Displacement energy of unit disk cotangent bundles

We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle $D^*M$ in a cotangent bundle $T^*M$, when the base manifold $M$ is an open Riemannian manifold. Our main result is that the displacement energy is not greater than $C r(M)$, where $r(M)$ is the inner radius of $M$, and $C$ is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.Comment: Title slightly changed. Close to the version published online in Math Zei

An exact sequence for contact- and symplectic homology

A symplectic manifold $W$ with contact type boundary $M = \partial W$ induces a linearization of the contact homology of $M$ with corresponding linearized contact homology $HC(M)$. We establish a Gysin-type exact sequence in which the symplectic homology $SH(W)$ of $W$ maps to $HC(M)$, which in turn maps to $HC(M)$, by a map of degree -2, which then maps to $SH(W)$. Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of $M$.Comment: Final version. Changes for v2: Proof of main theorem supplemented with detailed discussion of continuation maps. Description of degree -2 map rewritten with emphasis on asymptotic markers. Sec. 5.2 rewritten with emphasis on 0-dim. moduli spaces. Transversality discussion reorganized for clarity (now Remark 9). Various other minor modification

Predictors of unemployment status in people with relapsing multiple sclerosis: a single center experience

Background: Multiple sclerosis (MS) is the most common cause of nontraumatic chronic neurological disability affecting young adults during their crucial employment years. Objectives: To evaluate patients and disease related factors associated to unemployment in a cohort of relapsing–remitting (RR) MS patients. Methods: We included RRMS patients with a follow-up of at least 1 year. We collected data about years of school education and employment status. Patients underwent a neuropsychological evaluation using the Brief International Cognitive Assessment for Multiple Sclerosis (BICAMS). Demographic and clinical predictors of unemployment were assessed through a multivariable stepwise logistic regression model. Results: We evaluated 260 consecutive RRMS patients. Employed patients were less frequently female (68.4% vs 83.3%, p = 0.006), less disabled (median Expanded Disability Status Scale (EDSS) score: 2.0 (0–7.0) vs 2.5 (0–7.5), p < 0.001), with more years of school education (mean ± standard deviation (SD), years: 13.74 ± 0.30 vs 10.86 ± 3.47, p < 0.001). Female sex and a higher EDSS score resulted associated with a greater risk of unemployment (OR 3.510, 95% CI 1.654–7.448, p = 0.001; OR 1.366, 95% CI 1.074–1.737, p = 0.011, respectively), whereas a greater number of years of schooling and current disease-modifying therapy exposure resulted protective factors (OR 0.788, 95% CI 0.723–0.858, p < 0,001; OR 0.414, 95% CI 0.217–0.790, p = 0.008, respectively). Conclusions: Understanding work is pervasively influenced by consequences of MS, we confirmed the impact of demographic, physical, and cognitive factors on employment status in RRMS patients

Symplectic cohomology and q-intersection numbers

Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity. Equivariant Lagrangians mirror equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example 7.5, added discussion of sign

The Minimal Length of a Lagrangian Cobordism between Legendrians

To investigate the rigidity and flexibility of Lagrangian cobordisms between Legendrian submanifolds, we investigate the minimal length of such a cobordism, which is a $1$-dimensional measurement of the non-cylindrical portion of the cobordism. Our primary tool is a set of real-valued capacities for a Legendrian submanifold, which are derived from a filtered version of Legendrian Contact Homology. Relationships between capacities of Legendrians at the ends of a Lagrangian cobordism yield lower bounds on the length of the cobordism. We apply the capacities to Lagrangian cobordisms realizing vertical dilations (which may be arbitrarily short) and contractions (whose lengths are bounded below). We also study the interaction between length and the linking of multiple cobordisms as well as the lengths of cobordisms derived from non-trivial loops of Legendrian isotopies.Comment: 33 pages, 9 figures. v2: Minor corrections in response to referee comments. More general statement in Proposition 3.3 and some reorganization at the end of Section

Selected energy dark-field imaging using low energy electrons for optimal surface phase discrimination

We propose a general strategy for surface phase discrimination by dark-field imaging using low energy electrons, which maximizes contrast using diffraction spots, at selected optimal energies. The method can be automated to produce composite phase maps in real space and study the dynamics of complex phase transformations in real-time. To illustrate the capabilities of the technique, surface phases are mapped in the vicinity of liquid Ga droplets on the technologically important GaAs (001) surface