Deformations of Coisotropic Submanifolds in Jacobi Manifolds

In this paper, we attach an $L_\infty$-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), L\^e-Oh (locally conformal symplectic case). As a new special case, we attach an $L_\infty$-algebra to any coisotropic submanifold in a contact manifold. The $L_\infty$-algebra of a coisotropic submanifold $S$ governs the (formal) deformation problem of $S$.Comment: 41 pages, v2: several revisions, title and abstract slightly changed, mathematics unchanged; final version, to appear in J. Sympl. Geom. 16 (2018

Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $\alpha$ in the boundary condition, or equivalently of the volume $M$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.Comment: 16 page

Rare express saccades in elderly fallers

Q Yang1, T T Lê1, E Debay1, C Orssaud2, G Magnier3, Z Kapoula11Groupe IRIS Vision and Motricité Binoculaire, CNRS, Service d’Ophtalmologie-ORL-Stomatologie; 2Service d’Ophtalmologie, Hôpital Européen Georges Pompidou, Paris, France; 3Hôpital de gériatrie Henry Dunant, Paris, FranceObjective: To examine horizontal saccades in elderly subjects with falling history; prior extensive screening was done to recruit subjects with falling history in the absence of pathology.Methods: Twelve elderly with falling history were tested. Two testing conditions were used: the gap (fixation target extinguishes prior to target onset) and the overlap (fixation stays on after target onset) paradigms. Each condition was run at three viewing distances −20 cm, 40 cm, and 150 cm, corresponding to convergence angle at 17.1°, 8.6°, and 2.3°, respectively. Eye movements were recorded with the photoelectric IRIS (Skalar medical).Results: (i) like in healthy elderly subjects, elderly with falling history produce shorter latencies in the gap paradigm than in the overlap paradigm; (ii) their latencies are shorter at near distances (20 and 40 cm) relative to 150 cm for both paradigms; (iii) the novel result is that they fail to produce express latencies even in the conditions (near viewing distance and the gap task) known to promote high rates of express in adults (25%) or in healthy elderly (20%). Seven from the 10 healthy elderly produced express saccades at rates >12%, while 9 of the 12 older subjects with falling history showed no express saccades at all; the remaining 3 subjects showed low rates <12%.Conclusion: The quasi paucity of express saccades could be due to the disequilibrium of complex cortical/subcortical networks needed for making express saccades. The results support models suggesting specific network for express saccades; missing of such optomotor reflex may go along with missing other reflexes as well increasing the chances of falling.Keywords: elderly, falling, saccades, gap/overlap, express saccad

Polar Cremona Transformations and Monodromy of Polynomials

Consider the gradient map associated to any non-constant homogeneous polynomial f\in \C[x_0,...,x_n] of degree $d$, defined by \phi_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x)) where D(f)=\{x\in \CP^n; f(x)\neq 0\} is the principal open set associated to $f$ and $f_i=\frac{\partial f}{\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \ref{p1} we give a new lower bound for the degree $d(f)$ of $\phi_f$ under the assumption that the projective hypersurface $V:f=0$ has only isolated singularities. When $d(f)=1$, Theorem \ref{t4} yields very strong conditions on the singularities of $V$.Comment: 8 page

Synchronous colorectal liver metastasis: A network meta-analysis review comparing classical, combined, and liver-first surgical strategies.

BACKGROUND: In recent years, the management of synchronous colorectal liver metastasis has changed significantly. Alternative surgical strategies to the classical colorectal-first approach have been proposed. These include the liver-first and combined resections approaches. The objectives of this review were to compare the short- and long-term outcomes for all three approaches. METHODS: A systematic review of comparative studies was performed. Evaluated endpoints included surgical outcomes (5-year overall survival, 30-day mortality, and post-operative complications). Pair-wise and network meta-analysis (NMA) were performed to compare survival outcomes. RESULTS: Eighteen studies were included in this review, reporting on 3,605 patients. NMA and pair-wise meta-analysis of the 5-year overall survival did not show significant difference between the three surgical approaches: combined versus colorectal-first, mean odds ratio (OR) 1.02 (95% CI 0.8-1.28, P = 0.93); liver-first versus colorectal-first, mean OR 0.81 (95% CI 0.53-1.26, P = 0.37); liver-first versus combined, mean OR 0.80 (95% CI 0.52-1.24, P = 0.41). In addition NMA of the 30-day mortality among the three approaches also did not observe statistical difference. Analysis of variance showed that mean post-operative complications of all approaches were comparable (P = 0.51). CONCLUSION: There are considerable differences in the peri-operative management of synchronous CLM patients. This meta-analysis demonstrated no clear statistical surgical outcome or survival advantage towards any of the three approaches. J. Surg. Oncol. © 2014 Wiley Periodicals, Inc

Fibonacci numbers and self-dual lattice structures for plane branches

Consider a plane branch, that is, an irreducible germ of curve on a smooth complex analytic surface. We define its blow-up complexity as the number of blow-ups of points necessary to achieve its minimal embedded resolution. We show that there are $F_{2n-4}$ topological types of blow-up complexity $n$, where $F_{n}$ is the $n$-th Fibonacci number. We introduce complexity-preserving operations on topological types which increase the multiplicity and we deduce that the maximal multiplicity for a plane branch of blow-up complexity $n$ is $F_n$. It is achieved by exactly two topological types, one of them being distinguished as the only type which maximizes the Milnor number. We show moreover that there exists a natural partial order relation on the set of topological types of plane branches of blow-up complexity $n$, making this set a distributive lattice, that is, any two of its elements admit an infimum and a supremum, each one of these operations beeing distributive relative to the second one. We prove that this lattice admits a unique order-inverting bijection. As this bijection is involutive, it defines a duality for topological types of plane branches. The type which maximizes the Milnor number is also the maximal element of this lattice and its dual is the unique type with minimal Milnor number. There are $F_{n-2}$ self-dual topological types of blow-up complexity $n$. Our proofs are done by encoding the topological types by the associated Enriques diagrams.Comment: 21 pages, 16 page