Foliations on modular curves

It is proved, that a foliation on a modular curve given by the vertical trajectories of holomorphic differential corresponding to the Hecke eigenform is either the Strebel foliation or the pseudo-Anosov foliation.Comment: to appear Bulletin of the Brazilian Mathematical Society, New Serie

A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue 1-2, pp 405-42

Spherical averages in the space of marked lattices

A marked lattice is a $d$-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on ${\mathbb Z}^d$. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit

Impact of intravenous fluid composition on outcomes in patients with systemic inflammatory response syndrome

Introduction: Intravenous (IV) fluids may be associated with complications not often attributed to fluid type. Fluids with high chloride concentrations such as 0.9 % saline have been associated with adverse outcomes in surgery and critical care. Understanding the association between fluid type and outcomes in general hospitalized patients may inform selection of fluid type in clinical practice. We sought to determine if the type of IV fluid administered to patients with systemic inflammatory response syndrome (SIRS) is associated with outcome. Methods: This was a propensity-matched cohort study in hospitalized patients receiving at least 500 mL IV crystalloid within 48 hours of SIRS. Patient data was extracted from a large multi-hospital electronic health record database between January 1, 2009, and March 31, 2013. The primary outcome was in-hospital mortality. Secondary outcomes included length of stay, readmission, and complications measured by ICD-9 coding and clinical definitions. Outcomes were adjusted for illness severity using the Acute Physiology Score. Of the 91,069 patients meeting inclusion criteria, 89,363 (98 %) received 0.9 % saline whereas 1706 (2 %) received a calcium-free balanced solution as the primary fluid. Results: There were 3116 well-matched patients, 1558 in each cohort. In comparison with the calcium-free balanced cohort, the saline cohort experienced greater in-hospital mortality (3.27 % vs. 1.03 %, P <0.001), length of stay (4.87 vs. 4.38 days, P = 0.016), frequency of readmission at 60 (13.54 vs. 10.91, P = 0.025) and 90 days (16.56 vs. 12.58, P = 0.002) and frequency of cardiac, infectious, and coagulopathy complications (all P <0.002). Outcomes were defined by administrative coding and clinically were internally consistent. Patients in the saline cohort received more chloride and had electrolyte abnormalities requiring replacement more frequently (P <0.001). No differences were found in acute renal failure. Conclusions: In this large electronic health record, the predominant use of 0.9 % saline in patients with SIRS was associated with significantly greater morbidity and mortality compared with predominant use of balanced fluids. The signal is consistent with that reported previously in perioperative and critical care patients. Given the large population of hospitalized patients receiving IV fluids, these differences may confer treatment implications and warrant corroboration via large clinical trials. Trial registration: NCT02083198 clinicaltrials.gov; March 5, 201