The isolation number ι(G) of a graph G is the minimum cardinality of a set A⊂V(G) such that the subgraph induced by the vertices that are not in the union of the closed neighborhoods of vertices in A has no edges. The invariant, known also under the name vertex-edge domination number of G, has attracted a lot of interest in recent years. In this paper, we study the behavior of the isolation number under several graph operations, namely the Cartesian and the lexicographic product and the fractalization leading to generalized Sierpiński graphs. We prove several upper and lower bounds on the isolation number of the Cartesian product of two graphs. We prove a lower bound for the isolation number of the prism G□K2 over an arbitrary graph G, which in the case of bipartite graphs leads to the equality ι(G□K2)=γ(G), where γ(G) is the domination number of G. In particular, ι(Qn+1)=γ(Qn) holds for all positive integers n, where Qn is the n-dimensional hypercube. For the lexicographic product G∘H we prove that its isolation number, under certain mild restrictions, equals the total domination number of the first factor G. We also prove sharp lower and upper bounds on the isolation numbers of the generalized Sierpiński graphs SGt, where G is an arbitrary base graph. These bounds in the case of classical Sierpiński graphs, namely SKnt, coincide and lead to the exact values ι(SKnt)=(n−1)⋅nt−2 for all dimensions t≥2.
State supreme courts, long White and male, are increasingly diverse in recent years. As a result, a growing literature explores how various minority groups reach the bench. Interim appointments, occurring when governors in electoral states appoint someone to fill a midterm vacancy, are largely absent from this literature. This is problematic because, in states utilizing judicial elections, nearly half of all jurists initially receive an interim appointment. This subsequently conveys an incumbency advantage and may bypass women's and racial minorities' lower propensity to seek and secure office. Indeed, interim appointments are crucial to court diversification. Since appointment predictors vary by identity, we explore the path members of several groups take to the bench. We do so by drawing on a dataset of every jurist joining electoral state supreme courts from 1980 to 2023. We argue non-traditional jurists' interim appointments are conditional on the governor's political affiliation, institutional constraints, and the broader political context. We find Democrats are generally more likely to appoint non-traditional jurists, although both institutional and political factors are important moderators. Additionally, appointment predictors vary by identity group. Our findings underscore the nuanced ways state supreme courts become more diverse and the conditions under which governors appoint non-traditional jurists to the bench.
Behavioral traits such as risk-taking and agonistic behavior have been investigated in several turtle species because they are important for fitness and often correlate with one another. We measured risk-taking and active defense (attacks) of wild-caught Snapping Turtles (Chelydra serpentina) in an outdoor experimental setting to test predictions about how these traits relate. We quantified response to a threat (an approaching novel object) and measured movement latency, exploration, and activity in the arena. We did not find any correlations between risk-taking measurements, nor did risk-taking correlate with defensive response. There were no differences between sexes or sizes, which could be due to a limited behavioral repertoire on land. However, we found that time since handling was negatively correlated with the agonistic response, highlighting the importance of considering the effect of time since capture on behavioral assays. Overall, our study on risk-taking and agonistic behavior in wild Snapping Turtles revealed no correlation between these traits on land but emphasized the impact of time since capture on threat response behaviors.
Ethics education constitutes a fundamental component of nursing development, shaping practitioners capable of sound moral judgment, responsible action, and ethically grounded clinical decision-making. Despite long-standing written ethical codes within the nursing profession, substantial gaps persist between theoretical instruction and real-world practice, leading to fragmented moral competence among nursing students and nurses. This discussion paper synthesizes evidence from the “Promoting a Morally Competent Nurse” project, insights derived from an international blended intensive program to examine the multifaceted challenges that hinder effective nursing ethics education, and expert contributions. Identified challenges include complexity of ethics theories; inconsistencies in the nursing curricula; the misalignment between formal and hidden curriculum; lack of resources and time dedicated to ethics education; inconsistencies in the training methods; and issues related to ethical leadership. The paper argues that cultivating moral competence requires more than the transmission of theoretical knowledge -it demands experiential learning, role modelling, and ethically supportive environments. A roadmap is proposed outlining complementary roles for academic nurses and nurse managers in strengthening ethics education across educational and clinical settings. Enhancing ethical leadership, harmonizing curricula, and investing in educator preparation are critical steps toward building an ethically resilient nursing workforce The paper concludes that ethical competence is not a static achievement but a lifelong developmental process that must be nurtured through intentional, coordinated, and context-sensitive strategies.
Abstract Holographic thermal two-point functions can be analyzed using the operator product expansion which contains contributions from both multi-stress-tensor and double-trace operators. The former can be computed by analyzing the bulk equation of motion in a near-boundary expansion, but the latter has remained elusive — in practice, one resorts to solving a partial differential equation with limited accuracy. We show that imposing the Euclidean periodicity condition on the holographic correlator (also known as the KMS condition or thermal bootstrap), followed by Padé-Borel resummation, provides an efficient method for computing double-trace thermal coefficients. The resulting series converges rapidly and yields numerical values in excellent agreement with those obtained from solving the partial differential equation.
