in this paper some upper and lower bounds for the greatest eigenvalues of the pi and vertex pimatrices of a graph g are obtained. those graphs for which these bounds are best possible arecharacterized.

Let Mr denote the moduli space of semi-stable rank-r vector bundles with trivial determinant over a smooth projective curve C of genus g. In this paper we study the base locus Br ⊂ Mr of the linear system of the determinant line bundle L over Mr, i.e., the set of semi-stable rank-r vector bundles without theta divisor. We construct base points in Bg+2 over any curve C, and base points in B4 ove...

Suppose K = W (k)[1/p], where W (k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p 6= 2. We discuss an explicit construction of p-adic semi-stable representations of the absolute Galois group of K with Hodge-Tate weights from [0, p). This theory is applied to projective algebraic varieties over Q with good reduction outside 3 and semi-stable re...

A necessary and sufficient condition is given for semi-ampleness of a numerically effective (nef) and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over Mg, specifically, the semi-ampleness of the relative dualizing sheaf, in positive characteristic. An example is given which shows this and the semi-ampleness criterion fail in chara...

A necessary and sufficient condition is given for semi-ampleness of a nef and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over Mg , specifically, the semi-ampleness of the relative dualizing sheaf, in positive characteristic. An example is given which shows this, and the semi-ampleness criterion, fail in characteristic zero. A sec...

The real plane with its set of orientations or angles in $[0,\pi)$ is the simplest non trivial example a (projective) Hilbert space and provides nice illustrations quantum formalism. We present some them, namely covariant integral quantization, linear polarisation light as measurement, interpretation entanglement leading to violation Bell inequalities, spin one-half coherent states viewed two e...

Throughout this paper we work over the base field C. The basic problem in algebraic geometry treated here is the filling-in problem (or degeneration problem). Given a smooth projective family X× → ∆×, we would like to know when we can fill in a reasonably nice special fiber X0 to form a projective family X → ∆, perhaps up to a finite base change on ∆×. For example, when can X0 be smooth? When c...

In this paper we show that if q is a power of a prime p , then the projective special linear group PSL(2, q) and the stabilizer of a point of the projective line have maximum sum element orders among all proper subgroups of projective general linear group PGL(2, q) for q odd and even respectively