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T. Vidal

Publications and source records attributed to T. Vidal.

7 recordsLinked to original sources
arXiv2019

The converse of Bohr's equivalence theorem with Fourier exponents linearly independent over the rational numbers

Given two arbitrary almost periodic functions with associated Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip $V$, where both functions assume the same set of values on every open vertical substrip included in $V$, is a necessary and sufficient condition for both functions to have the same region of almost periodicity and to be $^*$-equivalent or Bohr-equivalent. This result represents the converse of Bohr's equivalence theorem for this particular case.

math.CA↗
arXiv2018

Gas phase Elemental abundances in Molecular cloudS (GEMS) I. The prototypical dark cloud TMC 1

GEMS is an IRAM 30m Large Program whose aim is determining the elemental depletions and the ionization fraction in a set of prototypical star-forming regions. This paper presents the first results from the prototypical dark cloud TMC 1. Extensive millimeter observations have been carried out with the IRAM 30m telescope (3mm and 2mm) and the 40m Yebes telescope (1.3cm and 7mm) to determine the fractional abundances of CO, HCO+, HCN, CS, SO, HCS+, and N2H+ in three cuts which intersect the dense filament at the well-known positions TMC 1-CP, TMC 1-NH3, and TMC 1-C, covering a visual extinction range from Av~3 to ~20mag. Two phases with differentiated chemistry can be distinguished: i) the translucent envelope with molecular hydrogen densities of (1-5)x10$^3$ cm$^{-3}$; and ii) the dense phase, located at Av>10mag, with molecular hydrogen densities >10$^4$ cm$^{-3}$. Observations and modeling show that the gas phase abundances of C and O progressively decrease along the C+/C/CO transition zone where C/H~8x10$^{-5}$ and C/O~0.8-1, until the beginning of the dense phase at Av~10 mag. This is consistent with the grain temperatures being below the CO evaporation temperature in this region. In the case of sulfur, a strong depletion should occur before the translucent phase where we estimate a S/H~(0.4 - 2.2) x10$^{-6}$, an abundance ~7-40 times lower than the solar value. A second strong depletion must be present during the formation of the thick icy mantles to achieve the values of S/H measured in the dense cold cores (S/H~8x10$^{-8}$). Based on our chemical modeling, we constrain the value of $\zeta_{\rm H_2}$ to ~(0.5 - 1.8) x10$^{-16}$ s$^{-1}$ in the translucent cloud.

astro-ph.GA↗
arXiv2018

On the real projections of zeros of almost periodic functions

This paper deals with the set of the real projections of the zeros of an arbitrary almost periodic function defined in a vertical strip $U$. It provides practical results in order to determine whether a real number belongs to the closure of such a set. Its main result shows that, in the case that the Fourier exponents $\{\lambda_1,\lambda_2,\lambda_3,\ldots\}$ of an almost periodic function are linearly independent over the rational numbers, such a set has no isolated points in $U$.

math.CV↗
arXiv2018

Almost Periodic Functions in terms of Bohr's Equivalence Relation

In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner's result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, $\zeta(s)$, can be uniformly approximated in $\{s=\sigma+it:\sigma>1\}$ by certain vertical translates of $\zeta(s)$.

math.CV↗
arXiv2017

A generalization of Bohr's Equivalence Theorem

Based on a generalization of Bohr's equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by certain classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a result like Bohr's equivalence theorem extended to the case of these functions.

math.CV↗
arXiv2017

Bohr's equivalence relation in the space of Besicovitch almost periodic functions

Based on Bohr's equivalence relation which was established for general Dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of Besicovitch, $B(\mathbb{R},\mathbb{C})$, defined in terms of polynomial approximations. From this, we show that in an important subspace $B^2(\mathbb{R},\mathbb{C})\subset B(\mathbb{R},\mathbb{C})$, where Parseval's equality and Riesz-Fischer theorem holds, its equivalence classes are sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class.

math.FA↗