Many 2-level polytopes from matroids

The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-level matroids generalize series-parallel graphs, which have been already successfully analyzed from the enumerative perspective. We bring to light some structural properties of 2-level matroids and exploit them for enumerative purposes. Moreover, the counting results are used to show that the number of combinatorially non-equivalent (n-1)-dimensional 2-level polytopes is bounded from below by $c \cdot n^{-5/2} \cdot \rho^{-n}$, where $c\approx 0.03791727$ and $\rho^{-1} \approx 4.88052854$.Comment: revised version, 19 pages, 7 figure

L’eponimo arabo Jurhum: La questione etimologica e delle fonti

The etymology of the Arab eponym Jurhum is here investigated on the basis of the contexts in which it occurs in historiographical sources and the meanings of the adjective jurhum, as attested in lexicographical sources. The resulting interpretation assigns Jurhum two original meanings (homonymy). Firstly, Jurhum means a tall palm, from the root H J R via metathesis (see nakhlah muhjirah) and, indeed, narrative texts describe a descendant of Jurhum precisely as a tall palm. The vegetal nature of Jurhum is likely to be an instance of totemism. Secondly, Jurhum means a benign serpent, from the root J R H (see Tigre gärha ‘good-natured’). However, it may refer to a malignant serpent by antiphrasis, as ḥubāb ‘friendly serpent/entity’ stands for shayṭān ‘hostile serpent/entity’. The animal nature of Jurhum is also an instance of totemism. A descendant of Jurhum is in fact described as a viper, i.e. a malignant serpent

The historical reality of the plural of paucity and the plural diminutive in Classical Arabic: Evidence from kalām al-‘arab (Part One)

This study investigates the semantics of the plural of paucity and the plural diminutive, based on their attestations in the non-literary source of Classical Arabic traditionally known as kalām al-‘arab. In noun plural marking, the meaning of the diminutive is as elusive as that of the plural of paucity. What is known of both kinds of meanings is mainly derived from the indirect description of early lexicographers and grammarians. To assess the historical reality of this traditional semantic description, attestations from the kalām al-‘arab are collected, then compared to data from Arabic dialects, and finally subjected to a distributional analysis. The grammatical categories of the collective, inherent plural, and the pseudo-dual are also considered in this assessment

The maximum likelihood degree of Fermat hypersurfaces

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.Comment: Final version. Accepted for publication on Journal of Algebraic Statistic

Genetic Basis of Self-Incompatibility in the Lichen-Forming Fungus Lobaria pulmonaria and Skewed Frequency Distribution of Mating-Type Idiomorphs: Implications for Conservation

Fungal populations that reproduce sexually are likely to be genetically more diverse and have a higher adaptive potential than asexually reproducing populations. Mating systems of fungal species can be self-incompatible, requiring the presence of isolates of different mating-type genes for sexual reproduction to occur, or self-compatible, requiring only one. Understanding the distribution of mating-type genes in populations can help to assess the potential of self-incompatible species to reproduce sexually. In the locally threatened epiphytic lichen-forming fungus Lobaria pulmonaria (L.) Hoffm., low frequency of sexual reproduction is likely to limit the potential of populations to adapt to changing environmental conditions. Our study provides direct evidence of self-incompatibility (heterothallism) in L. pulmonaria. It can thus be hypothesized that sexual reproduction in small populations might be limited by an unbalanced distribution of mating-type genes. We therefore assessed neutral genetic diversity (using microsatellites) and mating-type ratio in 27 lichen populations (933 individuals). We found significant differences in the frequency of the two mating types in 13 populations, indicating a lower likelihood of sexual reproduction in these populations. This suggests that conservation translocation activities aiming at maximizing genetic heterogeneity in threatened and declining populations should take into account not only presence of fruiting bodies in transplanted individuals, but also the identity and balanced representation of mating-type genes