A postscript on distances in convexn-gons

Letg(n) be the largest integerk such that every convex polygon withn vertices and sides has a vertexx such that the nextk vertices clockwise fromx, or the nextk vertices counterclockwise fromx, are successively farther fromx. We prove thatg(n)=[n/3]+1 forn≥4. An example givesg(n)≤[n/3]+1, and an extension of a 1952 construction of Leo Moser for a related planar problem shows thatg(n)≥[n/3]+1.     

LIMIT CYCLES AND CENTER-FOCUS OF NONLINEAR SYSTEMS

The authors consider the nonhnear systems x=h(y)-F(x),y=-g(x) in which g(x) may be not differentiabte and the system can be nonsymmetric. Some conditions which en-sure that there exists an infinite number of limit cycles are obtained. A problem about center-focus put forward by R. Conti has been answered.     

ON CIRCULANT DIGRAPHS WITH REGULAR AUTOMORPHISM GROUPS

Abstract. We prove that the cyclic group Zn(n≥3) has a k-regular digraph regular     

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n≥4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.     

Empirical likelihood confidence intervals for the differences of quantiles with missing data

Suppose that there are two nonparametric populations x and y with missing data on both of them. We are interested in constructing confidence intervals on the quantile differences of x and y. Random imputation is used. Empirical likelihood confidence intervals on the differences are constructed. Supported by the National Natural Science Foundation of China (No. 10661003) and Natural Science Foundation of Guangxi (No. 0728092).     

H m -modified wave operator for nonlinear Hartree equation in space of dimensionn≥2

We consider the scattering problem for the Hartree equation with potential |x|⁻¹ in a space of dimensionn≥2. We prove the existence ofH ^m -modified wave operator for Hartree equation on a dense set of a neighborhood of zero inH ^m (ℝ ⁿ ), meanwhile, we obtain also the global existence for the Cauchy problem of Hartree equation in a space of dimensionn≥2. This project is supported by the National Natural Science Foundation of China, 19601005     

Finite p-Groups Whose Abelian Subgroups Have a Trivial Intersection

A subgroup H of a finite group G is called a TI-subgroup if H ∩ H^x = 1 or H for all x ∈ G. In this paper, a complete classification for finite p-groups, in which all abelian subgroups are TI-subgroups, is given.     

An approximate expression related with RSA fixed points

Let T=T(n,e,a)be the number of fixed points of RSA(n,e)that are co-prime with n=pq,and A,B be sets of prime numbers in (1,x)and(1,y) respectively.An estimation on the mean-value M(A,B,e,a)=1/(#A)(#B)∑p∈A,q∈B,(p.q)=1 logT(pq,e,a)is given.     

On Cycles Containing a Given Arc in Regular Multipartite Tournaments

In this paper we prove that if T is a regular n-partite tournament with n≥4, then each arc of T lies on a cycle whose vertices are from exactly κ partite sets for κ=4,5,…,n. Our result, in a sense, generalizes a theorem due to Alspach.     

On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

LetΔ_1(x;φ) be the error term of the first Riesz mean of the Rankin-Selberg problem. We study the higher power moments ofΔ_1(x;φ) and derive an asymptotic formula for the 3-rd, 4-th and 5-th power moments by using Ivic's large value arguments and other techniques.     

SOLVING SYMMETRIC MONOTONE LINEAR VARIATIONAL INEQUALITIES BY SOME MODIFIED LEVITIN-POLYAK PROJECTION METHODS

Some modified Levitin-Polyak projection methods are proposed in this paper for solving monotone linear variational inequality x∈Ω，（x′-x)^T(Hx c)≤0，for any x′∈Ω.It is pointed out that there are similar methods for solving a general linear variational inequality.     

Semi-empirical likelihood confidence intervals for the differences of quantiles with missing data

Detecting population （group） differences is useful in many applications, such as medical research. In this paper, we explore the probabilistic theory for identifying the quantile differences .between two populations. Suppose that there are two populations x and y with missing data on both of them, where x is nonparametric and y is parametric. We are interested in constructing confidence intervals on the quantile differences of x and y. Random hot deck imputation is used to fill in missing data. Semi-empirical likelihood confidence intervals on the differences are constructed.     

THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS

51.IntroductionandNotationsLetD=(V,E)beadigraphandL(D)denotethesetofcyclelengthsofD.ForuEVandintegeri21,letfo(u):={vEVIthereedestsadirectedwalkoflengthifromutov}.WedelveRo(u):={u}.Letu,vEV.IfN (v)=N (v)andN--(v)=N--(v),thenwecanvacopyofu.LotDbeaprimitivedigraphand7(D)denotetheexponentofD.In1950,H.WielandtI61foundthat7(D)5(n--1)' 1andshowedthatthereisapiquedigraphthatattainsthisbound.In1964,A.L.DulmageandN.S.Mendelsohn[2]ObservedthattherearegapsintheexponentsetEd={ry(D)IDEPD.}…     

Outpaths of arcs in multipartite tournaments

1. IntroductionThroughout the paPer, we use the terminology and notation of [1] and [2]. Let D =(V(D), A(D)) be a digraPh. If xy is an arc of a digraPh D, then we say that x dominatesy, denoted by x - y. More generally, if A and B are two disjoint vertex sets of D such thatevery vertex of A dominates every vertex of B, then we say that A dominates B, denotedby A - B. The outset N (x) of a vertex x is the set of vertices dominated by x in D,and the inset N--(x) is the set of vertices d…     

L p approximation by Bernstein-Kantorovich quasi-interpolants

Bernstein-Kantorovich quasi-interpolants K^（2r-1）n（f, x） are considered and direct, inverse and equivalence theorems with Ditzian-Totik modulus of smoothness ω^2rφ（f, t）p （1 ≤ p ≤＋∞） are obtained.     

Moment inequalities and weak convergence for negatively associated sequences

A probability inequality for S_n and somepth moment (p⩾2) inequalities for |S_n| and max 1⩽k⩽n | S_k| are established. Here S_n is the partial sum of a negatively associated sequence. Based on these inequalities, a weak invariance principle for strictly stationary negatively associated sequences is proved under some general conditions. Project supported by the National Natural Science Foundation of China, the Doctoral Program Foundation of the State Education Commission of China and the High Eductional Natural Science Foundation of Guangdong Province.     

Existence of three HMOLS of type 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript><Emphasis Type="Italic">u</Emphasis><Superscript>1</Superscript>

A Latin squares of order v with ni missing sub-Latin squares （holes） of order hi （1 〈= i 〈 k）, which are disjoint and spanning （i.e. ∑k i=l1 nihi = v）, is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by （h1n1 h2n2…hknk ）. If any two PILS inaset of t PILS of type T are orthogonal, then we denote the set by t-HMOLS（T）. It has been proved that 3-HMOLS（2n31） exist for n ≥6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS（2nu1） with u ≥ 4, and prove that 3-HMOLS（2~u1） exist if n ≥ 54 and n ≥7/4u ＋ 7.     

Weakly stable constant mean curvature hypersurfaces

Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.     

A Conjecture Concerning the Pure Exponential Diophantine Equation a^x ＋ b^y = c^z

Let a, b, c, r be fixed positive integers such that a^2 ＋ b^2 = c^r, min（a, b, c, r） 〉 1 and 2 r. In this paper we prove that if a ≡ 2 （mod 4）, b ≡ 3 （mod 4）, c 〉 3.10^37 and r 〉 7200, then the equation a^x ＋ b^y = c^z only has the solution （x, y, z） = （2, 2, r）.     

Maximum distance holey packings and related codes

The notion of a maximum distance holey packing is introduced and used to construct optimal ternary (n, 3, 3) codes for all lengthsn=2 (mod 3) andn≥8. Combining this with Etzion’s result, the existence problem for an optimal ternary (n,3,3) code is solved completely. Project supported by the National Natural Science Foundation of China (Grant No. 19671064).