Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

The Harmonic Functional Calculus and Hyperreflexivity

A natural L∞ functional calculus for an absolutely continuous contraction is investigated. It is harmonic in the sense that for such a contraction and any bounded measurable function φ on the circle, the image can rightly be considered as φ̂(T ), where φ̂ is the solution of the Dirichlet problem for the disk with boundary values φ. The main result shows that if the functional calculus is isometri...

the effect of functional/notional approach on the proficiency level of efl learners and its evaluation through functional test

in fact, this study focused on the following questions: 1. is there any difference between the effect of functional/notional approach and the structural approaches to language teaching on the proficiency test of efl learners? 2. can a rather innovative language test referred to as "functional test" ge devised so so to measure the proficiency test of efl learners, and thus be as much reliable an...

On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

On the harmonic index and harmonic polynomial of Caterpillars with diameter four

The harmonic index H(G) , of a graph G is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in E(G), where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G. We present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in Caterpill...

On the harmonic index of bicyclic graphs

The harmonic index of a graph $G$, denoted by $H(G)$, is defined asthe sum of weights $2/[d(u)+d(v)]$ over all edges $uv$ of $G$, where$d(u)$ denotes the degree of a vertex $u$. Hu and Zhou [Y. Hu and X. Zhou, WSEAS Trans. Math. {bf 12} (2013) 716--726] proved that for any bicyclic graph $G$ of order $ngeq 4$, $H(G)le frac{n}{2}-frac{1}{15}$ and characterize all extremal bicyclic graphs.In this...