Index theorems for holomorphic self-maps

Let $M$ be a complex manifold and $S\subset M$ a (possibly singular) subvariety of $M$. Let $f\colon M\to M$ be a holomorphic map such that $f$ restricted to $S$ is the identity. We show that one can associate to $f$ a holomorphic section $X_f$ of a sheaf related to the embedding of $S$ in $M$ and that such a section reads the dynamical behavior of $f$ along $S$. In particular we prove that under generic hypotheses the canonical section $X_f$ induces a holomorphic action in the sense of Bott on the normal bundle of (the regular part of) $S$ in $M$ and this allows to obtain for holomorphic self-maps with non- isolated fixed points index theorems similar to Camacho-Sad, Baum-Bott and variation index theorems for holomorphic foliations. Finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self-maps of surfaces with a compact curve of fixed points.Comment: 46 pages, published versio

Efficacy of Operculectomy in the Treatment of 145 Cases with Unerupted Second Molars: A Retrospective Case–Control Study

The aim of this study is to assess whether operculectomy in patients with retained second molars eases spontaneous tooth eruption in respect to untreated controls. Two hundred and twenty-two patients with delayed eruption of at least one second molar were selected from the archives of the Department of Orthodontics, Milan, Italy. Eighty-eight patients, 40 males and 48 females (mean age 14.8 ± 1.3 years), met the inclusion criteria. Records were then divided into case and control groups. The case group consisted of patients that underwent removal of the overlaying mucosa over second molars (i.e., operculectomy) and the control group consisted of subjects who retained their operculum over an unerupted second molar and were followed for one year without performing any treatment. A total of 145 impacted second molars were considered (75 cases, 70 controls). A risk ratio with 95% confidence interval was used to compare the prevalence of eruption in the two groups. Spontaneous eruption occurred in 93.3% of cases in the operculectomy group (70/75), while in the control group, 10% teeth erupted spontaneously (7/70). Spontaneous eruption in the upper arch occurred in 95.2% of cases among treated patients (40 out of 42), while in the lower arch, spontaneous eruption occurred in 90.9% of cases (30 out of 33). Spontaneous eruption of the upper second molars in the control group occurred in 8.5% of cases (3 out of 35), while in the lower arch, it occurred in 8.5% (3 out of 35). Operculectomy can ease the spontaneous eruption of retained second molars and reduce the chances of inclusion

Mapping the Learning Pathways of Larval Zebrafish through Positively Stimulating Their Reward Pathways Using Paramecium

Zebrafish rely on visual acuity to hunt for prey all of their lives, for this reason, their eyesight develops when they are embryos. The zebrafish in this experiment are between 5 and 20 days old. Once their egg yolks are completely reabsorbed the zebrafish have a need to eat, and only then will they have an interest in eating Paramecium. The zebrafishes’ eyes will be able to see clearly by 5 days post fertilization as well, so when food is introduced to them, a certain colored light will be simultaneously shined in the same direction as the one the food is coming from. The purpose of this experiment is to use different colored visual cues to train larval zebrafish into expecting food whenever they are shone. The usage of lights also tests their extraordinary visual abilities. Red lights will be associated with a food reward in one group of larvae, and in the second group, green lights will be associated with a food reward, demonstrating associative learning. The zebrafishes’ unconditioned response (eating Paramecium when they appear) will be trained into a conditioned response (looking to eat Paramecium when the light is shined). After they have been conditioned to respond to these signals, the zebrafishes’ brains will be studied to find changes in their neural pathways. The expected results of this experiment should lead to the fish thinking Paramecium are coming at just the glimpse of a light