arXiv subjects
Publications and source records attributed to Kathleen Nowak.
We present a correspondence between multiplicity-free, self-dual, fusion rings and a digraph, hypergraph pair $(D,H)$. This correspondence is used to provide a complete characterization of all fusion rings corresponding to graphical properties of $D$. Further, we exploit this correspondence to provide a complete list of all non-isomorphic, self-dual, multiplicity-free fusion rings of rank at most 8.
Integration of heterogeneous sensors is a challenging problem across a range of applications. Prominent among these are multi-target tracking, where one must combine observations from different sensor types in a meaningful way to track multiple targets. Because sensors have differing error models, we seek a theoretically-justified quantification of the agreement among ensembles of sensors, both overall for a sensor collection, and also at a fine-grained level specifying pairwise and multi-way interactions among sensors. We demonstrate that the theory of mathematical sheaves provides a unified answer to this need, supporting both quantitative and qualitative data. The theory provides algorithms to globalize data across the network of deployed sensors, and to diagnose issues when the data do not globalize cleanly. We demonstrate the utility of sheaf-based tracking models based on experimental data of a wild population of black bears in Asheville, North Carolina. A measurement model involving four sensors deployed among the bears and the team of scientists charged with tracking their location is deployed. This provides a sheaf-based integration model which is small enough to fully interpret, but of sufficient complexity to demonstrate the sheaf's ability to recover a holistic picture of the locations and behaviors of both individual bears and the bear-human tracking system. A statistical approach was developed for comparison, a dynamic linear model which was estimated using a Kalman filter. This approach also recovered bear and human locations and sensor accuracies. When the observations are normalized into a common coordinate system, the structure of the dynamic linear observation model recapitulates the structure of the sheaf model, demonstrating the canonicity of the sheaf-based approach. But when the observations are not so normalized, the sheaf model still remains valid.
Since its popularization in the 1970s the Fiedler vector of a graph has become a standard tool for clustering of the vertices of the graph. Recently, Mendel and Noar, Dumitriu and Radcliffe, and Radcliffe and Williamson have introduced geometric generalizations of the Fiedler vector. Motivated by questions stemming from their work we provide structural characterizations for when a finite metric space can be isometrically embedded in a Hilbert space.
Partly in service of exploring the formal basis for Georgetown University's AvesTerra database structure, we formalize a recursive hypergraph data structure, which we call an ubergraph.
All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $\ell \in \{3,4,5,6,7\}$, a planar graph is 4-choosable if it is $\ell$-cycle-free. In terms of constraining the list assignment, one refinement of $k$-choosability is choosability with separation. A graph is $(k,s)$-choosable if the graph is colorable from lists of size $k$ where adjacent vertices have at most $s$ common colors in their lists. Every planar graph is $(4,1)$-choosable, but there exist planar graphs that are not $(4,3)$-choosable. It is an open question whether planar graphs are always $(4,2)$-choosable. A chorded $\ell$-cycle is an $\ell$-cycle with one additional edge. We demonstrate for each $\ell \in \{5,6,7\}$ that a planar graph is $(4,2)$-choosable if it does not contain chorded $\ell$-cycles.
The principal permanent rank characteristic sequence is a binary sequence $r_0 r_1 \ldots r_n$ where $r_k = 1$ if there exists a principal square submatrix of size $k$ with nonzero permanent and $r_k = 0$ otherwise, and $r_0 = 1$ if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.
In this paper, we show how certain three-class association schemes and orthogonal arrays give rise to partial geometric designs. We also investigate the connections between partial geometric designs and certain regular graphs having three or four distinct eigenvalues, three-class association schemes, orthogonal arrays of strength two and particular linear codes. We give various characterizations of these graphs, association schemes and orthogonal arrays in terms of partial geometric designs. We also give a list of infinite families of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.
Let $G$ be an additive group of order $v$. A $k$-element subset $D$ of $G$ is called a $(v, k, λ, t)$-almost difference set if the expressions $gh^{-1}$, for $g$ and $h$ in $D$, represent $t$ of the non-identity elements in $G$ exactly $λ$ times and every other non-identity element $λ+1$ times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. This paper reviews the recent work that has been done on almost difference sets and related topics. In this survey, we try to communicate the known existence and nonexistence results concerning almost difference sets. Further, we establish the link between certain almost difference sets and binary sequences with three-level autocorrelation. Lastly, we provide a thorough treatment of the tools currently being used to solve this problem. In particular, we review many of the construction methods being used to date, providing illustrative proofs and many examples.
Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the multiplicative cyclic subgroup $C$ of index twelve nor $C\cup \{0\}$ forms an almost difference set, but a union of cosets of $C$ provides us an almost difference set. We also calculate the intersection numbers and character tables of cyclotomic association schemes of class two, three and four obtained from these fields.