A Generalized Form of Whitney＇s Theorem

Whitney's theorem is a famous theorem in the local singularity theory.In this paper,as an application of Malgrange preparation theorem,a generalized form of Whitney's theorem will be derived.     

A Generalized Form of Whitney's Theorem

Whitney's theorem is a famous theorem in the local singularity theory.In this paper,as an application of Malgrange preparation theorem,a generalized form of Whitney's theorem will be derived.     

Sections of Serre fibrations with 2-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney's theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.     

Stability Zones in Whitney's Extension Problem for Ultradifferentiable Functions

We establish a connection between the growth rate of weight functions generating nonquasianalytic classes of ultradifferentiable functions of Beurling and Roumieu type and the validity of an analog of Whitney's extension theorem for these classes.     

Local sections of Serre fibrations with 3-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the authors (Brodsky et al. (2008) [2]). The main result of this paper proves the existence of local sections in a Serre fibration with all fibers homeomorphic to some fixed compact three-dimensional manifold.     

Many function-spaces are not normal if the domain is not compact

V. Neves [4] has proved that C(M, N) with Whitney's C-topology or Michor's extension of Schwartz's D-topology is not a normal topological space provided that M is not compact. This result was shown by giving a closed embedding of Van Douwen's non-normal space using means of non-standard analysis. In this paper we recover this theorem by standard-techniques and by working in the function-space itself instead of giving an embedding. A similar method is used to obtain the same result for various other function-spaces in the case that the domain is not compact: spaces of continuous functions and C ^k-functions with Whitney's topology and spaces of sections of arbitrary differentiability-classes. Even any subspace of these spaces with non-empty interior is not normal, for example the spaces of immersions, embeddings, Riemannian metrics and symplectic structures. This also answers an open problem posed by Hirsch [2].     

On the Continuability of Multivalued Analytic Functions to an Analytic Subset

In the paper, it is shown that a germ of a many-valued analytic function can be continued analytically along the branching set at least until the topology of this set is changed. This result is needed to construct the many-dimensional topological version of Galois theory. The proof heavily uses Whitney's stratification.     

On graph planarity and semi-duality

In this (partly expository) paper we describe some of our recent results on graph planarity. These results concern strengthenings of Kuratowski's planarity criterion for quasi-4-connected graphs, for bipartite quasi-4-connected graphs, and for cubic bipartite graphs as well as generalizations of matroid duality of graphs and strengthenings of Whitney's planarity criterion.     

Each maximal planar graph with exactly two separating triangles is Hamiltonian

A classical result of Whitney states that each maximal planar graph without separating triangles is Hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Chen [Any maximal planar graph with only one separating triangle is Hamiltonian J. Combin. Optim. 7 (2003) 79-86] proved that any maximal planar graph with only one separating triangle is still Hamiltonian. In this paper, it is shown that the conclusion of Whitney's Theorem still holds if there are exactly two separating triangles.     

A new approach to the linearity of testing planarity of graphs

In testing planarity of graphs, there are many criteria. The earliest one as known is the Kuratowski's theorem, then Whitney's, Maclane's, and so forth. Since the early sixties, people have begun researches on algorithms. Up to 1974, Hopcroft and Tarjan found an algorithm with a computing time being a linear function of the order of a graph. This is the linearity concerned here.This paper presents a new approach to the linearity by means of transforming the problem of testing planarity of a graphG into that of finding a spanning tree on another graphH, called an auxiliary graph ofG, with the order ofH being a linear function of that ofG. And moreover, we can also make the size ofH be a linear function of that ofG. The whole procedure is based on the building up of a theory of linear equations onGF (2) related toG.This was a report invited by RUTCOR, The State University of New Jersey, Rutgers, U. S. A. in 1984. And, the main part of this paper was completed during the author's stay at the Department of C & O, University of Waterloo, Canada.