Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let G_m,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a₁, …, a_m, each vertex b of B by an independent set b₁,…, b_n, and each edge ab of G by the complete bipartite graph with edges a_ib_j (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G_m,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.     

Approximate decidability in euclidean spaces

We study concepts of decidability (recursivity) for subsets of Euclidean spaces ?^k within the framework of approximate computability (type two theory of effectivity). A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by computability of the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces.     

Embeddings of chemical graphs in hypercubes

We study planar graphs embedded in the plane that have chemical applications: the degrees of all vertices are 3 or 2, all internal faces but one or two arer-gons, and each internal face is a simply connected domain. For wide classes of such graphs, we solve the existence problem for embeddings of the graph metric on the vertices in multidimensional cubes or cubical lattices preserving or doubling all the distances. Incidentally we present a complete classification of some interesting families of such graphs. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 339–352, September, 2000.     

Embedding of trees in euclidean spaces

It is proved that for any treeT the vertices ofT can be placed on the surface of a sphere inR ³ in such a way that adjacent vertices have distance 1 and nonadjacent vertices have distances less than 1.     

Embeddings of Gromov hyperbolic spaces

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant.?Another embedding theorem states that any -hyperbolic metric space embeds isometrically into a complete geodesic -hyperbolic space.?The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. Submitted: October 1998, Revised version: January 1999.     

Isometric approximation property in euclidean spaces

We give a necessary and sufficient quantitative geometric condition for a compact setA⊂R ⁿ to have the following property with a givenc≥1: For everyɛ>0 and for every mapf: A→R ⁿ such that there is an isometryS: A→R ⁿ such that |Sx−fx|≤cɛ for allx∈A.     

Embeddings of weighted Morrey spaces

We consider embedding theorems within the scale of weighted Morrey spaces , where the weight w belongs to the Muckenhoupt class and . This includes, in particular, the classical setting of weighted Lebesgue spaces. We study some typical examples for the weight like , but also deal with quite general assumptions.     

Embeddings of anisotropic Besov spaces into Sobolev spaces

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.     

Embeddings of covering projections of graphs

It is shown that any given k-fold covering projection of graphs p: G₁ → G₂ can be embedded in a k-fold covering projection of closed orientable surfaces π: S₁ → S₂ in the sense that there are embeddings of G₁ in S₁ and G₂ in S₂ such that p is the restriction of π. In the case of a regular covering projection p, which is the quotient map with respect to some group action on G₁, it is shown that there is a regular covering projection π of surfaces in which p can be embedded.     

EMBEDDINGS OF ONE KIND OF GRAPHS

The purpose of this paper is to display a new kind of simple graphs which belong to B. inwhich any graph has its orientable genus n,n≥3. Furthermore, for any integer k,1≤k≤n,there exists a graph B^kn of B. such that the non-orientable genus of B^kn is k.     

Infinitesimally homogeneous submanifolds of euclidean spaces

For a class of submanifolds of ^N, the infinitesimally homogeneous ones, the second fundamental form and itss-times iterated derivativessk+1 at any fixed point determine the immersion uniquely. The integerk>0 will be called the extrinsic Singer invariant. Any infinitesimally homogeneous submanifoldM (which is not necessarily complete) is an open part of a globally homogeneous (complete) submanifold. Indeed, the infinitesimal data at any pointp, determine, canonically, a Lie subgroupG of the isometry group of ^N , whose orbit atp is a complete submanifold that extendsM. Work partially supported by the GNSAGA of CNR and by the MURST of Italy     

Embeddings for anisotropic Besov spaces

We prove embedding theorems for fully anisotropic Besov spaces. In particular, inequalities between modulus of continuity in different metrics and of Sobolev type are obtained. Our goal is to get sharp estimates for some anisotropic cases previously unconsidered.