The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.     

混合幂丢番图不等式（英文）

It is proved that if λ_1, λ_2, ···, λ_7are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 σ 1/16, the inequality |λ_1x~2_1+ λ_2x~2_2+∑7 i=3λ_ix~4_i+ η| ( max1≤i≤7|x_i|)~(-σ)has infinitely many solutions in positive integers x_1, x_2, ···, x_7. Similar result is proved for |λ_1x~2_1+ λ_2x~2_2+ λ_3x~2_3+ λ_4x~4_4+ λ_5x~4_5+ λ_6x~4_6+ η| ( max1≤i≤6|xi|)-σ.These results constitute an improvement upon those of Shi and Li.     

L（1, 2）-edge-labelings for lattices

For a graph G and two positive integers j and k, an m-L(j, k)-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by kThe λ j,k-number of G is the minimum m such that an m-L(j, k)-edge-labeling is admitted by GIn this article, the L(1, 2)-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λ j,k-numbers of these graphs are obtained.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p~su=λa(x)|u|+~(p-2)u+λb(x)|u|~(α-2)|u|~βu+μ(x)/αδ|u|~(γ-2)|v|~δu in Ω,(-△)_p~su=λc(x)|v|+~(q-2)v+λb(x)|u|~α|u|~(β-2)v+μ(x)/βγ|u|~γ|v|~(δ-2)v in Ω,u=v=0 on R~N\Ω where λ 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 p ≤ q ∞, sq N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1~-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.     

Indecomposable large sets of Steiner triple systems with indices 5, 6

A family ( X, B1 ), (X, B2 ), . . . , (X, Bq ) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTS λ (v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTS λ (v) if there does not exist an LSTS λ'(v) contained in the collection for any λ' λ. In this paper, we show that for λ = 5, 6, there is an IDLSTS λ (v) for v ≡ 1 or 3 (mod 6) with the exception IDLSTS6 (7).     

“MANDELBROT SET” FOR A PAIROF LINEAR MAPS： THE LOCAL GEOMETRY

We consider the iterated function system {λz-1, λz 1} in the complex plane, for λ in the open unit disk. Let M be the set of λ such that the attractor of the IFS is connected. We discuss some topological and geometric properties of the set M and prove a new result about possible corners on its boundary.Some open problems and directions for further research are discussed as well.     

λ-fold indecomposable large sets of Steiner triple systems

A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTSλ(v) if there exists no LSTSλ (v) contained in the collection for any λ λ. In 1995, Griggs and Rosa posed a problem: For which values of λ 1 and orders v ≡ 1, 3 (mod 6) do there exist IDLSTSλ(v)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(v) (HLSTSλ(v)) as auxiliary designs to establish a recursive construction for IDLSTSλ(v) and show that there exists an IDLSTSλ(v) for λ = 2, 3, 4 and v ≡ 1, 3 (mod 6).     

A new pinching theorem for complete self-shrinkers and its generalization

In this paper,we firstly verify that if M~n is an n-dimensional complete self-shrinker with polynomial volume growth in R~(n+1),and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in R~(n+1) with λ≠0.Then we prove that there exists a positive constant γ,such that if |λ|≤γ and the squared norm of the second fundamental form of M satisfies0≤S-β_λ≤1/18,then S≡β_λ,λ 0 and M is a cylinder.Here β_λ=1/2(2+λ~2+|λ|(λ~2+4)~(1/2)).     

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|~(N-2) u + f(x, u), x ∈ ?,u ∈ W_0~(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R~N(N 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ λ_1, λ = λ_?(? = 2, 3, ···), and λ_? is the eigenvalues of the operator(-?_N, W_0~(1,N)(?)),which is defined by the Z_2-cohomological index.     

The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition.In the previous paper,we show that the Chern-Simons Higgs equation with parameter λ0 has at least two solutions(uλ1,uλ2) for λ sufficiently large,which satisfy that uλ1→u0 almost everywhere as λ→∞,and that uλ2→∞ almost everywhere as λ→∞,where u 0 is a(negative) Green function on M.In this paper,we study the asymptotic behavior of the solutions as λ→∞,and prove that uλ2-uλ2 converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary M is negative,or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.     

On estimating the hidden periodicities in linear time series models

In this paper the tollowing modelX(n)=sum from j=1 to p α_je~(inλj)+ξ_nis considered,where p,λ_1,λ_2,…,λ_p,are constants α=(α_1,α_2,…,α_p) is a random vector and {ξ_n;n=0,±1,±2,…} is a wide-sense stationary sequence with zero means.In [4],theorem about thestrong consistent estimates of λ_1,λ_2.…,λ_p and α are proved under the assumption that α is a constantvector and p and δ are known constants such that0<δ<(?){λ_i-λ_j}.The main purpose of the present paper is to prove theorems on the strong consistent estimates ofparameters p,λ_1,λ_2,…,λ_p and random vector α without knowing p and δ.Numerical examples arealso given to illustrate our method of estimation.     

On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.     

一类L-Fuzzy拓扑空间——1.L-Fuzzy点的邻近构造

Suppose (L₀,≤) is a poset (i.e. partially ordered set) with an order reversing involution λ→λ′ and has an all element 1. Let L={b|b is a subset of L₀ and satisfies the condition: if λ∈b,μ≤λ then μ∈b} then (L,?,∩,∪) is a complete and distributive lattice. L is called the lattice induced from poset L₀. There is a mapping α→a′={λ∈L₀|λ′(?)a} of a lattice L onto L. The mapping is clearly order reversing involution.     

LOWER BOUNDS FOR TNE FIRST GAP OF EIGENVALUES IN THE SCHRDINGER OPERATOR ON SIMPLE RIEMANNIAN MANIFOLDS

We give a lower bound for the first gap λ_2—λ_1 of the twolowerst eigenvalues of the Schr(o|¨)dinger operator-△+W(p) with the Dirichletboundary condition and a strictly convex potential W(p)on M in which M is acompact simple Riemannian manifold with smooth strictly convex boundary (?)MHere a compact Riemannian manifold M is said to be simple if M～(?)M istopologically R~2.We prove thatλ_2-λ_1≥(π~2)/(d~2)+min{0,-(n-1)K}where d is the diameter of M and-(n-1)K,(K≥0)the lower bound of theRicci curvature of M.This work generalizes the results in the classical Eucli-dean situation due to Singer,Wong and Yau,Yu and Zhong to a kind of curvedRiemannian manifold.     

Quasisymmetric geometry of the Julia sets of McMullen maps Dedicated to the Memory of Professor Lei Tan

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z~m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet(which is also standard in some sense).If the free critical points are not escaped, we give a sufficient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.     

"MANDELBROT SET" FOR A PAIR OF LINEAR MAPS:THE LOCAL GEOMETRY

We consider the iterated function system {λz-1, λz + 1} in the complex plane, for A in the open unit disk. Let M be the set of λ such that the attractor of the IFS is connected. We discuss some topological and geometric properties of the set M and prove a new result about possible corners on its boundary. Some open problems and directions for further research are discussed as well.     

单纯正交表OA$_\lambda(3, 5, v)$的存在性

An orthogonal array of strength t,degree k,order v and index λ,denoted by OAλ(t,k,v),is a λvt× k array on a v symbol set such that each λvt× t subarray contains each t-tuple exactly λ times.An OAλ(t,k,v) is called simple and denoted by SOAλ(t,k,v)if it contains no repeated rows.In this paper,it is proved that the necessary conditions for the existence of an SOAλ(3,5,v) with λ≥ 2 are also sufficient with possible exceptions where v = 6 and λ∈ {3,7,11,13,15,17,19,21,23,25,29,33}.     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.     

Existence of Simple OA_λ(3,5,v)′s

An orthogonal array of strength t,degree k,order v and index λ,denoted by OAλ(t,k,v),is a λvt× k array on a v symbol set such that each λvt× t subarray contains each t-tuple exactly λ times.An OAλ(t,k,v) is called simple and denoted by SOAλ(t,k,v)if it contains no repeated rows.In this paper,it is proved that the necessary conditions for the existence of an SOAλ(3,5,v) with λ≥ 2 are also sufficient with possible exceptions where v = 6 and λ∈ {3,7,11,13,15,17,19,21,23,25,29,33}.     

一类不确定薛定谔基尔霍夫方程解的存在性和集中性（英文）

This paper is concerned with the nonlinear Schrodinger-Kirchhoff system -(a+b∫_(R~3)|▽u|~2 dx)△u+λV(x)u=f(x,u) in R~3,where constants a 0,b≥ 0 and λ 0 is a parameter.We require that V(x) ∈C(R~3)and has a potential well V~(-1)(0).Combining this with other suitable assumptions on K and f,the existence of nontrivial solutions is obtained via variational methods.Furthermore,the concentration behavior of the nontrivial solution is also explored on the set V~(-1)(0) as λ→+∞ as well.It is worth noting that the(PS)-condition can not be directly got as done in the literature,which makes the problem more complicated.To overcome this difficulty,we adopt different method.