Straight line embeddings of cubic planar graphs with integer edge lengths

We prove that every simple cubic planar graph admits a planar embedding such that each edge is embedded as a straight line segment of integer length. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:270‐274, 2008     

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coeffcients in Banach-valued Lp-spaces are studied.To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations,the suffcient conditions are founded.By using these facts,the unique continuation properties are established.In the application part,the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.     

Addendum to “A definable nonstandard enlargement”

?o?'s theorem for bounded D ‐ultrapowers, D being the ultrafilter introduced by Kanovei and Shelah [4], is established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A definable nonstandard enlargement

This article establishes the existence of a definable (over ZFC), countably saturated nonstandard enlargement of the superstructure over the reals. This nonstandard universe is obtained as the union of an inductive chain of bounded ultrapowers (i.e. bounded with respect to the superstructure hierarchy). The underlying ultrafilter is the one constructed by Kanovei and Shelah [10]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

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…     

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.

     

Small Embeddings of Partial Steiner Triple Systems

It was proved in 2009 that any partial Steiner triple system of order u has an embedding of order v for each admissible . This result is best possible in the sense that, for each , there exists a partial Steiner triple system of order u that does not have an embedding of order v for any . Many partial Steiner triple systems do have embeddings of orders smaller than , but much less is known about when these embeddings exist. In this paper, we detail a method for constructing such embeddings. We use this method to show that each member of a wide class of partial Steiner triple systems has an embedding of order v for at least half (or nearly half) of the orders for which an embedding could exist. For some members of this class we are able to completely determine the set of all orders for which the member has an embedding.