We study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the Hopf equation whenever the classical solution of the equation develops a singularity. Modulation equations are derived that describe period-two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic. However, those equations have an elliptic region that may be entered by its solutions in a finite time, after which the corresponding period-two oscillations are seen to break down. This kind of phenomenon has not been observed for integrable schemes. The generation and propagation of period-two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions. Modulation equations are also presented for period-three oscillations. Numerical experiments are carried out that illustrate our analysis.

This paper is concerned with the problem of top- K ranking from pairwise comparisons. Given a collection of K items and a few pairwise comparisons across them, one wishes to identify the set of K items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model---the Bradley-Terry-Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress towards characterizing the performance (eg the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top- K ranking remains unsettled.

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al.[3] for bounded degree graphs, and for three-connected graphs of fixed genus g . Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the g th Laplacian eigenvalue is at most $2\left [

Let φ∈C^0∩W_{1,2}(Σ,X) where Σ is a compact Riemann surface, X is a compact locally CAT(1) space, and W_{1,2}(Σ,X) is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map u: Σ → X homotopic to φ or there exists a nontrivial conformal harmonic map v: S^2 → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.

A sensor pixel integrates optical intensity across its extent, and we explore the role that this integration plays in phase space tomography. The literature is inconsistent in its treatment of this integration—some approaches model this integration explicitly, some approaches are ambiguous about whether this integration is taken into account, and still some approaches assume pixel values to be point samples of the optical intensity. We show that making a point-sample assumption results in apodization of and thus systematic error in the recovered ambiguity function, leading to underestimating the overall degree of coherence. We explore the severity of this effect using a Gaussian Schell-model source and discuss when this effect, as opposed to noise, is the dominant source of error in the retrieved state of coherence.