Static and dynamic simulation in the classical two-dimensional anisotropic Heisenberg model

By using a simulated annealing approach, Monte Carlo and molecular-dynamics techniques we have studied static and dynamic behavior of the classical two-dimensional anisotropic Heisenberg model. We have obtained numerically that the vortex developed in such a model exhibit two different behaviors depending if the value of the anisotropy $\lambda$ lies below or above a critical value $\lambda_c$ . The in-plane and out-of-plane correlation functions ($S^{xx}$ and $S^{zz}$) were obtained numerically for $\lambda \lambda_c$ . We found that the out-of-plane dynamical correlation function exhibits a central peak for $\lambda > \lambda_c$ but not for $\lambda < \lambda_c$ at temperatures above $T_{BKT}$ .Comment: 7 page

Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4-manifolds

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.Comment: To appear in Mathematische Nachrichte

Stability of branched pull-back projective foliations

We prove that, if $n\geq 3$, a singular foliation $\mathcal{F}$ on $\mathbb P^n$ which can be written as pull-back, where $\mathcal{G}$ is a foliation in ${\mathbb P^2}$ of degree $d\geq2$ with one or three invariant lines in general position and $f:{\mathbb P^n}--->{\mathbb P^2}$, $deg(f)=\nu\geq2,$ is an appropriated rational map, is stable under holomorphic deformations. As a consequence we conclude that the closure of the sets $\{\mathcal {F}= f^{*}(\mathcal{G})\}$ are new irreducible components of the space of holomorphic foliations of certain degrees.Comment: arXiv admin note: substantial text overlap with arXiv:1503.07827, arXiv:1503.0071

Branched pull-back components of the space of codimension 1 foliations on Pn\mathbb P^n

Let $\mathcal{F}$ be written as $f^{*}\mathcal{G}$, where $\mathcal{G}$ is a foliation in ${\mathbb P^2}$ with three invariant lines in general position, say $(XYZ)=0$, and $f:{\mathbb P^n}--->{\mathbb P^2}$, $f=(F^\alpha_{0}:F^\beta_{1}:F^\gamma_{2})$ is a nonlinear rational map. Using local stability results of singular holomorphic foliations, we prove that: if $n\geq 3$, the foliation $\mathcal{F}$ is globally stable under holomorphic deformations. As a consequence we obtain new irreducible componentes for the space of codimension one foliations on $\mathbb P^n$. We present also a result which characterizes holomorphic foliations on ${\mathbb P^n}, n\geq 3$ which can be obtained as a pull back of foliations on ${\mathbb P^2}$ of degree $d\geq2$ with three invariant lines in general position.Comment: arXiv admin note: text overlap with arXiv:1503.0071

A possible analogy between the dynamics of a skydiver and a scalar field: cosmological consequences

The cosmological consequences of a slow rolling scalar field with constant kinetic term in analogy to the vertical movement of a skydiver after reaching terminal velocity are investigated. It is shown that the terminal scalar field hypothesis is quite realistic. In this approach, the scalar field potential is given by a quadratic function of the field. This model provides solutions in which the Universe was dominated in the past by a mixture of baryons and dark matter, is currently accelerating (as indicated by type Ia supernovae data), but will be followed by a contraction phase. The theoretical predictions of this model are consistent with current observations, therefore, a terminal scalar field is a viable candidate to dark energy.Comment: Accepted for publicatio

A description of several coordinate systems for hyperbolic spaces

This article simply presents several coordinate systems for 2 and 3-dimensional hyperbolic spaces, describing the general solutions of Helmholtz equation in each one of these systems.Comment: 26 pages, standard LaTeX article (uses packages graphics and fancyheadings), 3 figure

Irreducible components of the space of foliations by surfaces

Let $\mathcal{F}$ be written as $f^{*}(\mathcal{G})$, where $\mathcal{G}$ is a $1$-dimensional foliation on ${\mathbb P^{n-1}}$ and $f:{\mathbb P^n}--->{\mathbb P^{n-1}}$ a non-linear generic rational map. We use local stability results of singular holomorphic foliations, to prove that: if $n\geq 4$, a foliation $\mathcal{F}$ by complex surfaces on $\mathbb P^n$ is globally stable under holomorphic deformations. As a consequence, we obtain irreducible components for the space of two-dimensional foliations in $\mathbb P^n$. We present also a result which characterizes holomorphic foliations on ${\mathbb P^n}, n\geq 4$ which can be obtained as a pull back of 1- foliations in ${\mathbb P^{n-1}}$ of degree $d\geq2$

Some remarks on marginally trapped surfaces and geodesic incompleteness

In a recent paper, Eichmair, Galloway and Pollack have proved a Gannon-Lee-type singularity theorem based on the existence of marginally outer trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic spacetimes. However, one might wonder whether the corresponding incomplete geodesics could still be complete in a possible non-globally hyperbolic extension of spacetime. In this note, some variants of that result are given with weaker causality assumptions, thus suggesting that the answer is generically negative, at least if the putative extension has no closed timelike curves. However, unlike in the case of MOTS, on which only the outgoing family of normal geodesics is constrained, we have found it necessary in our proofs to impose also a weak convergence condition on the ingoing family of normal geodesics. In other words, we consider marginally trapped surfaces (MTS) in chronological spacetimes, introducing the natural notion of a generic MTS. In particular, a Hawking-Penrose-type singularity theorem is proven in chronological spacetimes with dimensions greater than 2 containing a generic MTS. Such surfaces naturally arise as cross-sections of quasi-local generalizations of black hole horizons, such as dynamical and trapping horizons. We end with some comments on the existence of MTS in initial data sets.Comment: 11 pages, no figures; erroneous argument corrected, results unchanged; one reference and a few remarks adde

The asphericity of random 2-dimensional complexes

We study random 2-dimensional complexes in the Linial - Meshulam model and prove that for the probability parameter satisfying $$p\ll n^{-46/47}$$ a random 2-complex $Y$ contains several pairwise disjoint tetrahedra such that the 2-complex $Z$ obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex $Z$ satisfies the Whitehead conjecture, i.e. any subcomplex $Z'\subset Z$ is aspherical. This implies that $Y$ is homotopy equivalent to a wedge $Z\vee S^2\vee...\vee S^2$ where $Z$ is a 2-dimensional aspherical simplicial complex. We also show that under the assumptions $$c/n3$and$0<\epsilon<1/47$, the complex$Z$is genuinely 2-dimensional and in particular, it has sizable 2-dimensional homology; it follows that in the indicated range of the probability parameter$p$the cohomological dimension of the fundamental group$\pi_1(Y)$ of a random 2-complex equals 2.Comment: 11 pages, 2 figure

Generalizing the MOND description of rotation curves

We present new mathematical alternatives for explaining rotation curves of spiral galaxies in the MOND context. For given total masses, it is shown that various mathematical alternatives to MOND, while predicting flat rotation curves for large galactic radii, predict curves with different peculiar features for smaller radii. They are thus testable against observational data.Comment: elsart style (uses packages graphics and amssymb), 9 pages, 3 figure