Singularities of nn-fold integrals of the Ising class and the theory of elliptic curves

We introduce some multiple integrals that are expected to have the same singularities as the singularities of the $n$-particle contributions $\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equation satisfied by these multiple integrals for $n=1, 2, 3, 4$ and only modulo some primes for $n=5$ and $6$, thus providing a large set of (possible) new singularities of the $\chi^{(n)}$. We discuss the singularity structure for these multiple integrals by solving the Landau conditions. We find that the singularities of the associated ODEs identify (up to $n= 6$) with the leading pinch Landau singularities. The second remarkable obtained feature is that the singularities of the ODEs associated with the multiple integrals reduce to the singularities of the ODEs associated with a {\em finite number of one dimensional integrals}. Among the singularities found, we underline the fact that the quadratic polynomial condition $1+3 w +4 w^2 = 0$, that occurs in the linear differential equation of $\chi^{(3)}$, actually corresponds to a remarkable property of selected elliptic curves, namely the occurrence of complex multiplication. The interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group, namely isogenies of elliptic curves, is sketched. Most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting an interpretation in terms of (motivic) mathematical structures beyond the theory of elliptic curves.Comment: 39 pages, 7 figure

Is laparoscopy still the gold standard in infertility assessment? A comparison of fertiloscopy versus laparoscopy in infertility. Results of an international multicentre prospective trial: the 'FLY' (Fertiloscopy-LaparoscopY) study.

BACKGROUND: The aim of this prospective multicentre study was to compare the two endoscopic techniques of laparoscopy and fertiloscopy in routine evaluation of the pelvis in infertile women. METHODS: A total of 92 women was selected in 14 University Hospitals to undergo fertiloscopy followed by transabdominal laparoscopy by a team of two surgeons in each hospital. RESULTS: A high degree of concordance was observed between these two techniques, in that if fertiloscopy did not detect any abnormalities, this was also confirmed by laparoscopy. Discordance was observed in similar numbers of cases: eight after laparoscopy and nine after fertiloscopy. The diagnostic index for fertiloscopy and laparoscopy was calculated; sensitivity (86 and 87% respectively) and negative predictive value (64 and 67% respectively) were similar. The kappa index was also calculated for each of the six structures/regions (right/left tube; right/left ovary; peritoneum of pouch of Douglas; posterior uterus), and concordance (0.78 to 0.91) was considered almost complete. CONCLUSIONS: These results confirm fertiloscopy as a minimally invasive safe procedure that may be considered as an alternative to diagnostic laparoscopy in the routine assessment of women without clinical or ultrasound evidence of pelvic disease. On the basis of the additional advantages of fertiloscopy, namely salpingoscopy or microsalpingoscopy, it is considered that fertiloscopy could replace laparoscopy as a routine procedure in such women

Diagnostic de l’acidose métabolique à la naissance par la détermination du pH eucapnique

International audienc

La césarienne peut-elle prévenir la paralysie cérébrale ? Implications médico-légales d’une étude écologique française

International audienceFor a long time, the benefit of a caesarean delivery in the prevention of cerebral palsy (CP) has been put forward, which was based on the assumption that CP is due to asphyxia in more than 50 % of the cases. However, from register-based data, this rate has been estimated less than 4 %. The aim of this study was to evaluate whether the rate of caesarean sections for fetal indication was correlated with the prevalence rate of CP in a French county