Several aspects of the dynamics of a granular two-dimensional (2D) packing of disks slowly tilted until the system loses stability and an avalanche takes place are discussed. The evolution of the system, constructed with monodisperse disks placed on a thin cell, is studied by image analysis. As in the 3D case (packing of spheres), the system undergoes several rearrangements of different magnitude before the avalanche takes place. For thick systems, not only are small rearrangements detected but also displacements of large clusters of disks are observed in the bulk and on the free surface of the packing. In particular, characteristic angles and the avalanche mass were determined for samples of different heights. On thick systems, velocity fields of large rearrangements are presented and changes in the internal structure of the packing produced by these rearrangements are analyzed. It is found that the main effects of rearrangements is to increase the disorder of the system. Also, as the disorder of the system increases its stability threshold decreases.

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The current health crisis brought about by the COVID-19 pandemic not only had a global impact, it also exacerbated the inequalities experienced by students of diverse backgrounds in the United States. Implementing inclusive and anti-racist pedagogical practices has gained a heightened and overdue sense of urgency, especially during the period of emergency remote teaching. At Lafayette College, a small liberal arts college in Pennsylvania, USA, the Inclusive Instructors Academy is a semester-long program aimed at supporting faculty from all disciplines to develop and incorporate inclusive practices that promote equity and belonging in their teaching. A critical aspect of the Inclusive Instructors Academy is its employment of student fellows under the Student-as-Partners model. The student fellows who participated in Fall 2020 and Spring 2021 provided feedback to their faculty partners on inclusive teaching approaches. This case study highlights how student-faculty partnerships can be a highly effective strategy for fostering more socially just learning environments.

Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω⊂B(H)d, where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity.

Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω ⊂ B(H) d , where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity.

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Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs -xy-convexity -is also considered. A characterization of xy-convex polynomials is given.