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In several countries, students learn basic mathematical skills first, followed by fractions before learning percentages. Previous longitudinal studies observed that basic mathematical skills (e.g., arithmetic) are significant predictors of understanding fractions, which is critical for later academic success. However, it is still unclear whether this predictive power generalizes to other types of rational numbers such as percentages. Here, we analyzed a large longitudinal dataset (2.798 students; 472.626 problem sets) from an online learning environment to predict performance in fraction understanding as well as processing percentages based on basic mathematical skills (i.e., arithmetic, measurement units, and geometry). As students learned fractions before percentages, we additionally assessed whether understanding fractions predicted processing percentages. We observed that performance in arithmetic, measurement units, and geometry significantly predicted performance on percentages and fraction understanding. We also found transfer effects from fractions to processing percentages. This indicates that processing percentages and understanding fractions similarly build on basic mathematical skills. Moreover, our results point towards the idea that knowledge attained on one format of rational numbers (fractions) may transfer to another format of rational numbers (percentages).

The work of McCloskey and Cohen popularized the concept of catastrophic interference. They used a neural network that tried to learn addition using two groups of examples as two different tasks. In their case, learning the second task rapidly deteriorated the acquired knowledge about the previous one. We hypothesize that this could be a symptom of a fundamental problem: addition is an algorithmic task that should not be learned through pattern recognition. Therefore, other model architectures better suited for this task would avoid catastrophic forgetting. We use a neural network with a different architecture that can be trained to recover the correct algorithm for the addition of binary numbers. This neural network includes conditional clauses that are naturally treated within the back-propagation algorithm. We test it in the setting proposed by McCloskey and Cohen and training on random additions one by one. The neural network not only does not suffer from catastrophic forgetting but it improves its predictive power on unseen pairs of numbers as training progresses. We also show that this is a robust effect, also present when averaging many simulations. This work emphasizes the importance that neural network architecture has for the emergence of catastrophic forgetting and introduces a neural network that is able to learn an algorithm. We acknowledge support from the CONEX-Plus programme funded by Universidad Carlos III de Madrid and the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 801538.

Flow networks can describe many natural and artificial systems. We present a model for a flow system that allows for volume accumulation, includes conduits with a nonlinear relation between current and pressure difference, and can be applied to networks of arbitrary topology. The model displays complex dynamics, including self-sustained oscillations in the absence of any dynamics in the inputs and outputs. In this work we analytically show the origin of self-sustained oscillations for the one-dimensional case. We numerically study the behavior of systems of arbitrary topology under different conditions: we discuss their excitability, the effect of different boundary conditions, and wave propagation when the network has regions of conduits with linear conductance.

At room temperature, micron-sized sheets of freestanding graphene are in constant motion, even in the presence of an applied bias voltage. We quantify the out-of-plane movement by collecting the displacement current using a nearby small-area metal electrode and present an Ito-Langevin model for the motion coupled to a circuit containing diodes. Numerical simulations show that the system reaches thermal equilibrium and the average rates of heat and work provided by stochastic thermodynamics tend quickly to zero. However, there is power dissipated by the load resistor, and its time average is exactly equal to the power supplied by the thermal bath. The exact power formula is similar to Nyquist's noise power formula, except that the rate of change of diode resistance significantly boosts the output power, and the movement of the graphene shifts the power spectrum to lower frequencies. We have calculated the equilibrium average of the power by asymptotic and numerical methods. Excellent agreement is found between experiment and theory.

Steady progress in the miniaturization of structures and devices has reached a scale where thermal fluctuations become relevant and it is thus important to understand how such fluctuations affect their mechanical stability. Here, we investigate the buckling of thermalized sheets and we demonstrate that thermal fluctuations increase the critical buckling load due to the enhanced scale-dependent bending rigidity for sheets that are much larger than a characteristic thermal length scale. The presented results are universal and apply to a wide range of microscopic sheets. These results are especially relevant for atomically thin 2D materials, where thermal fluctuations can significantly increase the critical buckling load because the thermal length scale is on the order of nanometers at room temperature., 10 pages, 4 figures

We explore the design parameter space of short (5--25 period), n-doped, Ga/(Al,Ga)As semiconductor superlattices (SSLs) in the sequential resonant tunneling regime. We consider SSLs at cool (77K) and warm (295K) temperatures, simulating the electronic response to variations in (a) the number of SSL periods, (b) the contact conductivity, and (c) the strength of disorder (aperiodicities). Our analysis shows that the chaotic dynamical phases exist on a number of sub-manifolds of codimension zero within the design parameter space. This result provides an encouraging guide towards the experimental observation of high-frequency intrinsic dynamical chaos in shorter SSLs., 18 pages, 6 figures

Suspended graphene exhibits ripples of size ranging from 50 to 100 {\AA} and height $\sim$10{\AA}, however, their origin remains undetermined. Previous theoretical works have proposed that rippling in graphene might be generated by the coupling between the bending modes and the density of electrons. These studies theoretically proposed that in the thermodynamic limit a membrane of single layer graphene displays a lattice instability for large enough electron-phonon coupling, undergoing a phase transition from a flat phase to a rippled one. In this work we solve the elasticity equations for a finite membrane of graphene coupled to the charge distribution, modeled in the tight-binding approximation, we find that the electron-phonon coupling controls a transition from a stable flat configuration to a stable rippled phase., Comment: 11 pages, 7 figures. arXiv admin note: text overlap with arXiv:1911.10536

Inspired by protein folding, we introduce funnels into the complex cost function landscapes of two processes, the tuning of networks, and the jamming of ideal spheres. In both processes, geometrical frustration plays a role -- tuning pressure differences between pairs of target nodes far from the source in a flow network impedes tuning of nearby pairs more than the reverse process, while unjamming the system in one region can make it more difficult to unjam elsewhere. By modifying the cost functions to control the order in which functions are tuned or regions unjam, we smooth out local minima while leaving global minima unaffected, significantly increasing the success rate of reaching global minima.

This work explores different mechanisms that induce thermal rectification in the nanoscale. The presence of interacting energy channels combined with simple asymmetries is sufficient for promoting the desired behavior. We use simple quantum dot configurations, identifying the basic properties that enhance rectification for each case: the size of a quantum dot state space (which suggests the use of scaled up systems with many interacting channels), tunneling asymmetries due to coherent tunneling in a double quantum dot, or quantum interference in a triangular triple quantum dot. An efficient and tunable thermal diode is proposed using a channel capacitively coupled to a mesoscopic switch., 12 pages, 13 figures. Published version

Collective electron transport causes a weakly coupled semiconductor superlattice under dc voltage bias to be an excitable system with $2N+2$ degrees of freedom: electron densities and fields at $N$ superlattice periods plus the total current and the field at the injector. External noise of sufficient amplitude induces regular current self-oscillations (coherence resonance) in states that are stationary in the absence of noise. Numerical simulations show that these oscillations are due to the repeated nucleation and motion of charge dipole waves that form at the emitter when the current falls below a critical value. At the critical current, the well-to-well tunneling current intersects the contact load line. We have determined the device-dependent critical current for the coherence resonance from experiments and numerical simulations. We have also described through numerical simulations how a coherence resonance triggers a stochastic resonance when its oscillation mode becomes locked to a weak ac external voltage signal. Our results agree with the experimental observations., 21 pages, 12 figures, preprint version including suppl. material

Noise-enhanced chaos in a doped, weakly coupled GaAs/Al_{0.45}Ga_{0.55}As superlattice has been observed at room temperature in experiments as well as in the results of the simulation of nonlinear transport based on a discrete tunneling model. When external noise is added, both the measured and simulated current-versus-time traces contain irregularly spaced spikes for particular applied voltages, which separate a regime of periodic current oscillations from a region of no current oscillations at all. In the voltage region without current oscillations, the electric field profile consist of a low-field domain near the emitter contact separated by a domain wall consisting of a charge accumulation layer from a high-field regime closer to the collector contact. With increasing noise amplitude, spontaneous chaotic current oscillations appear over a wider bias voltage range. For these bias voltages, the domain boundary between the two electric-field domains becomes unstable, and very small current or voltage fluctuations can trigger the domain boundary to move toward the collector and induce chaotic current spikes. The experimentally observed features are qualitatively very well reproduced by the simulations. Increased noise can consequently enhance chaotic current oscillations in semiconductor superlattices., Physical Review E 95, in press (2017)

Previous theoretical and experimental work has put forward 50-period semiconductor superlattices as fast, true random number generators at room temperature. Their randomness stems from feedback between nonlinear electronic dynamics and stochastic processes that are intrinsic to quantum transitions. This work theoretically demonstrates that shorter superlattices with higher potential barriers contain fully chaotic dynamics over several intervals of the applied bias voltage compared to the 50-periods device which presented a much weaker chaotic behavior. The chaos arises from deterministic dynamics, hence it persists even in the absence of additional stochastic processes. Moreover, the frequency of the chaotic current oscillations is higher for shorter superlattices. These features should allow for faster and more robust generation of true random numbers., 12 pages, 4 figures

In this work, we find an analytical flat-membrane solution to the saddle point equations, derived by Guinea et al. [Phys. Rev. B 89, 125428 (2014)], for the case of a suspended graphene membrane of circular shape. We also find how different buckled membrane solutions bifurcate from the flat membrane at critical temperatures and membrane radii. The saddle point equations take into account electron-phonon coupling and this coupling provides a residual stress even for a flat graphene layer. Below a critical temperature (which is exceedingly high for an infinite layer) or above a critical size that depend on boundary conditions, different buckling modes that may be the germ of rippling appear. Our results provide the opportunity to develop new feasible experiments dealing with buckling in small suspended graphene membranes that could verify them. These experiments may also be used to fit the phonon-electron coupling constant or the bending energy., 21 pages, 2 figures

We consider a simple spin-membrane model for rippling in graphene. The model exhibits transitions from a flat but rippled membrane to a buckled one. At high temperature the transition is second order but it is first order at low temperature for appropriate strength of the spin-spin coupling. Driving the system across the first order phase transition in nonequilibrium conditions that mimic interaction of the graphene membrane with a STM tip explains recent experiments. In particular, we observe a reversible behavior for small values of the STM current and an irreversible transition from flat rippled membrane to rigid buckled membrane when the current surpasses a critical value. This work opens the possibility to test mechanical properties of graphene under different temperature and electrostatic conditions., Comment: 20 pages, 8 figures

A system of atoms connected by harmonic springs to their nearest neighbors on a lattice is coupled to Ising spins that are in contact with a thermal bath and evolve under Glauber dynamics. Assuming a nearest-neighbor antiferromagnetic interaction between spins, we calculate analytically the equilibrium state. On a one-dimensional attice, the system exhibits first and second order phase transitions. The order parameters are the total magnetization and the number of spin pairs in an antiferromagnetic configuration. On a hexagonal two dimensional lattice, spins interact with their nearest-neighbors and next-nearest-neighbors. Together with the coupling to atoms, these interactions produce a complex behavior that is displayed on a phase diagram. There are: ordered phases associated to ripples with atomic wavelength and antiferromagnetic order, ordered phases associated to ripples with nanometer wavelengths and ferromagnetic order, disordered glassy phases, and other phases presenting stripes formed by different domains. These static phases are discussed in relation to existing experiments and results for other models found in the literature., accepted for publication in JSTAT

We analyze the so-called Kovacs effect in the one-dimensional Ising model with Glauber dynamics. We consider small enough temperature jumps, for which a linear response theory has been recently derived. Within this theory, the Kovacs hump is directly related to the monotonic relaxation function of the energy. The analytical results are compared with extensive Monte Carlo simulations, and an excellent agreement is found. Remarkably, the position of the maximum in the Kovacs hump depends on the fact that the true asymptotic behavior of the relaxation function is different from the stretched exponential describing the relevant part of the relaxation at low temperatures., accepted for publication in Phys. Rev. E