In this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. We present sufficient conditions for the existence of long exact extended d- homology sequence. Also we give some illustrative examples.

in this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. we present sufficient conditions for the existence of long exact extended d- homology sequence. also we give some illustrative examples.

In a companion paper ( arXiv:1910.03150 ) to this one, we proved that the Gromov–Witten theory of Fano orbifold line type D is governed by system Hirota Bilinear Equations. The goal prove every solution Equations determines new integrable hierarchy Lax equations. We suggest name extended D-Toda for equations, because it should be viewed as analogue Carlet’s bi-graded Toda hierarchy, which known...

the first extended zeroth-order connectivity index of a graph   g is defined as 0 1/2 1 ( ) ( ) ,   v   v v g      g d      where   v   (g) is the vertex set of g, and v d is the sum of degrees of neighbors of vertex v in g. we give a sharp lower bound for the first extended zeroth-order connectivity index of trees with given numbers of vertices and pendant vertices,...

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,yin R$ where $0neq bin R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. ...