We establish new uncertainty bounds for commutators and anti-commutators of observables in both finite- and infinite-dimensional quantum systems. Beginning with a short, elementary proof of Robertson’s standard uncertainty principle, we then derive inequalities whose lower bounds remain nonzero, regardless of the number of observables. The main result is a novel trace inequality comparing the block matrices of commutators and anti-commutators of several observables.

‘Teaching about Transatlantic Slavery’ was a three-year professional development project for teachers centring on the Historic Collections of Balliol College, University of Oxford, and the Museum of the American Revolution (MOAR) in Philadelphia. The project grew out of Balliol’s 2021 exhibition, ‘Slavery in the Age of Revolution’, which was inspired by the work of Fellows Sudhir Hazareesingh, Black Spartacus: the epic life of Toussaint Louverture (2020) and Marisa J. Fuentes, Dispossessed Lives: enslaved women, violence and the archive (2016). This paper discusses how we approached the aims of the project, telling a fuller history by foregrounding the role of Black people in the struggle for abolition and increasing awareness of the lasting impact of transatlantic slavery on the way we live and think today. We will address how some of the challenges of working with the material in Balliol’s collections became project strengths when we learnt to navigate bias and acknowledge and elaborate on the gaps and erasures in the historical record. We also acknowledge the diverse range of people who contributed and their importance to the consequent depth and effectiveness of the project. ‘Teaching about Transatlantic Slavery’ reached forty teachers and educators from the USA and the UK and has had lasting impact. The programme allowed teachers to practise teaching with objects, artwork, and primary source documents, deepen their subject knowledge, and develop skills for engaging in sensitive and difficult conversations around complex topics.

In this paper, we analyze the potential harms of everyday self-help discourse, particularly in contexts of systemic oppression. After analyzing the core principles of self-help, such as the power of positive thinking, eliminating negative thinking, gratitude, self-knowledge for self-improvement, and individual choice and responsibility, we argue that self-help discourse is likely to perpetuate different forms of harm in conditions of oppression. First, we argue that, when used in contexts of systemic oppression, self-help discourse contributes to victim-blaming. Secondly, we highlight that the harms of self-help discourse include epistemic and affective dimensions. On the epistemic front, self-help discourse carries epistemic risks, including testimonial smothering and hermeneutical injustice, by discouraging the expression of certain perspectives and lived experiences. On the affective front, self-help discourse reinforces oppression by adding emotional burdens to the oppressed, dismissing their emotional experiences, and imposing the emotional norms of the dominant group. Finally, we revisit the principles of self-help, and show how they contribute to the highlighted harms, particularly by reinforcing victim-blaming, and epistemic and affective forms of injustice.

We introduce and study Liouville number groups, namely additive subgroups of the real numbers in which every nonzero element is a Liouville number. Using continued-fraction methods and linear independence over the field of algebraic numbers, we establish the existence of large families of such groups with rich algebraic and topological structure. We prove that there exist $(2^{\mathfrak {c}}) $ pairwise distinct Liouville number groups generated by strong Liouville numbers and that, among these, there are $(2^{\mathfrak {c}})$ pairwise nonhomeomorphic groups. We further show that there exist continuum many countable Liouville number groups, each generated by countably many strong Liouville numbers and homeomorphic, as topological spaces, to the rational numbers. In addition, we prove that no subgroup of the real numbers is homeomorphic to the space of Liouville numbers, thereby highlighting a strong topological distinction between the space of Liouville numbers and Liouville number groups.

Spherical robots can conduct surveillance in hostile, cluttered environments without being damaged. Their protective shell can safely house sensors and sensitive components. The pendulum-based spherical robot provides a platform (yoke) for a camera to be housed inside the protective hull. However, lateral oscillations, also known as wobble, become prominent when operated at low speeds, leading to shaky camera feedback. These oscillations are caused by the coupling between the forward and steering motions due to nonholonomic constraints. Designing a controller to limit wobbling in these robots is challenging due to their underactuated nature. In this paper, we propose a model-based (wobble-free) controller to navigate a pendulum-steered spherical robot to perform wobble-free turning maneuvers consisting of circular arcs and straight lines. The model is developed using Lagrange-D’Alembert equations and accounts for the coupled forward and steering motions. Furthermore, we also propose quantifiable metrics for wobble by deriving expressions for radius of curvature, precession rate, wobble amplitude, and wobble frequency during circular motions and designed an input–output feedback linearization-based controller to control the robot’s heading direction and wobble. Additionally, we extend the controller’s capabilities to handle point-to-point navigation, enabling the robot to execute more complex and goal-oriented trajectories. This research ensures the robot maintains wobble-free motion across diverse paths, enhancing its navigational autonomy and the quality of camera feedback. Overall, the proposed controller enables a teleoperator to command a specific forward velocity and pendulum angle as per the desired turning radius or destination while effectively limiting the robot’s lateral oscillations.

This article explores the evolution of school dress code policies in the United States as a critical lens into the workings of public education, authority, and exclusion. Focusing on metropolitan Seattle as a case study, the research traces the shift from informal appearance norms to formalized and punitive dress code enforcement from the 1950s to the present. Rather than functioning as neutral guidelines, dress codes have historically reinforced gendered, racialized, and class-based hierarchies under the guise of order and safety. By situating dress policies within broader political and social transformations—including student rights movements, the rise of zero-tolerance policies, and recent federal investigations—this article demonstrates how dress codes have become tools of gatekeeping that shape access to learning and belonging in public schools. In highlighting moments of resistance and change, this study interrogates the role of schools in regulating identity and offers a historical foundation to understand contemporary inequities in educational discipline and policy.

Let K be a number field with ring of integers $\mathcal {O}_K$, and let $f(x)\in \mathcal {O}_K[x]$ be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of $f(x)$ for the iterates of $f(x)$ to be monogenic polynomials. More generally, we give necessary and sufficient conditions for the backward orbits of elements of $\mathcal {O}_K$ under $f(x)$ to be monogenerators. We apply our criteria to construct novel examples of dynamically monogenic polynomials, yielding infinite towers of monogenic number fields with the backward orbit of one monogenerator giving a monogenerator at the next level.