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 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 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 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 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 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 Suffridge's convolution theorem

We present an extension of Suffridge's convolution theorem for polynomials with restricted zeros on the unit circle. We also discuss a possible extension of the theorem of Laguerre for those polynomials and give an answer to a long-standing open question by Suffridge regarding an extension of the theorem of Gauß-Lucas.     

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 extension theorem to rough paths

We show that any continuous path of finite p-variation can be lifted to a geometric q  -rough path, where q>p$q>p$.     

An extension theorem in symplectic geometry

 We extend the ``Extension after Restriction Principle' for symplectic embeddings of bounded starlike domains to a large class of symplectic embeddings of unbounded starlike domains. Received: 21 January 2002 / Revised version: 5 July 2002 Mathematics Subject Classification (2000): Primary 53D35, Secondary 54C20     

An extension of a theorem of Kaplansky

A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.     

An extension of a theorem of Banach

Summary Via porosity a well-known theorem of Banach on bounded real valued continuous functions defined on [0, 1] is extended to higher dimensions. Summary. Via porosity a well-known theorem of Banach on bounded real valued continuous functions defined on [0, 1] is extended to higher dimensions.     

An extension of a non-commutative Choquet-Deny theorem

Let be a discrete group, and let be a normal subgroup of . Then the quotient map induces a group algebra homomorphism . It is shown that the kernel of this map may be decomposed as , where is a closed right ideal with a bounded left approximate identity and is a closed left ideal with a bounded right approximate identity. It follows from this fact that, if is a closed two-sided ideal in , then is closed in . This answers a question of Reiter.