BERNOULLI CONVOLUTIONS ASSOCIATED WITH CERTAIN NON——PISOT NUMBERS

The Bernoulli convolution Vλ measure is shown to be absolutely continuous with L^2 density for almost all 1/2＜λ＜1, and singular if λ^-1 is a Pisot number. It is an open question whether the Pisot type Bernoulli conuolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convo-lutions Vλ such that their density functions, if they exist, are not L^2. We also construct other Bernolulli convo-lutions whose density functions if they exist, behave rather badly.     

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)).     

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).     

The distance coloring of graphs

Let G be a connected graph with maximum degree Δ≥ 3.We investigate the upper bound for the chromatic number χγ(G) of the power graph Gγ.It was proved that χγ(G) ≤Δ(Δ-1)γ-1Δ-2+ 1 =:M + 1,where the equality holds if and only if G is a Moore graph.If G is not a Moore graph,and G satisfies one of the following conditions:(1) G is non-regular,(2) the girth g(G) ≤ 2γ- 1,(3)g(G) ≥ 2γ + 2,and the connectivity κ(G) ≥ 3 if γ≥ 3,κ(G) ≥ 4 but g(G) 6 if γ = 2,(4) Δis sufficiently larger than a given number only depending on γ,then χγ(G) ≤ M- 1.By means of the spectral radius λ1(G) of the adjacency matrix of G,it was shown that χ2(G) ≤λ1(G)2+ 1,where the equality holds if and only if G is a star or a Moore graph with diameter 2 and girth 5,and χγ(G)λ1(G)γ+1 ifγ≥3.     

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).     

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.     

Strong 3-commutativity preserving maps on standard operator algebras

Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ :A →A is said to be strong 3-commutativity preserving if [Φ(A), Φ(B)]3 = [A,B]3 for all A,B∈ A, where[A,B]3 is the 3-commutator of A,B defined by[A, B]3 = [[[A, B],B],B] with [A,B] = AB-BA. The main result in this paper is shown that.,if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h : A →F and a scalar λ∈ F with λ~4 = 1 such that Φ(A)=λ A+h(A)I for all A ∈ A.     

Representation type of local and two-vertex bocses

Let κ be an algebraically closed field. It has been proved by Zhang and Xu that if a bocs is of tame representation type, then the degree of the diffierential of the first solid arrow must be less than or equal to 3. We will prove in the present paper that: The bocs is still wild when the degree of the diffierential of the first arrow is equal to 3. Especially, the bocs with only one solid arrow is of tame type if and only if the degree of the diffierential of the arrow is less than or equal to 2. Moreover,...     

自伴算子空间上保持Jordan积投影的线性映射

Let Bs(H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dim H ≥ 2.It is proved that a linear surjective map on Bs (H) preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator U on H such that (X)=λU XU,X∈Bs(H) for some constant λ with λ∈{1,1}.     

The Criticality for Circular Coloring of Graph G_d~k

For two positive integers k and d such that k ≥ 2d, Gkd is the graph with vertex set {0,1, ...,k-1} in which ij is an edge if and only if d ≤ |i-j| ≤ k-d. Clearly, Gk1 is a complete graph of k vertices and we always assume d ≥ 2 in the following. It is easy to see (also [1]) that a graph G is (k, d)-colorable if and only if there exists a homomorphism from G to Gkd.     

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R)≤n (n≥2) if and only if the gl right Proj-dim MR≤n - 2 if and only if Extn-1(N, M) = 0 for all right R-modules N and M if and only if every (n - 2)th Proj-cosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R)≤n (n≥1) if and only if every (n-1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Vroj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R)≤2 are characterized.     

POSITIVE SOLUTIONS FOR PARAMETRIC EQUIDIFFUSIVE p-LAPLACIAN EQUATIONS

We consider a parametric Dirichlet problem driven by the p-Laplacian with a Carath′eodory reaction of equidiffusive type. Our hypotheses incorporate as a special case the equidiffusive p-logistic equation. We show that if λ1 0 is the principal eigenvalue of the Dirichlet negative p-Laplacian and λ λ1(λ being the parameter), the problem has a unique positive solution, while for λ∈(0,λ1], the problem has no positive solution.     

两个素数和一个素数平方和的丢番图逼近(英文)

We show that if λ1 , λ2 , λ3 are non-zero real numbers, not all of the same sign, η is real and λ1 /λ2 is irrational, then there are infinitely many ordered triples of primes (p1 , p2 , p3 ) for which |λ1 p1 + λ2 p2 + λ3 p2 3 + η| < (max pj )- 1/40 (log max pj ) 4 .     

On a problem of Sierpiński,Ⅱ

For any integer s≥ 2, let μsbe the least integer so that every integer l μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpi′nski asked for the determination of μs. Let pibe the i-th prime and let μs= p2 + p3 + + ps+1+ cs. Recently, the authors solved this problem. In particular,we have(1) cs=-2 if and only if s = 2;(2) the set of integers s with cs= 1100 has asymptotic density one;(3) cs∈ A for all s ≥ 3, where A is an explicit set with A ■[2, 1100] and |A| = 125. In this paper, we prove that,(1) for every a ∈ A, there exists an index s with cs= a;(2) under Dickson's conjecture, for every a ∈ A,there are infinitely many s with cs= a. We also point out that recent progress on small gaps between primes can be applied to this problem.     

Dense 3-uniform hypergraphs containing a large clique

An r-uniform graph C is dense if and only if every proper subgraph G' of G satisfies λ(G') λ(G).,where λ(G) is the Lagrangian of a hypergraph G. In 1980's, Sidorenko showed that π(F), the Turán density of an γ-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all dense F-hom-free γ-uniform hypergraphs. This connection has been applied in the estimating Turán density of hypergraphs. When γ=2 the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However,when r ≥ 3, it becomes much harder to estimate the Lagrangians of γ-uniform hypergraphs and to characterize the structure of all dense γ-uniform graphs. The main goal of this note is to give some sufficient conditions for3-uniform graphs with given substructures to be dense. For example, if G is a 3-graph with vertex set [t] and m edges containing [t-1]~(3),then G is dense if and only if m≥{t-2 3)+(t-2 2)+1. We also give a sufficient condition on the number of edges for a 3-uniform hypergraph containing a large clique minus 1 or 2 edges to be dense.     

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}.     

幺半群上的斜强可逆环

For a monoid M, we introduce the concept of skew strongly M-reversible rings which is a generalization of strongly M-reversible rings, and investigate their properties. It is shown that if G is a finitely generated Abelian group, then G is torsion-free if and only if there exists a ring R with |R| ≥ 2 such that R is skew strongly G-reversible. Moreover, we prove that if R is a right Ore ring with classical right quotient ring Q, then R is skew strongly M-reversible if and only if Q is skew strongly M-reversible.     

Optimization problems of the third edge-connectivity of graphs

The third edge-connectivity λ3(G) of a graph G is defined as the minimum cardinality over all sets of edges, if any, whose deletion disconnects G and each component of the resulting graph has at least 3 vertices. An upper bound has been established for λ3(G) whenever λ3(G) is well-defined. This paper first introduces two combinatorial optimization concepts, that is, maximality and superiority, of λ3(G), and then proves the Ore type sufficient conditions for G to be maximally and super third edge-connected. These concepts and results are useful in network reliability analysis.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.