A general strategy in vertebrate neural development is for neuronal precursors to stop dividing, to begin to differentiate, and then to migrate to their final destination. In this issue of Neuron, however, Godhino et al. provide evidence that some neuronal precursors undergo a terminal symmetric cell division at their final destination to form an entire neuronal layer.

The harmonic index of a graph G is defined as the sum of weights 2 deg(v)+deg(u) of all edges uv of E(G), where deg(v) denotes the degree of a vertex v in V (G). In this note we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561– 566] and establish some upper and lower bounds on the harmonic index of G.

The harmonic index ) (G H , of a graph G is defined as the sum of weights 1 )) deg( ) (deg( 2   v u of all edges in ) (G E , where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G as      ) ( 1 ) deg( ) deg( 2 ) , ( G E uv v u x x G H , where   1 0 ) ( ) , ( G H dx x G H . We present explicit formula for the values of harmonic polyn...

Let G be a connected graph with vertex set V (G) and edge set E(G). The eccentric connectivity index of G, denoted by ξc(G), is defined as ∑ v∈V (G) deg(v)ec(v), where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. In this paper, we propose the edge version of the above index, the edge eccentric connectivity index of G, denoted by ξc e(G), which is defined as ξc e(G) = ∑ f∈E(...

(0.0) Elliptic curves are perhaps the simplest 'non-elementary' mathematical objects. In this course we are going to investigate them from several perspectives: analytic (= function-theoretic), geometric and arithmetic. Let us begin by drawing some parallels to the 'elementary' theory, well-known from the undergraduate curriculum. Elementary theory This course arcsin, arccos elliptic integrals ...

The geometric-arithmetic index is another topological index was defined as 2 deg ( )deg ( ) ( ) deg ( ) deg ( ) G G uv E G G u v GA G u v     , in which degree of vertex u denoted by degG (u). We now define a new version of GA index as 4 ( ) 2 ε ( )ε ( ) ( ) ε ( ) ε ( ) G G e uv E G G G u v GA G   u v    , where εG(u) is the eccentricity of vertex u. In this paper we compute this new t...

Two long-standing paradigms in biology are that cells belonging to the same population exhibit little deviation from their average size and that symmetric cell division is size limited. Here, ultrastructural, morphometric and immunocytochemical analyses reveal that two Gammaproteobacteria attached to the cuticle of the marine nematodes Eubostrichus fertilis and E. dianeae reproduce by constrict...

The grain boundary character distribution in a commercial IF steel has been measured as a function of lattice misorientation and boundary plane orientation. The grain boundary plane distribution revealed a relatively low anisotropy with a tendency for grain boundaries to terminate on low index planes having relatively low surface energy and large interplanar spacings. Although the most common g...

It is easy to see that there is at most one pair of polynomials (q(x), r(x)) satisfying (1); for if (q1(x), r1(x)) and (q2(x), r2(x)) both satisfy the relation with respect to the same polynomial u(x) and v(x), then q1(x)v(x)+r1(x) = q2(x)v(x)+r2(x), so (q1(x)− q2(x))v(x) = r2(x)−r1(x). Now if q1(x)− q2(x) is nonzero, we have deg((q1 − q2) · v) = deg(q1 − q2)+deg(v) ≥ deg(v) > deg(r2 − r1), a c...

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...