The Electric Field of a Uniformly Charged Non-Conducting Cubic Surface

As an integrative and insightful example for undergraduates learning about electrostatics, we discuss how to use symmetry, Coulomb's Law, superposition, Gauss's law, and visualization to understand the electric field produced by a non-conducting cubic surface that is covered with a uniform surface charge density. We first discuss how to deduce qualitatively, using only elementary physics, the surprising fact that the electric field inside the cubic surface is nonzero and has a complex structure, pointing inwards towards the cube center from the midface of each cube and pointing outwards towards each edge and corner. We then discuss how to understand the quantitative features of the electric field by plotting an analytical expression for E along symmetry lines and on symmetry surfaces. This example would be a good choice for group problem solving in a recitation or flipped classroom.Comment: 40 pages, 17 figures, appendix included, submitted to American Journal of Physic

Spatially Localized Unstable Periodic Orbits

Using an innovative damped-Newton method, we report the first calculation of many distinct unstable periodic orbits (UPOs) of a large high-dimensional extensively chaotic partial differential equation. A majority of the UPOs turn out to be spatially localized in that time dependence occurs only on portions of the spatial domain. With a particular weighting of 127 UPOs, the Lyapunov fractal dimension D=8.8 can be estimated with a relative error of 2%. We discuss the implications of these spatially localized UPOs for understanding and controlling spatiotemporal chaos.Comment: 16 pages (total), 3 eps figures (Includes two new references and a new footnote) Submitted to Physical Review Letter

Fractal Basins of Attraction Associated with a Damped Newton's Method

An intriguing and unexpected result for students learning numerical analysis is that Newton's method, applied to the simple polynomial z^3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may help to stimulate student interest in numerical analysis, we investigate the question of whether a damping method, which is designed to increase the likelihood of convergence for Newton's method, modifies the fractal structure of the basin boundaries. The overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting.Comment: 11 pages of LateX output with five 8-bit color .ps figures. To appear in SIAM Review, 199

Comment on "Optimal Periodic Orbits of Chaotic Systems"

In a recent Letter, Hunt and Ott argued that SHORT-period unstable periodic orbits (UPOs) would be the invariant sets associated with a chaotic attractor that are most likely to optimize the time average of some smooth scalar performance function. In this Comment, we show that their conclusion does not hold generally and that optimal time averages may specifically require long-period UPOs. This situation can arise when long-period UPOs are able to spend substantial amounts of time in a region of phase space that is close to large values of the performance function.Comment: One Page, 1 Figure, Double Column format. Submitted to Physical Review Letter

Dependence of extensive chaos on the spatial correlation length (substantial revision)

We consider spatiotemporal chaotic systems for which spatial correlation functions decay substantially over a length scale xi (the spatial correlation length) that is small compared to the system size L. Numerical simulations suggest that such systems generally will be extensive, with the fractal dimension D growing in proportion to the system volume for sufficiently large systems (L >> xi). Intuitively, extensive chaos arises because of spatial disorder. Subsystems that are sufficiently separated in space should be uncorrelated and so contribute to the fractal dimension in proportion to their number. We report here the first numerical calculation that examines quantitatively how one important characterization of extensive chaos---the Lyapunov dimension density---depends on spatial disorder, as measured by the spatial correlation length xi. Surprisingly, we find that a representative extensively chaotic system does not act dynamically as many weakly interacting regions of size xi.Comment: 14 pages including 3 figures (Postscript files separate from the main text), uses equations.sty and aip.sty macros. Submitted to Natur

Characterization of the transition from defect- to phase-turbulence

For the complex Ginzburg-Landau equation on a large periodic interval, we show that the transition from defect- to phase-turbulence is more accurately described as a smooth crossover rather than as a sharp continuous transition. We obtain this conclusion by using a powerful parallel computer to calculate various order parameters, especially the density of space-time defects, the Lyapunov dimension density, and the correlation lengths of the field phase and amplitude. Remarkably, the correlation length of the field amplitude is, within a constant factor, equal to the length scale defined by the dimension density. This suggests that a correlation measurement may suffice to estimate the fractal dimension of some large homogeneous chaotic systems.Comment: 18 pages including 4 figures, uses REVTeX macros. Submitted to Phys. Rev. Let

Spatial Variation of Correlation Times for 1D Phase Turbulence

For one-dimensional phase-turbulent solutions of the Kuramoto-Sivashinsky equation with rigid boundary conditions, we show that there is a substantial variation of the correlation time~$\tau_c(x)$ with spatial position $x$ in moderately large systems of size $L$. These results suggest that some time-averaged properties of spatiotemporal chaos do not become homogeneous away from boundaries for large systems and for long times.Comment: 17 pages including 5 figures (Postscript files separate from the main text), uses revtex macros. Submitted to Physics Letters

Not Just a Black Box: Learning Important Features Through Propagating Activation Differences

Note: This paper describes an older version of DeepLIFT. See https://arxiv.org/abs/1704.02685 for the newer version. Original abstract follows: The purported "black box" nature of neural networks is a barrier to adoption in applications where interpretability is essential. Here we present DeepLIFT (Learning Important FeaTures), an efficient and effective method for computing importance scores in a neural network. DeepLIFT compares the activation of each neuron to its 'reference activation' and assigns contribution scores according to the difference. We apply DeepLIFT to models trained on natural images and genomic data, and show significant advantages over gradient-based methods.Comment: 6 pages, 3 figures, this is an older version; see https://arxiv.org/abs/1704.02685 for the newer versio

Stationarity and Redundancy of Multichannel EEG Data Recorded During Generalized Tonic-Clonic Seizures

A prerequisite for applying some signal analysis methods to electroencephalographic (EEG) data is that the data be statistically stationary. We have investigated the stationarity of 21-electrode multivariate EEG data recorded from ten patients during generalized tonic-clonic (GTC) seizures elicited by electroconvulsive therapy (ECT). Stationarity was examined by calculating probability density functions (pdfs) and power spectra over small equal-length non-overlapping time windows and then by studying visually and quantitatively the evolution of these quantities over the duration of the seizures. Our analysis shows that most of the seizures had time intervals of at least a few seconds that were statistically stationary by several criteria and simultaneously for different electrodes, and that some leads were delayed in manifesting the statistical changes associated with seizure onset evident in other leads. The stationarity across electrodes was further examined by studying redundancy of the EEG leads and how that redundancy evolved over the course of the GTC seizures. Using several different measures, we found a substantial redundancy which suggests that fewer than 21 electrodes will likely suffice for extracting dynamical and clinical insights. The redundancy analysis also demonstrates for the first time posterior-to-anterior time delays in the mid-ictal region of GTC seizures, which suggests the existence of propagating waves. The implications of these results are discussed for understanding GTC seizures and ECT treatment.Comment: 44 pages, 10 pages of figure

A Hierarchical Approach to Scaling Batch Active Search Over Structured Data

Active search is the process of identifying high-value data points in a large and often high-dimensional parameter space that can be expensive to evaluate. Traditional active search techniques like Bayesian optimization trade off exploration and exploitation over consecutive evaluations, and have historically focused on single or small (<5) numbers of examples evaluated per round. As modern data sets grow, so does the need to scale active search to large data sets and batch sizes. In this paper, we present a general hierarchical framework based on bandit algorithms to scale active search to large batch sizes by maximizing information derived from the unique structure of each dataset. Our hierarchical framework, Hierarchical Batch Bandit Search (HBBS), strategically distributes batch selection across a learned embedding space by facilitating wide exploration of different structural elements within a dataset. We focus our application of HBBS on modern biology, where large batch experimentation is often fundamental to the research process, and demonstrate batch design of biological sequences (protein and DNA). We also present a new Gym environment to easily simulate diverse biological sequences and to enable more comprehensive evaluation of active search methods across heterogeneous data sets. The HBBS framework improves upon standard performance, wall-clock, and scalability benchmarks for batch search by using a broad exploration strategy across coarse partitions and fine-grained exploitation within each partition of structured data.Comment: Presented at the 2020 ICML Workshop on Real World Experiment Design and Active Learnin