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Autoresonance is a phenomenon of physical interest that may take place when a nonlinear oscillator is forced at a frequency that varies slowly. The stroboscopic averaging method (SAM), which provides an efficient numerical technique for the integration of highly oscillatory systems, cannot be used directly to study autoresonance due to the slow changes of the forcing frequency. We study how to modify SAM to cater for such slow variations. Numerical experiments show the computational advantages of using SAM.

The Orion Molecular Cloud complex, one of the nearest (D = 406 pc) and most extensively studied massive star-forming regions, is ideal for constraining the physics of stellar feedback, but its ~12 deg diameter on the sky requires a dedicated approach to mapping ionized gas structures within and around the nebula. The Sloan Digital Sky Survey (SDSS-V) Local Volume Mapper (LVM) is a new optical integral field unit (IFU) that will map the ionized gas within the Milky Way and Local Group galaxies, covering 4300 deg^2 of the sky with the new LVM Instrument. We showcase optical emission line maps from LVM covering 12 deg^2 inside of the Orion belt region, with 195,000 individual spectra combined to produce images at 0.07 pc (35.3") resolution. This is the largest IFU map made (to date) of the Milky Way, and contains well-known nebulae (the Horsehead Nebula, Flame Nebula, IC 434, and IC 432), as well as ionized interfaces with the neighboring dense Orion B molecular cloud. We resolve the ionization structure of each nebula, and map the increase in both the [SII]/Ha and [NII]/Ha line ratios at the outskirts of nebulae and along the ionization front with Orion B. [OIII] line emission is only spatially resolved within the center of the Flame Nebula and IC 434, and our ~0.1 pc scale line ratio diagrams show how variations in these diagnostics are lost as we move from the resolved to the integrated view of each nebula. We detect ionized gas emission associated with the dusty bow wave driven ahead of the star sigma Orionis, where the stellar wind interacts with the ambient interstellar medium. The Horsehead Nebula is seen as a dark occlusion of the bright surrounding photo-disassociation region. This small glimpse into Orion only hints at the rich science that will be enabled by the LVM., Comment: 13 pages, 10 figures, accepted for publication in A&A

We present a framework that allows for the non-asymptotic study of the $2$-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the strongly log-concave case. This allows us to study in a unified way a number of different integrators proposed in the literature for the overdamped and underdamped Langevin dynamics. In addition, we analyse a novel splitting method for the underdamped Langevin dynamics which only requires one gradient evaluation per time step. Under an additional smoothness assumption on a $d$--dimensional strongly log-concave distribution with condition number $\kappa$, the algorithm is shown to produce with an $\mathcal{O}\big(\kappa^{5/4} d^{1/4}\epsilon^{-1/2} \big)$ complexity samples from a distribution that, in Wasserstein distance, is at most $\epsilon>0$ away from the target distribution., Comment: 29 pages, 2 figures

We construct integrators to be used in Hamiltonian (or Hybrid) Monte Carlo sampling. The new integrators are easily implementable and, for a given computational budget, may deliver five times as many accepted proposals as standard leapfrog/Verlet without impairing in any way the quality of the samples. They are based on a suitable modification of the processing technique first introduced by J.C. Butcher. The idea of modified processing may also be useful for other purposes, like the construction of high-order splitting integrators with positive coefficients.

Yebes 40\,m radio telescope is the main and largest observing instrument at Yebes Observatory and it is devoted to Very Long Baseline Interferometry (VLBI) and single dish observations since 2010. It has been covering frequency bands between 2\,GHz and 90\,GHz in discontinuous and narrow windows in most of the cases, to match the current needs of the European VLBI Network (EVN) and the Global Millimeter VLBI Array (GMVA). Nanocosmos project, a European Union funded synergy grant, opened the possibility to increase the instantaneous frequency coverage to observe many molecular transitions with single tunnings in single dish mode. This reduces the observing time and maximises the output from the telescope. We present the technical specifications of the recently installed 31.5-50 GHz (Q band) and 72-90.5 GHz (W band) receivers along with the main characteristics of the telescope at these frequency ranges. We have observed IRC+10216, CRL 2688 and CRL 618, which harbour a rich molecular chemistry, to demonstrate the capabilities of the new instrumentation for spectral observations in single dish mode. The results show the high sensitivity of the telescope in the Q band. The spectrum of IRC+10126 offers a signal to noise ratio never seen before for this source in this band. On the other hand, the spectrum normalised by the continuum flux towards CRL\,618 in the W band demonstrates that the 40~m radio telescope produces comparable results to those from the IRAM 30~m radio telescope, although with a smaller sensitivity. The new receivers fulfil one of the main goals of Nanocosmos and open the possibility to study the spectrum of different astrophysical media with unprecedented sensitivity., Comment: 22 pages, 27 figures, 6 tables. Refereed manuscript

In this manuscript, we study the properties of a family of second-order differential equations with damping, its discretizations and their connections with accelerated optimization algorithms for $m$-strongly convex and $L$-smooth functions. In particular, using the Linear Matrix Inequality LMI framework developed by \emph{Fazlyab et. al. $(2018)$}, we derive analytically a (discrete) Lyapunov function for a two-parameter family of Nesterov optimization methods, which allows for the complete characterization of their convergence rate. In the appropriate limit, this family of methods may be seen as a discretization of a family of second-order ordinary differential equations for which we construct(continuous) Lyapunov functions by means of the LMI framework. The continuous Lyapunov functions may alternatively, be obtained by studying the limiting behaviour of their discrete counterparts. Finally, we show that the majority of typical discretizations of the family of ODEs, such as the Heavy ball method, do not possess Lyapunov functions with properties similar to those of the Lyapunov function constructed here for the Nesterov method., Comment: 21 pages, 1 figure

This paper is devoted to investigate whether the popular No U-turn (NUTS) sampling algorithm is correct, i.e.\ whether the target probability distribution is \emph{exactly} conserved by the algorithm. It turns out that one of the Gibbs substeps used in the algorithm cannot always be guaranteed to be correct., Comment: Some statements in the paper are misleading. It is possible to think of NUTS at not being a slice/Gibbs sampler and, with an alternative interpretation, it may be be possible to prove that the algorithm is correct. In addition the experiment reported in Figure 2 should have had many initial states drawn from the target rather than using a single value

Industrial symbiosis involves creating integrated cycles of by-products and waste between networks of industrial actors in order to maximize economic value, while at the same time minimizing environmental strain. In such a network, the global environmental strain is no longer equal to the sum of the environmental strain of the individual actors, but it is dependent on how well the network performs as a whole. The development of methods to understand, manage or optimize such networks remains an open issue. In this paper we put forward a simulation model of by-product flow between industrial actors. The goal is to introduce a method for modelling symbiotic exchanges from a macro perspective. The model takes into account the effect of two main mechanisms on a multi-objective optimization of symbiotic processes. First it allows us to study the effect of geographical properties of the economic system, said differently, where actors are divided in space. Second, it allows us to study the effect of clustering complementary actors together as a function of distance, by means of a spatial correlation between the actors' by-products. Our simulations unveil patterns that are relevant for macro-level policy. First, our results show that the geographical properties are an important factor for the macro performance of symbiotic processes. Second, spatial correlations, which can be interpreted as planned clusters such as Eco-industrial parks, can lead to a very effective macro performance, but only if these are strictly implemented. Finally, we provide a proof of concept by comparing the model to real world data from the European Pollutant Release and Transfer Register database using georeferencing of the companies in the dataset. This work opens up research opportunities in interactive data-driven models and platforms to support real-world implementation of industrial symbiosis., Comment: 23 pages, 9 figures, 2 tables

The leapfrog integrator is routinely used within the Hamiltonian Monte Carlo method and its variants. We give strong numerical evidence that alternative, easy to implement algorithms yield fewer rejections with a given computational effort. When the dimensionality of the target distribution is high, the number of accepted proposals may be multiplied by a factor of three or more. This increase in the number of accepted proposals is not achieved by impairing any positive features of the sampling. We also establish new non-asymptotic and asymptotic results on the monotonic relationship between the expected acceptance rate and the expected energy error. These results further validate the derivation of one of the integrators we consider and are of independent interest., Comment: 37 pages, 8 figures

We consider the application of Runge-Kutta (RK) methods to gradient systems $(d/dt)x = -\nabla V(x)$, where, as in many optimization problems, $V$ is convex and $\nabla V$ (globally) Lipschitz-continuous with Lipschitz constant $L$. Solutions of this system behave contractively, i.e. the Euclidean distance between two solutions $x(t)$ and $\widetilde{x}(t)$ is a nonincreasing function of $t$. It is then of interest to investigate whether a similar contraction takes place, at least for suitably small step sizes $h$, for the discrete solution. Dahlquist and Jeltsch results' imply that (1) there are explicit RK schemes that behave contractively whenever $Lh$ is below a scheme-dependent constant and (2) Euler's rule is optimal in this regard. We prove however, by explicit construction of a convex potential using ideas from robust control theory, that there exists RK schemes that fail to behave contractively for any choice of the time-step $h$., Comment: 13 pages, 2 figures

We show that, for appropriate combinations of the values of the delay and the forcing frequency, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word-series techniques to obtain high-order averaged equations for differential equations without delay.

We introduce and analyze a family of heterogeneous multiscale methods for the numerical integration of highly oscillatory systems of delay differential equations with constant delays. The methodology suggested provides algorithms of arbitrarily high accuracy.

This article is an abridged and commented translation into Spanish of the 1815 memoir where Gauss introduced the quadrature rules now associated with his name. Gauss' work does not resemble at all the stardard text-book treatment of Gaussian quadrature. The original memoir is an example of mathematical virtuosity, based on a superb use of series, where the problem is reformulated as a problem in functional approximation that is solved by means of continued fractions., Comment: in Spanish

We present an analysis based on word combinatorics of splitting integrators for Ito or Stratonovich systems of stochastic differential equations. In particular we present a technique to write down systematically the expansion of the local error; this makes it possible to easily formulate the conditions that guarantee that a given integrator achieves a prescribed strong or weak order. This approach bypasses the need to use the Baker-Campbell-Hausdorff (BCH) formula and shows the existence of an order barrier of two for the attainable weak order. The paper also provides a succinct introduction to the combinatorics of words.

The implementation of multi-stage splitting integrators is essentially the same as the implementation of the familiar Strang/Verlet method. Therefore multi-stage formulas may be easily incorporated into software that now uses the Strang/Verlet integrator. We study in detail the two-parameter family of palindromic, three-stage splitting formulas and identify choices of parameters that may outperform the Strang/Verlet method. One of these choices leads to a method of effective order four suitable to integrate in time some partial differential equations. Other choices may be seen as perturbations of the Strang method that increase efficiency in molecular dynamics simulations and in Hybrid Monte Carlo sampling., Comment: 20 pages, 8 figures, 2 tables

We propose and analyze a heterogenous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method (SAM) suggested here may provide approximations with $\(\mathcal{O}(H^2+1/\Omega^2)\)$ errors with a computational effort that grows like $\(H^{-1}\)$ (the inverse of the stepsize), uniformly in the forcing frequency Omega.

In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, {\em universal} version of it which is solved algebraically; then, the results are tranferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.

We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the tech- nique applied here, the tasks of constructing the averaged system and the associ- ated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of sim- ple recursions. As distinct from the situation with other approaches, with word- series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approxima- tions to the true dynamics.

Observations of 28SiO v=0 J=1-0 line emission (7-mm wavelength) from AGB stars show in some cases peculiar profiles, composed of a central intense component plus a wider plateau. Very similar profiles have been observed in CO lines from some AGB stars and most post-AGB nebulae and, in these cases, they are clearly associated with the presence of conspicuous axial symmetry and bipolar dynamics. We present systematic observations of 28SiO v=0 J=1-0 emission in 28 evolved stars, performed with the 40~m radio telescope of the IGN in Yebes, Spain. We find that the composite core plus plateau profiles are almost always present in O-rich Miras, OH/IR stars, and red supergiants. They are also found in one S-type Mira ($\chi$ Cyg), as well as in two semiregular variables (X Her and RS Cnc) that are known to show axial symmetry. In the other objects, the profiles are simpler and similar to those of other molecular lines. The composite structure appears in the objects in which SiO emission is thought to come from the very inner circumstellar layers, prior to dust formation. The central spectral feature is found to be systematically composed of a number of narrow spikes, except for X Her and RS Cnc, in which it shows a smooth shape that is very similar to that observed in CO emission. These spikes show a significant (and mostly chaotic) time variation, while in all cases the smooth components remain constant within the uncertainties. The profile shape could come from the superposition of standard wide profiles and a group of weak maser spikes. Alternatively, we speculate that the very similar profiles detected in objects that are axisymmetric may be indicative of the systematic presence of a significant axial symmetry in the very inner circumstellar shells around AGB stars; the presence of such symmetry would be independent of the probable weak maser effects in the central spikes., Comment: 12 pages, 28 figures. To be published in Astronomy and Astrophysics

We present a technique, based on so-called word series, to write down in a systematic way expansions of the strong and weak local errors of splitting algorithms for the integration of Stratonovich stochastic differential equations. Those expansions immediately lead to the corresponding order conditions. Word series are similar to, but simpler than, the B-series used to analyze Runge-Kutta and other one-step integrators. The suggested approach makes it unnecessary to use the Baker-Campbell-Hausdorff formula. As an application, we compare two splitting algorithms recently considered by Leimkuhler and Matthews to integrate the Langevin equations. The word series method bears out clearly reasons for the advantages of one algorithm over the other.

In the first part of the present work we consider periodically or quasiperiodically forced systems of the form $(d/dt)x = \epsilon f(x,t \omega )$, where $\epsilon\ll 1$, $\omega\in\mathbb{R}^d$ is a nonresonant vector of frequencies and $f(x,\theta)$ is $2\pi$-periodic in each of the $d$ components of $\theta$ (i.e.\ $\theta\in\mathbb{T}^d$). We describe in detail a technique for explicitly finding a change of variables $x = u(X,\theta;\epsilon)$ and an (autonomous) averaged system $(d/dt) X = \epsilon F(X;\epsilon)$ so that, formally, the solutions of the given system may be expressed in terms of the solutions of the averaged system by means of the relation $x(t) = u(X(t),t\omega;\epsilon)$. Here $u$ and $F$ are found as series whose terms consist of vector-valued maps weighted by suitable scalar coefficients. The maps are easily written down by combining the Fourier coefficients of $f$ and the coefficients are found with the help of simple recursions. Furthermore these coefficients are {\em universal} in the sense that they do not depend on the particular $f$ under consideration. In the second part of the contribution, we study problems of the form $(d/dt) x = g(x)+f(x)$, where one knows how to integrate the "unperturbed" problem $(d/dt)x = g(x)$ and $f$ is a perturbation satisfying appropriate hypotheses. It is shown how to explicitly rewrite the system in the "normal form" $(d/dt) x = \bar g(x)+\bar f(x)$, where $\bar g$ and $\bar f$ are {\em commuting} vector fields and the flow of $(d/dt) x = \bar g(x)$ is conjugate to that of the unperturbed $(d/dt)x = g(x)$. In Hamiltonian problems the normal form directly leads to the explicit construction of formal invariants of motion. Again, $\bar g$, $\bar f$ and the invariants are written as series consisting of known vector-valued maps and universal scalar coefficients that may be found recursively.

We introduce a new Adaptive Integration Approach (AIA) to be used in a wide range of molecular simulations. Given a simulation problem and a step size, the method automatically chooses the optimal scheme out of an available family of numerical integrators. Although we focus on two-stage splitting integrators, the idea may be used with more general families. In each instance, the system-specific integrating scheme identified by our approach is optimal in the sense that it provides the best conservation of energy for harmonic forces. The AIA method has been implemented in the BCAM-modified GROMACS software package. Numerical tests in molecular dynamics and hybrid Monte Carlo simulations of constrained and unconstrained physical systems show that the method successfully realises the fail-safe strategy. In all experiments, and for each of the criteria employed, the AIA is at least as good as, and often significantly outperforms the standard Verlet scheme, as well as fixed parameter, optimized two-stage integrators. In particular, the sampling efficiency found in simulations using the AIA is up to 5 times better than the one achieved with other tested schemes.

We show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The manipulations required involve complex numbers rather than vector fields or diffeomorphisms. More precisely we construct a group G and a Lie algebra g in such a way that the elements of G and g are families of complex numbers; the operations to be performed involve the multiplication F in G and the bracket of g and result in universal coefficients that are then applied to write the normal form or the invariants of motion of the specific problem under consideration.

Cyanogen (NCCN) is the simplest member of the series of dicyanopolyynes. It has been hypothesized that this family of molecules can be important constituents of interstellar and circumstellar media, although the lack of a permanent electric dipole moment prevents its detection through radioastronomical techniques. Here we present the first solid evidence of the presence of cyanogen in interstellar clouds through the detection of its protonated form toward the cold dark clouds TMC-1 and L483. Protonated cyanogen (NCCNH+) has been identified through the J=5-4 and J=10-9 rotational transitions using the 40m radiotelescope of Yebes and the IRAM 30m telescope. We derive beam averaged column densities for NCCNH+ of (8.6+/-4.4)e10 cm-2 in TMC-1 and (3.9+/-1.8)e10 cm-2 in L483, which translate to fairly low fractional abundances relative to H2, in the range (1-10)e-12. The chemistry of protonated molecules in dark clouds is discussed, and it is found that, in general terms, the abundance ratio between the protonated and non protonated forms of a molecule increases with increasing proton affinity. Our chemical model predicts an abundance ratio NCCNH+/NCCN of 1e-4, which implies that the abundance of cyanogen in dark clouds could be as high as (1-10)e-8 relative to H2, i.e., comparable to that of other abundant nitriles such as HCN, HNC, and HC3N., Comment: Accepted for publication in A&A Letters

It is well known that symplectic Runge-Kutta and Partitioned Runge-Kutta methods exactly preserve {\em quadratic} first integrals (invariants of motion) of the system being integrated. While this property is often seen as a mere curiosity (it does not hold for arbitrary first integrals), it plays an important role in the computation of numerical sensitivities, optimal control theory and Lagrangian mechanics, as described in this paper, which, together with some new material, presents in a unified way a number of results now scattered or implicit in the literature. Some widely used procedures, such as the direct method in optimal control theory and the computation of sensitivities via reverse accumulation imply "hidden" integrations with symplectic Partitioned Runge-Kutta schemes.

We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves by means of a simple rule. While word series have appeared before in the literature, extended word series are introduced in this paper. We exemplify the use of extended word series by studying the reduction to normal form and averaging of some perturbed integrable problems. We also provide a detailed analysis of the behaviour of splitting numerical methods for those problems.

We study a method, Extra Chance Generalized Hybrid Monte Carlo, to avoid rejections in the Hybrid Monte Carlo method and related algorithms. In the spirit of delayed rejection, whenever a rejection would occur, extra work is done to find a fresh proposal that, hopefully, may be accepted. We present experiments that clearly indicate that the additional work per sample carried out in the extra chance approach clearly pays in terms of the quality of the samples generated., Comment: 13 pages, 8 figures (including 1 diagram, 3 pseudocodes), 1 table, updated (corrections, typos and new appendix)

We construct numerical integrators for Hamiltonian problems that may advantageously replace the standard Verlet time-stepper within Hybrid Monte Carlo and related simulations. Past attempts have often aimed at boosting the order of accuracy of the integrator and/or reducing the size of its error constants; order and error constant are relevant concepts in the limit of vanishing step-length. We propose an alternative methodology based on the performance of the integrator when sampling from Gaussian distributions with not necessarily small step-lengths. We construct new splitting formulae that require two, three or four force evaluations per time-step. Limited, proof-of-concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard Verlet method, especially in problems with high dimensionality., Comment: 30 pages, 5 figures