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vignette|Coordonnées géographiques sur un globe : la latitude correspond à la mesure de l’angle marqué phi (φ) ; la mesure de l’angle marqué lambda (λ) par rapport au méridien de référence donne la longitude.|194x194px lang=fr|vignette|upright=1.5|Latitude et longitude sur la Terre. Par coordonnés géographiques (ou encore « repères géographiques ») d'un lieu sur la Terre, on entend un système de trois coordonnées qui sont le plus souvent : la latitude, la longitude et l'altitude (ou l'élévation) par rapport au niveau moyen de la mer (élévation orthométrique) ou par rapport à une surface de référence, en général ellipsoïde (élévation ellipsoïdale). Ces coordonnées géographiques découlent d'un système géodésique qui modélise la forme de la Terre. Pour se repérer à la surface de la planète, on peut utiliser des systèmes de représentation graphique appelés « repères cartographiques du pays ». vignette|upright=2|Carte du monde montrant la latitude et la longitude, suivant une projection

Trace coordinates of Teichmüller spaces of Riemann surfaces

Thomas Gauglhofer

Using an algebraic formalism based on matrices in SL(2,R), we explicitly give the Teichmüller spaces of Riemann surfaces of signature (0,4) (X pieces), (1,2) ("Fish" pieces) and (2,0) in trace coordinates. The approach, based upon gluing together two building blocks (Q and Y pieces), is then extended to tree-like pants decomposition for higher signatures (g,n) and limit cases such as surfaces with cusps or cone-like singularities. Given the Teichmüller spaces, we establish a set of generators of their modular groups for signatures (0,4), (1,2) and (2,0) in trace coordinates using transformations acting separately on the building blocks and an algorithm on dividing geodesics. The fact that these generators act particularly nice in trace coordinates gives further motivation to this choice (rather then the one of Fenchel-Nielsen coordinates). This allows us to solve the Riemann moduli problem for X pieces, "Fish" pieces and surfaces of genus 2; i.e. to give the moduli spaces as the fundamental domains for the action of the modular groups on the Teichmüller spaces. In this context, we also give an algorithm deciding whether two Riemann surfaces of signatures (0,4), (1,2) or (2,0) given by points in the Teichmüller space are isometric or not. As a consequence, we show the following two results concerning simple closed geodesics: On any purely hyperbolic Riemann surface (containing neither cusps nor cone-like singularities), the longest of two simple closed geodesics that intersect one another n times is of length at least ln, a sharp constant independent of the surface. We explicitly give ln for n = 1,2,3 and study its behaviour when n goes to infinity. X pieces are spectrally rigid with respect to the length spectrum of simple closed geodesics.

EPFL2006

Trace coordinates of Teichmüller space of Riemann surfaces of signature $(0,4)$

Thomas Gauglhofer, Klaus-Dieter Semmler

Summary: We explicitly give

calT

, the Teichmüller space of four-holed spheres (which we call

X

pieces) in trace coordinates, as well as its modular group and a fundamental domain for the action of this group on

calT

which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than

2operatornamearccosh(2)

intersect at most once; two closed geodesics of lengths smaller than

2operatornamearccosh(3)

are both non-dividing or intersect at most once.

2005

Persistent spatial clusters of high body mass index in a Swiss urban population as revealed by the 5-year GeoCoLaus longitudinal study

Murielle Bochud, Joël Chételat, Idris Guessous, Stéphane Joost, Fred Michel Paccaud

OBJECTIVE Body mass index (BMI) may cluster in space among adults and be spatially dependent. Whether and how BMI clusters evolve over time in a population is currently unknown. We aimed to determine the spatial dependence of BMI and its 5-year evolution in a Swiss general adult urban population, taking into account the neighborhood- and individual-level characteristics. DESIGN Cohort study SETTING Swiss general urban population PARTICIPANTS 6,481 geo-referenced individuals from the CoLaus cohort at baseline (age range 35–74 years, period=2003-2006) and 4,460 at follow-up (period=2009-2012). OUTCOME MEASURES Body weight and height were measured by trained health care professionals with participants standing without shoes in light indoor clothing. BMI was calculated as weight (kg) divided by height squared (m2). Participants were geocoded using their postal address (geographic coordinates of the place of residence). Getis-Ord Gi statistic was used to measure the spatial dependence of BMI values at baseline and its evolution at follow-up. RESULTS BMI was not randomly distributed across the city. At baseline and at follow-up, significant clusters of high versus low BMIs were identified and remained stable during the two periods. These clusters were meaningfully attenuated after adjustment for neighborhood-level income but not individual-level characteristics. Similar results were observed among participants who developed obesity. CONCLUSIONS To our knowledge, this is the first study to report longitudinal changes in BMI clusters in adults from a general population. Spatial clusters of high BMI persisted over a 5-year period and were mainly influenced by neighborhood-level income. ARTICLE SUMMARY STRENGTHS AND LIMITATIONS OF THE STUDY • As far as we know, this is the first study to report the persistence of spatial clusters of high BMI values over a 5-year period in adults from a general population • The observed east to west pattern of BMI clustering fits known socio-economic and ethno-cultural differences distinguishing these opposite regions of the city of Lausanne, Switzerland • A consequence of the social policy applied by the city is likely to fix populations with modest income in subsidized housing located in specific areas • While recruitment methods of the CoLaus study aimed at collecting information on a representative sample of the general population, adult participants and non-participants to the CoLaus study may differ and participation bias cannot be excluded • We considered several individual-level covariates but data on individual income was missing. We used instead the median income of the including city statistical sector.

Bmj Publishing Group2016