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The observational counterparts of theoretically predicted first hydrostatic cores (FHSC) have been searched for in the interstellar medium for nearly two decades now. Distinguishing them from other types of more evolved but still embedded objects remains a challenge because these objects have a short lifetime, are small, and embedded in a dense cocoon. One possible lead to finding them is the characterization of the outflows that are launched by these objects. We observed the L1451-mm FHSC candidate with the NOEMA interferometer in order to study the emission of several molecules. A nonlocal thermodynamic equilibrium analysis of the CH$_3$OH detected lines was performed to retrieve the physical conditions of the emitting region around the central source, together with the CH$_3$OH, SiO, CS, and H$_2$CO column densities. Of the targeted molecules, we detected lines of c-C$_3$H$_2$, CH$_3$OH, CS, C$^{34}$S, SO, DCN, DCO$^+$, H$_2$CO, HC$_3$N, HDCO, and SiO. One of the methanol lines appears to be a maser line. The detection of this class I maser and the SiO line in L1451-mm support the presence of a low-velocity and compact outflow. The excitation conditions of the thermal lines of methanol are also compatible with shocks (H$_2$ density of $\sim 3\times 10^6$~cm$^{-3}$ and a temperature higher than 40~K). Although these low-velocity outflows are theoretically predicted by some models of FHSC, these models also predict the shock temperature to be below 20~K, that is, not evaporating methanol. In addition, the predicted velocities would not erode the grains and release silicon in the gas phase. We therefore conclude that these new observations favor the hypothesis that L1451-mm would be at a very early protostellar stage, launching an outflow nearly on the plane of the sky with a higher velocity than is observed., Comment: Accepted for publication in A&A

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., Comment: 13 pages

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., Comment: 26 pages, 20 figures, A&A in press

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 $U$.

In this paper we establish a new equivalence relation on the spaces of almost periodic functions which allows us to prove a result like Bohr's equivalence theorem extended to the case of all these functions., Comment: In this paper we get a result like Bohr's equivalence theorem extended to the case of all almost periodic functions, not only for the case of existence of integral basis. arXiv admin note: substantial text overlap with arXiv:1711.04112

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 $\zeta(s)$., Comment: This is a modified version of our homonymous paper published in Ramanujan J. (2018). As we pointed out in our corrigendum (2019), we correct a gap found in the original version which is due to a property that does not generally hold, but it does under the assumption of existence of an integral basis. The earlier equivalence relation is revised to adapt correctly the situation in the general case. arXiv admin note: text overlap with arXiv:1711.04122

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., Comment: Because of a mistake detected in one of the references, the equivalence relation which is inspired by that of Bohr is revised to adapt correctly the situation in the general case. arXiv admin note: text overlap with arXiv:1801.08035

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., Comment: Because of a mistake detected in one of the references, the previous version of this paper has been modified by the authors to restrict the scope of its application to the case of existence of an integral basis