The non-symmetric strong multiplicity property for sign patterns

We develop a non-symmetric strong multiplicity property for matrices that may or may not be symmetric. We say a sign pattern allows the non-symmetric strong multiplicity property if there is a matrix with the non-symmetric strong multiplicity property that has the given sign pattern. We show that this property of a matrix pattern preserves multiplicities of eigenvalues for superpatterns of the pattern. We also provide a bifurcation lemma, showing that a matrix pattern with the property also allows refinements of the multiplicity list of eigenvalues. We conclude by demonstrating how this property can help with the inverse eigenvalue problem of determining the number of distinct eigenvalues allowed by a sign pattern

Barycenter of the arithmetic-harmonic quantum divergence

A notion of divergence is a very important and useful tool to measure the difference between probability distributions or between data (information). We consider a quantum divergence constructed by the difference of two-variable weighted arithmetic and harmonic means on the open convex cone of positive definite Hermitian matrices, called the arithmetic-harmonic quantum divergence. We see its invariance properties and study the barycenter minimizing the weighted sum of arithmetic-harmonic quantum divergences to given variables. We provide the lower bound for the barycenter of the arithmetic-harmonic quantum divergence in terms of Loewner order and its upper bound in terms of operator norm

GLT sequences and normal matrices

The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(\kappa)$, where $\kappa$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices

Schur functions and immanantal identities

Littlewood developed the theory of symmetric functions and immanants. It is known that some identities for immanants correspond to the ordinary products of Schur functions via the Littlewood-Richardson rule. We discuss the relations between immanants and plethysm, another type of products of Schur functions. We present immanantal identities corresponding to the most basic formula of plethysm. As an application, we show some inequalities for positive semidefinite Hermitian matrices

Artificial Intelligence-Enhanced Digital Storytelling: Empowering Young Creators in a Summer STEM Camp

This lesson engages students in grades 3–6 in creating digital stories enhanced by artificial intelligence (AI), encouraging logical thinking, creativity, and problem-solving. Using a structured, project-based format, students developed multimedia-rich narratives with support from AI tools for idea generation, writing refinement, and visual content creation. The students explored STEM role models, wrote story drafts, and edited videos using WeVideo. The project concluded with the production of AI-supported digital stories, assessed through a structured rubric

The Plot Thickens: Literacy Through Computational Thinking

In this lesson, students analyzed word frequency, character interactions, and recurring themes in Frankenstein to predict how the novel unfolds before reading the next chapter. Students mapped a decision tree for Victor Frankenstein’s choices and predict alternative endings. This lesson utilized Plotting Plots (n.d.-a), created Dr. Tom Liam Lynch as a classroom resource that blends literature with data analysis. This tool leverages computational and quantitative approaches to understanding books by creating visual representations of literary data. The Much Ado About K-12 Computer Science: A Crash Course for ELA and English Teachers YouTube video playlist videos (Plotting Points, n.d.-c) are designed for secondary English Language Arts teachers to strategically map keywords and to support students’ exploration and inferences of what they could mean for the plot of the text

Puzzling Our Way into Computational Thinking

This unplugged activity is a brief, practical introduction to computational thinking that uses an accessible and decidedly low-tech approach: solving a jigsaw puzzle. The skills needed to collaboratively solve a jigsaw puzzle illustrate the key concepts of computational thinking in a straightforward way that makes the basics of decomposition, pattern recognition, abstraction, and algorithmic thinking come to life. The learning representation included here was taught as part of a professional development workshop for PK-12 teachers but could easily be adapted to use with learners from upper elementary grades through middle school, high school, or university

Erratum for the paper "Solution of symmetric positive semidefinite Procrustes problem'' [Electron. J. Linear Algebra 35 (2019) 543-554]

We give an example showing that the main result, i.e., Theorem 2.12, in the paper by Peng, Wang, Peng, and Chen [Electron. J. Linear Algebra 35 (2019) 543-554] is not correct. We also indicate where the mistake in the proof occurs

Construction of a solution to the rank 2 Horn problem

Given three sets of $n$ real eigenvalues satisfying the trace equality and the Horn inequalities, we know that there are $n\times n$ real symmetric matrices $A$ and $B$ so that $A$ has the first set of eigenvalues, $B$ has the second set of eigenvalues, and $A+B$ has the last set of eigenvalues. Under the condition that $B$ is a rank 2 matrix, we give a construction for the matrices $A$ and $B$. This construction is based on performing two orthogonal rank 1 updates on $A$. We end with a discussion of the relationship between this rank 2 Horn problem and the following similar problem: given a set of $n$ real eigenvalues, a set of $2$ real eigenvalues, and a set of $n+2$ real eigenvalues satisfying certain conditions, find an $(n+2)\times(n+2)$ real symmetric matrix such that the top left principal submatrix has the first set of eigenvalues, the bottom right principal submatrix has the second set of eigenvalues, and the full matrix has the last set of eigenvalues

Unplugged to Plugged: An Introduction to Coding for Elementary School Children

This course introduced 4th and 5th graders to coding concepts like sequencing, loops, and decomposition through an unplugged-to-plugged learning sequence. Over four after-school learning sessions, students explored programming first through their bodies (i.e., unplugged), then in the block-based programming environment, Scratch (i.e., plugged). The goal was for learners to transition from concrete forms of programming to an abstract understanding needed for block-based programming. To achieve this goal, the plugged activities were intentionally designed around the concept of concreteness fading, where the unplugged programming challenge mirrored the plugged Scratch environment, allowing students to move from a concrete to more abstract understanding of computer science. The activities in the course drew from ready-to-use materials for computer science education (e.g., Code.org) as well as free, block-based coding websites (e.g., Scratch)