＊

Let(M,T) be a closed manifold with an involution T.The fixed point set of T is F.In this article,bordism classes of the involutions with fixed point set F = ∪(from i=1 to m) CP i(1) × HP i(n) are determined,where CP(1) and H P(n) denote the 1-dimensional complex projective space and n-dimensional quaternionic projective space respectively,and n = 2p-2 or n = 2p-1(p > 1).     

Confidence Intervals of Variance Functions in Generalized Linear Model

In this paper we introduce an appealing nonparametric method for estimating variance and conditional variance functions in generalized linear models （GLMs）, when designs are fixed points and random variables respectively, Bias-corrected confidence bands are proposed for the （conditional） variance by local linear smoothers. Nonparametric techniques are developed in deriving the bias-corrected confidence intervals of the （conditional） variance. The asymptotic distribution of the proposed estimator is established and show that the bias-corrected confidence bands asymptotically have the correct coverage properties. A small simulation is performed when unknown regression parameter is estimated by nonparametric quasi-likelihood. The results are also applicable to nonparamctric autoregressive times series model with heteroscedastic conditional variance.     

FORCED ALMOST PERIODIC OSCILLATION OF SECOND ORDER NONLINEAR EQUATION

1 IntroductionConsider equation of the Duffing fOrmX" cx' g(x) = p(t) (1)where c is a constant, p and g are continuous functions defined on (--oo, oc).In the early years of the l990's, some authors investigated the existence andstability of periodic solution of system (1), when p(t) is a periodic function(see [1],[2]).In this paper, we consider the situation in where p(t) is an almost periodicfunction. Under some conditions, we prove the existence and uniqueness ofalmost periodic soluti…     

On the GF(p) Linear Complexity of Hall's Sextic Sequences and Some Cyclotomic-Set-Based Sequences

Klapper(1994) showed that there exists a class of geometric sequences with the maximal possible linear complexity when considered as sequences over GF(2), but these sequences have very low linear complexities when considered as sequences over GF(p)(p is an odd prime). This linear complexity of a binary sequence when considered as a sequence over GF(p) is called GF(p) complexity. This indicates that the binary sequences with high GF(2) linear complexities are inadequate for security in the practical application, while,their GF(p) linear complexities are also equally important, even when the only concern is with attacks using the Berlekamp-Massey algorithm [Massey, J. L., Shift-register synthesis and bch decoding, IEEE Transactions on Information Theory, 15(1), 1969, 122–127]. From this perspective, in this paper the authors study the GF(p) linear complexity of Hall's sextic residue sequences and some known cyclotomic-set-based sequences.     

APPROXIMATION OF A CAUCHY-JENSEN ADDITIVE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES＊

Using the fixed point and direct methods,we prove the Hyers-Ulam stability of the following Cauchy-Jensen additive functional equation 2f p i=1 xi + q j=1 yj + 2 d k=1 zk 2 = p i=1 f(xi) + q j=1 f(yj) + 2 d k=1 f(zk),where p,q,d are integers greater than 1,in non-Archimedean normed spaces.     

Coordinating pricing and inventory control in a fluctuating environment

This paper addresses the simultaneous determination of pricing and inventory replenishment strate- gies under a fluctuating environment. Specifically, we analyze the single item, periodic review model. The demand consists of two parts： the deterministic component, which is influenced by the price, and the stochastic component （perturbation）. The distribution of the stochastic component is determined by the current state of an exogenous Markov chain. The price that is charged in any given period can be specified dynamically. A replenishment order may be placed at the beginning of some or all of the periods, and stockouts are fully backlogged. Ordering costs that are lower semicontinuous, and inventory/backlog （or surplus） costs that are continuous with polynomial growth. Finite-horizon and infinite-horizon problems are addressed. Existence of optimal policies is established. Furthermore, optimality of （s,S,p）-type policies is proved when the ordering cost consists of fixed and proportional cost components and the surplus cost （these costs are all state-dependent） is convex.     

Liouville Type Results for a p-Laplace Equation with Negative Exponent

Positive entire solutions of the equation Δpu=u~(-q) in R~N(N ≥ 2)where 1p≤N,q0,are classified via their Morse indices.It is seen that there is a critical power q=q_c such that this equation has no positive radial entire solution that has finite Morse index when qq c but it admits a family of stable positive radial entire solutions when 0q≤q_c.Proof of the stability of positive radial entire solutions of the equation when 1p2 and 0q≤q_c relies on Caffarelli–Kohn–Nirenberg's inequality.Similar Liouville type result still holds for general positive entire solutions when 2p≤N and qq_c.The case of 1p2 is still open.Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p=2 obtained previously.Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.