Continuity of the Peierls barrier and robustness of laminations

We study the Peierls barrier for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start with the case of a fixed local potential and derive an estimate for the difference of the periodic Peierls barrier and the Peierls barrier of a general rotation number in a given point. A similar estimate was obtained by Mather in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that the Peierls barrier is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers for a given rotation number, is open in the C1-topology.Comment: 20 pages, submitted to Ergodic Theory and Dynamical System

Ghost circles in lattice Aubry-Mather theory

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multidimensional counterpart of monotone twist maps. They often admit a variational structure, so that the solutions are the stationary points of a formal action function. Classical Aubry-Mather theory establishes the existence of a large collection of solutions of any rotation vector. For irrational rotation vectors this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the gradient flow of the formal action function and we prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called "ghost circle". The existence of ghost circles is first proved for rational rotation vectors and Morse action functions. The main technical result is a compactness theorem for ghost circles, based on a parabolic Harnack inequality for the gradient flow, which implies the existence of ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry-Mather set has a gap, then this gap must be parametrized by minimizers, or contain a non-minimizing solution.Comment: 39 pages, 1 figur

On the destruction of minimal foliations

Monotone variational recurrence relations such as the Frenkel-Kontorova lattice, arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number. They are the action minimizers that constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected or a Cantor set. A connected Aubry-Mather set is called a minimal foliation. In the case of twist maps, it describes an invariant circle, while in solid state physics it corresponds to a continuum of ground states. A Cantor Aubry-Mather set is called a minimal lamination. In this paper we prove that when the rotation number of a minimal foliation is either rational or easy to approximate by rational numbers, then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation. This generalizes a theorem of Mather for twist maps to general recurrence relations

Simple and Effective Visual Models for Gene Expression Cancer Diagnostics

In the paper we show that diagnostic classes in cancer gene expression data sets, which most often include thousands of features (genes), may be effectively separated with simple two-dimensional plots such as scatterplot and radviz graph. The principal innovation proposed in the paper is a method called VizRank, which is able to score and identify the best among possibly millions of candidate projections for visualizations. Compared to recently much applied techniques in the field of cancer genomics that include neural networks, support vector machines and various ensemble-based approaches, VizRank is fast and finds visualization models that can be easily examined and interpreted by domain experts. Our experiments on a number of gene expression data sets show that VizRank was always able to find data visualizations with a small number of (two to seven) genes and excellent class separation. In addition to providing grounds for gene expression cancer diagnosis, VizRank and its visualizations also identify small sets of relevant genes, uncover interesting gene interactions and point to outliers and potential misclassifications in cancer data sets

Ghost circles in lattice Aubry-Mather theory

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multidimensional counterpart of monotone twist maps. They often admit a variational structure, so that the solutions are the stationary points of a formal action function. Classical Aubry-Mather theory establishes the existence of a large collection of solutions of any rotation vector. For irrational rotation vectors this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the gradient flow of the formal action function and we prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called "ghost circle". The existence of ghost circles is first proved for rational rotation vectors and Morse action functions. The main technical result is a compactness theorem for ghost circles, based on a parabolic Harnack inequality for the gradient flow, which implies the existence of ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry-Mather set has a gap, then this gap must be parametrized by minimizers, or contain a non-minimizing solution.Comment: 39 pages, 1 figur

Medical Team Evaluation: Effect on Emergency Department Waiting Time and Length of Stay

<div><p>Emergency Departments (ED) are trying to alleviate crowding using various interventions. We assessed the effect of an alternative model of care, the Medical Team Evaluation (MTE) concept, encompassing team triage, quick registration, redesign of triage rooms and electronic medical records (EMR) on door-to-doctor (waiting) time and ED length of stay (LOS). We conducted an observational, before-and-after study at an urban academic tertiary care centre. On July 17^th 2014, MTE was initiated from 9:00 a.m. to 10 p.m., 7 days a week. A registered triage nurse was teamed with an additional senior ED physician. Data of the 5-month pre-MTE and the 5-month MTE period were analysed. A matched comparison of waiting times and ED LOS of discharged and admitted patients pertaining to various Emergency Severity Index (ESI) triage categories was performed based on propensity scores. With MTE, the median waiting times improved from 41.2 (24.8–66.6) to 10.2 (5.7–18.1) minutes (min; <i>P</i> < 0.01). Though being beneficial for all strata, the improvement was somewhat greater for discharged, than for admitted patients. With a reduction from 54.3 (34.2–84.7) to 10.5 (5.9–18.4) min (<i>P</i> < 0.01), in terms of waiting times, MTE was most advantageous for ESI4 patients. The overall median ED LOS increased for about 15 min (<i>P</i> < 0.01), increasing from 3.4 (2.1–5.3) to 3.7 (2.3–5.6) hours. A significant increase was observed for all the strata, except for ESI5 patients. Their median ED LOS dropped by 73% from 1.2 (0.8–1.8) to 0.3 (0.2–0.5) hours (<i>P</i> < 0.01). In the same period the total orders for diagnostic radiology increased by 1,178 (11%) from 10,924 to 12,102 orders, with more imaging tests being ordered for ESI 2, 3 and 4 patients. Despite improved waiting times a decrease of ED LOS was only seen in ESI level 5 patients, whereas in all the other strata ED LOS increased. We speculate that this was brought about by the tendency of triage physicians to order more diagnostic radiology, anticipating that it may be better for the downstream physician to have more information rather than less.</p></div