In this paper, we describe the indices of the top and the least nonvanihing cohomologies H (X(w), Lλ) of line budles on Schubert varieties X(w) given by nondominant weights in the Kac-Moody setting. We also prove some surjective Theorem for maps between some cohomology modules.

We investigate the parameterized complexity of Maximum Independent Set and Dominating Set restricted to certain geometric graphs. We show that Dominating Set is W[1]-hard for the intersection graphs of unit squares, unit disks, and line segments. For Maximum Independent Set, we show that the problem is W[1]-complete for unit segments, but fixed-parameter tractable if the segments are axis-paral...

A generalized quadrangle of order s ≥ 2 is isomorphic to W (s) if and only if there is a hyperbolic line every point of which is regular. This is a characterization of the symplectic generalized quadrangle W (s) which only needs the existence of s + 1 regular points (in a nice position).

We present optical spectroscopy of a sample of metal-rich extragalactic H II regions in the spiral galaxies M83 (NGC 5236), NGC 3351 and NGC 6384. The metal abundance, estimated from semi-empirical methods using strong emission lines, is found to be above solar for most of the objects. The sample includes a number of circumnuclear H II regions (hot spots), which are found in general to have spe...

Total length (L(T)) and mass measurements of 28,596 specimens of European chub Leuciscus cephalus, collected from a variety of waterways across Italy, were used to compute standard mass (W(s)) equations by both empirical percentile (EmP) and regression line percentile (RLP) methods. The use of the EmP W(s) equation [log(10) W(s) = -4·79 + 2·68log(10) L(T) + 0·10(log(10) L(T))(2)] to compute rel...

We give a tight analysis of an old and popular sweep-line heuristic for constructing a spanning tree of a set of n points in the plane. The algorithm sweeps a vertical line across the input points from left to right, and each point is connected by a straight line segment to the closest point left of (or on) the sweep-line. If W denotes the weight the Euclidean minimum spanning tree (EMST), the ...

For a graph G, denote by L i (G) its i-iterated line graph and denote by W (G) its Wiener index. We prove that the function W (L i (G)) is convex in variable i. Moreover, this function is strictly convex if G is different from a path, a claw K 1,3 and a cycle. As an application we prove that W (L i (T)) = W (T) for every i ≥ 3 if T is a tree in which no leaf is adjacent to a vertex of degree 2,...

Graph theory is the study of mathematical structures called graphs. We define a graph as a pair (V,E), where V is a nonempty set, and E is a set of unordered pairs of elements of V . V is called the set of vertices of G, and E is the set of edges. Two vertices a and b are adjacent provided (a, b) ∈ E. If a pair of vertices is adjacent, the vertices are referred to as neighbors. We can represent...

Let f be a bounded function on the real line IF!. One may characterize the structural properties off by the modulus of smoothness w(t,f) = sup{lf (4 -f( y)l; x, y E 08, I x y I < t>, and, if w(t) is a continuous nondecreasing function of t > 0 such that w(O) = 0, by the Lipschitz class Lip(w) which is the set of all continuous functions such that su~~<~<i w(t, f)/o(t) < 00. It is possible to ex...

The geometric structure of theories with gauge fields of spins two and higher should involve a higher spin generalisation of Riemannian geometry. Such geome-tries are discussed and the case of W ∞-gravity is analysed in detail. While the gauge group for gravity in d dimensions is the diffeomorphism group of the space-time, the gauge group for a certain W-gravity theory (which is W ∞-gravity in ...