# Publicaciones del CMAT

Publicaciones de los docentes del Centro de Matemática

Armentano, D (2013).

## Complexity of Path-Following Methods for the Eigenvalue Problem

Foundation of Computational Mathematics, (submitted).

A unitarily invariant projective framework is introduced to analyze the complexity of path–following methods for the eigenvalue problem. A condition number, and its relation to the distance to ill–posedness, is given. A Newton map appropriate for this context is defined, and a version of Smale’s γ-Theorem is proven. The main result of this paper bounds the complexity of path–following methods in terms of the length of the path in the condition metric.

Armentano, D and Cucker, F (2013).

## A Randomized Homotopy for the Hermitian Eigenpair Problem

Foundation of Computational Mathematics, (submitted).

We describe and analyze a randomized homotopy algorithm for the Hermitian eigenvalue problem. Given an n × n Hermitian matrix A the algorithm returns, almost surely, a pair (λ, v) which approximates, in a very strong sense, an eigenpair of A. We prove that the expected cost of this algorithm, where the expectation is both over the random choices of the algorithm and a probability distribution on the input matrix A, is O(n^4 ), that is, quadratic on the input size. Our result relies on a cost assumption for some pseudo-random number generators whose rationale is argued by us.

Potrie, R (2013).

## On the work of Jorge Lewowicz on expansive homeomorphisms

Publicaciones Matemáticas de Uruguay, 14:1-25.

Fisher, T, Potrie, R, and Sambarino, M (2013).

## Dynamical coherence for partially hyperbolic diffeomorphisms isotopic to Anosov in tori

Math. Z. :1-31.

Maderna, E (2013).

## Translation invariance of weak KAM solutions of the Newtonian N-body problem

Proc. Amer. Math. Soc., 141(8):2809-2816.

We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c ≥ 0, of the classical N -body problem in an Euclidean space E of dimension k ≥ 2. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant (c = 0) also called weak KAM solutions. We show that all these solutions are invariant under the action of E by translations on the space of configurations. We deduce the existence of non-invariant solutions for the super-critical equations (c > 0).

Maderna, E (2013).

## Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

Regular and Chaotic Dynamics, 18(6):661-678.

For N-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in 1/r^a with 0<a<2 we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.

Treibich, A (2013).

## Systèmes d'équations polynomiales pour les revêtements hyperelliptiques d-osculants

Comptes Rendus de l'Académie des Sciences Serie I Mathématiques.

Abella, A, Ferrer Santos, W, and Haim, M (2012).

## Some constructions of compact quantum groups

Sao Paulo Journal of Mathematical Sciences, 6(1):1-40.

Haim, M, Iovanov, M, and Torrecilla, B (2012).

## On two conjectures of Faith

Journal of Algebra, 367:166-175.

Potrie, R (2012).

## Recurrence of non-resonant homeomorphisms on the torus

Proceedings of the American Mathematical Society, 140(11):3973–3981.

Maderna, E (2012).

## On weak KAM theory for N-body problems

Ergodic Theory Dynam. Systems, 32(3):1019-1041.

We consider N-body problems with (1/r)^2κ potential where κ ∈ (0, 1), including the Newtonian case (κ = 1/2). Given R > 0 and T > 0, we find a uniform upper bound for the minimal action of paths binding in time T any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding in free time two configurations). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax-Oleinik semigroup and we show that they are global viscosity solutions of the corresponding Hamilton-Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

Avritzer, D, Pan, I, and González-Sprinberg, G (2012).

## On singular quadratic complexes, quintic curves and Cremona transformations

Rend. Circolo Mat. di Palermo, 61(2):201-240.

Ferrer, W, Haim, M, and Abella, A (2012).

## Some constructions of compact quantum groups

Sao Paulo Journal of mathematical sciences, 6(1):1-40.

The purpose of this paper is to consider some basic constructions in the category of compact quantum groups --for example de case of extensions, of Drinfeld twists, of matched pairs, of extensions, of linked pairs and of cocycle Singer pairs -- with special emphasis in the finite dimensional situation. We give conditions, in some cases necessary and sufficient, to extend to the new objects the original compact structure. We illustrate the results in the case of matched pairs of groups.

Armentano, D (2011).

## A review of some recent results on Random Polynomials over R and over C

Publicaciones Matemáticas del Uruguay, 12:1-14.

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the distribution of the roots of certain complex random polynomials. We discuss a recent result in this direction, which shows that the associated points in the sphere (via the stereographic projection) are surprisingly well-suited with respect to the minimal logarithmic energy on the sphere.

Pan, I (2011).

## On the untwisting of general de Jonquières and cubo-cubic Cremona transformations of P^3

Contributions to Algebra and Geometry, 52:1-21.

Pan, I (2011).

## On Cremona transformations of \P^3 which factorize in a minimal form

Rev. Unión Mat. Angentina.

We consider Cremona transformations of the complex projective space of dimension 3 which factorize as a product of at most two elementary links of type II, without small contractions, connecting two Fano 3-folds. We show that there are essentially eight classes of such transformations and we give a geometric description of elements in each of these classes.

Rittatore, A and Renner, L (2011).

## The ring of regular functions of an algebraic monoid

Transactions of the American Mathematical Society, 363:6671 - 6683.

Bourel, M, Rittatore, A, and Dickenstein, A (2011).

## Self-dual projective toric varieties

Journal of the London Mathematical Society-Second Series, 84(2):514-540.

Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T-module P(V). We determine when a projective toric subvariety X of P(V) is self-dual, in terms of the configuration of weights of V.

Treibich, A (2011).

## Non-linear evolution equations and hyperelliptic covers of elliptic curves

Regular and Chaotic Dynamics , 16(3-4):290-310.