A Postscript on Erdoes－Szekeres Theorem

Let g(n) denote the least integer such that among any g(n) points in general position in the plane there are always n points in convex position.In this paper we show that g(n)≤(n-2 ↑2n-5) 2 by a new method.     

Subcategories of fixed points of mutations in root categories with type _(n-1)

For any n 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n-1) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in R(n). As an application, it is shown that the affine Kac-Moody algebra of type n-2 is isomorphic to a Lie subalgebra of the Kac-Moody algebra of type n-1.     

BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY

In this paper,we consider the following nonlinear elliptic problem:△~2u=|u|~(8/(n-4))u+μ|u|~(q-1)u,in Ω,△u = u = 0 on δΩ,where Ω is a bounded and smooth domain in R~n,n ∈ {5,6,7},μ is a parameter and q ∈]4/(n- 4),(12- n)/(n- 4)[.We study the solutions which concentrate around two points of Ω.We prove that the concentration speeds are the same order and the distances of the concentration points from each other and from the boundary are bounded.For Ω =(Ω_α)α a smooth ringshaped open set,we establish the existence of positive solutions which concentrate at two points of Ω.Finally,we show that for μ 0,large enough,the problem has at least many positive solutions as the LjusternikSchnirelman category of Ω.     

Generalized Steiner Triple Systems with Group Size g ≡0, 3 (mod 6)

Generalized Steirier triple systems, GS(2,3,n,g), are equivalent to maximum constant weight codes over an alphabet of size g 1 with distance 3 and weight 3 in which each codeword has length n. The necessary conditions for the existence of a GS(2,3,n,g) are (n-1)g≡0 (mod 2), n(n-1)g2≡0 (mod 6), and n≥g 2. These necessary conditions are shown to be sufficient by several authors for 2≤g≤11. In this paper, three new results are obtained. First, it is shown that for any given g, g≡0 (mod 6) and g≥12, if there exists a GS(2.3.n.g) for all n, g 2≤n≤7g 13. then the necessary conditions are also sufficient. Next, it is also shown that for any given g, g≡3 (mod 6) and g≥15, if there exists a GS(2,3,n,g) for all n, n≡1 (mod 2) and g 2≤n≤7g 6, then the necessary conditions are also sufficient. Finally, as an application, it is proved that the necessary conditions for the existence of a GS(2,3,n,g) are also sufficient for g=12,15.     

The eventually distance minimizing rays in moduli spaces

The eventually distance minimizing ray(EDM ray) in moduli spaces of the Riemann surfaces of analytic finite type with 3 g + n-3 0 is studied, which was introduced by Farb and Masur [5]. The asymptotic distance of EDM rays in a moduli space and the distance of end points of EDM rays in the boundary of the moduli space in the augmented moduli space are discussed in this article. A relation between the asymptotic distance of EDM rays and the distance of their end points is established. It is proved also that the distance of end points of two EDM rays is equal to that of end points of two Strebel rays in the Teichmu¨ller space of a covering Riemann surface which are leftings of some representatives of the EDM rays. Meanwhile, simpler proofs for some known results are given.     

On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere

Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constant γ(n) depending only on n such that if |H| ≤γ(n) and β(n,H)≤S ≤β(n,H)+n/18,then S≡β(n,H) and M is a Clifford torus.Here,β(n,H)=n+n~3/2(n-1)H~2+n(n-2)/2(n-1)(1/2)n~2H~4+4(n-1)H~2.     

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = (σ+m)n/(n-σ-2) is its critical exponent provided max{-1, [(1-m)n-2]/(n+1)} σ n-2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Furthermore, we demonstrate that if max{1, σ + m} p ≤ pc, then every positive solution of the equations blows up in finite time; whereas for ppc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n ≤σ+2.     

ON GROWTH OF MEROMORPHIC SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS AND TWO CONJECTURES OF C.C. YANG

In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equationsf(z)n+ P_(n-1)(f) = 0,where n ≥ 2 and P_(n-1)(f) is a difference polynomial of degree at most n- 1 in f with small functions as coefficients. Moreover, we give two examples to show that one conjecture proposed by Yang and Laine [2] does not hold in general if the hyper-order of f(z) is no less than 1.     

The spectrum of compact hypersurface in sphere

Let M be a compact minimal hypersurface of sphere Sn 1(1). Let (M) be H (r)-torus of sphere Sn 1 (1).Assume they have the same constant mean curvature H, the result in [1] is that ifSpec0(M, g) =Spec0((M), g),then for 3≤ n ≤ 6, r2≤n-1/n or n ≥ 6, r2 ≥ n-1, then M is isometric to (M). We improved the result and prove that: if Spec0(M,g) =Spec0((M),g), then M is isometric to (M). Generally, if Specp(M,g) =Specp((M),g), here p is fixed and satisfies that n(n - 1) ≠ 6p(n - p), then M is isometric to (M).     

Restricted Fault Diameter of Hypercube Networks

This paper studies restricted fault diameter of the n-dimensional hypercube networks Qn (n ≥ 2).It is shown that for arbitrary two vertices x and y with the distance d in Qn and any set F with at most 2n-3 vertices in Qn - {x, y}, if F contains neither of neighbor-sets of x and y in Qn, then the distance between x andy in Qn - F is given by D(Qn-F;x,y){=1 , for=1;≤d 4 , for 2≤d≤n-2,n≥4;≤n 1, for d=n-1,n≥3; =n, for d=n. Furthermore, the upper bounds are tight. As an immediately consequence, Qn can tolerate up to 2n-3 vertices failures and remain diameter 4 if n = 3 and n 2 if n ≥ 4 provided that for each vertex x in Qn, all the neighbors of x do not fail at the same time. This improves Esfahanian‘s result.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].