A polynomial upper bound for the mixing time of edge rotations on planar maps

We consider a natural local dynamic on the set of all rooted planar maps with $n$ edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the $n$-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this "edge rotation" chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times $n^{-11/2}$. In doing so, we provide a partially new proof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulations, which makes it possible to generalise a recent result of the author and Stauffer to a chain that relates to edge rotations via Tutte's bijection

Growing uniform planar maps face by face

We provide “growth schemes” for inductively generating uniform random -angulations of the sphere with faces, as well as uniform random simple triangulations of the sphere with faces. In the case of -angulations, we provide a way to insert a new face at a random location in a uniform -angulation with faces in such a way that the new map is precisely a uniform -angulation with faces. Similarly, given a uniform simple triangulation of the sphere with faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer

Lagrangians of Hypergraphs

The Lagrangian of a hypergraph is a function that in a sense seems to measure how ‘tightly packed’ a subset of the hypergraph one can find. Lagrangians were first used by Motzkin and Straus to obtain a new proof of a classic theorem of Turán, and subsequently found a number of very valuable applications in Extremal Hypergraph Theory; one remarkable result they yield is the disproof of a famous "jumping conjecture" of Erdos, which we reprove entirely; we will also introduce a very recent method based on Razborov's flag algebras to show that, though the jumping conjecture is false in general, hypergraphs "do jump" in some cases

Rare presentation of a testicular angiofibroma treated with testis sparing surgery

Introduction: Testicular benign tumors are very rare (< 5%). Testicular Angiofibroma (AF) is one of those, however the gold standard of treatment and follow-up is still unclear. Case report: A 47 years-old man with only one functioning testis was referred to our clinic for a palpable right testicular mass and atrophic contralateral testis. Patient underwent testis-sparing surgery with inguinal approach and intraoperative frozen sections examination with diagnosis of AF. Final histology confirmed AF. Post-operative follow-up was uneventful. Clinical and ultrasonographic follow-up was negative after 8 months. Conclusion: We report a conservative surgery in a patient with AF of the solitary testis. AF is a benign para-testicular fibrous neoplasm that could be misinterpreted as malignant tumor and treated with orchiectomy. Testis-sparing surgery is recommended in this case with intraoperative pathological examination. The excision of the mass is enough but in front of a possible recurrence a long follow-up is advisable

The use of albumin in the complications of cirrhosis: evidence and future perspectives

The therapeutic use of albumin in cirrhosis dates back to the 50s, when hypoalbuminemia was thought to play a crucial role in the pathophysiology of ascites. Today, while its efficacy in the treatment of ascites is still under investigation, it has been proved that albumin is able to improve patient outcome and survival in some specific complications of cirrhosis, such as the prevention of post-paracentesis circulatory dysfunction and the treatment of spontaneous bacterial peritonitis and hepatorenal syndrome. Beside its oncotic power, albumin carries other biological properties, the so called non-oncotic properties, including transportation and detoxification of several molecules, free radical scavenging, modulation of vascular permeability, activity on the immune system and on the haemostatic balance. Some experimental evidences indicate that not only albumin concentration but also its function is reduced in patients with cirrhosis. However, the clinical implications of such functional abnormalities is still unclear. We here present the available evidence on the use of albumin in cirrhosis and future perspectives

An Innovative Hyperbaric Hypothermic Machine Perfusion Protects the Liver from Experimental Preservation Injury

Purpose. Hypothermic machine perfusion systems seem more effective than the current static storage to prevent cold ischemic liver injury. Thus, we test an innovative hyperbaric hypothermic machine perfusion (HHMP), which combines hyperbaric oxygenation of the preservation solution and continuous perfusion of the graft. Methods. Rat livers were preserved with Celsior solution according to 4 different modalities: normobaric static preservation; hyperbaric static preservation at 2 atmosphere absolute (ATA); normobaric dynamic preservation, with continuous perfusion; hyperbaric dynamic preservation, with continuous perfusion at 2 ATA. After 24 h cold preservation, we assessed different parameters. Results. Compared to baseline, livers preserved with the current static storage showed severe ultrastructural damage, glycogen depletion and an increased oxidative stress. Normobaric perfused livers showed improved hepatocyte ultrastructure and ameliorated glycogen stores, but they still suffered a significant oxidative damage. The addition of hyperbaric oxygen produces an extra benefit by improving oxidative injury and by inducing endothelial NO synthase (eNOS) gene expression. Conclusions. Preservation by means of the present innovative HHMP reduced the liver injury occurring after the current static cold storage by lowering glycogen depletion and oxidative damage. Interestingly, only the use of hyperbaric oxygen was associated to a blunted oxidative stress and an increased eNOS gene expression