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

A Hardy inequality in twisted waveguides

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.Comment: LaTeX, 20 page

Spectrum of the Schr\"odinger operator in a perturbed periodically twisted tube

We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of "slowing down" the twisting in the mean gives rise to a non-empty discrete spectrum.Comment: LaTeX2e, 10 page

Dark energy domination in the Virgocentric flow

The standard \LambdaCDM cosmological model implies that all celestial bodies are embedded in a perfectly uniform dark energy background, represented by Einstein's cosmological constant, and experience its repulsive antigravity action. Can dark energy have strong dynamical effects on small cosmic scales as well as globally? Continuing our efforts to clarify this question, we focus now on the Virgo Cluster and the flow of expansion around it. We interpret the Hubble diagram, from a new database of velocities and distances of galaxies in the cluster and its environment, using a nonlinear analytical model which incorporates the antigravity force in terms of Newtonian mechanics. The key parameter is the zero-gravity radius, the distance at which gravity and antigravity are in balance. Our conclusions are: 1. The interplay between the gravity of the cluster and the antigravity of the dark energy background determines the kinematical structure of the system and controls its evolution. 2. The gravity dominates the quasi-stationary bound cluster, while the antigravity controls the Virgocentric flow, bringing order and regularity to the flow, which reaches linearity and the global Hubble rate at distances \ga 15 Mpc. 3. The cluster and the flow form a system similar to the Local Group and its outflow. In the velocity-distance diagram, the cluster-flow structure reproduces the group-flow structure with a scaling factor of about 10; the zero-gravity radius for the cluster system is also 10 times larger. The phase and dynamical similarity of the systems on the scales of 1-30 Mpc suggests that a two-component pattern may be universal for groups and clusters: a quasi-stationary bound central component and an expanding outflow around it, due to the nonlinear gravity-antigravity interplay with the dark energy dominating in the flow component.Comment: 7 pages, 2 figures, Astronomy and Astrophysics (accepted

The observed infall of galaxies towards the Virgo cluster

We examine the velocity field of galaxies around the Virgo cluster induced by its overdensity. A sample of 1792 galaxies with distances from the Tip of the Red Giant Branch, the Cepheid luminosity, the SNIa luminosity, the surface brightness fluctuation method, and the Tully-Fisher relation has been used to study the velocity-distance relation in the Virgocentric coordinates. Attention was paid to some observational biases affected the Hubble flow around Virgo. We estimate the radius of the zero-velocity surface for the Virgo cluster to be within (5.0 - 7.5) Mpc corresponding to (17 - 26)^\circ at the mean cluster distance of 17.0 Mpc. In the case of spherical symmetry with cosmological parameter \Omega_m=0.24 and the age of the Universe T_0= 13.7 Gyr, it yields the total mass of the Virgo cluster to be within M_T=(2.7 - 8.9) * 10^{14} M_\sun in reasonable agreement with the existing virial mass estimates for the cluster.Comment: 22 pages, 11 figures, 2 tables. Accepted for publication in MNRA

Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States

We prove general comparison theorems for eigenvalues of perturbed Schrodinger operators that allow proof of Lieb--Thirring bounds for suitable non-free Schrodinger operators and Jacobi matrices.Comment: 11 page

Binary Models for Marginal Independence

Log-linear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical log-linear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in contingency tables. The approach taken is graphical and draws on analogies to multivariate Gaussian models for marginal independence. For the graphical model representation we use bi-directed graphs, which are in the tradition of path diagrams. We show how the models can be parameterized in a simple fashion, and how maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. Finally we consider combining these models with symmetry restrictions

Dark energy and key physical parameters of clusters of galaxies

We study physics of clusters of galaxies embedded in the cosmic dark energy background. Under the assumption that dark energy is described by the cosmological constant, we show that the dynamical effects of dark energy are strong in clusters like the Virgo cluster. Specifically, the key physical parameters of the dark mater halos in clusters are determined by dark energy: 1) the halo cut-off radius is practically, if not exactly, equal to the zero-gravity radius at which the dark matter gravity is balanced by the dark energy antigravity; 2) the halo averaged density is equal to two densities of dark energy; 3) the halo edge (cut-off) density is the dark energy density with a numerical factor of the unity order slightly depending on the halo profile. The cluster gravitational potential well in which the particles of the dark halo (as well as galaxies and intracluster plasma) move is strongly affected by dark energy: the maximum of the potential is located at the zero-gravity radius of the cluster.Comment: 8 pages, 1 figur

Stein structures: existence and flexibility

This survey on the topology of Stein manifolds is an extract from our recent joint book. It is compiled from two short lecture series given by the first author in 2012 at the Institute for Advanced Study, Princeton, and the Alfred Renyi Institute of Mathematics, Budapest.Comment: 29 pages, 11 figure

Exploring control-value motivational profiles of mathematics anxiety, self-concept, and interest in adolescents

In this study, we identified multidimensional profiles in students’ math anxiety, math self-concept, and math interest using data from a large generalizable sample of 16,547 9th grade students in the United States who participated in the National Study of Learning Mindsets. We also analyzed the extent that students’ profile memberships are associated with related measures such as prior mathematics achievement, academic stress, and challenge-seeking behavior. Five multidimensional profiles were identified: two profiles which demonstrated relatively high levels of interest and self-concept, along with low math anxiety, in line with the tenets of the control-value theory of academic emotions (C-VTAE); two profiles which demonstrated relatively low levels of interest and self-concept, and high levels of math anxiety (again in accordance with C-VTAE); and one profile, comprising more than 37% of the total sample, which demonstrated medium levels of interest, high levels of self-concept, and medium levels of anxiety. All five profiles varied significantly from one another in their association with the distal variables of challenge seeking behavior, prior mathematics achievement, and academic stress. This study contributes to the literature on math anxiety, self-concept, and interest by identifying and validating student profiles that mainly align with the control-value theory of academic emotions in a large, generalizable sample