Matroids arising from electrical networks

This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main results. First, we exhibit a matroid quotient formed by the dual of a network embedded in a surface with boundary and the dual of the associated Dirichlet matroid. This generalizes an analogous result for graphic matroids of cellularly embedded graphs. Second, we characterize the Bergman fans of Dirichlet matroids as explicit subfans of graphic Bergman fans. In doing so, we generalize the connection between Bergman fans of complete graphs and phylogenetic trees. Third, we use the half-plane property of Dirichlet matroids to prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the chromatic polynomial of the underlying graph.Comment: 27 pages, 14 figure

Higher discrete homotopy groups of graphs

This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$ is trivial for all $n\geq 2$. Second we exhibit for each $n\geq 1$ a natural homomorphism $\psi:A_n(G)\to \mathcal{H}_n(G)$, where $\mathcal{H}_n(G)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $\mathcal{H}_n(G)$ is nontrivial and $\psi$ is surjective. It follows that for each $n\geq 1$ there are graphs $G$ for which $A_n(G)$ is nontrivial.Comment: 21 pages, 10 figure

Four quotient set gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.Comment: 9 pages, to appear in the American Mathematical Monthl

Two remarks about nilpotent operators of order two

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.Comment: 7 pages. To appear in Proceedings of the AM

The Graphic Nature of Gaussian Periods

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework with which to study the properties of certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, this approach reveals that these classical objects display a dazzling array of visual patterns of great complexity and remarkable subtlety

JPSS Science Data Services for the Direct Readout Community

The Suomi National Polar-orbiting Partnership (S-NPP) and Joint Polar Satellite System (JPSS) High Rate Data (HRD) link provides Direct Broadcast data to users in real-time, utilizing their own remote field terminals. The Field Terminal Support (FTS) provides the resources needed to support the Direct Readout communities by providing software, documentation, and periodic updates to enable them to produce data products from SNPP and JPSS. The FTS distribution server will also provide the necessary ancillary and auxiliary data needed for processing the broadcasts, as well as making orbital data available to assist in locating the satellites of interest. In addition, the FTS provides development support for the algorithm and software through GSFC Direct Readout Laboratory (DRL) International Polar Orbiter Processing Package (IPOPP) and University of Wisconsin (UWISC) Community Satellite Processing Package (CSPP), to enable users to integrate the algorithms into their remote terminals. The support the JPSS Program provides to the institutions developing and maintaining these two software packages, will demonstrate the ability to produce ready-to-use products from the HRD link and provide risk reduction effort at a minimal cost. This paper discusses the key functions and system architecture of FTS

Four Quotient Set Gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the MONTHLY, despite its intense coverage of quotient sets over the years