An extension theorem for planar semimodular lattices

We prove that every finite distributive lattice \(D\) can be represented as the congruence lattice of a rectangular lattice \(K\) in which all congruences are principal. We verify this result in a stronger form as an extension theorem.     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.     

An extension of the factorization theorem to the non-positive case

This paper presents a method of determining joint distributions by known conditional distributions. A generalization of the Factorization Theorem is proposed. The generalized theorem is proved under the assumption that the support of unknown joint distribution may be divided into a countable number of sets, which all satisfy the relative weak positivity condition. This condition is defined in the paper and it generalizes the positivity condition introduced by Hammersley and Clifford. The theorem is illustrated with three examples. In the first example we determine a joint density in the case when the support of an unknown density is a continuous nonproduct set from Euclidean space . In the second example we seek the joint probability for the number of trials and the number of successes in Bernoulli's scheme. We also examine a simple example given by Kaiser and Cressie (J. Multivariate Anal. 73 (2000) 199).     

An existence theorem for planar maps

The main result of this paper is a theorem concerning possible cubic maps on the plane or sphere. The dual approach of spherical triangulations will be used. Every such triangulation must contain a vertex of degree less than 8 and other than 5, or else contain one of a list of 5 configurations. Due to the occurrence of these five configurations in four color reduction arguments, this implies that a minimal five color map must have at least one face with six or seven neighbors. The theorem is given in Heesch [1, p. 129 ff.], but the proof here is much shorter, due to a modification of Heesch's principle of “discharging”. The principle itself is obtained by exploiting the Euler formula, and should be compared with the theory of Euler contributions as developed by Ore in [3], and by Ore and Stemple [4].     

An extension of the fenchel theorem

We give an extension of the Fenchel-Borsuk result by introducing the total absolute torsion-curvature KT() for regular curves whose tangent indicatrix is a piecewise regular curve (-closed curves). We prove that KT() is low bounded by 2 and we give a geometric characterization for the -closed curves whose KT() is minimal.     

An extension of the Lindeberg-Feller theorem

A new version is presented of the necessary and sufficient condition of convergence, to a normal law, of sums of independent variables in a nonclassical situation (i.e., absence of limiting negligibility of variables). The obtained condition differs from previously obtained conditions by the fact that it does not use Levy's metric and that is is closer to classical formulations. A similar condition is sufficient for the closeness of two convolutions when the number of components of the convolutions increases without bounds.     

An extension of the Dulmage-Mendelsohn theorem

We prove a local Dulmage-Mendelsohn theorem, a generalization of their Theorem 18[1] to matrices which are decomposable or have period greater than one. This can be used to estimate the index of a wide variety of matrices and has applications to finite automata theory, stochastic processes, graph theory, and other fields.     

An embedding theorem for finite planar spaces

LetS be a finite planar space such that any two distinct planes intersect in a line. We show thatk≤n ²+1 for anyk-cap ofS, wheren is the order ofS. Moreover, if a (n ²+1)-cap exists inS, a necessary and sufficient condition is provided forS to be embeddable in a 3-dimensional projective space. Work supported by the National Research Project “Strutture geometriche, Combinatoria e loro applicazioni” of the italian M.U.R.S.T.     

An extension of Lucas' theorem

Let be a prime. A famous theorem of Lucas states that if are nonnegative integers with . In this paper we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with initial values and .

     

An extension of Vizing's theorem

Let G = (V (G), E (G)) be a simple graph of maximum degree δ ≤ D such that the graph induced by vertices of degree D is either a null graph or is empty. We give an upper bound on the number of colours needed to colour a subset S of V (G) ∪ E (G) such that no adjacent or incident elements of S receive the same colour. In particular, if S = E (G), we have the chromatic index χ′(G) ≤ D whereas if S = V (G) ∪ E (G) and for some positive integer k, we have the total chromatic number _χT(G) ≤ D + k. © 1997 John Wiley & Sons, Inc.     

An extension of Takahashi's theorem

A classical result of T. Takahashi [8] is generalized to the case of hypersurfaces in the Euclidean space E ^m . More concretely, we classify Euclidean hypersurfaces whose coordinate functions in E ^m are eigenfunctions of their Laplacian.Partially supported by a CAICYT Grant PR84-1242-C02-02 Spain.     

An extension of Jellett's theorem

The aim of this work is to show that a star-shaped hypersurface of constant mean curvature into the Euclidean sphere Sn+1 must be a geodesic sphere. This result extends the one obtained by Jellett in 1853 for such type of surfaces in the Euclidean space R³. In order to do that we will compute a useful formula for the Laplacian of a new support function defined over a hypersurface M of a Riemannian manifold .     

An extension of Radner's theorem

Radner's theorem states that the optimal solution for a static linear quadratic Gaussian (LQG) team is linear. In this paper, we find the optimal solution for a static LQG team in which each player knows which observations he has, but in which the observation set that a player receives (how many, which measuring device,etc.) is random before the team acts. Via the concept of nesting, the result extends to the dynamic case and includes teams in which the order of play of the team members as well as their sets of data are random. We also include random changes in the cost function which depend on the randomness of the observation system but are independent of the stochastic process that the team is observing and controlling.     

An isomorphism extension theorem for Landau-Ginzburg B-models

Landau-Ginzburg mirror symmetry studies isomorphisms between A- and B-models, which are graded Frobenius algebras that are constructed using a weighted homogeneous polynomial W and a related symmetry group G. Given two polynomials W₁, W₂ with the same weights and same group G, the corresponding A-models built with (W₁,G) and (W₂,G) are isomorphic. Though the same result cannot hold in full generality for B-models, which correspond to orbifolded Milnor rings, we provide a partial analogue. In particular, we exhibit conditions where isomorphisms between unorbifolded B-models (or Milnor rings) can extend to isomorphisms between their corresponding orbifolded B-models (or orbifolded Milnor rings).     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.