On continuation of a topology from a system of generators of a group to a topology on the group

In this paper we establish conditions under which a topology is continued from a subset of a group to a topology on the group coordinated with the group operation.     

Homological Invariants and Quasi-Isometry

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cd_R over a commutative ring R satisfies the inequality if Λ embeds uniformly into Γ and holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry. Received: November 2004 Revision: April 2004 Accepted: April 2004     

() does not coarsely embed into a Hilbert space

We show that a Banach space with a normalized symmetric basis behaving like that of () cannot coarsely embed into a Hilbert space.

     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

极大平面图在不可定向曲面上强嵌入的一个注记

§ 1　IntroductionA strong embeddingμ( G) of a graph G in a surface S is such an embedding thateachface boundary of the surface is a circuit.( A strong embedding is also sometimes called acircular embedding,see[1 ] orclosed2 -cell embedding[2 ] ) .Graphsconsidered here are sim-ple( that is,they have no loops or multiple edges) .Terminology here follows those in[3] .In[1 ] ,Richter,Seymour and Siran proved that every3-connected planar graph canbe strongly embedded on some non-orientable sur…     

Maps of manifolds into the plane which lift to standard embeddings in codimension two

Let be a smooth map of a closed n-dimensional manifold (n2) into the plane and let be an orthogonal projection. We say that f has the standard lifting property, if every embedding with is standard in a certain sense. In this paper we give some sufficient conditions for a generic smooth map f to have the standard lifting property when M is a closed surface or an n-dimensional homotopy sphere.     

The Bott-Borel-Weil Theorem for direct limit groups

We show how highest weight representations of certain infinite dimensional Lie groups can be realized on cohomology spaces of holomorphic vector bundles. This extends the classical Bott-Borel-Weil Theorem for finite-dimensional compact and complex Lie groups. Our approach is geometric in nature, in the spirit of Bott's original generalization of the Borel-Weil Theorem. The groups for which we prove this theorem are strict direct limits of compact Lie groups, or their complexifications. We previously proved that such groups have an analytic structure. Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. One of the technical problems here is that, in general, the limit Lie algebras will have root systems but need not have root spaces, so we need to develop machinery to handle this somewhat delicate situation.

     

On the Generation of Some Embeddable GF(2) Geometries

The generating rank is determined for several GF(2)-embeddable geometries and it is demonstrated that their generating and embedding ranks are equal. Specifically, we prove that each of the two generalized hexagons of order (2, 2) has generating rank 14, that the central involution geometry of the Hall-Janko sporadic group has generating rank 28, and that the dual polar space DU(6,2) has generating rank 22. We also include a survey of all instances in which either the generating or embedding rank of an embeddable GF(2) geometry is known.     

Spanning embeddings of arrangeable graphs with sublinear bandwidth

The Bandwidth Theorem of Böttcher, et al. [Mathematische Annalen 343 (2009), 175–205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a‐arrangeable graphs H with , where n is the number of vertices of H. Our result implies that sufficiently large n‐vertex graphs G with minimum degree at least contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, et al. [Combinatorica 33 (2013), 125–160] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most . We obtain corresponding results for graphs embeddable on different orientable surfaces. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 270–289, 2016