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

Phenological and yield response of primed carrot (Daucus carota L) seeds under deficit irrigation

Seedling emergence and stand establishment of carrot seeds are often slow and erratic which results in low productivity. Poor seed quality together with lack of pre-sowing seed treatments and improper irrigation management can be mentioned as the major factors that influence the yield and productivity of carrot. The present study was carried out with the objective to evaluate the effects of different seed priming techniques on early seedling establishment, growth and yield of carrot (Daucus carota L. cultivated variety Nantes) exposed to different irrigation intervals, under field conditions at Gerado, South Wollo, Ethiopia. Four seed priming treatments (no priming, hydro priming, halo priming and hormonal priming) and three irrigation intervals (4, 7 and 10 days) were combined as factorial RCBD in split plot arrangement with three replications. The irrigation intervals were assigned to main plots and the seed priming techniques to sub plots. Result indicated that the interaction effects of priming techniques and irrigation intervals significantly affected the phenological and yield parameters. Distilled water treatment in seven and ten days irrigation interval recorded the highest marketable carrot root yields of 33.73 t h-1 and 30.63 t h-1, respectively. Hence, hydro priming and seven days irrigation interval can be recommended for the production of carrot in the study area and similar agro-ecologies. Given the promising results obtained, further repetitions of the study are recommended to validate the use of these techniques in further locations and in different seasons