arXiv subjects

Pamela Harris

Publications and source records attributed to Pamela Harris.

7 recordsLinked to original sources

arXiv2025

Restricted Fubini Rankings and Restricted Unit Interval Parking Functions

We study three natural types of restrictions on Fubini rankings and unit interval parking functions, which are motivated by their correspondence with ordered set partitions. For each restriction type, we define the corresponding subset of Fubini rankings and unit interval parking functions, establish enumerative results, and provide bijections between the restricted families. We also obtain exponential generating functions and combinatorial interpretations, including connections with exceedances in permutations and with the absence of cyclical adjacencies in set partitions.

math.CO↗

arXiv2022

CASA, the Common Astronomy Software Applications for Radio Astronomy

CASA, the Common Astronomy Software Applications, is the primary data processing software for the Atacama Large Millimeter/submillimeter Array (ALMA) and the Karl G. Jansky Very Large Array (VLA), and is frequently used also for other radio telescopes. The CASA software can handle data from single-dish, aperture-synthesis, and Very Long Baseline Interferometery (VLBI) telescopes. One of its core functionalities is to support the calibration and imaging pipelines for ALMA, VLA, VLA Sky Survey (VLASS), and the Nobeyama 45m telescope. This paper presents a high-level overview of the basic structure of the CASA software, as well as procedures for calibrating and imaging astronomical radio data in CASA. CASA is being developed by an international consortium of scientists and software engineers based at the National Radio Astronomical Observatory (NRAO), the European Southern Observatory (ESO), the National Astronomical Observatory of Japan (NAOJ), and the Joint Institute for VLBI European Research Infrastructure Consortium (JIV-ERIC), under the guidance of NRAO.

astro-ph.IM↗

arXiv2020

Tipsy cop and drunken robber: a variant of the cop and robber game on graphs

Motivated by a biological scenario illustrated in the YouTube video \url{ https://www.youtube.com/watch?v=Z_mXDvZQ6dU} where a neutrophil chases a bacteria cell moving in random directions, we present a variant of the cop and robber game on graphs called the tipsy cop and drunken robber game. In this game, we place a tipsy cop and a drunken robber at different vertices of a finite connected graph $G$. The game consists of independent moves where the robber begins the game by moving to an adjacent vertex from where he began, this is then followed by the cop moving to an adjacent vertex from where she began. Since the robber is inebriated, he takes random walks on the graph, while the cop being tipsy means that her movements are sometimes random and sometimes intentional. Our main results give formulas for the probability that the robber is still free from capture after $m$ moves of this game on highly symmetric graphs, such as the complete graphs, complete bipartite graphs, and cycle graphs. We also give the expected encounter time between the cop and robber for these families of graphs. We end the manuscript by presenting a general method for computing such probabilities and also detail a variety of directions for future research.

math.CO↗

arXiv2019

Kostant's Weight Multiplicity Formula and the Fibonacci and Lucas Numbers

Consider the weight $λ$ which is the sum of all simple roots of a simple Lie algebra. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of the zero weight in the representation with highest weight $λ$. We prove that in Lie algebras of type $A$ and $B$, the number of contributing terms to the multiplicity of the zero-weight space in the representation with highest weight $λ$ is given by a Fibonacci number, and that in Lie algebras of type $C$ and $D$, the analogous result is given by a multiple of a Lucas number.

math.RT↗

arXiv2017

The Graph of Critical Pairs of a Crown

There is a natural way to associate with a poset $P$ a hypergraph $H$, called the hypergraph of critical pairs, so that the dimension of $P$ is exactly equal to the chromatic number of $H$. The edges of $H$ have variable sizes, but it is of interest to consider the graph $G$ formed by the edges of $H$ that have size~2. The chromatic number of $G$ is less than or equal to the dimension of $P$ and the difference between the two values can be arbitrarily large. Nevertheless, there are important instances where the two parameters are the same, and we study one of these in this paper. Our focus is on a family $\{S_n^k:n\ge 3, k\ge 0\}$ of height two posets called crowns. We show that the chromatic number of the graph $G_n^k$ of critical pairs of the crown $S_n^k$ is the same as the dimension of $S_n^k$, which is known to be $\lceil 2(n+k)/(k+2)\rceil$. In fact, this theorem follows as an immediate corollary to the stronger result: The independence number of $G_n^k$ is $(k+1)(k+2)/2$. We obtain this theorem as part of a comprehensive analysis of independent sets in $G_n^k$ including the determination of the second largest size among the maximal independent sets, both the reversible and non-reversible types.

math.CO↗

arXiv2014

Generalizing Zeckendorf's Theorem: The Kentucky Sequence

By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be written uniquely as a sum of non-adjacent elements; this is called a legal decomposition. Previous work examined the distribution of the number of summands and the spacings between them, in legal decompositions arising from the Fibonacci numbers and other linear recurrence relations with non-negative integral coefficients. Many of these results were restricted to the case where the first term in the defining recurrence was positive. We study a generalization of the Fibonacci numbers with a simple notion of legality which leads to a recurrence where the first term vanishes. We again have unique legal decompositions, Gaussian behavior in the number of summands, and geometric decay in the distribution of gaps.

math.NT↗

arXiv2014

A new characterization of the exceptional Lie algebras

For a simple Lie algebra, over $\mathbb{C}$, we consider the weight which is the sum of all simple roots and denote it $\tildeα$. We formally use Kostant's weight multiplicity formula to compute the "dimension" of the zero-weight space. In type $A_r$, $\tildeα$ is the highest root, and therefore this dimension is the rank of the Lie algebra. In type $B_r$, this is the defining representation, with dimension equal to 1. In the remaining cases, the weight $\tildeα$ is not dominant and is not the highest weight of an irreducible finite-dimensional representation. Kostant's weight multiplicity formula, in these cases, is assigning a value to a virtual representation. The point, however, is that this number is nonzero if and only if the Lie algebra is classical. This gives rise to a new characterization of the exceptional Lie algebras as the only Lie algebras for which this value is zero.

math.RA↗