Asymptotically maximal families of hypersurfaces in toric varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $\Delta$ there are families of hypersurfaces with the Newton polytopes $(\lambda\Delta)_{\lambda \in \mathbb{N}}$ that are asymptotically maximal when $\lambda$ tends to infinity. We also show that these results generalize to complete intersections.Comment: 18 pages, 1 figur

Combinatorial simplex algorithms can solve mean payoff games

A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games. Mean payoff games are known to be in NP and co-NP; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a non-degenerate linear program over Hahn series.Comment: v1: 15 pages, 3 figures; v2: improved presentation, introduction expanded, 18 pages, 3 figure

Homology class of a Lagrangian Klein bottle

It is shown that an embedded Lagrangian Klein bottle represents a non-trivial mod 2 homology class in a compact symplectic four-manifold $(X,\omega)$ with $c_1(X)\cdot[\omega]>0$. (In versions 1 and 2, the last assumption was missing. A counterexample to this general claim and the first proof of the corrected result have been found by Vsevolod Shevchishin.) As a corollary one obtains that the Klein bottle does not admit a Lagrangian embedding into the standard symplectic four-space.Comment: Version 3 - completely rewritten to correct a mistake; Version 4 - minor edits, added references; AMSLaTeX, 6 page

Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology

This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram colorings (1956). In the second section we describe how Fox work was generalized to distributive colorings (racks and quandles) and eventually in the work of Jones and Turaev to link invariants via Yang-Baxter operators, here the importance of statistical mechanics to topology will be mentioned. Finally we describe recent developments which started with Mikhail Khovanov work on categorification of the Jones polynomial. By analogy to Khovanov homology we build homology of distributive structures (including homology of Fox colorings) and generalize it to homology of Yang-Baxter operators. We speculate, with supporting evidence, on co-cycle invariants of knots coming from Yang-Baxter homology. Here the work of Fenn-Rourke-Sanderson (geometric realization of pre-cubic sets of link diagrams) and Carter-Kamada-Saito (co-cycle invariants of links) will be discussed and expanded. Dedicated to Lou Kauffman for his 70th birthday.Comment: 35 pages, 31 figures, for Knots in Hellas II Proceedings, Springer, part of the series Proceedings in Mathematics & Statistics (PROMS

Equilibrium shapes of flat knots

We study the equilibrium shapes of prime and composite knots confined to two dimensions. Using rigorous scaling arguments we show that, due to self-avoiding effects, the topological details of prime knots are localised on a small portion of the larger ring polymer. Within this region, the original knot configuration can assume a hierarchy of contracted shapes, the dominating one given by just one small loop. This hierarchy is investigated in detail for the flat trefoil knot, and corroborated by Monte Carlo simulations.Comment: 4 pages, 3 figure

WhoLoDancE: Towards a methodology for selecting Motion Capture Data across different Dance Learning Practice

<p>In this paper we present the objectives and preliminary work of WhoLoDancE a Research and Innovation Action funded under the European Union‘s Horizon 2020 programme, aiming at using new technologies for capturing and analyzing dance movement to facilitate whole-body interaction learning experiences for a variety of dance genres. Dance is a diverse and heterogeneous practice and WhoLoDancE will develop a protocol for the creation and/or selection of dance sequences drawn from different dance styles for different teaching and learning modalities. As dance learning practice lacks standardization beyond dance genres and specific schools and techniques, one of the first project challenges is to bring together a variety of dance genres and teaching practices and work towards a methodology for selecting the appropriate shots for motion capturing, to acquire kinetic material which will provide a satisfying proof of concept for Learning scenarios of particular genres. The four use cases we are investigating are 1) classical ballet, 2) contemporary dance, 3) flamenco and 4) Greek folk dance.</p

Randomized crossover pharmacokinetic evaluation of subcutaneous versus intravenous granisetron in cancer patients treated with platinum-based chemotherapy

BACKGROUND: 5-HT3-receptor antagonists are one of the mainstays of antiemetic treatment, and they are administered either i.v. or orally. Nevertheless, sometimes neither administration route is feasible, such as in patients unable to admit oral intake managed in an outpatient setting. Our objective was to evaluate the bioavailability of s.c. granisetron. PATIENTS AND METHODS: Patients receiving platinum-based chemotherapy were randomized to receive 3 mg of granisetron either s.c. or i.v. in a crossover manner during two cycles. Blood and urine samples were collected after each cycle. Pharmacokinetic parameters observed with each administration route were compared by analysis of variance. RESULTS: From May to November 2005, 31 patients were included and 25 were evaluable. Subcutaneous granisetron resulted in a 27% higher area under the concentration-time curve for 0-12 hours (AUC(0-12h)) and higher levels at 12 hours, with similar values for AUC(0-24h). The maximum concentration was lower with the s.c. than with the i.v. route and was observed 30 minutes following s.c. administration. CONCLUSION: Granisetron administered s.c. achieves complete bioavailability. This is the first study that shows that s.c. granisetron might be a valid alternative to i.v. delivery. Further trials to confirm clinical equivalence are warranted. This new route of administration might be especially relevant for outpatient management of emesis in cancer patients