In this paper, we express the generating function of a combination of cranks of partitions that belong to congruence classes 3, 4, 5, 6, and 7 modulo 15, in terms of the Eisenstein series. Our results rely on the observation that the generating function of the combination we consider is a weight 1 modular form on Gamma 0(225) with the non-trivial modulo 3 primitive Dirichlet character. We obtain the modularity result by analyzing the transformation properties of related Klein forms. The main results are proven using the fact that the corresponding space is generated by Eisenstein series. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Research demonstrates that prospective secondary teachers benefit from connections from their advanced tertiary mathematics courses to secondary teaching practice. Less is known, however, about the instructional factors that prospective secondary teachers themselves see as beneficial to their future teaching. To address this gap, we report prospective secondary teachers’ self-reported benefits—in terms of confidence and value—as well as the features of instructional activity that the teachers attribute to these benefits. We draw on survey responses of 70 prospective secondary teachers enrolled in tertiary mathematics courses for teachers using MODULE(S2) materials in one of four content areas. Participants of this study were part of a larger pool of 192 prospective secondary teachers who completed surveys on their perception of instructional practices used in their tertiary course. Through a cultural-historical activity theory lens, findings suggest that simulations of practice and opportunities for class discussion were key activities and mediating factors reported by participants for the benefits gained from their tertiary courses.
Abstract Construction fences can become unstable during high-wind events, particularly when attached objects such as signage, banners, or shade cloths increase the surface area exposed to wind. However, current inspection and preparedness practices often rely on manual observation, limited time and labor, and practitioners’ field experience, making it difficult to quickly assess fence stability and determine appropriate preparedness actions before high-wind events. To address this challenge, this study developed a vision-language model-based system that integrates prompt-based information extraction, computer vision-based fence condition assessment, engineering-based stability analysis, and large language model-based practical guideline generation. The system extracts key variables from user input, identifies fence panels and attached objects using instance segmentation, estimates wind-exposed surface areas, performs engineering-based stability analysis, and generates practical guidelines based on the stability analysis results. Technical evaluation showed that the instance segmentation model achieved an overall mAP50 of 0.866 in detecting fence-related objects. For practical guideline generation, the Construction Material and Incomplete and Temporary Structure 3-30E model with a temperature of 0.5 and top-p of 0.7 achieved the highest bidirectional encoder representations from transformers score F1 value of 0.853 among the tested configurations. In addition, a practitioner-based evaluation was conducted to examine the practical value of the proposed system from the perspective of potential users. The results indicated positive perceptions, particularly for practical guideline generation, system usefulness, and future adoption intention. This study contributes to automated inspection and proactive construction site management for severe weather preparedness by demonstrating how image-based condition assessment, engineering-informed stability evaluation, and natural-language decision support can be integrated into an artificial intelligence–based decision-support tool.
Land management practices dictate soil physico-chemical and biological functions, and consequently, the agroecological benefits of soils. While several studies exist detailing the effects of cover crops on soil health parameters in annual systems, there is no clear understanding of microbial responses under perennial systems, especially after conversion to annual systems. A 20-year perennial fescue (Schendonorus arundinaceus) field was converted to annual systems including two levels of cover crops (cover crop vs no cover crop) and two levels of tillage (tillage vs no tillage). The cover crop included a mix of winter wheat (Triticum aestivum) and crimson clover (Trifolium incarnatum), while the tillage included disc plowing. Three and four years after conversion, soil samples were collected from no-till with cover crop (NT-CC), no-till with no cover crop (NT-NC), tillage with cover crop (Till-CC), and tillage with no cover crop (Till-NC) plots. Additional samples were also collected from an adjacent field under perennial fescue grass (PER). Soil physico-chemical properties were greater (p < 0.05) under PER and NT-CC than other management practices evaluated. As a result, total microbial biomass and the composition of microbial community were greater under PER and NT-CC than other management practices. Four years after conversion, the Gr- bacteria and fungi populations under PER were 49% and 59% greater than the average of all CC, and 51% and 133% greater under the average of all tillage managements, respectively. Further, the predato:prey ratio was 38% greater under PER than Till-CC and Till-NC managements. While a conversion from perennial to annual systems can result in soil health parameter degradation, NT-CC is a better annual alternative to perennial systems than other annual practices evaluated.
This paper develops a theoretical model to study how cross-country productivity asymmetry affects the endogenous formation of global free trade. The main theoretical prediction is that (i) when productivity levels are sufficiently similar, the equilibrium is global free trade; and (ii) when productivity levels are sufficiently asymmetric, the equilibrium outcome is the status quo regime. We empirically confirm the main theoretical prediction using panel data from 195 & times; 194 country pairs, along with information on all FTAs formed in the world from 1950 to 2017. The empirical results suggest that country pairs with higher productivity asymmetry are less likely to sign a reciprocal trade agreement. Therefore, this paper rationalizes the inclusion of productivity differences in empirical studies on the formation of trade agreements.
