On suspensions of noncontractible compacta of trivial shape

We prove that: (i) There exists a 2-dimensional noncontractible cohomologically locally connected cell-like compactum whose reduced suspension is a contractible ANR; (ii) If the suspension of a compactum is contractible, then is weakly contractible.

     

Positive Solutions of {Delta u}+u^{(n+2)/(n-2)}=0 on Contractible Domains

We show that there are bounded contractible domains in R^n, n ≥ 3, on which the Dirichlet problem for the equation Δu +u^{(n+2)/(n-2)} = 0 admit positive solutions. This result, combining with the well-known nonexistence result of Pohozaev, implies that geometry of the domains plays a crucial role in the solvability of the problem.     

On the dynamics of a family of entire functions

We prove that for any bounded type irrational number 0θ1,the boundary of the Siegel disk of fα(z)=e2πiθsin(z)+αsin3(z),α∈C,which centered at the origin,is a quasicircle passing through 2,4 or 6 critical points of fαcounted with multiplicity.     

關於Borsuk的絕對同倫擴充性質的討論

<正> 序言.在同倫論中,常常需要考慮滿足這種性質的拓撲空間X設Y為任意的一個正規空間,B為Y的任何一個非空閉集,任何一個由B×(0,1)+Y×(0)到X的映像都可以扩充為一個由Y×(0,1)到X的映像,我們稱這種性質為絕對同倫扩充性質,具有這種性質的空間以及用AHE表示.Borsuk曾經介紹這樣一個重要的定理:     

The number of contractible edges in 3-connected graphs

An edge of a 3-connected graph is calledcontractible if its contraction results in a 3-connected graph. Ando, Enomoto and Saito proved that every 3-connected graph of order at least five has |G|/2 or more contractible edges. As another lower bound, we prove that every 3-connected graph, except for six graphs, has at least (2|E(G)| + 12)/7 contractible edges. We also determine the extremal graphs. Almost all of these extremal graphsG have more than |G|/2 contractible edges.     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.     

On contractible polyhedra that are not simply contractible

In answer to a question of Michael, Dydak, Segal and Spiez have constructed a contractible polyhedron that is not strictly contractible. In the present note we prove a related result; by using alternative methods we show that there exist contractible polyhedra that are not simply (hence not strictly) contractible.

     

On the Structure of Contractible Vertex Pairs in Chordal Graphs

An edge/non-edge in a k-connected graph is contractible if its contraction does not result in a graph of lower connectivity. We focus our study on contractible edges and non-edges in chordal graphs. Firstly, we characterize contractible edges in chordal graphs using properties of tree decompositions with respect to minimal vertex separators. Secondly, we show that in every chordal graph each non-edge is contractible. We also characterize non-edges whose contraction leaves a k-connected chordal graph.     

不含某些子图的k连通图中的k可收缩边

最近Ando等证明了在一个$k$($k\geq 5$ 是一个整数) 连通图 $G$ 中,如果 $\delta(G)\geq k+1$, 并且 $G$ 中既不含 $K^{-}_{5}$,也不含 $5K_{1}+P_{3}$, 则$G$ 中含有一条 $k$ 可收缩边.对此进行了推广,证明了在一个$k$连通图$G$中,如果 $\delta(G)\geq k+1$,并且 $G$ 中既不含$K_{2}+(\lfloor\frac{k-1}{2}\rfloor K_{1}\cup P_{3})$,也不含 $tK_{1}+P_{3}$ ($k,t$都是整数,且$t\geq 3$),则当 $k\geq 4t-7$ 时, $G$ 中含有一条 $k$ 可收缩边.     

ℓ-homology of right-angled Coxeter groups based on barycentric subdivisions

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Σ_L on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a triangulation of , then Σ_L is a contractible n-manifold. We establish vanishing (in a certain range) of the reduced ℓ²-homology of Σ_L in the case where L is the barycentric subdivision of a cellulation of a manifold. In particular, we prove the Singer Conjecture (on the vanishing of the reduced ℓ²-homology except in the middle dimension) in the case of Σ_L where L is the barycentric subdivision of a cellulation of , n=6,8.     

Nonuniform FDM for Poisson-Type Equations on a Disk with Singular Solutions

We are concerned with the Dirichlet boundary value problem of Poisson-type equations on a disk. Matsunaga and Yamamoto [8 N. Matsunaga and T. Yamamoto ( 1999 ). Convergence of Swartztrauber–Sweet's approximation for the Poisson-type equation on a disk . Numer. Funct. Anal. Optimz. 20 : 917 – 928 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] proved that if the exact solution u is very smooth over the closure of the disk, then the approximate solution by the Swartztrauber–Sweet scheme with uniform partition is second order accurate. In this article, it is assumed that the exact solution performs singular properties such that its derivatives go to infinity at the boundary of the disk. We use a stretching polynomial-like function with a parameter to construct a local grid refinement and consider the Swartztrauber–Sweet scheme over the non-uniform partition. The effects of the parameter are analyzed completely by carrying out convergence analysis and numerical results show that there exists an optimal value for the parameter to have a best approximated solution.     

Contractible elements in k‐connected graphs not containing some specified graphs

In 15 , Thomassen proved that any triangle‐free k‐connected graph has a contractible edge. Starting with this result, there are several results concerning the existence of contractible elements in k‐connected graphs which do not contain specified subgraphs. These results extend Thomassen's result, cf., 2 , 3 , 9 - 13 . In particular, Kawarabayashi 12 proved that any k‐connected graph without K subgraphs contains either a contractible edge or a contractible triangle. In this article, we further extend these results, and prove the following result. Let k be an integer with k ≥ 6. If G is a k‐connected graph such that G does not contain as a subgraph and G does not contain as an induced subgraph, then G has either a contractible edge which is not contained in any triangle or a contractible triangle. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:97–109, 2008     

单位圆到凸区域上的调和拟共形映照

设F（x）=p（x）e^ir（x）为单位圆周到约当凸曲线Γ上的保向同胚映照.本文证明:若ess inf|F’（x）|>0且对于一切的φ∈R有|F（φ+x）+F（φ-x）-2F（φ）|≤M|x|^α,这里α>1,M为正常数,则ω=P[F]（z）为单位圆到凸区域Ω=int（Γ）上为调和拟共形映照.     

On the strict contractibility of polyhedra

In this paper we study conditions under which a compact contractible polyhedron is strictly contractible to a point.     

On the fourth and fifth coefficients of strongly starlike functions

For 0 < α ≤ 1, analytic functions f(z) = z + a₂z² + a₃z³ + … in the unit disk U are strongly starlike of order α if ¦arg {zf′ (z)/f(z)}¦ < πα / 2, z ∈ U. We find sharp estimates on the fourth and fifth coefficients of functions in this class.     

An Extremal Problem on Contractible Edges in 3-Connected Graphs

An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The existence of contractible edges is a very useful induction tool. Let G be a simple 3-connected graph with at least five vertices. Wu [7] proved that G has at most vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly vertices that are not incident to contractible edges. We show that all such graphs can be constructed from either a single vertex or a 3-edge-connected graph (multiple edges are allowed, but loops are not allowed) by a simple graph operation. Research partially supported by an ONR grant under grant number N00014-01-1-0917     

Symplectic homology and periodic orbits near symplectic submanifolds

We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values.     

Some notes about positive solutions of Δu+a(x)u+u (N +2)/(N−2)=0

In this paper, we prove that on a large class of bounded contractible and noncontractible domains with piecewise smooth boundary, there is at least one positive solution to Δu+a(x)u+u ^{(N+2)/(N−2)} =0 (N≥3) with zero Dirichlet boundary condition.     

Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations

Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs.     

Covering contractible edges in 3-connected graphs. I: Covers of size three are cutsets

An edge of a 3-connected graph is said to be contractible if its contraction results in a 3-connected graph. In this paper, a covering of contractible edges is studied. We give an alternative proof to the result of Ota and Saito (Scientia (A) 2 (1988) 101–105) that the set of contractible edges in a 3-connected graph cannot be covered by two vertices, and extended this result to a three-vertex covering. We also study the existence of a contractible edge whose contraction preserves a specified cycle, and show that a non-hamiltonian 3-connected graph has a contractible edge whose contraction preserves the circumference.