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We show how machine learning methods can unveil the fractional and delayed nature of discrete dynamical systems. In particular, we study the case of the fractional delayed logistic map. We show that given a trajectory, we can detect if it has some delay effect or not, and also to characterize the fractional component of the underlying generation model., Comment: 14 pages, 9 figures

Anomalous diffusion is present at all scales, from atomic to large scales. Some exemplary systems are; ultra-cold atoms, telomeres in the nucleus of cells, moisture transport in cement-based materials, the free movement of arthropods, and the migration patterns of birds. The characterization of the diffusion gives critical information about the dynamics of these systems and provides an interdisciplinary framework with which to study diffusive transport. Thus, the problem of identifying underlying diffusive regimes and inferring the anomalous diffusion exponent {$\alpha$} with high confidence is critical to physics, chemistry, biology, and ecology. Classification and analysis of raw trajectories combining machine learning techniques with statistics extracted from them have widely been studied in the Anomalous Diffusion Challenge ge (Munoz-Gil et al., 2021). Here we present a new data-driven method for working with diffusive trajectories. This method utilizes Gramian Angular Fields (GAF) to encode one-dimensional trajectories as images (Gramian Matrices), while preserving their spatiotemporal structure for input to computer-vision models. This allows us to leverage two well-established pre-trained computer-vision models, ResNet and MobileNet, to characterize the underlying diffusive regime, and infer the anomalous diffusion exponent {$\alpha$}. Short raw trajectories, of lengths between 10 and 50, are commonly encountered in single-particle tracking experiments and are the most difficult to characterize. We show that by using GAF images, we can outperform the current state-of-the-art while increasing accessibility to machine learning methods in an applied setting., Comment: 19 pages, 15 figures

Taming the instabilities inherent to many nonlinear optical phenomena is of paramount importance for modern photonics. In particular, the so-called snake instability is universally known to severely distort localized wave stripes, leading to the occurrence of transient, short-lived dynamical states that eventually decay. The phenomenon is ubiquitous in nonlinear science, from river meandering to superfluids, and to date it remains apparently uncontrollable. However, here we show that optical snake instabilities can be harnessed by a process that leads to the formation of stationary and robust two-dimensional zigzag states. We find that such new type of nonlinear waves exists in the hyperbolic regime of cylindrical micro-resonators and it naturally corresponds to two-dimensional frequency combs featuring spectral heterogeneity and intrinsic synchronization. We uncover the conditions of the existence of such spatiotemporal photonic snakes and confirm their remarkable robustness against perturbations. Our findings represent a new paradigm for frequency comb generation, thus opening the door to a whole range of applications in communications, metrology, and spectroscopy., Comment: 6 figures, 11 pages

The 2030 Agenda of the United Nations (UN) revolves around the Sustainable Development Goals (SDGs). A critical step towards that objective is identifying whether scientific production aligns with the SDGs' achievement. To assess this, funders and research managers need to manually estimate the impact of their funding agenda on the SDGs, focusing on accuracy, scalability, and objectiveness. With this objective in mind, in this work, we develop ASDG, an easy-to-use artificial-intelligence (AI)-based model for automatically identifying the potential impact of scientific papers on the UN SDGs. As a demonstrator of ASDG, we analyze the alignment of recent aerospace publications with the SDGs. The Aerospace data set analyzed in this paper consists of approximately 820,000 papers published in English from 2011 to 2020 and indexed in the Scopus database. The most-contributed SDGs are 7 (on clean energy), 9 (on industry), 11 (on sustainable cities) and 13 (on climate action). The establishment of the SDGs by the UN in the middle of the 2010 decade did not significantly affect the data. However, we find clear discrepancies among countries, likely indicative of different priorities. Also, different trends can be seen in the most and least cited papers, with clear differences in some SDGs. Finally, the number of abstracts the code cannot identify is decreasing with time, possibly showing the scientific community's awareness of SDG.

The results of the Anomalous Diffusion Challenge (AnDi Challenge) have shown that machine learning methods can outperform classical statistical methodology at the characterization of anomalous diffusion in both the inference of the anomalous diffusion exponent alpha associated with each trajectory (Task 1), and the determination of the underlying diffusive regime which produced such trajectories (Task 2). Furthermore, of the five teams that finished in the top three across both tasks of the AnDi challenge, three of those teams used recurrent neural networks (RNNs). While RNNs, like the long short-term memory (LSTM) network, are effective at learning long-term dependencies in sequential data, their key disadvantage is that they must be trained sequentially. In order to facilitate training with larger data sets, by training in parallel, we propose a new transformer based neural network architecture for the characterization of anomalous diffusion. Our new architecture, the Convolutional Transformer (ConvTransformer) uses a bi-layered convolutional neural network to extract features from our diffusive trajectories that can be thought of as being words in a sentence. These features are then fed to two transformer encoding blocks that perform either regression or classification. To our knowledge, this is the first time transformers have been used for characterizing anomalous diffusion. Moreover, this may be the first time that a transformer encoding block has been used with a convolutional neural network and without the need for a transformer decoding block or positional encoding. Apart from being able to train in parallel, we show that the ConvTransformer is able to outperform the previous state of the art at determining the underlying diffusive regime in short trajectories (length 10-50 steps), which are the most important for experimental researchers., Comment: 20 pages, 13 figures

Many of today's problems require techniques that involve the solution of arbitrarily large systems $A\mathbf{x}=\mathbf{b}$. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor decomposition. The numerical experiments support the fact that this algorithm converges especially fast when the matrix of the linear system is Laplacian-Like. These matrices that follow the tensor structure of the Laplacian operator are formed by sums of Kronecker product of matrices following a particular pattern. Moreover, this set of matrices is not only a linear subspace it is a a Lie sub-algebra of a matrix Lie Algebra. In this paper, we characterize and give the main properties of this particular class of matrices. Moreover, the above results allow us to propose an algorithm to explicitly compute the orthogonal projection onto this subspace of a given square matrix $A \in \mathbb{R}^{N\times N}.$

We present a method to study the dynamics of a quasi-two dimensional Bose-Einstein condensate which contains initially many vortices at arbitrary locations. We present first the analytical solution of the dynamics in a homogeneous medium and in a parabolic trap for the ideal non-interacting case. For the homogeneous case this was introduced in the context of photonics. Here we discuss this case in the context of Bose-Einstein condensates and extend the analytical solution to the trapped case, for the first time. This linear case allows one to obtain the trajectories of the position of phase singularities present in the initial condensate along with time. Also, it allows one to predict some quantities of interest, such as the time at which a vortex and an antivortex contained in the initial condensate will merge. Secondly, the method is complemented with numerical simulations of the non-linear case. We use a numerical split-step simulation of the non-linear Gross-Pitaevskii equation to determine how these trajectories and quantities of interest are changed by the presence of interactions. We illustrate the method with several simple cases of interest both in the homogeneous and parabolically trapped systems., Comment: 11 pages, 4 figures. Published in the Special Issue "Computational Methods for Quantum Matter", https://www.mdpi.com/journal/condensedmatter/special_issues/Comput_Quantum_Matter

We theoretically investigate the dynamics, bifurcation structure and stability of localized states in Kerr cavities driven at the pure fourth-order dispersion point. Both the normal and anomalous group velocity dispersion regimes are analysed, highlighting the main differences with the standard second order dispersion case. In the anomalous regime, single and multi-peak localised states exist and are stable over a much wider region of the parameter space. In the normal dispersion regime, stable narrow bright solitons exist. Some of our findings can be understood by a new scenario reported here for the spatial eigenvalues, which imposes oscillatory tails to all localised states. To be published in Optics Letters., Comment: To be published in Optics Letters

Anomalous diffusion occurs at very different scales in nature, from atomic systems to motions in cell organelles, biological tissues or ecology, and also in artificial materials, such as cement. Being able to accurately measure the anomalous exponent associated with a given particle trajectory, thus determining whether the particle subdiffuses, superdiffuses or performs normal diffusion is of key importance to understand the diffusion process. Also, it is often important to trustingly identify the model behind the trajectory, as this gives a large amount of information on the system dynamics. Both aspects are particularly difficult when the input data are short and noisy trajectories. It is even more difficult if one cannot guarantee that the trajectories output in experiments is homogeneous, hindering the statistical methods based on ensembles of trajectories. We present a data-driven method able to infer the anomalous exponent and to identify the type of anomalous diffusion process behind single, noisy and short trajectories, with good accuracy. This model was used in our participation in the Anomalous Diffusion (AnDi) Challenge. A combination of convolutional and recurrent neural networks were used to achieve state-of-the-art results when compared to methods participating in the AnDi Challenge, ranking top 4 in both classification and diffusion exponent regression., Comment: 20 pages, 11 figures

Deviations from Brownian motion leading to anomalous diffusion are ubiquitously found in transport dynamics, playing a crucial role in phenomena from quantum physics to life sciences. The detection and characterization of anomalous diffusion from the measurement of an individual trajectory are challenging tasks, which traditionally rely on calculating the mean squared displacement of the trajectory. However, this approach breaks down for cases of important practical interest, e.g., short or noisy trajectories, ensembles of heterogeneous trajectories, or non-ergodic processes. Recently, several new approaches have been proposed, mostly building on the ongoing machine-learning revolution. Aiming to perform an objective comparison of methods, we gathered the community and organized an open competition, the Anomalous Diffusion challenge (AnDi). Participating teams independently applied their own algorithms to a commonly-defined dataset including diverse conditions. Although no single method performed best across all scenarios, the results revealed clear differences between the various approaches, providing practical advice for users and a benchmark for developers., Comment: 63 pages, 5 main figures, 1 table, 28 supplementary figures. Website: http://www.andi-challenge.org

We uncover a novel and robust phenomenon that causes the gradual self-replication of spatiotemporal Kerr cavity patterns in cylindrical microresonators. These patterns are inherently synchronised multi-frequency combs. Under proper conditions, the axially-localized nature of the patterns leads to a fundamental drift instability that induces transitions amongst patterns with a different number of rows. Self-replications, thus, result in the stepwise addition or removal of individual combs along the cylinder's axis. Transitions occur in a fully reversible and, consequently, deterministic way. The phenomenon puts forward a novel paradigm for Kerr frequency comb formation and reveals important insights into the physics of multi-dimensional nonlinear patterns., Comment: 13 pages, 9 figures, to appear in Physical Review Letters

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains., Comment: 12 pages

Dendrograms are a way to represent evolutionary relationships between organisms. Nowadays, these are inferred based on the comparison of genes or protein sequences by taking into account their differences and similarities. The genetic material of choice for the sequence alignments (all the genes or sets of genes) results in distinct inferred dendrograms. In this work, we evaluate differences between dendrograms reconstructed with different methodologies and obtained for different sets of organisms chosen at random from a much larger set. A statistical analysis is performed in order to estimate the fluctuation between the results obtained from the different methodologies. This analysis permit us to validate a systematic approach, based on the comparison of the organisms' metabolic networks for inferring dendrograms. It has the advantage that it allows the comparison of organisms very far away in the evolutionary tree even if they have no known ortholog gene in common., Comment: 22 pages, 5 figures, 3 tables

In recent years, the topic of car-following has experimented an increased importance in traffic engineering and safety research. This has become a very interesting topic because of the development of driverless cars \cite{google_driverless_cars}. Driving models which describe the interaction between adjacent vehicles in the same lane have a big interest in simulation modeling, such as the Quick-Thinking-Driver model. A non-linear version of it can be given using the logistic map, and then chaos appears. We show that an infinite-dimensional version of the linear model presents a chaotic behavior using the same approach as for studying chaos of death models of cell growth., Comment: To Appear in Semigroup Forum

A wide range of applications and research has been done with genome-scale metabolic models. In this work we describe a methodology for comparing metabolic networks constructed from genome-scale metabolic models and how to apply this comparison in order to infer evolutionary distances between different organisms. Our methodology allows a quantification of the metabolic differences between different species from a broad range of families and even kingdoms. This quantification is then applied in order to reconstruct phylogenetic trees for sets of various organisms., Comment: 3 figures, 8 tables