in this paper, at first we mention to some results related to pi and vertex co-pi indices and then we introduce the edge versions of co-pi indices. then, we obtain some properties about these new indices.

BACKGROUND AND PURPOSE Graves' disease (GD) is the most common hyperthyroid disorder, but the therapeutic strategy for choosing between medical or ablative therapy has not been standardized. Thyroid hypervascularity is an important diagnostic feature in GD. This study collected Doppler ultrasonography data from patients with GD, Hashimoto's thyroiditis, and simple goiter to develop a hemodynami...

A 2-cell embedding of a graph G into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs Kn,n into nonorientable surfaces. Such a regular embedding of Kn,n exists only when n = 2p a1 1 p a2 2 · · · p ak k (a...

Let V be a finite dimensional real vector space, let g be the real span of a finite set of commuting endomorphisms of V , and G = exp g. We study the orbit structure in elements of a finite partition of V into explicit G-invariant connected sets. In particular, we prove that either there is an open conull G-invariant subset of V in which every G-orbit is regular, or there is a G-invariant, conu...

We obtain an estimate of free entropy of generators in a type II1-factor M which has a subfactor N of finite index with a subalgebra P = P1∨P2 ⊂ N where P1 = R1∩P , P2 = R2∩P are diffuse, R1,R2 ⊂ P are mutually commuting hyperfinite subfactors, and an abelian subalgebra A ⊂ N such that the correspondence PL 2(N , τ)A is M-weakly contained in a subcorrespondence PHA of PL 2(M, τ)A, generated by ...

in this paper, we present some inequalities for the co-pi index involving the some topological indices, the number of vertices and edges, and the maximum degree. after that, we give a result for trees. in addition, we give some inequalities for the largest eigenvalue of the co-pi matrix of g.

Representations of nite languages as a product (catenation) of languages are investigated, where the factor languages are \prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions are not unique even the number of factors can vary exponentially. The paper investigates the uniqueness of prime decompositions, as well as the commuting of the facto...

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras [R, FFT]. We prove that generically their action on finitedimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free G -opers on P with regular singularity at one point and irregular singula...

BACKGROUND Active transportation has been associated with favorable health outcomes. Previous research highlighted the influence of neighborhood educational level on active transportation. However, little is known regarding the effect of commuting distance on social disparities in active commuting. In this regard, women have been poorly studied. The objective of this paper was to evaluate the r...