非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

Selection from poisson processes

Summary There are givenk Poisson processes with mean arrival times 1/λ₁,...1/λ _k . Let λ_[1]≦λ_[2]≦...≦λ_[k] denote the ordered set of values λ₁...,λ_[k]. We consider three procedures for selecting the process corresponding to λ_[k]. The processes are observed until there areN arrivals from any of the given processes, when the processes are observed continuously, or until there are at leastN arrivals, when the processes are observed at successive intervals of time whereN is a pre-determined positive integer. In the continuous case, the process for which theNth arrival time is shortest, is selected. In the discrete case, the selection involves certain randomization. Given (λ_[k]/λ_[k-1])≧0>1, it is shown that the probability of a correct selection (Pcs) is minimized whenθλ_[1]=θλ_[2]=...=θλ_[k-1]=θλ_[k]=θλ, say. The Pcs for this configuration is independent of λ for two of the given procedures, and monotone increasing in λ for the third. The value ofN is determined by a lower bound placed on the value of the Pcs. The problem of selecting from given Poisson processes for the discrete case is related to the problem of selecting from given Poisson populations. An application of the given procedures to a problem of selecting the “most probable event” from a multinomial population, is considered.     

The total reducibility order of a polynomial in two variables

LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ _i=1 ^n(λ) A _iλ ^k _μ whereA _iλ are distinct primes. Let ϱ_λ(A) =n(λ) − 1 and let ρ(A) = Σ_λɛσ(A)ϱ_λ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.     

Riemann surfaces in fibered polynomial hulls

Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and letA(Δ) be the algebra of complex functions continuous on Δ and analytic in int Δ. LetK be a compact set in C² such that Π(K)=Γ, and letK _λ≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,K _λ is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))²-S(λ) quadratic in w withR,S∈A(Δ) such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ intK _λ, whereS has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component ofK _λ. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull ofK. Furthermore, we show that БК/K is the disjoint union of such disks.     

Normal Subgroups of Hecke Groups <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">λ</Emphasis>)

Let λ ≥ 2 and let H(λ) be the Hecke group associated to λ. Also let H(λ)\U be the Riemann surface associated to the Hecke group H(λ). In this article, we study the even subgroup H _e (λ) and the power subgroups H ^m (λ) of the Hecke groups H(λ). We also study some genus 0 normal subgroups of finite index of H(λ). Finally, we discuss free normal subgroups of H(λ).     

Two cardinal versions of diamond

Roughly speaking, ◇_K,λ asserts the existence of a sequence of size <κ sets that captures every subset ofλ on a stationary set. The paper is devoted to the study of ◇_K,λ and related principles, which are for instance obtained by considering sequences of larger sets, or by requesting the simultaneous capture of many subsets ofλ. Our main result is that ◇_K,λ holds in caseλ>2^<K .     

Large sets of pure directed triple systems with index λ

A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.     

Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u _λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu _λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu _λ as λ→∞. This research has been supported by the Japan Society for the Promotion of Science.     

On Closed Weingarten Surfaces

We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ²ⁿ⁺¹. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ³.     

Iterated grafting and holonomy lifts of Teichmüller space

Let X be a closed hyperbolic surface and λ, η be weighted geodesic multicurves which are short on X. We show that the iterated grafting along λ and η is close in the Teichmüller metric to grafting along a single multicurve which can be given explicitly in terms of λ and η. Using this result, we study the holonomy lifts gr _λ ρ _X,λ of Teichmüller geodesics ρ _X,λ for integral laminations λ and show that all of them have bounded Teichmüller distance to the geodesic ρ _X,λ. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences gr _nλ X and show that they converge geometrically to a punctured surface.     

Splitting kernels into small summands

Let λ be a regular cardinal. An epimorphism between abelian groups is λ -pure if it is projective with respect to abelian groups of size less than λ. We show that cotorsion groups A have λ-pure projective dimension greater than 1 for all uncountable λ ≤ |A/tA|, where tA denotes the torsion subgroup of A. For λ > |A/tA|, cotorsion groups A are λ-pure projective.     

The Hausdorff Centred measure of the symmetry Cantor sets

Let 0＜λ≤1/3,K (λ) be the attractor of an iterated function system { ψ1,ψ2 } on the line, where ψ1 (x ) =λx, ψ2(x)=1-λ+λx, x∈ [0,1]. We call K (λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K (λ).     

On Closed Weingarten Surfaces

We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ²ⁿ⁺¹. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ³.     

Almost-all results on the p λ problem. II

Summary Suppose that 1/2 ≦ λ < 1. Balog and Harman proved that for any real θ there exist infinitely many primes p satisfying p^λ-θ < p^{-(1-λ)/2+ ε} (with an asymptotic result). In the present paper we establish that for almost all θ in the interval 0 ≦ θ < 1 there exist infinitely many primes p such that {p^λ-θ} < p^{-min{(2-λ)/6,(14-9λ)/32}+ε}. Thus we obtain a better result for almost all θ than for a single θ if λ>1/2.     

On the embeddings of mendelsohn triple systems

It is proved in this paper that there exists an incomplete Mendelsohn triple system IMTS(u,v; λ) if and only ifλ(u-v)(u-2v-1)≡0(mod 3),u≥2v+1 and (u, v, λ) ≠ (6, 1, 1). As a consequence, it is proved that for any given λ≥1, a Mendelsohn triple system MTS (v, λ) can be embedded in an MTS (u, λ) if and only ifλu(u-1)≡0(mod 3) andu≥2v+1. Project supported by the National Natural Science Foundation of China.     

“Gap 1” two-cardinal principles and the omitting types theorem for ℒ(Q)

Forλ a strong limit singular cardinal, and more generally forλ > 2^cofλ , we prove the equivalence of a number of model theoretic and combinatorial conditions, including the ℒ(Q)-completeness theorem for theλ ⁺-interpretation, an omitting types theorem for ℒ(Q) in theλ ⁺-interpretation, and a weak form of Jensen’s principle □_λ. Research supported in part by the United States-Israel Binational Science Foundation. The author thanks the referee for a thorough revision of the paper. Publication #269.     

Connectedness of certain random graphs

L. Dubins conjectured in 1984 that the graph on vertices {1, 2, 3, ...} where an edge is drawn between verticesi andj with probabilityp _ij=λ/max(i, j) independently for each pairi andj is a.s. connected forλ=1. S. Kalikow and B. Weiss proved that the graph is a.s. connected for anyλ>1. We prove Dubin’s conjecture and show that the graph is a.s. connected for anyλ>1/4. We give a proof based on a recent combinatorial result that forλ≦1/4 the graph is a.s. disconnected. This was already proved forλ<1/4 by Kalikow and Weiss. Thusλ=1/4 is the critical value for connectedness, which is surprising since it was believed that the critical value is atλ=1.     

The Existence of (ν,6, λ)-Perfect Mendelsohn Designs with λ > 1

The basic necessary conditions for the existence of a (v, k, λ)-perfect Mendelsohn design (briefly (v, k, λ)-PMD) are v ≥ k and λ v(v − 1) ≡ 0 (mod k). These conditions are known to be sufficient in most cases, but certainly not in all. For k = 3, 4, 5, 7, very extensive investigations of (v, k, λ)-PMDs have resulted in some fairly conclusive results. However, for k = 6 the results have been far from conclusive, especially for the case of λ = 1, which was given some attention in papers by Miao and Zhu [34], and subsequently by Abel et al. [1]. Here we investigate the situation for k = 6 and λ > 1. We find that the necessary conditions, namely v ≥ 6 and λ v(v − 1)≡0 (mod 6) are sufficient except for the known impossible cases v = 6 and either λ = 2 or λ odd. Researcher F.E. Bennett supported by NSERC Grant OGP 0005320.     

Remarks on enlargements of generalized topologies

The concept of enlargement of a generalized topology was introduced by á. Császár [4]. He also introduced the concepts of (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity on enlargements. In this paper, we characterize the (κ,λ)-continuity and introduce the concept of strong (κ,λ)-continuity on enlargements. In particular, we study characterizations for the strong (κ,λ)-continuity and the relationships among (μ,ν)-continuity, (κ,λ)-continuity, strong (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity.     

Equality of multiplicity free skew characters

In this paper we show that two skew diagrams λ/μ and α/β can represent the same multiplicity free skew character [λ/μ]=[α/β] only in the the trivial cases when λ/μ and α/β are the same up to translation or rotation or if λ=α is a staircase partition λ=(l,l−1,…,2,1) and λ/μ and α/β are conjugate of each other.