Unbounded regions of Infinitely Logconcave Sequences

We study the properties of a logconcavity operator on a symmetric, unimodal subset of finite sequences. In doing so we are able to prove that there is a large unbounded region in this subset that is $\infty$-logconcave. This problem was motivated by the conjecture of Moll and Boros in that the binomial coefficients are $\infty$-logconcave.Comment: 12 pages, final version incorporating referee's comments. Now published by the Electronic Journal of Combinatorics http://www.combinatorics.org/index.htm

Generalized Spectral Analysis for Large Sets of Approval Voting Data

Generalized Spectral analysis of approval voting data uses representation theory and the symmetry of the data to project the approval voting data into orthogonal and interpretable subspaces. Unfortunately, as the number of voters grows, the data space becomes prohibitively large to compute the decomposition of the data vector. To attack these large data sets we develop a method to partition the data set into equivalence classes, in order to drastically reduce the size of the space while retaining the necessary characteristics of the data set. We also make progress on the needed statistical tools to explain the results of the spectral analysis. The standard spectral analysis will be demonstrated, and our partitioning technique is applied to U.S. Senate roll call data

Multiclass Total Variation Clustering

Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a general framework for multiclass total variation clustering that does not rely on recursion. The results greatly outperform previous total variation algorithms and compare well with state-of-the-art NMF approaches

Identifying group contributions in NBA lineups with spectral analysis

We address the question of how to quantify the contributions of groups of players to team success. Our approach is based on spectral analysis, a technique from algebraic signal processing, which has several appealing features. First, our analysis decomposes the team success signal into components that are naturally understood as the contributions of player groups of a given size: individuals, pairs, triples, fours, and full five-player lineups. Secondly, the decomposition is orthogonal so that contributions of a player group can be thought of as pure: Contributions attributed to a group of three, for example, have been separated from the lower-order contributions of constituent pairs and individuals. We present detailed a spectral analysis using NBA play-by-play data and show how this can be a practical tool in understanding lineup composition and utilization

An Adaptive Total Variation Algorithm for Computing the Balanced Cut of a Graph

We propose an adaptive version of the total variation algorithm proposed in [3] for computing the balanced cut of a graph. The algorithm from [3] used a sequence of inner total variation minimizations to guarantee descent of the balanced cut energy as well as convergence of the algorithm. In practice the total variation minimization step is never solved exactly. Instead, an accuracy parameter is specified and the total variation minimization terminates once this level of accuracy is reached. The choice of this parameter can vastly impact both the computational time of the overall algorithm as well as the accuracy of the result. Moreover, since the total variation minimization step is not solved exactly, the algorithm is not guarantied to be monotonic. In the present work we introduce a new adaptive stopping condition for the total variation minimization that guarantees monotonicity. This results in an algorithm that is actually monotonic in practice and is also significantly faster than previous, non-adaptive algorithms

Vorticity Dynamics and Sound Generation in Two-Dimensional Fluid Flow

An approximate solution to the two-dimensional incompressible fluid equations is constructed by expanding the vorticity field in a series of derivatives of a Gaussian vortex. The expansion is used to analyze the motion of a corotating Gaussian vortex pair, and the spatial rotation frequency of the vortex pair is derived directly from the fluid vorticity equation. The resulting rotation frequency includes the effects of finite vortex core size and viscosity and reduces, in the appropriate limit, to the rotation frequency of the Kirchhoff point vortex theory. The expansion is then used in the low Mach number Lighthill equation to derive the far-field acoustic pressure generated by the Gaussian vortex pair. This pressure amplitude is compared with that of a previous fully numerical simulation in which the Reynolds number is large and the vortex core size is significant compared to the vortex separation. The present analytic result for the far-field acoustic pressure is shown to be substantially more accurate than previous theoretical predictions. The given example suggests that the vorticity expansion is a useful tool for the prediction of sound generated by a general distributed vorticity field