Diagonalizably realizable implies universally realizable

A spectrum $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix $A$. The spectrum $\Lambda$ is diagonalizably realizable ($\mathcal{DR}$) if the realizing matrix $A$ is diagonalizable, and $\Lambda$ is universally realizable ($\mathcal{UR}$) if it is realizable for each possible Jordan canonical form allowed by $\Lambda.$ In 1981, Minc proved that if $\Lambda$ is the spectrum of a diagonalizable positive matrix, then $\Lambda$ is universally realizable. One of the main open questions about the problem of universal realizability of spectra iswhether $\mathcal{DR}$ implies $\mathcal{UR}$. Here, we prove a surprisingly simple result, which shows how diagonalizably realizable implies universally realizable

Order relations of the Wasserstein mean and the spectral geometric mean

On the space of positive definite matrices, several operator means are popular and have been studied extensively. In this paper, we investigate the near order and the Löwner order relations on the curves defined by the Wasserstein mean and the spectral geometric mean. We show that the near order $\preceq$ is stronger than the eigenvalue entrywise order and that $A\natural_t B \preceq A\diamond_t B$ for $t\in [0,1]$. We prove the monotonicity properties of the curves originated from the Wasserstein mean and the spectral geometric mean in terms of the near order. The Löwner order properties of the Wasserstein mean and the spectral geometric mean are also explored

Numerical range for weighted Moore-Penrose inverse of tensor

This article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define the notions of weighted normal tensor for an even-order square tensor and weighted tensor norm. Finally, we apply these to study the theory of numerical range for the weighted Moore-Penrose inverse of an even-order square tensor and exploit its several properties. We also obtain a few new results in matrix setting

AI-Enhanced Writing Self- Assessment : Empowering Student Revision with AI Tools

This technology-rich, three-day lesson for 9th-12th grade English Language Arts students leverages artificial intelligence (AI)-enhanced rubrics and writing analysis tools (Ouyang & Jiao, 2021) to improve student writing and self-assessment skills. Students explore AI-enhanced digital rubrics, use AI tools to analyze their writing, and apply AI-generated feedback to revise their work. This approach enhances current writing assignments and develops critical thinking skills and digital literacy. Assessments include AI-generated feedback reports, peer evaluations, and final revised writing samples demonstrating improvement based on AI and peer input, equipping students with valuable skills for future writing tasks

Introduction

As the third year of JTILT ends, I find myself reflecting on what the journal provides, how it was established, and where it is heading. Past issues and articles capture various teaching styles, strategies, delivery approaches, and purposes. Each article presents candid details regarding their use of technology and how that influenced learning. I am constantly reminded that teaching is a science and an art. I relish detailed procedures and steps that authors articulate in their work. I also marvel at the art of teaching and the broad possibilities that exist in developing instruction. JTILT has helped me broaden my perspectives and consider new facets of the field

Preparing Educators to Navigate the Social-Emotional Terrain: A Game-Based Approach

The FIXIT Social-Emotional Learning (SEL) game was created by the authors as an analog card game to help educators prepare for social-emotional challenges in the classroom. In the game, players work together as a team to overcome obstacles and practice addressing challenging scenarios while thinking through support beyond academic needs. This game-based approach provides a low-risk environment for educators to practice and prepare for real-life situations. Learning assessment is completed through gameplay discussions, end-of-game debriefs, and personal learning reflections. This version of FIXIT is for educators, but the game can be adapted to fit any content that includes problem-solving for all ages and grades ECE-12+

Synthesizing Magnesium Oxide through a 3D Experience

This lesson is from the Types of Reactions unit in a High School Chemistry class and focuses on the synthesis of magnesium oxide. The purpose of this lesson is for students to conduct a lab experiment, and understand, at an atomic level, how the reaction occurs. In collaboration between a Chemistry teacher and Educational Technology faculty member, one three-dimensional (3D) experience was created to support the visualization of the reaction and minimize the students’ misconceptions.

Motion to Immersion: Combining Computer Science, Virtual Asset Design, and Motion Capture for High School Students

There are many opportunities for learning under the umbrella of computer science. Coding and programming experiences can overshadow other computational learning programs. Digital asset design for virtual environments is applicable to game design and motion capture laboratories in the movement sciences. This summer camp for high school students featured coding and digital asset development as applied to motion capture and the prevention of musculoskeletal injuries

Working-class Academics: Challenging Deficit Narratives Through Cultural Wealth

When navigating higher education (HE), working-class academics (WCAs) encounter persistent socioeconomic, cultural, and personal barriers throughout their academic careers. This study, grounded in the theoretical frameworks of Pierre Bourdieu and Tara Yosso, sought to illuminate the cultural wealth of WCAs. Employing a mixed-methods approach over a five-year period, the research engaged with 244 WCAs from various institutions across the United Kingdom (UK). The findings challenged the dominant deficit narratives surrounding WCAs, revealing that WCAs actively cultivate and leverage a rich array of cultural assets, encompassing examples of aspirational, navigational, linguistic, familial, social, and resistant capital. Two overarching themes emerged from the data: the profound impact of shared lived experiences in empowering marginalised students, and the crucial role of WCAs as change agents. The study demonstrates how WCAs employ their backgrounds as pedagogical assets while advocating for structural reforms. These findings suggest institutions should formally recognise working-class cultural wealth through revised hiring criteria, targeted support programmes, and inclusive decision-making frameworks. This research advances understanding of how marginalised groups can transform rather than simply adapt to institutional cultures