Applications of canonical relations in generic differential geometry

Conflict sets are loci of intersecting wavefronts emanating from $l$ different surfaces. We show that generically conflict sets are Legendrian: locally they admit the structure of wavefronts. Simple stable singularities for this problem in $\R^n$ occur when $0 \leq n-l \leq 4$. Other related sets, such as kite curves and centre sets are also defined and discussed. Throughout canonical relations are used as an essential tool to carry out several of these geometrical constructions.Comment: 20 pages, 12 figure

La production céramique de Pont de Roque-Haute : synthèse et comparaisons

La production céramique du gisement languedocien de Pont de Roque-Haute est replacée dans le cadre des plus anciennes productions céramiques de Méditerranée occidentale. Des liens avec l'Italie centro-méridionale sont privilégiés

La production céramique : caractérisation technique et typologique

La production céramique du gisement languedocien de Pont de Roque-Haute fait l'objet d'une caractérisation typologique, technologique et spatiale

Henri Temianka Correspondence; (manen)

https://digitalcommons.chapman.edu/temianka_correspondence/2344/thumbnail.jp

On dangerous self-tangencies in families of conflict sets

In this note we observe an interesting link between an intersection of two Legendrian manifolds in $ST^\ast\R^2$ - or dangerous self-tangencies, on the one hand and hyperbolic Morse transition of immersed curves on a torus on the other hand. The connection between these two arises naturally in the study of one-parameter families of conflict sets

Are both sympatric species Ilex perado and Ilex canariensis secretly hybridizing? Indication from nuclear markers collected in Tenerife

BACKGROUND: Intra-specific and intra-individual polymorphism is frequently observed in nuclear markers of Ilex (Aquifoliaceae) and discrepancy between plastid and nuclear phylogenies is the rule in this genus. These observations suggest that inter-specific plastid or/and nuclear introgression played an important role in the process of evolution of Ilex. With the aim of a precise understanding of the evolution of this genus, two distantly related sympatric species collected in Tenerife (Canary Islands), I. perado and I. canariensis, were studied in detail. Introgression between these two species was previously never reported. One plastid marker (the atpB-rbcL spacer) and two nuclear markers, the ribosomal internal transcribed spacer (ITS) and the nuclear encoded plastid glutamine synthetase (nepGS) were analyzed for 13 and 27 individuals of I. perado and I. canariensis, respectively. RESULTS: The plastid marker is intra-specifically constant and correlated with species identity. On the other hand, whereas the nuclear markers are conserved in I. perado, they are highly polymorphic in I. canariensis. The presence of pseudogenes and recombination in ITS sequences of I. canariensis explain this polymorphism. Ancestral sequence polymorphism with incomplete lineage sorting, or past or recent hybridization with an unknown species could explain this polymorphism, not resolved by concerted evolution. However, as already reported for many other plants, past or recent introgression of an alien genotype seem the most probable explanation for such a tremendous polymorphism. CONCLUSIONS: Data do not allow the determination with certitude of the putative species introgressing I. canariensis, but I. perado is suspected. The introgression would be unilateral, with I. perado as the male donor, and the paternal sequences would be rapidly converted in highly divergent and consequently unidentifiable pseudogenes. At least, this study allows the establishment of precautionary measures when nuclear markers are used in phylogenetic studies of genera having experienced introgression such as the genus Ilex