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.

We prove that if $$G(R)=G_\pi (\Phi ,R)$$ $$(E(R)=E_{\pi }(\Phi , R))$$ is an (elementary) Chevalley group of rank $$> 1$$ R a local ring (with $$\frac{1}{2}$$ for the root systems $${{\textbf{A}}}_2, {{\textbf{B}}}_l, {{\textbf{C}}}_l, {{\textbf{F}}}_4, {{\textbf{G}}}_2$$ and with $$\frac{1}{3}$$ $${{\textbf{G}}}_{2})$$ then G(R) (or (E(R)) regularly bi-interpretable R. As consequence this the...

One o f the ma in aims o f [5] is to obta in bounds for asr(A) for var ious classes of noncommuta t i ve rings A; in part icular , they show that: (i) asr(A) ~< 1 + d whenever A is a module finite a lgebra over a commuta t ive Noe ther ian ring R with d im(maxspec R) = d, and (ii) asr(A) = 1 if A is a semi-local ring (see [5], Theorems 3.1 and 2.4], respectively). The a im o f this note is to s...

NADPH-cytochrome P450 oxidoreductase catalyzes transfer of electrons from NADPH, via two flavin cofactors, to various cytochrome P450s. The crystal structure of the rat reductase complexed with NADP(+) has revealed that nicotinamide access to FAD is blocked by an aromatic residue (Trp-677), which stacks against the re-face of the isoalloxazine ring of the flavin. To investigate the nature of in...

In this paper, we study an irreducible decomposition structure of the $$\mathcal {D}$$ -module direct image $$\pi _+(\mathcal {O}_{\mathbb {c}^n})$$ for finite map : \mathbb {C}^n \rightarrow {C}^n/ ({\mathcal {S}_{n_1}\times \cdots \times \mathcal {S}_{n_r}}).$$ We explicitly construct simple components {C}^n})$$ by providing their generators and multiplicities. Using equivalence categories hi...

Cyclobutane dimer photolyases are proteins that bind to UV-damaged DNA containing cyclobutane pyrimidine dimer lesions. They repair these lesions by photo-induced electron transfer. The electron donor cofactor of a photolyase is a two-electron-reduced flavin adenine dinucleotide (FADH(-)). When FADH(-) is photo-excited, it transfers an electron from an excited pi --> pi* singlet state to the py...