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Carlos Gutierrez

Publications and source records attributed to Carlos Gutierrez.

14 recordsLinked to original sources
arXiv2007

The Implicit Function Theorem for continuous functions

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily C^1) maps.

math.AT↗
arXiv2007

Planar maps whose second iterate has a unique fixed point

Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most one element, where Fix(F) denotes the set of fixed points of F. (b) If Spec(F) is disjoint of the real line R, then Fix(F^2) has at most one element. (c) If F is a C^1 map and, for all p belonging to R^2, the derivative F'(p) is neither a homothety nor has simple real eigenvalues, then Fix(F^2) has at most one element, provided that Spec(F) is disjoint of either (c1) the union of the number 0 with the intervals ]-\infty, -1] and [1,\infty[, or (c2) the interval [-1-a, 1+a]. Conditions under which Fix(F^n), with n>1, is at most unitary are considered.

math.DS↗
arXiv2006

Foliations and Polynomial Diffeomorphisms of $\mathbb{R}^{3}$

Let $Y=(f,g,h):\mathbb{R}^{3} \to \mathbb{R}^{3}$ be a $C^{2}$ map and let $\spec(Y)$ denote the set of eigenvalues of the derivative $DY_p$, when $p$ varies in $\mathbb{R}^3$. We begin proving that if, for some $ε>0,$ $\spec(Y)\cap (-ε,ε)=\emptyset,$ then the foliation $\mathcal{F}(k),$ with $k\in \{f,g,h\},$ made up by the level surfaces $\{k={\rm constant}\},$ consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of $\mathbb{R}^n.$

math.DS↗
arXiv2006

Injectivity of differentiable maps R^2 --> R^2 at infinity

The main result given in Theorem~1.1 is a condition for a map $X$, defined on the complement of a disk $D$ in R^2 with values in R^2, to be extended to a topological embedding of R^2, not necessarily surjective. The map $X$ is supposed to be just differentiable with the condition that, for some $e>0,$ at each point the eigenvalues of the differential do not belong to the real interval $(-e,\infty).$ The extension is obtained by restricting X to the complement of some larger disc. The result has important connections with the property of asymptotic stability at infinity for differentiable vector fields.

math.DS↗
arXiv2006

On Peixoto's Conjecture for Flows on Non-orientable 2-Manifolds

Let M be a non-orientable compact 2-manifold of genus 4. Then there exists a family of quasi-minimal, Kupka-Smale smooth vector fields X_r in M, depending smoothly on 0<=r<e, such that, for some flow box V in M of X_0, and for all 0<=r,v<e, (1) X_r restricted to V is a flow box; (2) If r is not equal to v, X_r is a C^\infty--twist perturbation of X_v localized in V; (3) X_r and X_v are topologically equivalent.

math.DS↗
arXiv2005

Iterated Images and the Plane Jacobian Conjecture

We show that the iterated images of a Jacobian pair stabilize; that is, the k-th iterates of a polynomial map of complex two-space to itself with a nonzero constant Jacobian determinant all have the same image for sufficiently large k. More generally, we obtain the same result for open polynomial maps of a closed algebraic subset X of complex N-space to itself that have finite coimage, and for cofinite subsets of such an X invariant under the map. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.

math.AG↗
arXiv2004

Very Small Array observations of the Sunyaev-Zel'dovich effect in nearby galaxy clusters

We present VSA observations (~34GHz) on scales ~20 arcmin towards a complete, X-ray-flux-limited sample of seven clusters at redshift z<0.1. Four have significant SZ detections in the presence of CMB primordial anisotropy. We use a bayesian MCMC method for inference from the VSA data, with X-ray priors on cluster positions and temperatures, and radio priors on sources. We make assumptions of beta-model gas distributions and of hydrostatic equilibrium, to evaluate probability densities for the gas mass and total mass out to r_200. Our combined estimate of the gas fraction is 0.08^{+0.06}_{-0.04}h^{-1} The random errors are poor (note that the errors are higher than would have been obtained with the usual chi-squared method) but the control of bias is good. We have described the MCMC analysis method specifically in terms of SZ but hope the description will be of more general use. We find that the effects of primordial CMB contamination tend to be similar in the estimates of both the gas mass and total mass over our narrow range of angular scales, so that there is little effect of primordials on the gas fraction determination. Using our total mass estimates we find a normalisation of the mass-temperature relation based on the profiles from the VSA cluster pressure maps that is in good agreement with recent M-T determinations from X-ray cluster measurements.

astro-ph↗
arXiv2004

The Markus-Yamabe Conjecture for Differentiable vector fields of R^2

(a) Let X=(f,g) be a differentiable map in the plane (not necessarily C^1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DX(p) when p varies in R^2. If, for some ε>0, the set Spec(X) is disjoint of [0,ε) then X is injective. (b) Let X be a differentiable vector field such that X(0)=0 and $Re(z)< 0$ for all z in Spec(X). Then, for all p in R^2, there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to {0}.

math.DS↗
arXiv2003

Bifurcations of Umbilic Points and Related Principal Cycles

The simplest patterns of qualitative changes on the configurations of lines of principal curvature} around umbilic points on surfaces whose immersions into $\mathbb R^3$ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper. Global effects, due to umbilic bifurcations, on these configurations such as the appearance and annihilation of periodic principal lines, called also principal cycles, are also studied here.

math.DG↗
arXiv2003

First results from the Very Small Array -- III. The CMB power spectrum

We present the power spectrum of the fluctuations in the cosmic microwave background detected by the Very Small Array (VSA) in its first season of observations in its compact configuration. We find clear detections of first and second acoustic peaks at l~200 and l~550, plus detection of power on scales up to l=800. The VSA power spectrum is in very good agreement with the results of the Boomerang, Dasi and Maxima telescopes despite the differing potential systematic errors.

astro-ph↗
arXiv1999

On $C^r-$closing for flows on 2-manifolds

For some full measure subset B of the set of iet's (i.e. interval exchange transformations) the following is satisfied: Let X be a $C^r$, $1\le r\le \infty$, vector field, with finitely many singularities, on a compact orientable surface M. Given a nontrivial recurrent point $p\in M$ of X, the holonomy map around p is semi-conjugate to an iet $E :[0,1) \to [0,1).$ If $E\in B$ then there exists a $C^r$ vector field Y, arbitrarily close to X, in the $C^r-$topology, such that Y has a closed trajectory passing through p.

math.DS↗
arXiv1999

Two-Dimensional Line Strength Maps in Three Well-studied Early-Type Galaxies

Integral field spectroscopy has been obtained for the nuclear regions of 3 large, well-studied, early-type galaxies. From these spectra we have obtained line strength maps for about 20 absorption lines, mostly belonging to the Lick system. An extensive comparison with multi-lenslet spectroscopy shows that accurate kinematic maps can be obtained, and also reproducible line strength maps. Comparison with long-slit spectroscopy also produces good agreement. We show that Mg is enhanced with respect to Fe in the inner disk of one of the three galaxies studied, the Sombrero. [Mg/Fe] there is larger than in the rest of the bulge. The large values of Mg/Fe in the central disk are consistent with the centres of other early-type galaxies, and not with large disks, like the disk of our Galaxy, where [Mg/Fe] is approximately 0. We confirm with this observation a recent result of Worthey (1998) that Mg/Fe is determined by the central kinetic energy, or escape velocity, of the stars, only, and not by the formation time scale of the stars. A stellar population analysis using the models of Vazdekis et al. (1996) shows that our observed H gamma agrees well with what is predicted based on the other lines. Using the line strength of the Ca II IR triplet as an indicator of the abundance of Ca, we find that Ca follows Fe, and not Mg, in these galaxies. This is peculiar, given the fact that Ca is an alpha-element. Finally, by combining the results of this paper with those of Vazdekis et al. (1997) we find that the line strength gradients in the three galaxies are primarily caused by variations in metallicity.

astro-ph↗