Contact structures on M \times S^2

We show that if a manifold $M$ admits a contact structure, then so does $M \times S^2$ . Our proof relies on surgery theory, a theorem of Eliashberg on contact surgery and a theorem of Bourgeois showing that if $M$ admits a contact structure then so does $M \times T^2$ .     

Metrics with nonnegative curvature on {S^2 \times \mathbb{R}^4}

We study nonnegatively curved metrics on S²×\mathbbR⁴{S^2\times\mathbb{R}^4}. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking’s almost-positively curved metric on S ² × S ³ extends to a nonnegatively curved metric on S²×\mathbbR⁴{S^2\times\mathbb{R}^4} (so that Wilking’s space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.     

Half-flat structures on indecomposable Lie groups

This article can be viewed as a continuation of the articles [SH] and [FS] in which the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradicals which admit a half-flat SU(3)-structure. As an important step of the proof, a considerable refinement of the classification of six-dimensional Lie algebras with five-dimensional non-Abelian nilradicals is established. Additionally, it is proved that all non-solvable six-dimensional Lie algebras admit half-flat SU(3)-structures.     

Half-flat structures on decomposable lie groups

Half-at SU(3)-structures are the natural initial values for Hitchin’s evolution equations whose solutions define parallel G₂-structures. Together with the results of [SH], the results of this article completely solve the existence problem of left-invariant half-at SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras, which refines and clarifies the existing classification of five-dimensional Lie algebras.     

On the Homflypt skein module of $S^1 \times S^2$

Let k be a subring of the field of rational functions in x, v,s which contains . If M is an oriented 3-manifold, let denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) ; (2) L with a positiv e twist ; (3) where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of given by J. Hoste and M. Kidwell and also V. Turaev; the second basis is related to a Young idempotent basis for based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements , for n a nonzero integer, and the elements , for any integer m, are invertible ink, then -torsion module . Here the free part is generated by the empty link . In addition, if the elements , for m an integer, are invertible in k, then has no torsion. We also obtain some results for more general k. Received January 7, 2000; in final form September 20, 2000 / Published online April 12, 2001     

Lagrangian isotopy of tori in $${S^2\times S^2}$$ and $${{\mathbb{C}}P^2}$$CP2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space \({{\mathbb{R}}^4}\), the projective plane \({{\mathbb{C}}P^2}\), and the monotone \({S^2 \times S^2}\). The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for \({T^*{\mathbb{T}}^2}\), i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.     

图$P_m\times P_n, P_m\times C_n$ 和C_m\times C_n$}的邻点可区别全色数的注记

设 $G$ 是一个简单图. 设$f$是从$V(G) \cup E(G)$到 $\{1, 2,\ldots, k\}$的一个映射.对任意的 $v\in V(G)$, 设$C_f(v)=\{f(v)\}\cup \{f (vw)|w\in V(G),vw\in E(G)\}$ . 如果 $f$ 是一个 $k$-正常全染色, 且对 $u, v\in V(G),uv\in E(G)$, 有 $C_f(u)\neq C_f(v)$, 那么称 $f$ 为$k$-邻点可区别全染色 (简记为$k$-$AVDTC$). 设     

Morse Index of Saddle Equilibria of Gradient-Like Flows on Connected Sums of $$mathbb S^{n-1}times mathbb S^1$$

Mathematical Notes -     

L-Factor of Irreducible x1×x2(×)σ

The L-factor of irreducible x1×x2(×)σ defined by Piatetski-Shapiro is computed by using non-split Bessel functional.     

$K_v$的最大$K_2K_3$填充设计

Abstract Let Kv be the complete graph on v vertices, and G a finite simple undirected graph without isolated vertices. A G-packing of Kv, denoted by （v, G, 1）-packing, is a pair （X,A） where X is the vertex set of K＋ and ＋4 is a family of edge-disjoint subgraphs isomorphic to G in Kv. In this paper, the maximum number of subgraphs in a （v, G, 1）-packing is determined when G is K2 x K3, the Cartesian product of K2 and K3, leaving two orders undetermined. This design originated from the use of DNA library screening.     

The rigidity of Clifford torusS^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS ⁿ⁺¹ (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n ²(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.     

Magische 4\times 4-Quadrate

Zusammenfassung. Diese Note behandelt zwei Typen von magischen -Quadraten und beschreibt mittels der Symmetriegruppen eines 4- bzw. 5-dimensionalen Würfels die Gesamtheit aller solchen Quadrate. Eingegangen 15.1.1997, angenommen 28.2.1997     

关于$3\times 3$阶三角矩阵环上模的研究

文章对$3\times 3$阶三角矩阵环$$\Gamma = \left(\begin{array}{ccc}T & 0 & 0 \\M & U & 0\\{N \otimes _U M} & N & V \\\end{array}\right)$$上的模作了研究,其中T,U,V均是环, M,N分别是U-T, V-U双模.通过用一个五元组$(A,B,C;f,g)$来描述一个左$\Gamma$-模 (其中$A \in \mod T, B\in {\rm mod} U, C \in {\rm mod} V$, $f:M \otimes _T A \to B \in {\rm mod} U, g:N \otimes _U B \to C \in {\rm mod} V$), 文章分别刻画了$\Gamma$上的一致模、空的模、有限嵌入模,并且确定了${ }_\Gamma (A \oplus B \oplus C)$的根和基座．     

Deep Reduction of Darboux Transformation to a $3\times3$ Spectral Problem

The Darboux transformation of a 3 × 3 spectral problem which is associated with the higherorder nonlinear SchrSdinger equation is given. Some solutions of the higher-order nonlinear Schroedinger equation are provided by taking different “seeds“.     

Minimal graphs in {M \times \mathbb{R}}

We study minimal graphs in . First, we establish some relations between the geometry of the domain and the existence of certain minimal graphs. We then discuss the problem of finding the maximal number of disjoint domains Ω ⊂ M that admit a minimal graph that vanishes on ∂Ω. When M is two-dimensional and has non-negative sectional curvature, we prove that this number is 3. This was proved by Tkachev in . Maria Fernanda Elbert was partially supported by CNPq and Faperj.     

Global convergence of an isentropic Euler-Poisson system in $\R^+ \times \R^d$

We prove the global-in-time convergence of an Euler-Poisson system near a constant equilibrium state in the whole space $\R^d$, as physical parameters tend to zero. The result follows from the uniform global existence of smooth solutions by means of energy estimates together with compactness arguments. For this purpose, we establish uniform estimates for $\dive \,u$ and $\curle \,u$ instead of $\nabla u$.