A Hadamard product involving N-matrices

We show that for any pair M,N of n by n M-matrices, the Hadamard (entry-wise) product M°N _-1 is again an M-matrix. For a single M-matrix M, the matrix M°M ^-1 is also considered.     

A Hadamard product involving N-matrices

We show that for any pair M,N of n by n M-matrices, the Hadamard (entry-wise) product M°N_-1 is again an M-matrix. For a single M-matrix M, the matrix M°M^-1 is also considered.     

Ky Fan's N-matrices and linear complementarity problems

We consider the linear complementarity problem (LCP),w=Az + q, w0,z0,w ^T z=0, when all the off-diagonal entries ofA are nonpositive (the class of Z-matrices), all the proper principal minors ofA are positive and the determinant ofA is negative (the class of almost P-matrices). We shall call this the class of F-matrices. We show that ifA is a Z-matrix, thenA is an F-matrix if and only if LCP(q, A) has exactly two solutions for anyq0,q0, and has at most two solutions for any otherq. Research supported by AFOSR-89-0512.     

A generalization ofL-splines

A theory of generalized splines is developed for all regular formally self adjoint differential operatorsL with real coefficients. A special case of such operators are those which may be factored in the formL =L ₁ ^* L ₁, such as those related to the generalized splines of Ahlberg, Nilson, and Walsh [1, 2], and theL-splines of Schultz and Varga [6]. Theorems giving unique interpolation, integral relations, and convergence rates are established. IfL has a certain positivity property, a useful extremal result is proven.This research was supported in part by a NASA Traineeship, at the Georgia Institute of Technology.     

A generalization of finiteness

We explore an analogy between the family ₁, of finite/cofinite languages and the family ₁ of languages whose syntactic monoids are J-trivial. It is shown that (a) J-trivial monoids, (b)L-trivial monoids, (c) R-trivial monoids, and (d) a recently studied family, that we callK, of aperiodic monoids are natural generalization of the families of syntactic semigroups of (a) finite/cofinite languages, (b) definite languages, (c) reverse definite languages, and (d) generalized definite languages, respectively. In the case of alphabets of one and two letters, the languages corresponding to the familyK of monoids are characterized, illustrating the above-mentioned analogy explicitly.This work was done partly at the University of Paris VI and VII under the scientific exchange program between Canada and France, and partly at the Institut für Rechner- und Programmstrukturen, Gesellschaft für Mathematik und Datenverarbeitung mbH. Bonn, Germany.     

A generalization of Lee codes

Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set (PDDS) in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and dominating sets. In this paper we initiate a systematic study of PDDSs. PDDSs related to the application will be constructed and the non-existence of some PDDSs will be shown. In addition, an extension of the long-standing Golomb–Welch conjecture, in terms of PDDS, will be stated. We note that all constructed PDDSs are lattice-like which is a very important feature from the practical point of view as in this case decoding algorithms tend to be much simpler.     

A generalization of Calkin's identity

The purpose of this short note is to give a generalization of Calkin's identity.     

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.     

A generalization of theKohlberg criterion

TheKohlberg criterion, which characterizes the nucleolus of a game (over the set of all imputations) in terms of balanced collections, is here extended to the sets of payoff vectors for various coalition structures. It is shown that one generalization yields a necessary condition; another yields a sufficient condition. A necessary and sufficient condition — which is unfortunately somewhat difficult to verify — is also given.     

A generalization of outerplanar graphs

Let G be a planar graph and W a set of vertices, G is W-outerplanar if it can be embedded in the plane so that all vertices of W lie on the exterior face. We give a characterization of these graphs by forbidden subgraphs, an upper bound on the number of edges, and other properties which lead to an algorithm of W-outerplanarity testing.     

A generalization of parallel addition

Summary Parallel addition of positive operators, a concept introduced by W. N. Anderson and R. J. Duffin [1] in connection with network theory, has already been studied by several authors. We specifically mention W. N. Anderson and G. E. Trapp [2] and [3], T. Ando [4], K. Nishio [12], E. L. Pekarev and J. L. Smul'jan [13], as well as the article [10] by the present authors.The purpose of this note is to study a generalization of parallel addition. In particular, it will be shown (Theorem 3.2) that the corresponding quasi-units, a concept introduced in [10], are again the extreme points of the convex sets, formed by the positive operators less than or equal to some fixed operator.     

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.