Energy methods have shown promise as measures for quantifying the vulnerability of power systems to problems of voltage instability and collapse. However, to make such measures more useful as a security assessment tool in an operational environment, it is important to provide physical interpretations of the quantitative measure. This paper will demonstrate that for certain basic load models, th...

The standard radius of curvature at a point q(s) on a smooth curve can be defined as the limiting radius of circles through three points that all coalesce to q(s). In the study of ideal knot shapes it has recently proven useful to consider a global radius of curvature of the curve at q(s) defined as the smallest possible radius amongst all circles passing through this point and any two other po...

Some connections between the rational curve K := {(1, t, t q+1 | t ∈ GF (q 2))}∪ {(0, 0, 1)} and regular symplectic spreads of P G(3, q), q even, are presented by using some geometry and combinatorics of symplectic polarities commuting with unitary polarities. Some results on the geometry of commuting orthogonal and unitary polarities will be also discussed.

where C(Fq) denotes the set of Fq-rational points of the curve C. Here we will be interested in maximal(resp. minimal) curves over Fq2 , that is, we will consider curves C attaining Hasse-Weil’s upper (resp. lower) bound: #C(Fq2) = q + 1 + 2gq (resp. q + 1− 2gq). Here we are interested to consider the hyperelliptic curve C given by the equation y = x + 1 over Fq2 . We are going to determine whe...

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ a fixed algebraic closure of $\mathbb{Q}$, and $G\_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ the absolute Galois group $\mathbb{Q}$. The action $G\_{\mathbb{Q}}$ on adelic Tate module $E$ induces representation $\rho\_E\colon G\_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).$ goal this paper is to explain...

Let X be a projective geometrically irreducible non-singular algebraic curve defined over a finite field Fq2 of order q . If the number of Fq2 -rational points of X satisfies the Hasse-Weil upper bound, then X is said to be Fq2 -maximal. For a point P0 ∈ X (Fq2 ), let π be the morphism arising from the linear series D := |(q + 1)P0|, and let N := dim(D). It is known that N ≥ 2 and that π is ind...

Here we study rank 2 arithmetically Cohen-Macaulay vector bundles on a a ruled surface over a smooth genus q curve, essentially proving their non-existence if q ≥ 2 and the ruled surface is rather balanced.

Abstract Let C be a complex smooth projective algebraic curve endowed with an action of finite group G such that the quotient has genus at least 3. We prove if -curve is very general for these properties, then natural map from algebra $${{\mathbb {Q}}}G$$ Q G to {Q}}}$$...

For an elliptic curve E/Q, we determine the maximum number of twists E/Q it can have such that E(Q)tors ) E(Q)[2]. We use these results to determine the number of distinct quadratic fields K such that E(K)tors ) E(Q)tors. The answer depends on E(Q)tors and we give the best possible bound for all the possible cases.