We define and investigate two families of dual polynomials associated with the Gauss–Legendre polynomials, which have recently found interesting applications in computer graphics. Using the presented results, one can derive representations of the Gauss–Legendre polynomials, construct the dual bases for Lagrange bases and solve certain approximation problems arising, for example, in CAGD.
Given a set V=v1,…,vn of n elements and a family S=S1,S2,…,Sm of (possibly intersecting) subsets of V, we consider a scheduling problem involving the perpetual monitoring (or attending) of these subsets. At each time step, one element of V is visited, and all subsets in S containing that element are considered attended during that step. Each subset Sj is assigned an urgency factor hj, which reflects how frequently it should be attended relative to other subsets. Let ti(j) denote the time slot when subset Sj is attended for the ith time. The goal is to devise a perpetual schedule for visiting the elements of V that minimizes the maximum value of hj(ti+1(j)−ti(j)). We refer to this problem as the fair hitting sequence (FHS problem), noting its connection to the minimum hitting set problem. Specifically, in the uniform FHS case, where all urgency factors hj are equal, the problem reduces to the minimum hitting set problem, which is NP-hard.We show that schedules based on a single hitting set, even an optimal one, can yield poor approximation ratios. To address this, we propose a deterministic algorithm that partitions the family S into sub-families and combines hitting sets for these sub-families, achieving O(log2m)-approximate schedules. Additionally, we establish a lower bound on the optimal objective value of FHS using linear programming (LP) techniques. Leveraging this bound, we develop a randomized algorithm that, with high probability, computes O(logm)-approximate schedules.
This study examines the determinants of green purchase behavior (GPB) among young consumers in a digital context by integrating drive for environmental responsibility (DER), social media attachment (SMA), green product knowledge (GPK), and green consumption values (GCV) within the Theory of Reciprocal Determinism (TRD). In addition, it investigates the moderating role of SMA in the relationship between DER and GPB, as well as the role of attitude in the link between GCV and GPB. Survey data from 285 participants aged 18 to 30 collected using a non-probability sampling method. Partial Least Squares Structural Equation Modeling (PLS-SEM) were utilized in the study. The findings reveal that DER, SMA, GPK, and GCV positively influence GPB. However, the moderating effects of SMA and attitude are not supported, indicating that these factors act as independent predictors rather than conditional influencers of GPB. These results provide theoretical insights into the stability of behavioral drivers and reinforce the importance of cognitive and environmental determinants within the TRD framework, even without significant interaction effects. The findings emphasize the significance of individual DER, digital engagement, GPK, and values among urbanized and digitally connected youth, promoting GPB in the digital era. The research offers an original contribution by integrating DER and SMA within the TRD framework in the context of Malaysian digital sustainability. It extends green consumption beyond the Western perspective and provides novel empirical evidence from an emerging economy. The study highlights the theoretical, practical, and policy implications of these findings.
Amphibole in peridotite xenoliths in Cenozoic volcanic rocks of the Massif Central, France, is generally sparse but widely distributed. Here, we discuss its origin in the xenolith suite of Marais de Limagne (Devès volcanic field) and its bearing on evolution of lithospheric mantle in the southern part of the Massif Central. The xenoliths are lherzolites containing (1) no amphibole, (2) 2–5 vol.
Planning education in Poland has developed within a fragmented institutional setting, creating uncertainty about where design-related competences belong and how they are taught. This paper verifies the place of urban design-focused training within planning programmes by examining how curricula and learning outcomes frame graduates’ capacity to make informed decisions on the built environment. Using a framework that integrates contextual, systemic, and pedagogical challenges, the study combines a quantitative review of 28 university programmes, curriculum analysis, and insights from exploratory expert interviews. The findings show a structural mismatch in the classification of planning as a scientific field, a marginal presence of design-oriented content, and weak alignment between intended learning outcomes and their actual delivery, largely due to acute staffing shortages. The study provides new insight into planning education in Central Europe and helps clarify the evolving role and disciplinary boundaries of design within planning training.
