Artículos con la etiqueta ‘análisis’

Metric number theory, lacunary series and systems of dilated functions

Por • 20 jul, 2013 • Category: Leyes

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{k \geq 1}$ the system $(\{n_k x\})_{k \geq 1}$ (or, more general, $(f(n_k x))_{k \geq 1}$ for a 1-periodic function $f$) shows many asymptotic properties which are typical for the behavior of systems of \emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena.

The Foundations of Analysis

Por • 31 mar, 2013 • Category: Crítica

This is a detailed and self-contained introduction to the real number system from a categorical perspective. We begin with the categorical definition of the natural numbers, review the Eudoxus theory of ratios as presented in Book V of Euclid, and then use these classical results to define the positive real numbers categorically.

¿Para qué nos endeudamos?

Por • 26 ago, 2010 • Category: Economía

El argumento preferido de quienes siempre acuden al canto de sirena para que Venezuela se endeude es que con la deuda se van a financiar proyectos de desarrollo que permiten pagar la deuda contraída y que el país tiene un bajo nivel de deuda respecto al tamaño de la economía. Suena cautivante esta conseja pero al mismo tiempo es muy peligrosa