Divergence operators and odd Poisson brackets

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the "odd laplacian", $\Delta$, of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).Comment: 27 pages; new Section 1, introduction and conclusion re-written, typos correcte

Curves with constant curvature ratios

Curves in Rn for which the ratios between two consecutive curvatures are constant are characterized by the fact that their tangent indicatrix is a geodesic in a flat torus. For n = 3, 4, spherical curves of this kind are also studied and compared with intrinsic helices in the sphere

Explicit Bézier control net of a PDE surface

The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface. In other words, we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points. The set of scalar coefficients of these combinations works like a mould for PDE surfaces. Thus, once this mould has been computed for a given degree, real-time manipulation of the resulting surfaces becomes possible by modifying the prescribed information.The work was partially supported by Spanish Ministry of Economy and Competitiveness DGICYT grant MTM2015-64013

Bézier surfaces with prescribed diagonals

The affine space of all tensor product Bézier patches of degree n × n with prescribed main diagonal curves is determined. First, the pair of Bézier curves which can be diagonals of a Bézier patch is characterized. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves or tangent planes along boundary curves are also prescribed.Funding for open access charge: CRUE-Universitat Jaume

Algunas consideraciones acerca de las SSTC 75/1984, 53/1985 y 70/1985 (Homenaje al profesor Tomás y Valiente en su X Aniversario, 1996-2006)

In this work is approached a comparative study of the STC 75/1984, 53/1985 and 70/1985 and of the perspective of the same ones in relation to the thorny question of abortion. At that moment (around the middle of the Eighty), the reformation of the legislation on the interruption of the pregnancy was object of public debate in the Spanish life, not only as an initiative of the Government, but also of some social organizations which aimed to increase the suppositions in those this delicate matter could not be punishable in Spain. In this sense, one of the main defenders of the decriminalization of abortion in Spain was Francisco Tomás y Valiente (1932- 1996)

Incidencias en la trayectoria de la Comisión General Penal de Codificación (1858-1863). La actuación del ministro Fernández Negrete y sus consecuencias

Este artículo aborda las incidencias en la Sección Penal de la Comisión General de Codificación entre 1858 y 1863, etapa en la que nuestro incipiente Estado liberal pedía con urgencia una nueva organización judicial, el cambio de su normativa criminal y la creación de un procedimiento penal. Junto a la Comisión, el Ministro Fernández Negrete fue uno de los principales defensores de la reforma de la justicia criminal española. En los trabajos de ambos se basaron las Leyes Orgánica del Poder Judicial de 1870 y Provisional de Enjuiciamiento Criminal de 1872, y el Código Penal de 1870.This article approaches the incidences in the Penal Section of the General Commission of Code between 1858 and 1863, stage in which our incipient liberal State requested with urgency a new judicial organization, its normative criminal’s change and the creation of a penal procedure. Next to the Commission, the Minister Fernández Negrete was one of the main defenders of the reformation of the Spanish criminal justice. In the works of both the Organic Laws of the Judicial Power of 1870 were based and Provisional of Criminal Prosecution of 1872, and the Penal Code of 1870.peerReviewe