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A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases., Comment: 27 pages

We model cloud-to-ground lightning strike impacts in the French Alps over the period 2011-2021 (approximately 1.4 million of events) using spatio-temporal point processes. We investigate first and higher-order structure for this point pattern and address the questions of homogeneity of the intensity function, first-order separability and dependence between events. The tuning of nonparametric methods and the different tests we consider in this study make the computational cost very expensive. We therefore suggest different subsampling strategies to achieve these tasks.

Only 21% of employees consider themselves engaged at work. Moreover, disengagement has been shown to be even more problematic when work is repetitive in nature. Lack of engagement has been linked to variety of negative outcomes for employees and companies (e.g., turnover, absenteeism, well-being, safety incidents, productivity). Gamification, i.e., integrating game elements into work systems, has been successfully used to increase engagement and motivation, even when work tasks were mundane and repetitive. In the current study, we focused on a commonly used game element, goal setting, through the lens of self-determination theory and goal-setting theory. We argue that goals given by an external source (e.g., company, experimenter) produce extrinsic motivation, which improves engagement and performance only in the short term. We posit that self-set goals lead to more autonomous motivation, and therefore long-term engagement and performance. One hundred two participants completed a repetitive material-handling task in one of three conditions (assigned goal, self-set goal, no goal). Results showed that perceived autonomy (autonomous motivation) and performance were best when goals were self-set. Engagement, however, was equal between self-set and assigned goals. The results indicate that self-set goals have the greatest potential to generate long-term positive outcomes both for employees and companies., Comment: in French language, CIGI QUALITA MOSIM 2023, Jun 2023, Trois Rivi{\`e}res, Canada

The ability to tailor with a high accuracy the inter-site connectivity in a lattice is a crucial tool for realizing novel topological phases of matter. Here, we report the experimental realization of photonic dimer chains with long-range hopping terms of arbitrary strength and phase, providing a rich generalization of the celebrated Su-Schrieffer-Heeger model. Our experiment is based on a synthetic dimension scheme involving the frequency modes of an optical fiber loop platform. This setup provides direct access to both the band dispersion and the geometry of the Bloch wavefunctions throughout the entire Brillouin zone allowing us to extract the winding number for any possible configuration. Finally, we highlight a topological phase transition solely driven by a time-reversal-breaking synthetic gauge field associated with the phase of the long-range hopping, providing a route for engineering topological bands in photonic lattices belonging to the AIII symmetry class.

We provide a $O(k^2 \mathrm{log} k)$ vertex kernel for cograph edge editing. This improves a cubic kernel found by Guillemot, Havet, Paul and Perez [1] which involved four reduction rules. We generalize one of their rules, based on packing of induced paths of length four, by introducing t-modules, which are modules up to t edge modifications. The key fact is that large t-modules cannot be edited more than t times, and this allows to obtain a near quadratic kernel. The extra $\mathrm{log} k$ factor seems tricky to remove as it is necessary in the combinatorial lemma on trees which is central in our proof. Nevertheless, we think that a quadratic bound should be reachable.

A subset $I$ of the vertex set $V(G)$ of a graph $G$ is called a $k$-clique independent set of $G$ if no $k$ vertices in $I$ form a $k$-clique of $G$. An independent set is a $2$-clique independent set. Let $\pi_k(G)$ denote the number of $k$-cliques of $G$. For a function $w: V(G) \rightarrow \{0, 1, 2, \dots\}$, let $G(w)$ be the graph obtained from $G$ by replacing each vertex $v$ by a $w(v)$-clique $K^v$ and making each vertex of $K^u$ adjacent to each vertex of $K^v$ for each edge $\{u,v\}$ of $G$. For an integer $m \geq 1$, consider any $w$ with $\sum_{v \in V(G)} w(v) = m$. For $U \subseteq V(G)$, we say that $w$ is uniform on $U$ if $w(v) = 0$ for each $v \in V(G) \setminus U$ and, for each $u \in U$, $w(u) = \left\lfloor m/|U| \right\rfloor$ or $w(u) = \left\lceil m/|U| \right\rceil$. Katona asked if $\pi_k(G(w))$ is smallest when $w$ is uniform on a largest $k$-clique independent set of $G$. He placed particular emphasis on the Sperner graph $B_n$, given by $V(B_n) = \{X \colon X \subseteq \{1, \dots, n\}\}$ and $E(B_n) = \{\{X,Y\} \colon X \subsetneq Y \in V(B_n)\}$. He provided an affirmative answer for $k = 2$ (and any $G$). We determine graphs for which the answer is negative for every $k \geq 3$. These include $B_n$ for $n \geq 2$. Generalizing Sperner's Theorem and a recent result of Qian, Engel and Xu, we show that $\pi_k(B_n(w))$ is smallest when $w$ is uniform on a largest independent set of $B_n$. We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs., Comment: 14 pages, minor corrections made

This brief is the third of a four-part series that focuses on bilingual education, bilingual educators, and addressing the bilingual teacher shortage in contexts across the United States. This research was commissioned by the New Jersey State Department of Education, which is committed to providing quality bilingual education to its linguistically diverse student population. More than half of U.S. states are experiencing bilingual teacher shortages, with shortages most acute in states that provide bilingual education to the greatest numbers of English learners (ELs) and those that require teachers to obtain specific bilingual endorsements (for more information on the term "English learner," please refer to our previous brief, Bilingual Education Across the United States). These states experience a persistent tension between: (1) requiring educators to comply with certification requirements and (2) meeting the increasing demand for bilingual teachers, driven by the growing population of ELs and the increasing popularity of dual language programs. States have created various policy solutions to broaden their bilingual teacher pool by diversifying the ways in which teachers may gain certification. In this brief, we share strategies that state education agencies may use to provide more flexibility in teacher certification policies and thereby mitigate bilingual teacher shortages, including: (1) providing educators multiple pathways to certification; (2) offering prospective teachers a menu of certification requirements; (3) allowing educators to earn probationary certifications; and (4) creating flexible reciprocity policies.

This brief is the first of a four-part series that focuses on bilingual education, bilingual educators, and addressing the bilingual teacher shortage in contexts across the United States. This research was commissioned by the New Jersey State Department of Education, which is committed to providing quality bilingual education to its linguistically diverse student population. Multilingual learners and students classified as English learners (ELs), a growing population nationwide, come to school with tremendous assets, among which are proficiency in a language other than English and the potential to become bilingual. Yet without school-based opportunities to develop biliteracy--that is, literacy in both their home language and English--their linguistic potential can remain underdeveloped. Many states recognize these students' multilingual potential and have prioritized bilingual education: Some states require bilingual education, while others encourage or allow it in certain contexts. In this brief, we define some of the key terms in bilingual education and then provide an overview of the different state policies for the provision of bilingual education to English learners.

In an effort to reduce power system-caused wildfires, utilities carry out public safety power shutoffs (PSPS) in which portions of the grid are de-energized to mitigate the risk of ignition. The decision to call a PSPS must balance reducing ignition risks and the negative impact of service interruptions. In this work, we consider three PSPS scheduling scenarios, which we model as dynamic programs. In the first two scenarios, we assume that N PSPSs are budgeted as part of the investment strategy. In the first scenario, a penalty is incurred for each PSPS declared past the Nth event. In the second, we assume that some costs can be recovered if the number of PSPSs is below $N$ while still being subject to a penalty if above N. In the third, the system operator wants to minimize the number of PSPS such that the total expected cost is below a threshold. We provide optimal or asymptotically optimal policies for each case, the first two of which have closed-form expressions. Lastly, we establish the applicability of the first PSPS model's policy to critical-peak pricing, and obtain an optimal scheduling policy to reduce the peak demand based on weather observations.