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We prove that Bertrand's property cannot occur in a special-relativistic scenario using the properties of the period function of planar centres. We also explore some integrability properties of the relativistic Coulomb problem and the asymptotic behavior of collision solutions., Comment: 18 pages

Consider, for any integer $n\ge3$, the set $\text{Pos}_n$ of all $n$-periodic tree patterns with positive topological entropy and the set $\text{Irr}_n\subset\text{Pos}_n$ of all $n$-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\text{Pos}_n$, $\text{Irr}_n$ and $\text{Pos}_n\setminus\text{Irr}_n$. Let $\lambda_n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty)$. We explicitly construct an irreducible $n$-periodic tree pattern $\mathcal{Q}_n$ whose entropy is $\log(\lambda_n)$. We prove that this entropy is minimum in $\text{Pos}_n$. Since the pattern $\mathcal{Q}_n$ is irreducible, $\mathcal{Q}_n$ also minimizes the entropy in the family $\text{Irr}_n$. We also prove that the minimum positive entropy in the set $\text{Pos}_n\setminus\text{Irr}_n$ (which is nonempty only for composite integers $n\ge6$) is $\log(\lambda_{n/p})/p$, where $p$ is the least prime factor of $n$., Comment: 39 pages, 21 figures

In this paper we study the appearance of bifurcations of limit cycles in an epidemic model with two types of aware individuals. All the transition rates are constant except for the alerting decay rate of the most aware individuals and the rate of creation of the less aware individuals, which depend on the disease prevalence in a non-linear way. For the ODE model, the numerical computation of the limit cycles and the study of their stability are made by means of the Poincar\'e map. Moreover, sufficient conditions for the existence of an endemic equilibrium are also obtained. These conditions involve a rather natural relationship between the transmissibility of the disease and that of awareness. Finally, stochastic simulations of the model under a very low rate of imported cases are used to confirm the scenarios of bistability (endemic equilibrium and limit cycle) observed in the solutions of the ODE model., Comment: 18 pages

We report phenomenal yield strengths, up to one fourth of the theoretical strength of silver, recorded in microcompression testing of initially dislocation free silver micro and nanocubes synthesized from a multistep seed growth process. These high strengths and the massive strain bursts that occur upon yield are results of the initially dislocation free single crystal structure of the pristine samples that yield through spontaneous nucleation of dislocations. When the pristine samples are exposed to a focused ion beam to fabricate pillars and then compressed, the dramatic strain burst does not occur, and they yield at a quarter of the strength of their pristine counterparts. Regardless of the defect state of the samples prior to testing, a size effect is apparent, where the yield strength increases as the sample size decreases. Since dislocation starvation and the single arm source mechanisms cannot explain a size effect on yield strength in dislocation free samples, we investigate the dislocation nucleation mechanisms controlling the size effect through careful experimental observations and molecular statics simulations. We find that intrinsic or extrinsic symmetry breakers such as surface defects, edge roundness, external sample shape, or a high vacancy concentration can influence dislocation nucleation, and thus contribute to the size effect on yield strength in initially dislocation-free samples.

We consider the model of cooperative learning via distributed non-Bayesian learning, where a network of agents tries to jointly agree on a hypothesis that best described a sequence of locally available observations. Building upon recently proposed weak communication network models, we propose a robust cooperative learning rule that allows asynchronous communications, message delays, unpredictable message losses, and directed communication among nodes. We show that our proposed learning dynamics guarantee that all agents in the network will have an asymptotic exponential decay of their beliefs on the wrong hypothesis, indicating that the beliefs of all agents will concentrate on the optimal hypotheses. Numerical experiments provide evidence on a number of network setups., Comment: Submitted to ACC2021

Oxygen activity and surface stability are two key parameters in the search for advanced materials for intermediate temperature solid oxide electrochemical cells, as overall device performance depends critically on them. In particular $in$ $situ$ and $operando$ characterisation techniques have accelerated the understanding of degradation processes and the identification of active sites, motivating the design and synthesis of improved, nanoengineered materials. In this short topical review we report on the latest developments of various sophisticated $in$ $situ$ and $operando$ characterization techniques, including Transmission and Scanning Electron Microscopy (TEM and SEM), surface-enhanced Raman spectroscopy (SERS), Electrochemical Impedance Spectroscopy (EIS), X-ray Diffraction (XRD) and synchrotron based X-ray photoelectron spectroscopy (XPS) and X-ray absorption spectroscopy (XAS), among others. We focus on their use in three emerging topics, namely: (i) the analysis of general electrochemical reactions and the surface defect chemistry of electrode materials; (ii) the evolution of electrode surfaces achieved by nanoparticle exsolution for enhanced oxygen activity and (iii) the study of surface degradation caused by Sr segregation, leading to reduced durability. For each of these topics we highlight the most remarkable examples recently published. We anticipate that ongoing improvements in the characterisation techniques and especially a complementary use of them by multimodal approaches will lead to improved knowledge of $operando$ processes, hence allowing a significant advancement in cell performance in the near future.

This paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ODE model defined on contact networks. The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a non-oscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters., Comment: 17 pages, 11 figures

Spatial Atomic Layer Deposition (SALD) is a recent approach that is up to two orders of magnitude faster than conventional ALD, and that can be performed at atmospheric pressure and even in the open air. Previous works have exploited these assets to focus on the possibility of high-rate, large-area deposition for scaling up into mass production. Conversely, here we show that SALD indeed represents an ideal platform for the selective deposition of functional materials by proper design and miniaturization of SALD close-proximity heads. In particular, we have used the potential offered by 3D printing to fabricate custom close-proximity SALD injection heads. By using 3D printing, the heads can be easily designed and readily modified to obtain different deposition areas, free-form patterns, and even complex multimaterial structures. The heads can be printed in different materials to adjust to the chemistry of the precursors and the deposition conditions used. Polymeric heads can be used as cheap (even disposable) heads that are both used for performing deposition and for prototyping and optimization purposes. Finally, by designing a miniaturized head with circular concentric gas channels, 3D printing of functional materials can be performed with nanometric resolution in Z. This constitutes a new 3D printing approach based on gaseous precursors. Because the selective deposition strategies presented here are based on the SALD process, conformal and continuous thin films of functional materials can be printed at low temperatures and with high deposition rate in the open air. Our approach represents a new versatile way of printing functional materials and devices with spatial and topological control, thus extending the potential of SALD and ALD in general, and opening a new avenue in the field of area-selective deposition of functional materials., Comment: 20 pages, 4 figures

We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be catalogued according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincar\'e-Birkhoff Theorem.

Temporal action detection is a fundamental yet challenging task in video understanding. Video context is a critical cue to effectively detect actions, but current works mainly focus on temporal context, while neglecting semantic context as well as other important context properties. In this work, we propose a graph convolutional network (GCN) model to adaptively incorporate multi-level semantic context into video features and cast temporal action detection as a sub-graph localization problem. Specifically, we formulate video snippets as graph nodes, snippet-snippet correlations as edges, and actions associated with context as target sub-graphs. With graph convolution as the basic operation, we design a GCN block called GCNeXt, which learns the features of each node by aggregating its context and dynamically updates the edges in the graph. To localize each sub-graph, we also design an SGAlign layer to embed each sub-graph into the Euclidean space. Extensive experiments show that G-TAD is capable of finding effective video context without extra supervision and achieves state-of-the-art performance on two detection benchmarks. On ActivityNet-1.3, it obtains an average mAP of 34.09%; on THUMOS14, it reaches 51.6% at IoU@0.5 when combined with a proposal processing method. G-TAD code is publicly available at https://github.com/frostinassiky/gtad., Comment: Accepted by CVPR2020. 8 pages, 9 figures, 2 pages appendix

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the phenomenon of resonance may appear. We give a sufficient condition on the perturbation in order that resonance occurs when the period annulus of the isochronous oscillator is bounded. In this context, resonance means that all solutions escape from the period annulus., Comment: 10 pages, 1 figure

We study the asymptotic development at infinity of an integral operator. We use this development to give sufficient conditions in order to upper bound the number of critical periodic orbits that bifurcate from the outer boundary of the period function of planar potential centers. We apply the main results to two different families: the power-like potential family $\ddot x=x^p-x^q$, $p,q\in\mathbb{R}$, $p>q$; and the family of dehomogenized Loud's centers., Comment: 33 pages

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context, resonance means that all solutions are unbounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators., Comment: 28 pages

We give an alternative proof for the celebrated Bertrand's theorem as a corollary of the isochronicity of a certain family of centers., Comment: 10 pages, 1 figure

This paper is concerned with the study of the criticality of families of planar centers. More precisely, we study sufficient conditions to bound the number of critical periodic orbits that bifurcate from the outer boundary of the period annulus of potential centers. In the recent years, the new approach of embedding the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary has shown to be fruitful in this issue. In this work, we tackle with a remaining case that was not taken into account in the previous studies in which the Roussarie-Ecalle compensator plays an essential role. The theoretical results we develop are applied to study the bifurcation diagram of the period function of two different families of centers: the power-like family $\ddot x=x^p-x^q$, $p,q\in\mathbb{R}$ with $p>q$; and the family of dehomogenized Loud's centers., Comment: 4 figures

In a central force system the angle between two successive passages of a body through pericenters is called the apsidal angle. In this paper we prove that for central forces of the form $f(r)\sim \lambda r^{-(\alpha+1)}$ with $\alpha<2$ the apsidal angle is a monotonous function of the energy, or equivalently of the orbital eccentricity., Comment: 4 pages

We consider the family of dehomogenized Loud's centers $X_{\mu}=y(x-1)\partial_x+(x+Dx^2+Fy^2)\partial_y,$ where $\mu=(D,F)\in\mathbb{R}^2,$ and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu},\mu\in\mathbb{R}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set $\Gamma_{B}$ of codimension 1 in $\mathbb{R}^2$. In the present paper we succeed in proving that a subset of $\Gamma_{B}$ has criticality equal to one.

We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number.