On nef and semistable hermitian lattices, and their behaviour under tensor product

We study the behaviour of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces. One approach to show that semistable vector bundles in characteristic zero are preserved by tensor product is based on the notion of nef vector bundles. We revisit this approach and show how far it can be transferred to hermitian lattices. J.-B. Bost conjectured that semistable hermitian lattices are preserved by tensor product. Using properties of nef hermitian lattices, we establish an inequality which improves on earlier results in that direction. On the other hand, we show that, in contrast to nef vector bundles, nef hermitian lattices are not preserved by tensor product. We axiomatize our method in the general context of monoidal categories, and give an elementary proof of the fact that semistable multifiltered vector spaces (which play a role in diophantine approximation) are preserved by tensor product.Comment: revised versio

Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2

We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities of fluid flows and magnetic fields. To each three-component link in Euclidean 3-space, we associate a geometrically natural generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers, but patterned after J.H.C. Whitehead's integral formula for the Hopf invariant. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of 3-space, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we did this for three-component links in the 3-sphere. Komendarczyk has applied this approach in special cases to derive a higher order helicity for magnetic fields whose ordinary helicity is zero, and to obtain from this nonzero lower bounds for the field energy.Comment: 22 pages, 8 figures. arXiv admin note: text overlap with arXiv:1101.337

The Rabin cryptosystem revisited

The Rabin public-key cryptosystem is revisited with a focus on the problem of identifying the encrypted message unambiguously for any pair of primes. In particular, a deterministic scheme using quartic reciprocity is described that works for primes congruent 5 modulo 8, a case that was still open. Both theoretical and practical solutions are presented. The Rabin signature is also reconsidered and a deterministic padding mechanism is proposed.Comment: minor review + introduction of a deterministic scheme using quartic reciprocity that works for primes congruent 5 modulo

Entropies from coarse-graining: convex polytopes vs. ellipsoids

We examine the Boltzmann/Gibbs/Shannon $\mathcal{S}_{BGS}$ and the non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \ $\mathcal{S}_q$ \ and the Kaniadakis $\kappa$-entropy \ $\mathcal{S}_\kappa$ \ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between two generally different approaches: the Euclidean/Riemannian metric one that reflects independence and picks cubes as the fundamental cells and the symplectic/canonical one that picks spheres/ellipsoids for this role. Our discussion is motivated by and confined to the behaviour of Hamiltonian systems of many degrees of freedom. We see that Dvoretzky's theorem provides asymptotic estimates for the minimal dimension beyond which these two approaches are close to each other. We state and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe

Convexity and the Euclidean metric of space-time

We address the question about the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.Comment: 41 pages. No figures. Standard LaTeX2e. Change of affiliation of the author and mostly superficial changes in this version. Accepted for publication by "Universe" in a Special Issue with title: "100 years of Chronogeometrodynamics: the Status of Einstein's theory of Gravitation in its Centennial Year

A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments

The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equation for neutral and ionic solutions, respectively. In the present work solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented to the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of a ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency, and allow for the treatment of different boundary conditions, as for example surface systems. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes

Quintessential Maldacena-Maoz Cosmologies

Maldacena and Maoz have proposed a new approach to holographic cosmology based on Euclidean manifolds with disconnected boundaries. This approach appears, however, to be in conflict with the known geometric results [the Witten-Yau theorem and its extensions] on spaces with boundaries of non-negative scalar curvature. We show precisely how the Maldacena-Maoz approach evades these theorems. We also exhibit Maldacena-Maoz cosmologies with [cosmologically] more natural matter content, namely quintessence instead of Yang-Mills fields, thereby demonstrating that these cosmologies do not depend on a special choice of matter to split the Euclidean boundary. We conclude that if our Universe is fundamentally anti-de Sitter-like [with the current acceleration being only temporary], then this may force us to confront the holography of spaces with a connected bulk but a disconnected boundary.Comment: Much improved exposition, exponent in Cai-Galloway theorem fixed, axionic interpretation of scalar explained, JHEP version. 33 pages, 3 eps figure