A generalization of a theorem of Foxby

In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated theorem of Foxby.      

A generalization of a theorem of Edwards

In this note we extend a theorem of Edwards on the characterization of topological manifolds for polyhedra to a more general class of stratified spaces. We show that a cone-like space of dimension is a topological manifold if and only if the base space of every point in is a simply connected cone-like sphere.

     

A generalization of a theorem of Heins

Let be the family of holomorphic selfmaps of the unit disk in the complex plane . Heins established the continuity of the functional which assigns to ( denotes the identity map) either (i) the fixed point of or (ii) the limit of its iterations or (iii) if ( represents the boundary of ). Using an Abate extension of the Denjoy-Wolff lemma to strongly convex domains, we extend this result of Heins to selfmaps of strongly convex domains in with boundary.

     

A generalization of a theorem of Kruskal

LetA be a set ofk-subsets of {1,...,n} with . We determine the minimal cardinality of thel-shadow ofA. This theorem generalizes a theorem of Kruskal [10], Katona [8], Lindström [11] and other which gives a solution for the problem without the condition . Furthermore we give a necessary and sufficient condition for the uniqueness of the solution in the KKL theorem.     

A generalization of a theorem of Neumaier

In this paper, we show that for given positive integer p, the number of connected biregular graphs with exactly three distinct eigenvalues, and with smallest eigenvalue at least \(-p\), is finite.     

A generalization of a theorem of Ore

Let (K,v)$(K,v)$ be a discrete rank one valued field with valuation ring _Rv$R_v$. Let L/K$L/K$ be a finite extension such that the integral closure S   of _Rv$R_v$ in L   is a finitely generated _Rv$R_v$-module. Under a certain condition of v  -regularity, we obtain some results regarding the explicit computation of _Rv$R_v$-bases of S, thereby generalizing similar results that had been obtained for algebraic number fields in El Fadil et al. (2012) [7]. The classical Theorem of Index of Ore is also extended to arbitrary discrete valued fields. We give a simple counter example to point out an error in the main result of Montes and Nart (1992) [12] related to the Theorem of Index and give an additional necessary and sufficient condition for this result to be valid.     

A generalization of Plantholt's theorem

Let K denote the complete graph K_2n+1 with each edge replicated r times and let χ′(G) denote the chromatic index of a multigraph G. A multigraph G is critical if χ′(G) > χ′(G/e) for each edge e of G. Let S be a set of sn – 1 edges of K. We show that, for 0 < s ≦ r, G/S is critical and that χ′ (G/(S ∪{e})) = 2rn + r – s for all e ∈ E(G/S). Plantholt [M. Plantholt, The chromatic index of graphs with a spanning star. J. Graph Theory 5 (1981) 5–13] proved this result in the case when r = 1.     

A generalization of Jung's theorem

Jung's theorem establishes a relation between circumradius and diameter of a convex body. Half of the diameter can be interpreted as the maximum of circumradii of all 1-dimensional sections or 1-dimensional orthogonal projections of a convex body. This point of view leads to two series of j-dimensional circumradii, defined via sections or projections. In this paper we study some relations between these circumradii and by this we find a natural generalization of Jung's theorem.I would like to thank Prof. Dr J. M. Wills, who called my attention to these generalized circumradii.