Braids of entangled particle trajectories

In many applications, the two-dimensional trajectories of fluid particles are available, but little is known about the underlying flow. Oceanic floats are a clear example. To extract quantitative information from such data, one can measure single-particle dispersion coefficients, but this only uses one trajectory at a time, so much of the information on relative motion is lost. In some circumstances the trajectories happen to remain close long enough to measure finite-time Lyapunov exponents, but this is rare. We propose to use tools from braid theory and the topology of surface mappings to approximate the topological entropy of the underlying flow. The procedure uses all the trajectory data and is inherently global. The topological entropy is a measure of the entanglement of the trajectories, and converges to zero if they are not entangled in a complex manner (for instance, if the trajectories are all in a large vortex). We illustrate the techniques on some simple dynamical systems and on float data from the Labrador sea.Comment: 24 pages, 21 figures. PDFLaTeX with RevTeX4 macros. Matlab code included with source. Fixed an inconsistent convention problem. Final versio

A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.Comment: We give a new definition of Lax-Oleinik type operator; add some reference

Weak KAM for commuting Hamiltonians

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct geometrical method (Stoke's theorem). We also obtain a "generalization" of a theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G and for H are the same. As a corrolary we obtain the equality of the Aubry sets, of the Peierls barrier and of flat parts of Mather's $\alpha$ functions. This is also related to works of Sorrentino ([Sor09]) and Bernard ([Ber07b]).Comment: 23 pages, accepted for publication in NonLinearity (january 29th 2010). Minor corrections, fifth part added on Mather's $\alpha$ function (or effective Hamiltonian

Analysis of Droplet Train/Moving Substrate Interactions in Ink-Jetting Processes

Ink-jetting technology has been applied to several processes in solid free-form fabrication (SFF) wherein droplets impinge onto a substrate to deposit the build material. Droplet impact behaviour on a surface has been the interest of many researchers; however, few studies have been undertaken to investigate the interaction of droplets with the moving substrate. This paper reports the impact behaviour of the droplets jetted at different frequencies onto a substrate moving over a range of velocities. The phenomena associated with the interaction were classified into three main regimes.Mechanical Engineerin

Periodic orbits of period 3 in the disc

Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page

Improving the Complementary Methods to Estimate Evapotranspiration Under Diverse Climatic and Physical Conditions

Reliable estimation of evapotranspiration (ET) is important for the purpose of water resources planning and management. Complementary methods, including complementary relationship areal evapotranspiration (CRAE), advection aridity (AA) and Granger and Gray (GG), have been used to estimate ET because these methods are simple and practical in estimating regional ET using meteorological data only. However, prior studies have found limitations in these methods especially in contrasting climates. This study aims to develop a calibration-free universal method using the complementary relationships to compute regional ET in contrasting climatic and physical conditions with meteorological data only. The proposed methodology consists of a systematic sensitivity analysis using the existing complementary methods. This work used 34 global FLUXNET sites where eddy covariance (EC) fluxes of ET are available for validation. A total of 33 alternative model variations from the original complementary methods were proposed. Further analysis using statistical methods and simplified climatic class definitions produced one distinctly improved GG-model-based alternative. The proposed model produced a single-step ET formulation with results equal to or better than the recent studies using data-intensive, classical methods. Average root mean square error (RMSE), mean absolute bias (BIAS) and R2 (coefficient of determination) across 34 global sites were 20.57 mm month−1, 10.55 mm month−1and 0.64, respectively. The proposed model showed a step forward toward predicting ET in large river basins with limited data and requiring no calibration

Detection of Lying Electrical Vehicles in Charging Coordination Application Using Deep Learning

The simultaneous charging of many electric vehicles (EVs) stresses the distribution system and may cause grid instability in severe cases. The best way to avoid this problem is by charging coordination. The idea is that the EVs should report data (such as state-of-charge (SoC) of the battery) to run a mechanism to prioritize the charging requests and select the EVs that should charge during this time slot and defer other requests to future time slots. However, EVs may lie and send false data to receive high charging priority illegally. In this paper, we first study this attack to evaluate the gains of the lying EVs and how their behavior impacts the honest EVs and the performance of charging coordination mechanism. Our evaluations indicate that lying EVs have a greater chance to get charged comparing to honest EVs and they degrade the performance of the charging coordination mechanism. Then, an anomaly based detector that is using deep neural networks (DNN) is devised to identify the lying EVs. To do that, we first create an honest dataset for charging coordination application using real driving traces and information revealed by EV manufacturers, and then we also propose a number of attacks to create malicious data. We trained and evaluated two models, which are the multi-layer perceptron (MLP) and the gated recurrent unit (GRU) using this dataset and the GRU detector gives better results. Our evaluations indicate that our detector can detect lying EVs with high accuracy and low false positive rate

Expansive homeomorphisms of the plane

This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques involve topological and metric aspects of the plane. The use of a Lyapunov metric function which defines the same topology as the one induced by the usual metric but that, in general, is not equivalent to it is an example of such techniques. The discovery of a hypothesis about the behavior of Lyapunov functions at infinity allows us to generalize some results that are valid in the compact context. Additional local properties allow us to obtain another classification theorem.Comment: 29 pages, 22 figure