Untold Stories: A Phenomenological Study of Parent and Educator Perspectives of Parental Engagement in Title One Elementary Schools

This qualitative, phenomenological study explored parent and educator perceptions of parental engagement in Title One elementary schools. Twelve participants were included in this study. Six parents and six educators from the Southeastern, Southwestern, and Western regions of the United States defined parental engagement and how parental engagement impacts student achievement. The following research questions guided this study: (a) What are parent perceptions of parental engagement in Title One schools? (b) What are teacher perceptions of parental engagement in Title One schools? (c) What are the similarities and differences in parent and teacher perception of parental engagement? (d) How can these perceptions be utilized to improve parental engagement to benefit student achievement? Joyce Epstein’s six types of involvement was the framework used in this study. Six themes emerged: (a) Communication, (b) Partnership/Relationship, (c) Methods of Engagement, (d) Achievement, (e) Mental Health, and (f) Resources. Themes one through five emerged for parents, while all six themes emerged for educators. This study concludes with a discussion of findings, implications for future research, and recommendations for practice

Mind your Language (Model): Fact-Checking LLMs and their Role in NLP Research and Practice

Much of the recent discourse within the NLP research community has been centered around Large Language Models (LLMs), their functionality and potential -- yet not only do we not have a working definition of LLMs, but much of this discourse relies on claims and assumptions that are worth re-examining. This position paper contributes a definition of LLMs, explicates some of the assumptions made regarding their functionality, and outlines the existing evidence for and against them. We conclude with suggestions for research directions and their framing in future work

On Minimizing the Energy of a Spherical Graph Representation

Graph representations are the generalization of geometric graph drawings from the plane to higher dimensions. A method introduced by Tutte to optimize properties of graph drawings is to minimize their energy. We explore this minimization for spherical graph representations, where the vertices lie on a unit sphere such that the origin is their barycentre. We present a primal and dual semidefinite program which can be used to find such a spherical graph representation minimizing the energy. We denote the optimal value of this program by $\rho(G)$ for a given graph $G$. The value turns out to be related to the second largest eigenvalue of the adjacency matrix of $G$, which we denote by $\lambda_2$. We show that for $G$ regular, $\rho(G) \leq \frac{\lambda_{2}}{2} \cdot v(G)$, and that equality holds if and only if the $\lambda_{2}$ eigenspace contains a spherical 1-design. Moreover, if $G$ is a random $d$-regular graph, $\rho(G)=\left(\sqrt{(d-1)} +o(1)\right)\cdot v(G)$, asymptotically almost surely.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

PI Degree and Irreducible Representations of Quantum Determinantal Rings and their Associated Quantum Schubert Varieties

We study quantum determinantal rings at roots of unity and calculate the PI degree using results of Lenagan-Rigal and Haynal to reduce the problem to finding properties of their associated matrices. These matrices, in turn, correspond to Cauchon-Le diagrams from which we can calculate the required matrix properties. In particular, we show that any matrix corresponding to an $m\times n$ diagram has invariant factors which are powers of 2. Our calculations allow us to state an explicit expression for the PI degree of quantum determinantal rings when the deformation parameter $q$ is a primitive $\ell^{\text{th}}$ root of unity with $\ell$ odd. Using this newly calculated PI degree we present a method to construct an irreducible representation of maximal dimension. Building on these results, we use the strong connection between quantum determinantal rings and certain quantum Schubert varieties through noncommutative dehomogenisation to obtain expressions for the PI degree of such quantum Schubert varieties under the same conditions on $q$.Comment: 36 page

A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

This paper extends an algorithm and canonical embedding by Cauchon to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon's original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space $A'$ from the original quantum algebra $A$ via a series of change of variables within the division ring of fractions $\mathrm{Frac}(A)$. The canonical embedding takes a completely prime ideal $P\lhd A$ to a completely prime ideal $Q\lhd A'$ such that when $A$ is a PI algebra, ${\rm PI}\text{-}{\rm deg}(A/P) = {\rm PI}\text{-}{\rm deg}(A'/Q)$. When the quantum parameter is a root of unity we can state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of $A/P$ given a suitable irreducible representation of $A'/Q$ when $A$ is PI.Comment: 27 page

Representations of Quantum Nilpotent Algebras at Roots of Unity, and Their Completely Prime Quotients

This thesis studies algebras contained in a large class of iterated Ore extensions, as well as their quotient algebras by completely prime ideals, and develops methods for computing their \emph{polynomial identity (PI)} degree and constructing irreducible representations of maximal dimension. This class contains quantum nilpotent algebras, including many examples of quantised coordinate rings and quantised enveloping algebras. When the deformation parameters are allowed to be roots of unity these algebras often become PI algebras. We focus our attention on such algebras in this work. By extending Cauchon's deleting derivations algorithm in the generic setting we are able, given a suitable PI algebra $A$ and completely prime ideal $P\lhd A$, to construct a quantum affine space $A'$ and completely prime ideal $Q\lhd A'$, such that the quotient algebras $A/P$ and $A'/Q$ share the same PI degree. This extends a result of Haynal, where existence of $Q$ was proved but no method of construction was provided. The PI degree of several small examples are then calculated. For completely prime quotients of quantum matrices the PI degree is shown to be closely related to properties of \emph{Cauchon-Le diagrams}. We prove that given any Cauchon-Le diagram, the invariant factors of its associated matrix are all powers of $2$. Furthermore, we compute the \emph{toric permutation} of Cauchon-Le diagrams corresponding to \emph{quantum determinantal rings}, which then allows us to state an explicit formula for the PI degree of a quantum determinantal ring at a root of unity. Finally, we show how certain irreducible representations of the quotient $A'/Q$ may be passed through the deleting derivations algorithm to give an irreducible representation of $A/P$, and we construct an irreducible representation of a general quantum determinantal ring

Prospective memory in schizophrenia: Relationship to medication management skills, neurocognition and symptoms in individuals with schizophrenia [pre-print]

Objective: Impaired adherence to medication regimens is a serious concern for individuals with schizophrenialinked to relapse and poorer outcomes. One possible reason for poor adherence to medication ispoor ability to remember future intentions, labeled prospective memory skills. It has been demonstratedin several studies that individuals with schizophrenia have impairments in prospective memory that arelinked to everyday life skills. However, there have been no studies, to our knowledge, examining therelationship of a clinical measure of prospective memory to medication management skills, a key elementof successful adherence. Methods: In this Study 41 individuals with schizophrenia and 25 healthy adultswere administered a standardized test battery that included measures of prospective memory, medicationmanagement skills, neurocognition, and symptoms. Results: Individuals with schizophrenia demonstratedimpairments in prospective memory (both time and event-based) relative to healthy controls.Performance on the test of prospective memory was correlated with the standardized measure ofmedication management in individuals with schizophrenia. Moreover, the test of prospective memorypredicted skills in medication adherence even after measures of neurocognition were accounted for.Conclusions: This suggests that prospective memory may play a key role in medication managementskills and thus should be a target of cognitive remediation programs