On Turán type inequality with doubling weights andA * weights

LetH _n be the set of real algebraic polynomials of degreen, whose zeros all lie in the interval [−1,1]. The well known Turán type inequalities tell us that forf(x)∈H _n , it holds ‖f′‖≥C√n‖f‖. This note deals with the weighted Turán type inequalities with the weights having inner singularities underL ^p norm for 0<p≤∞. Our results essentially extend the result of Wang and Zhou (2002), and the method used in this paper is simpler and more direct than that of Wang and Zhou (2002). The results and methods have their own values in approximation theory and computation.     

A class of not max-stable extreme value distributions

The sequences {Zi,n, l≤i≤n}, n≥l have multi-nomial distribution among i.i.d. random variables {X1,i, i≥1}, {X2,i,u≥l }, …, {Xm,i, i≥1 }. The extreme value distribution Gz(x) of this particular triangular array of i.i.d, random variables Z1,n, Z2Zn,n is discussed in this paper. We found a new type of not max-stable extreme value distributions, i) Gz(x) = r-1∏i=1ФAiαi(x) × Фαr (x);ii) Gz (x) = r-1∏i=1ψAiαi (x) × ψαr (x); iii) Gz (x) = r-1∏i=1 ∧Ai (λix) × A(x), r≥2, 0＜α1≤α2≤…≤αr and λi∈ (0,1] for i, l≤i≤r-1 which occur if Fj, …, Fm belong to the same MDA.     

Precise asymptotics in the law of the logarithm for random fields in Hilbert space

Consider the positive d-dimensional lattice d (d≥2) with partial ordering ≤, let {XK; K ∈ d } be i.i.d. random vari- ables taking values in a real separable Hilbert space (H, ||·||) with mean zero and covariance operator ∑, and set partial sums SN =∑K≤NXK, K , N ∈ d . Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε 0 by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of (Gut and Spǎtaru, 2003).     

Extreme value distributions of mixing two sequences with the same MDA

Suppose {Xi,i≥1} and {Yi, i≥1} are two independent sequences with distribution functions and , respectively. Zi,n is the combination of Xi and Yi with a probability for each I with 1≤ i≤n. The extreme value distribution GZ(x) of this particular triangular array of the i.i.d. Random variables Z1,n,… Z2,n, Zn,n is discussed. We found a new form of the extreme value distributions i)ΦAα1(x)Φα2 and ii) ψAα(x)ψα2(x)α1<α2), which are not max-stable. It occurs if FX and FY belong to the same MDA(Φ) or MDA(Ψ).     

A general version of the Morse-Sard theorem

Let k,m,n be positive integers,and k≥2,α∈(0,1],0＜r＜min {m,n} an integer,d=r (m-r)/(k α),and iff∈ Ck,α(Rm,R″),A =Cr(f)= {x ∈ Rm ,rank(Df(x))≤r},thenf(A)is d-null.Thus the statement posed by Arthur Sard in 1965 can be completely solved when k≥2.     

Congruences for finite triple harmonic sums

Zhao (2003a) first established a congruence for any odd prime p>3, S(1,1,1;p)≡-2Bp-3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β,γ;p) (mod p) is considered for all positive integers α,β,γ. We refer to w=α β γ as the weight of the sum, and show that if w is even, S(α,β,γ;p)≡0 (mod p) for p≥w 3; if w is odd, S(α,β,γ;p)≡rBp≥w (mod p) for p≥w, here r is an explicit rational number independent of p. A congruence of Catalan number is obtained as a special case.     

Monotone routing in multirate rearrangeable logd（N,m，p） network

The construction of multirate rearrangeable network has long been an interesting problem. Of many results published, all were achieved on 3-stage Clos network. The monotone routing algorithm proposed by Huet al. (2001) was also first applied to 3-stage Clos network. In this work, we adopt this algorithm and apply it to log _d (N,m,p) networks. We first analyze the properties of log _d (N,m,p) networks. Then we use monotone algorithm in log _d (N,0,p) network. Furthermore we extend the result to construct multirate rearrangeable networks based on log _d (N,m,p) network (1≤m≤n−1). Project (No. 10371028) supported by the National Natural Science Foundation of China     

A class of Marcinkiewicz integral operators on product domains

In this paper, the author proves that theL ^p-boundedness of the Marcinkiewicz integral μ_Ω on product domainsR ⁿ×R^m; for Ω∈(1)∩(5) improves the result of Chen et al. (2000). Project supported by Major Project of NSFC (No.19631080) and NSF of Zhejiang province (No. RC97017).     

Regular-singular crossings on nonlinear systems with turning point

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On sums of powers of integers

For non-negative integers k, n, let P _k (n) denote the sum {fx27-1}. We show by two different means that if k ≥ 3 and odd, then n ²(n+1)² iss a factor of the polynomial P _k (n); and if k ≥ 2 and even, then n (n+1) (2n+1) is a factor of the polynomial P _k (n). We also derive a relatively unknown result first obtained by Johann Faulhaber in the 17th century. Shailesh Shirali has been at Rishi Valley School, Andhra Pradesh (Krishnamurti Foundation India) since the 1980’s. He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and on historical aspects of the subject. He has been involved in the Mathematics Olympiad movement at the national and international level for the past two decades. He is the author of several expository books and articles aimed at interested high school students.     

Science Letters： New solutions of shear waves in piezoelectric cubic crystals

Acoustic wave propagation in piezoelectric crystals of classes?43m and 23 is studied. The crystals Tl3VS4 and Tl3TaSe4 (43m) of the Chalcogenide family and the crystal Bi12TiO20 (23) possess strong piezoelectric effect. Because the surface Bleustein-Gulyaev waves cannot exist in piezoelectric cubic crystals, it was concluded that new solutions for shear-horizontal surface acoustic waves (SH-SAWs) are found in the monocrystals using different electrical boundary conditions such as electri- cally “short” and “open” free-surfaces for the unique [101] direction of wave propagation. For the crystal Tl3TaSe4 with coefficient of electromechanical coupling (CEMC) Ke2=e2/(C×g)～1/3, the phase velocity Vph for the new SH-SAWs can be calculated with the following formula: Vph=(Va Vt)/2, where Vt is the speed of bulk SH-wave, Vt=Vt4(1 Ke2)1/2, Va=aKVt4, aK=2[Ke(1 Ke2)1/2-Ke2]1/2, and Vt4=(C44/ρ)1/2. It was found that the CEMC K2 evaluation for Tl3TaSe4 gave the value of K2=2(Vf–Vm)/Vf～0.047 (～4.7%), where Vf～848 m/s and Vm~828 m/s are the new-SAW velocities for the free and metallized surfaces, respectively. This high value of K2(Tl3TaSe4) is significantly greater than K2(Tl3VS4)～3% and about five times that of K2(Bi12TiO20).     

在n维赋范空间中单位球体积及其渐进性质

Let Bnp={x ∈ Rn‖|x ‖p≤l} be the unit ball ofp norm in the n-dimensional normed space p. The formula for the volume of Bnp was obtained and its asymptotic properties were found out as n→∞and p→∞.     

Scaling behavior of nucleotide cluster in DNA sequences

In this paper we study the scaling behavior of nucleotide cluster in 11 chromosomes of Encephalitozoon cuniculi Genome. The statistical distribution of nucleotide clusters for 11 chromosomes is characterized by the scaling behavior of P（S） ∝ e^-αs＇ where S represents nucleotide cluster size. The cluster-size distribution P（S1＋S2） with the total size of sequential C-G cluster and A-T cluster S1＋S2 were also studied. P（S1＋S2） follows exponential decay. There does not exist the case of large C-G cluster following large A-T cluster or large A-T cluster following large C-G cluster. We also discuss the relatively random walk length function L（n） and the local compositional complexity of nucleotide sequences based on a new model. These investigations may provide some insight into nucleotide cluster of DNA sequence.     

Projectively flat Asanov Finsler metric

In this work, we study the Asanov Finsler metric F=α(β2/α2 gβ/α 1)1/2exp{(G/2)arctan[β/(hα) G/2]}, where α=(αijyiyj)1/2 is a Riemannian metric and β=biyj is a 1-form, g∈(-2,2), h=(1-g2/4)1/2, G=g/h. We give the necessary and sufficient condition for Asanov metric to be locally projectively flat, i.e., α is projectively flat and β is parallel with respect to α. Moreover, we proved that the Douglas tensor of Asanov Finsler metric vanishes if and only if β is parallel with respect to α.     

受外部周期激励的弱阻尼受迫KdV方程的吸引子

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A rigidity theorem for submanifolds in S^n＋p with constant scalar curvature

Let M‘ be a closed submanifold isometrically immersed in a unit sphere S^n p. Denote by R, H and S, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form of M‘, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for M‘ immersed in S^n p with parallel normalized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.     

Semi On-line Scheduling Problem for Maximizing the Minimum Machine Completion Time on m Identical Machines

In this paper, a semi on-line version on m identical machines M1 , M2, …, Mm ( m ≥ 3 ) was considered, where the processing time of the largest job is known in advance. Our goal is to maximize the minimum machine load, an NPLS algorithm was presented and its worst-case ratio was proved to be equal to m - 1 which is the best possible value. It is concluded that if the total processing time of jobs is also known to be greater than (2 m - 1 )Pmax where pmax is the largest job‘s processing time, then the worstcase ratio is 2 - 1/m.     

Viscosity approximation methods with weaMy contractive mappings for nonexpansive mappings

Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K be the unique fixed point of the weak contraction x1→tf（x）＋（1-t）Tx. If T has a fixed point and E admits a weakly sequentially continuous duality mapping from E to E^＊, then it is shown that {xt} converges to a fixed point of T as t→0. The results presented here improve and generalize the corresponding results in （Xu, 2004）.     

具有转向点的非线性系统的奇异摄动

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混合流体完备方程组的局部解析解

With the straification theory we have proved the transversal layer s03,k(D) of complete equations for mixed fluid is not an empty set: s03,k(D)≠φfor all k(k≥1). Based on this conclusion and the "secondary equation" of s03,k(D), this paper fully presents the expressions of coefficients in all local analytic solutions of the equations. Therefore we provide the calculation formulas by which we can get the numerical solutions to any desired accuracy.