Let A and B be positive operators 0 < q ? 1. In this paper, we shall show that if Aq?0 (A?0/2B?0A?0/2) q?0/?0+?0 (B?0/2A?0B?0/2) Bq?0 hold for fixed ?0 > 0. Then the following inequalities hold: Aq1? (A?/2B?A?/2) q1?/?+? (B?/2A?B?/2) Bq1? all ?0, q1 q. Also, a normality of class p-A(s, t) s 0, t p Moreover, T or T* belongs to p-wA(s, some 1 S is an operator which W(S) ST = T*S, then self-...

In this note first we define a BCK‐algebra on the states of a deterministic finite automaton. Then we show that it is a BCK‐algebra with condition (S) and also it is a positive implicative BCK‐algebra. Then we find some quotient BCK‐algebras of it. After that we introduce a hyper BCK‐algebra on the set of all equivalence classes of an equivalence relation on the states of a deterministic finite...

Trellis Graphics is implemented in S-PLUS using the traditional S graphics engine. In contrast, lattice, the implementation of Trellis Graphics in R, uses grid graphics as the underlying mechanism. Being an independent implementation, lattice differs from the original in several other aspects as well. Most of the literature available on Trellis Graphics describes the implementation in S-PLUS. T...

J. A. Bondy and U. S. R. Murty, Graph theory with Applications, London: MacMillan (1976). R. M. Gera, On Dominator Colorings in Graphs, Graph Theory Notes of New York LIT, 25-30 (2007) F. Harary, Graph Theory, Narosa Publishing 1969. D. Michalak, On Middle and Total graphs with Coarseness Number Equal 1, Lecture Notes in Mathematics, Volume 1018: Graph Theory, Springer-Verlag, Berlin, 139 – 150...

These notes are an informal, first installment in an ongoing project to develop a convenient template for computational experimentation in econometrics. The approach is illustrated by means of an example based on some current research with Steve Portnoy on improving the speed of quantile regression algorithms. The computations are carried out in splus, but similar techniques could be adapted fo...