29 research outputs found
Graph-Theoretic Simplicial Complexes, Hajos-type Constructions, and k-Matchings
A graph property is monotone if it is closed under the removal of edges and vertices. Given a graph G and a monotone graph property P, one can associate to the pair (G,P) a simplicial complex, which serves as a way to encode graph properties within faces of a topological space. We study these graph-theoretic simplicial complexes using combinatorial and topological approaches as a way to inform our understanding of the graphs and their properties.
In this dissertation, we study two families of simplicial complexes: (1) neighborhood complexes and (2) k-matching complexes. A neighborhood complex is a simplicial complex of a graph with vertex set the vertices of the graph and facets given by neighborhoods of each vertex of the graph. In 1978, Lov\\u27asz used neighborhood complexes as a tool for studying lower bounds for the chromatic number of graphs. In Chapter 2, we will prove results about the connectivity of neighborhood complexes in relation to Haj\\u27os-type constructions and analyze randomly generated graphs arising from two Haj\\u27os-type stochastic algorithms using SageMath. Chapter 3 will focus on k-matching complexes. A k-matching complex of a graph is a simplicial complex with vertex set given by edges of the graph and faces given sets of edges in the graph such that each vertex of the induced graph has degree at most k. We pursue the study of k-matching complexes and investigate 2-matching complexes of wheel graphs and caterpillar graphs
2-Matching Complexes
A -matching complex is a simplicial complex which captures the
relationship between -matchings of a graph. In this paper, we will use
discrete Morse Theory and the Matching Tree Algorithm to prove homotopical
results. We will consider a class of graphs for which the homotopy type of the
-matching complex transforms from a sphere to a point with the addition of
leaves. We end the paper by defining -matching sequences and looking at the
- and -matching complexes of wheel graphs and perfect caterpillar graphs
General polygonal line tilings and their matching complexes
A (general) polygonal line tiling is a graph formed by a string of cycles,
each intersecting the previous at an edge, no three intersecting. In 2022,
Matsushita proved the matching complex of a certain type of polygonal line
tiling with even cycles is homotopy equivalent to a wedge of spheres. In this
paper, we extend Matsushita's work to include a larger family of graphs and
carry out a closer analysis of lines of triangle and pentagons, where the
Fibonacci numbers arise.Comment: 22 page
A Reflection on Growth Mindset and Meritocracy
As mathematicians working in higher education we reflect on meritocracy and growth mindset with a focus on the relationship between the two. We also note the subtle differences between growth mindset and grit. Our reflection ends with suggestions for how to move forward in the math classroom and throughout the collegiate level
Triangulations, order polytopes, and generalized snake posets
This work regards the order polytopes arising from the class of generalized
snake posets and their posets of meet-irreducible elements. Among generalized
snake posets of the same rank, we characterize those whose order polytopes have
minimal and maximal volume. We give a combinatorial characterization of the
circuits in these order polytopes and then conclude that every regular
triangulation is unimodular. For a generalized snake word, we count the number
of flips for the canonical triangulation of these order polytopes. We determine
that the flip graph of the order polytope of the poset whose lattice of filters
comes from a ladder is the Cayley graph of a symmetric group. Lastly, we
introduce an operation on triangulations called twists and prove that twists
preserve regular triangulations.Comment: 39 pages, 26 figures, comments welcomed
Lattice polytopes from Schur and symmetric Grothendieck polynomials
Given a family of lattice polytopes, two common questions in Ehrhart Theory
are determining when a polytope has the integer decomposition property and
determining when a polytope is reflexive. While these properties are of
independent interest, the confluence of these properties is a source of active
investigation due to conjectures regarding the unimodality of the
-polynomial. In this paper, we consider the Newton polytopes arising
from two families of polynomials in algebraic combinatorics: Schur polynomials
and inflated symmetric Grothendieck polynomials. In both cases, we prove that
these polytopes have the integer decomposition property by using the fact that
both families of polynomials have saturated Newton polytope. Furthermore, in
both cases, we provide a complete characterization of when these polytopes are
reflexive. We conclude with some explicit formulas and unimodality implications
of the -vector in the case of Schur polynomials.Comment: 37 pages, 3 tables, 4 figures; Comments Welcome; Version 2: updated
references to acknowledge one result was previously known, corrected values
in Table 1 and reference correct OEIS sequence; Version 3: Final Version. To
appear in Electronic Journal of Combinatoric
Height and body-mass index trajectories of school-aged children and adolescents from 1985 to 2019 in 200 countries and territories: a pooled analysis of 2181 population-based studies with 65 million participants
Summary Background Comparable global data on health and nutrition of school-aged children and adolescents are scarce. We aimed to estimate age trajectories and time trends in mean height and mean body-mass index (BMI), which measures weight gain beyond what is expected from height gain, for school-aged children and adolescents. Methods For this pooled analysis, we used a database of cardiometabolic risk factors collated by the Non-Communicable Disease Risk Factor Collaboration. We applied a Bayesian hierarchical model to estimate trends from 1985 to 2019 in mean height and mean BMI in 1-year age groups for ages 5–19 years. The model allowed for non-linear changes over time in mean height and mean BMI and for non-linear changes with age of children and adolescents, including periods of rapid growth during adolescence. Findings We pooled data from 2181 population-based studies, with measurements of height and weight in 65 million participants in 200 countries and territories. In 2019, we estimated a difference of 20 cm or higher in mean height of 19-year-old adolescents between countries with the tallest populations (the Netherlands, Montenegro, Estonia, and Bosnia and Herzegovina for boys; and the Netherlands, Montenegro, Denmark, and Iceland for girls) and those with the shortest populations (Timor-Leste, Laos, Solomon Islands, and Papua New Guinea for boys; and Guatemala, Bangladesh, Nepal, and Timor-Leste for girls). In the same year, the difference between the highest mean BMI (in Pacific island countries, Kuwait, Bahrain, The Bahamas, Chile, the USA, and New Zealand for both boys and girls and in South Africa for girls) and lowest mean BMI (in India, Bangladesh, Timor-Leste, Ethiopia, and Chad for boys and girls; and in Japan and Romania for girls) was approximately 9–10 kg/m2. In some countries, children aged 5 years started with healthier height or BMI than the global median and, in some cases, as healthy as the best performing countries, but they became progressively less healthy compared with their comparators as they grew older by not growing as tall (eg, boys in Austria and Barbados, and girls in Belgium and Puerto Rico) or gaining too much weight for their height (eg, girls and boys in Kuwait, Bahrain, Fiji, Jamaica, and Mexico; and girls in South Africa and New Zealand). In other countries, growing children overtook the height of their comparators (eg, Latvia, Czech Republic, Morocco, and Iran) or curbed their weight gain (eg, Italy, France, and Croatia) in late childhood and adolescence. When changes in both height and BMI were considered, girls in South Korea, Vietnam, Saudi Arabia, Turkey, and some central Asian countries (eg, Armenia and Azerbaijan), and boys in central and western Europe (eg, Portugal, Denmark, Poland, and Montenegro) had the healthiest changes in anthropometric status over the past 3·5 decades because, compared with children and adolescents in other countries, they had a much larger gain in height than they did in BMI. The unhealthiest changes—gaining too little height, too much weight for their height compared with children in other countries, or both—occurred in many countries in sub-Saharan Africa, New Zealand, and the USA for boys and girls; in Malaysia and some Pacific island nations for boys; and in Mexico for girls. Interpretation The height and BMI trajectories over age and time of school-aged children and adolescents are highly variable across countries, which indicates heterogeneous nutritional quality and lifelong health advantages and risks
Robust estimation of bacterial cell count from optical density
Optical density (OD) is widely used to estimate the density of cells in liquid culture, but cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count. We address this with an interlaboratory study comparing three simple, low-cost, and highly accessible OD calibration protocols across 244 laboratories, applied to eight strains of constitutive GFP-expressing E. coli. Based on our results, we recommend calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals <1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements: in our study, fluorescence per cell measurements showed only a 1.07-fold mean difference between plate reader and flow cytometry data
Heterogeneous contributions of change in population distribution of body mass index to change in obesity and underweight NCD Risk Factor Collaboration (NCD-RisC)
From 1985 to 2016, the prevalence of underweight decreased, and that of obesity and severe obesity increased, in most regions, with significant variation in the magnitude of these changes across regions. We investigated how much change in mean body mass index (BMI) explains changes in the prevalence of underweight, obesity, and severe obesity in different regions using data from 2896 population-based studies with 187 million participants. Changes in the prevalence of underweight and total obesity, and to a lesser extent severe obesity, are largely driven by shifts in the distribution of BMI, with smaller contributions from changes in the shape of the distribution. In East and Southeast Asia and sub-Saharan Africa, the underweight tail of the BMI distribution was left behind as the distribution shifted. There is a need for policies that address all forms of malnutrition by making healthy foods accessible and affordable, while restricting unhealthy foods through fiscal and regulatory restrictions