Almost-all results on the <Emphasis Type=”Italic”>p</Emphasis><Superscript>&;#955;</Superscript> problem

Summary  We describe a procedure to construct a 4-coloured graph representing a closed, connected 3-manifold Italic>M starting from a Heegaard diagram of Italic>M. As a consequence, we prove that, to each Heegaard diagram of a (closed) 3-manifold Italic>M, canonically corresponds a spine (Italic>Heegaard spine) of Italic>M.     

“Minimal Surfaces in ℍ<Superscript>2</Superscript> × ℝ”

In ² × ℝ one has catenoids, helicoids and Scherk-type surfaces. A Jenkins-Serrin type theorem holds here. Moreover there exist complete minimal graphs in ² with arbitrary continuous asymptotic values. Finally, a graph on a domain of ² cannot have an isolated singularity. Received: 20 June 2002     

“断块”与 “板块”

从“断块”学说的观点看,“板块”是断块的一种特殊类型,它是被巨大的深切整个岩石圈的断裂带所围限的断块,是最大一级的断块,称做“岩石圈断块”. 板块学说主要是在对海洋的研究与地球物理研究的基础上发展起来的,它提出了一些有价值的新论点.但大多数研究板块的人,对大陆地区的地质情况还缺乏深入研究,不能充分利用本世纪以来在大陆地质构造研究方面所获得的科学结论与思想方法,这就使板块学说的进一步发展受到了阻碍。 本文先对断块学说的历史发展与现阶段理论作一简单介绍,然后应用断块构造的力学分析与历史分析方法,对板块构造研究中出现的一些主要问题谈一些看法.作者认为板块边界的力学机制主要是受基底锯齿状断裂的活动方式控制的,板块内部的应力分布也主要与板块内部次一级强度不同的断块的复杂活动有关。关于板块运动的驱动力,不但应考虑地球内部的热运动,而且应考虑到重力作用和地球作为一个旋转天体所具有的各种动力学行为.     

Addendum to “Complete rotation hypersurfaces with <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript> constant in space forms”*

Here we study complete rotation hypersurfaces with constant k-th mean curvature H_k in even and 2 < k < n. We prove the existence of a constant such that there are no such hypersurfaces for . We have only one compact hypersurface of this kind with . For each there is a corresponding family of complete immersed rotation hypersurfaces, each family containing two isoparametric hypersurfaces. For H_k ≥ 0, there is also such a family, now containing only one isoparametric hypersurface. Finally, we prove the existence of compact hypersurfaces with arbitrarily large H_k , neither isometric to a sphere nor to a product of spheres. *Bull. Braz. Math. Soc. 30 (2), 1999, 139–161. **Partially supported by FUNCAP, Brazil. ***Partially supported by CNPq, Brazil and DGAPA-UNAM, México.     

Hypercyclic and chaotic convolution operators associated with the Dunkl operator on <Emphasis Type=”Bold”>C</Emphasis>

Summary We show that there exist infinitely many positive integers m not of the form n-&ohgr;(n) for any positive integer n. Here, &ohgr;(n) stands for the number of distinct prime factors of n. A similar result holds with &ohgr;(n) replaced by the total number of prime factors of n (counting multiplicities), or by the number of divisors of n.     

Quincunx multiresolution analysis for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℚ<Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.     

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.     

传承&#183;探索&#183;创新

金风送爽,丹桂飘香。在这收获的季节里,中国珠算心算协会和世界珠算心算联合会相继在天津召开会员大会,组成新一届理事会和新的领导机构,这标志着我国和世界珠算心算事业将开启崭新的一页!     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

On the Crossing Number of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> □ <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(K_m □ P_n), the crossing number of the Cartesian product K_m □ P_n. We prove that for m ≥ 3,n ≥ 1 and cr(K_m □ P_n)≥ (n − 1)cr(K_m+2 − e) + 2cr(K_m+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for m = 6, i.e., cr(K₆ □ P_n) = 15n + 3. Research supported by NFSC (60373096, 60573022).     

An extremal problem in the Hardy space <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for 0&lt;<Emphasis Type="Italic">p</Emphasis>&lt;∞

We prove that if the function determining a linear functional over the Hardy space is analytic on the disk of radius greater than 1 then the extremal function of this functional is analytic on the same disk.     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

The number of S-precomplete classes in the functional system <Emphasis Type="Italic">P</Emphasis><Stack><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript><Superscript><Emphasis Type="Italic">τ</Emphasis></Superscript></Stack>

The problem on the number of precomplete classes in the functional system P _κ ^τ is considered, elements of P _κ ^τ are deterministic S-functions defined on words of length τ composed from letters of an alphabet of cardinality κ. An asymptotics for the number of S-precomplete classes in P _κ ^τ is obtained for arbitrary fixed κ and τ tending to infinity.     

Equitable total coloring of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> ⋁ <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ _t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ _et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).     

On Dirichlet Type Spaces <Emphasis Type="Italic">A</Emphasis><Stack><Subscript><Emphasis Type="Italic">ω</Emphasis></Subscript><Superscript>2</Superscript></Stack> Over the Half–Plane

Some extensions of the results of the first author related with the Hilbert spaces A _ω,0 ² of functions holomorphic in the half–plane are proved. Some new Hilbert spaces A _ω ² of Dirichlet type are introduced, which are included in the Hardy space H2 over the half–plane. Several results on representations, boundary properties, isometry, interpolation, biorthogonal systems and bases are obtained for the spaces A _ω ² ? H₂.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

The critical group of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> × <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

In this paper, the structure of the critical group of the graph K _m × C _n is determined, where m, n ≥ 3.     

A Ring Theoretic Construction of Hadamard Difference Sets in ℤ<Subscript>8</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let be the Galois ring of characteristic 2³ and rank n and let . We give an explicit construction of Hadamard difference sets in .}Research supported by NSA grant MDA 904-02-1-0080.     

Seas of squares with sizes from a Π<Stack><Subscript>1</Subscript><Superscript>0</Superscript></Stack> set

A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+o(1))( _n/2 ⁿ ). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number α(n) of such families, and they proved that \({2^{\left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)}} \leqslant \alpha \left( n \right) \leqslant {2^{2\sqrt 2 \left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)\left( {1 + o\left( 1 \right)} \right)}}\) They conjectured that the constant \(2\sqrt 2 \) can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.