Ergodicity of polygonal slap maps

Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflections laws. We study the absolutely continuous invariant probabilities of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps having more than one ergodic absolutely continuous invariant probability.Comment: 17 pages, 6 figure

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

We study polygonal billiards with reflection laws contracting the reflected angle towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic SRB measures whose basins cover a set of full Lebesgue measure.Comment: 26 pages, 2 figure

Characterizing videos, audience and advertising in Youtube channels for kids

Online video services, messaging systems, games and social media services are tremendously popular among young people and children in many countries. Most of the digital services offered on the internet are advertising funded, which makes advertising ubiquitous in children's everyday life. To understand the impact of advertising-based digital services on children, we study the collective behavior of users of YouTube for kids channels and present the demographics of a large number of users. We collected data from 12,848 videos from 17 channels in US and UK and 24 channels in Brazil. The channels in English have been viewed more than 37 billion times. We also collected more than 14 million comments made by users. Based on a combination of text-analysis and face recognition tools, we show the presence of racial and gender biases in our large sample of users. We also identify children actively using YouTube, although the minimum age for using the service is 13 years in most countries. We provide comparisons of user behavior among the three countries, which represent large user populations in the global North and the global South

Portal Cavernoma Cholangiopathy Secondary to Polycythemia Vera: Case Report and Echoendoscopic Findings

info:eu-repo/semantics/publishedVersio

Ergodic properties of polygonal billiards with strongly contracting reflection laws

We consider convex polygonal billiards with a reflection law that contracts the reflection angle towards the normal. Polygonal billiards with orthogonal reflections are called slap maps. For polygons without parallel sides facing each other, the slap map has a finite number of ergodic absolutely continuous invariant probability measures. We show that any billiard with a strongly contracting reflection law has the same number of ergodic SRB measures with the same mixing periods as the ergodic absolutely continuous invariant probabilities of its corresponding slap map. The case of billiards in regular polygons and triangles is studied in detail

Demystifying endoscopic retrograde cholangiopancreatography (ERCP) during pregnancy

Background: For many years, ERCP was avoided in pregnancy given the concerns regarding the adverse effects that, with special focus on radiation, could occur in the developing fetus. However, the postponement or rejection of ERCP in pregnant women, may lead to a higher risk for mother and fetus, especially when the indication is unequivocal, namely cholangitis, biliary pancreatitis and symptomatic choledocholithiasis. Summary and key messages: This review aims to summarize the scarce literature on the subject in order to plan ERCP in pregnancy with the highest safety. The use of techniques that reduce radiation and increase the protection of pregnant women allow radiation levels far below the safety limits. We also discuss the various alternatives of ERCP without radiation. EUS can eliminate the need for ERCP with doubtful choledocholithiasis and plan the best approach in those with previous evidence. The possibility of performing “ERCP” with a linear echoendoscope uniquely under ultrasound control has been described. Conversely, the two-step strategy (initial sphincterotomy with stent placement without fluoroscopy and after delivery, ERCP with lithiasis extraction) proved to be safe obviating fluoroscopy. In conclusion, ERCP can be performed in pregnancy safely and effectively with minimal radiation or even no-radiation at all.info:eu-repo/semantics/publishedVersio