Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)

This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs, additional examples & figures, additional comparison between classical LTL/schemata algorithms up to the provided translations, and an example of how to do model checking with schemata; 36 pages, 8 figure

The boundary of the outer space of a free product

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space $P\mathcal{O}(G,\{G_1,\dots,G_k\})$ for the axes topology with the space of projective minimal, \emph{very small} $(G,\{G_1,\dots,G_k\})$-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the $G_i$'s, and whose tripod stabilizers are trivial. Its topological dimension is equal to $3N+2k-4$, and the boundary has dimension $3N+2k-5$. We also prove that any very small $(G,\{G_1,\dots,G_k\})$-tree has at most $2N+2k-2$ orbits of branch points.Comment: v3: Final version, to appear in the Israel Journal of Mathematics. Section 3, regarding the definition and properties of geometric trees, has been rewritten to improve the exposition, following a referee's suggestio

A Cartan-Hadamard type result for relatively hyperbolic groups

In this article, we prove that if a finitely presented group has an asymptotic cone which is tree-graded with respect to a precise set of pieces then it is relatively hyperbolic. This answers a question of M. Sapir.Comment: 33 pages, 2 figure

Thrombolytic removal of intraventricular haemorrhage in treatment of severe stroke: results of the randomised, multicentre, multiregion, placebo-controlled CLEAR III trial

Background: Intraventricular haemorrhage is a subtype of intracerebral haemorrhage, with 50% mortality and serious disability for survivors. We aimed to test whether attempting to remove intraventricular haemorrhage with alteplase versus saline irrigation improved functional outcome. Methods: In this randomised, double-blinded, placebo-controlled, multiregional trial (CLEAR III), participants with a routinely placed extraventricular drain, in the intensive care unit with stable, non-traumatic intracerebral haemorrhage volume less than 30 mL, intraventricular haemorrhage obstructing the 3rd or 4th ventricles, and no underlying pathology were adaptively randomly assigned (1:1), via a web-based system to receive up to 12 doses, 8 h apart of 1 mg of alteplase or 0·9% saline via the extraventricular drain. The treating physician, clinical research staff, and participants were masked to treatment assignment. CT scans were obtained every 24 h throughout dosing. The primary efficacy outcome was good functional outcome, defined as a modified Rankin Scale score (mRS) of 3 or less at 180 days per central adjudication by blinded evaluators. This study is registered with ClinicalTrials.gov, NCT00784134. Findings: Between Sept 18, 2009, and Jan 13, 2015, 500 patients were randomised: 249 to the alteplase group and 251 to the saline group. 180-day follow-up data were available for analysis from 246 of 249 participants in the alteplase group and 245 of 251 participants in the placebo group. The primary efficacy outcome was similar in each group (good outcome in alteplase group 48% vs saline 45%; risk ratio [RR] 1·06 [95% CI 0·88–1·28; p=0·554]). A difference of 3·5% (RR 1·08 [95% CI 0·90–1·29], p=0·420) was found after adjustment for intraventricular haemorrhage size and thalamic intracerebral haemorrhage. At 180 days, the treatment group had lower case fatality (46 [18%] vs saline 73 [29%], hazard ratio 0·60 [95% CI 0·41–0·86], p=0·006), but a greater proportion with mRS 5 (42 [17%] vs 21 [9%]; RR 1·99 [95% CI 1·22–3·26], p=0·007). Ventriculitis (17 [7%] alteplase vs 31 [12%] saline; RR 0·55 [95% CI 0·31–0·97], p=0·048) and serious adverse events (114 [46%] alteplase vs 151 [60%] saline; RR 0·76 [95% CI 0·64–0·90], p=0·002) were less frequent with alteplase treatment. Symptomatic bleeding (six [2%] in the alteplase group vs five [2%] in the saline group; RR 1·21 [95% CI 0·37–3·91], p=0·771) was similar. Interpretation: In patients with intraventricular haemorrhage and a routine extraventricular drain, irrigation with alteplase did not substantially improve functional outcomes at the mRS 3 cutoff compared with irrigation with saline. Protocol-based use of alteplase with extraventricular drain seems safe. Future investigation is needed to determine whether a greater frequency of complete intraventricular haemorrhage removal via alteplase produces gains in functional status

Unimodality Problems in Ehrhart Theory

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This version updated October 2017 to correct an error in the original versio