arXiv subjects

Katherine Thompson

Publications and source records attributed to Katherine Thompson.

14 recordsLinked to original sources

arXiv2024

Implementation and Assessment of Machine Learning Models for Forecasting Suspected Opioid Overdoses in Emergency Medical Services Data

We present efforts in the fields of machine learning and time series forecasting to accurately predict counts of future suspected opioid overdoses recorded by Emergency Medical Services (EMS) in the state of Kentucky. Forecasts help government agencies properly prepare and distribute resources related to opioid overdoses. Our approach uses county and district level aggregations of suspected opioid overdose encounters and forecasts future counts for different time intervals. Models with different levels of complexity were evaluated to minimize forecasting error. A variety of additional covariates relevant to opioid overdoses and public health were tested to determine their impact on model performance. Our evaluation shows that useful predictions can be generated with limited error for different types of regions, and high performance can be achieved using commonly available covariates and relatively simple forecasting models.

cs.LG↗

arXiv2023

Quartic Integral Polynomial Pell Equations

In this paper we classify all monic, quartic, polynomials $d(x)\in\mathbb{Z}[x]$ for which the Pell equation $$f(x)^2-d(x)g(x)^2=1$$ has a non-trivial solution with $f(x),g(x)\in\mathbb{Z}[x]$.

math.NT↗

arXiv2023

Densities of integer sets represented by quadratic forms

Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least $3$, we show that there is a $\delta(f) \in \mathbb{Q} \cap (0,1)$ such that $D_f(X) \sim \delta(f) X$ and call $\delta(f)$ the density of $f$. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density $0$ within the set of locally represented integers.

math.NT↗

arXiv2022

Palindromic Polynomials over Finite Fields

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application is given for systems of linear equations over $\mathbb{F}$ of index $2$.

math.NT↗

arXiv2022

Quaternary quadratic forms with prime discriminant

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Petersson norm $\langle C, C \rangle$ of the cuspidal part of the theta series of $Q$. We derive an upper bound on $\langle C, C \rangle$ that depends on the smallest positive integer not represented by the dual form $Q^{*}$. In addition, we give a non-trivial upper bound on the sum of the integers $n$ excepted by $Q$.

math.NT↗

arXiv2022

Elementary Proofs of Representation by Ternary Quadratic Forms

Mordell in 1958 gave a new proof of the three squares theorem. Those techniques were generalized by Blackwell, et al., in 2016 to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries". We continue the generalization of these techniques to four additional forms.

math.NT↗

arXiv2021

Trends on 3d Transition Metal Coordination on Monolayer MoS$_2$

Two-dimensional materials (2DM) have attracted much interest due to their distinct optical, electronic, and catalytic properties. These properties can be by tuned a range of methods including substitutional doping or, as recently demonstrated, by surface functionalization with single atoms, increasing even further 2DM portfolio. Here we theoretically and experimentally describe the coordination reaction between MoS$_2$ monolayers with 3d transition metals (TMs), exploring the nature and the trend of MoS$_2$-TMs interaction. Density Functional Theory calculations, X-Ray Photoelectron Spectroscopy (XPS), and Photoluminescence (PL) point to the formation of MoS$_2$-TM coordination complexes, where the adsorption energy trend for 3d TM resembles the crystal-field (CF) stabilization energy for weak-field complexes. Pearson's theory for hard-soft acid-base and Ligand-field theory were applied to discuss the periodic trends on 3d TM coordination on the MoS$_2$ surface. We found that softer acids with higher ligand field stabilization energy, such as Ni$^{2+}$, tend to form bonds with more covalent character with MoS$_2$, which can be considered a soft base. On the other hand, harder acids, such as Cr$^{3+}$, tend to form bonds with more ionic character. Additionally, we studied the trends in charge transfer and doping observed in the XPS and PL results, where metals such as Ni led to an n-type of doping, while Cu functionalization results in p-type doping. Therefore, the formation of coordination complexes on TMD's surface is demonstrated to be a promising and effective way to control and to understand the nature of the single-atom functionalization of TMD.

cond-mat.mtrl-sci↗

arXiv2018

The Sum of Four Squares Over Real Quadratic Number Fields

Well-known results of Lagrange and Jacobi prove that the every $m \in \mathbb N$ can be expressed as a sum of four integer squares, and the number $r(m)$ of such representations can be given by an explicit formula in $m$. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is $\mathbb Q(\sqrt{5})$. We provide explicit formulas for $r(m)$ for $K= \mathbb Q (\sqrt{2})$ and $K= \mathbb Q(\sqrt{5})$. We then consider the theta series of the sum of four squares over any real quadratic number field, providing explicit upper and lower bounds for the Eisenstein coefficients. Last, we include examples of the complete theta series decomposition of the sum of four squares over $\mathbb Q (\sqrt{3})$, $\mathbb Q (\sqrt{13})$ and $\mathbb Q (\sqrt{17})$.

math.NT↗

arXiv2017

Phylogenetic Derivative: A Tool for Assessing Local Tree Reconstruction in the Presence of Recombination

Recently, much attention has been given to understanding recombination events along a chromosome in a variety of field. For instance, many population genetics problems are limited by the inaccuracy of inferred evolutionary histories of chromosomes sampled randomly from a population. This evolutionary history differs among genomic locations as an artifact of recombination events along a chromosome. Thus, much recent attention has been focused on identifying these recombination points. However, many proposed methods either make simplifying, but unrealistic, assumptions about recombination along a chromosome, or are unable to scale to large genome-wide data like what has become commonplace in statistical genetics. Here, we introduce a \emph{phylogenetic derivative} to describe the relatedness of neighboring trees along a chromosome. This phylogenetic derivative is a computationally efficient, flexible metric that can be also be used assess the prevalence of recombination across a chromosome. These proposed methods are tested and perform well in analyzing both simulated data and a real mouse data set.

q-bio.QM↗

arXiv2017

Numbers Represented by a Finite Set of Binary Quadratic Forms

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than just the "trivial" 0? We address this question in the case of integral, positive- definite, reduced, binary quadratic forms. For forms of the same discriminant, we can use the structure of the underlying class group. If, however, the forms have different discriminants, we must apply class field theory.

math.NT↗

arXiv2016

Classically Integral Quadratic Forms Excepting at Most Two Values

Let $S \subseteq \mathbb{N}$ be finite. Is there a positive definite quadratic form that fails to represent only those elements in $S$? For $S = \emptyset$, this was solved (for classically integral forms) by the $15$-Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the $290$-Theorem of Bhargava-Hanke in the mid-2000s. In 1938 Halmos attempted to list all weighted sums of four squares that failed to represent $S=\{m\}$; of his $88$ candidates, he could provide complete justifications for all but one. In the same spirit, we ask, "for which $S = \{m, n\}$ does there exist a quadratic form excepting only the elements of $S$?" Extending the techniques of Bhargava and Hanke, we answer this question for quaternary forms. In the process, we prove what Halmos could not; namely, that $x^2+2y^2+7z^2+13w^2$ represents all positive integers except $5$. We develop new strategies to handle forms of higher dimensions, yielding an enumeration of and proofs for the $73$ possible pairs that a classically integral positive definite quadratic form may except.

math.NT↗

arXiv2015

A Generalization of Mordell to Ternary Quadratic Forms

Mordell in 1958 gave a new proof of the three squares theorem. We generalize those techniques to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries" as well as three other ternary forms.

math.NT↗

arXiv2011

The embedding structure for linearly ordered topological spaces

In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality $κ$ it is proved that this quasi-order has no maximal element for $κ$ at least the size of the continuum and that in fact the dominating number for such quasi-orders is maximal, i.e. $2^κ$. Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal structure of their respective embedding quasi-order. The basis problem for uncountable LOTS is also considered; assuming the Proper Forcing Axiom there is an eleven element basis for the class of uncountable LOTS and a six element basis for the class of dense uncountable LOTS in which all points have countable cofinality and coinitiality.

math.LO↗

arXiv2010

Arithmetic Structure in Sparse Difference Sets

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration $\{v_1,...,v_\ell\}$ of vectors in $\mathbb{Z}^d$, we show that if $A\subset[1,N]^d$ with $|A|/N^d\geq C N^{-1/\ell}$, then there necessarily exists $r\ne0$ such that $\{rv_1, ...,rv_\ell\}\subseteq A-A$.

math.NT↗