Filtrations on the knot contact homology of transverse knots

We construct a new invariant of transverse links in the standard contact structure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle S^*R^3 of R^3, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.Comment: 23 pages, v2: minor corrections suggested by refere

Weak perturbations of the p-Laplacian

We consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities.Comment: 20 page

Log-mean linear models for binary data

This paper introduces a novel class of models for binary data, which we call log-mean linear models. The characterizing feature of these models is that they are specified by linear constraints on the log-mean linear parameter, defined as a log-linear expansion of the mean parameter of the multivariate Bernoulli distribution. We show that marginal independence relationships between variables can be specified by setting certain log-mean linear interactions to zero and, more specifically, that graphical models of marginal independence are log-mean linear models. Our approach overcomes some drawbacks of the existing parameterizations of graphical models of marginal independence

Eigenvalue estimates for Schroedinger operators on metric trees

We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains

We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semi-classical parameter. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and estimate this decay rate. This extends our results, discussing the results of a recent paper by Ekholm--Kova\v{r}\'ik--Portmann, to include also non-simply connected domains.Comment: 15 pages, 4 figure

Rational Symplectic Field Theory for Legendrian knots

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.Comment: 58 pages, many figures; v3: minor corrections; final version, to appear in Inventiones Mathematica

An Input-output Model of the North Central Region of Texas

The primary objective was to estimate the structural interrelationships of the North Central Texas Economy in 1967. This region is a thirty-one county area with a 1970 population of 3,064,560. Economic interdependencies were estimated by Input-Output analysis. The regional Input-Output Model consists of transactions, input coefficients, and interdependence coefficients tables. Monetary values of transactions among 108 processing sectors, of sales to final demand (including regional household consumption and exports), and of purchases in addition to local interindustry transactions (including household payments and imports) are estimated in the transactions table from primary and secondary data. Input coefficients estimate the value of inputs required from each processing sector to produce one dollar of output for a sector. Interdependence coefficients show the total required expansion of output in all regional processing sectors as a result of a dollar of output for a sector

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