A probabilistic approach to a classical result of ore

The subgroup commutativity degree sd(G) of a finite group G was introduced almost ten years ago and deals with the number of commuting subgroups in the subgroups lattice L(G) of G. The extremal case sd(G) = 1 detects a class of groups classified by Iwasawa in 1941 (in fact sd(G) represents a probabilistic measure which allows us to understand how far is G from the groups of Iwasawa). Among them we have sd(G) = 1 when L(G) is distributive, that is, when G is cyclic. The characterization of a cyclic group by the distributivity of its lattice of subgroups is due to a classical result of Ore in 1938. Therefore sd(G) is strongly related to structural properties of L(G). Here we introduce a new notion of probability gsd(G) in which two arbitrary sublattices S(G) and T(G) of L(G) are involved simultaneously. In case S(G) = T(G) = L(G), we find exactly sd(G). Upper and lower bounds in terms of gsd(G) and sd(G) are among our main contributions, when the condition S(G) = T(G) = L(G) is removed. Then we investigate the problem of counting the pairs of commuting subgroups via an appropriate graph. Looking at the literature, we noted that a similar problem motivated the permutability graph of non–normal subgroups ΓN (G) in 1995, that is, the graph where all proper non– normal subgroups of G form the vertex set of ΓN (G) and two vertices H and K are joined if HK = KH. The graph ΓN (G) has been recently generalized via the notion of permutability graph of subgroups Γ(G), extending the vertex set to all proper subgroups of G and keeping the same criterion to join two vertices. We use gsd(G), in order to introduce the non–permutability graph of subgroups ΓL(G) ; its vertices are now given by the set L(G) − CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and we join two vertices H, K of ΓL(G) if HK 6= KH. We finally study some classical invariants for ΓL(G) and find numerical relations between the number of edges of ΓL(G) and gsd(G)

Mixed-Species Effects on Soil C and N Stocks, C/N Ratio and pH Using a Transboundary Approach in Adjacent Common Garden Douglas-Fir and Beech Stands

Mixed forest of Douglas-fir and beech has been suggested as one of the possible future forest types in Northwest Europe but the effects of this mixed forest on soil properties relative to monoculture stands are unknown. In a transboundary investigation of adjacent common garden Douglas-fir and beech stands, we determined the effects on topsoil properties. However, responses of C and N stocks, the C/N ratio and pH were site- and soil layer-specific and were mainly single-sided and without synergistic effects. Beech reduced the soil C and N stocks in Douglas-fir at the nutrient-poor site, caused an increase in the C/N ratio in the forest floor and mineral soil at both nutrient-poor and -rich sites, and reduced the acidifying effect of Douglas-fir at the nutrient-poor site. These results do not support the hypothesis that mixture effects would be consistent across sites and soil layers. The lack of synergistic effects may be attributed to the relatively similar litter quality or rooting depth that prevented any larger niche differentiation and complementarity. The results indicate that the transboundary approach within a mature common garden proved useful as a platform to test tree species interactions, and this approach could be explored in soil studies until dedicated mixed-species common gardens reach maturity

Word Complexity is in the Eye of the Beholder

Lexical complexity is a highly subjective notion, yet this factor is often neglected in lexical simplification and readability systems which use a "one-size-fits-all" approach. In this paper, we investigate which aspects contribute to the notion of lexical complexity in various groups of readers, focusing on native and non-native speakers of English, and how the notion of complexity changes depending on the proficiency level of a non-native reader. To facilitate reproducibility of our approach and foster further research into these aspects, we release a dataset of complex words annotated by readers with different backgrounds

Network of the Day: Aggregating and Visualizing Entity Networks from Online Sources

This software demonstration paper presents a project on the interactive visualization of social media data. The data presentation fuses German Twitter data and a social relation network extracted from German online news. Such fusion allows for comparative analysis of the two types of media. Our system will additionally enable users to explore relationships between named entities, and to investigate events as they develop over time. Cooperative tagging of relationships is enabled through the active involvement of users. The system is available online for a broad user audience

Shelf-life and post-harvest quality of tomato (Lycopesicon esculentum Mill.) varieties to different packaging materials at Mersa, North Wollo, Ethiopia

Tomato (Lycopersicon esculentum Mill.) has a short shelf life at ambient conditions and is a highly perishable crop. Extreme post-harvest losses occur as a result of the wrong packaging materials. However, by employing the right packing materials, tomato varietals can have longer shelf lives. Globally rising fresh tomato demand has forced the development of essential mechanisms, including packaging materials, to improve shelf life. The current study was initiated to evaluate the quality and shelf-life of tomato varieties in response to packaging materials at Mersa, North Wollo, Ethiopia, during 2021 cropping season. Three replications of a completely randomized design were used to test three tomato varieties (Roma VF, Oval red and Woyno) and seven packing materials [closed carton (CC), open carton (OC), closed wooden box (CWB), open wooden box (OWB), perforated polyethylene bag (PPB), non-perforated polyethylene bag (NPPB) and control (C)] at room temperature (20-22°C). According to the findings, there is a significant (P<0.05) interaction effect between packaging materials and varieties on a number of parameters, including physiological weight loss, decay percentage, disease incidence, total soluble solids (TSS), tomato fruit PH, juice color score, overall acceptability, marketability percentage, and shelf life. Non-perforated polyethylene plastic experienced the highest physiological weight losses of 79.88% and 79.63% after 18 days of storage. Roma VF variety showed the greatest weight loss. In addition, PPB showed the lowest decay percentage (20%) and maximum marketability (20%) during the 18th day of storage. At the end of storage, NPPB with Roma VF and Woyno varieties had a substantially (100%) larger decay loss of tomato fruits. NPPB has been linked to the highest disease incidence (20%). Roma VF and Oval red recorded the highest pH tomato fruit’s color and overall acceptability score on PPB. It can, thus, be concluded that packaging of tomato fruits in PPB can extend shelf-life with better-quality of the produce. However, to develop plausible recommendation, the study should be repeated in multi-location with more packaging methods and varieties over seasons