Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps.Such recurrence relations often admit a variational structure, so that the solutions x : Z d → R are the stationary points of a formal action function W (x). Given any rotation vector ω ∈ R d , classical Aubry-Mather theory establishes the existence of a large collection of solutions of ∇W (x) = 0 of rotation vector ω. For irrational ω, this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps.In this paper, we study the parabolic gradient flow dx dt = −∇W (x) and we will prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called 'ghost circle'. The existence of these ghost circles is known in dimension d = 1, for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice 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 filled with minimizers, or contain a non-minimizing solution.
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 17th 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; P < 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 (P < 0.01), in terms of waiting times, MTE was most advantageous for ESI4 patients. The overall median ED LOS increased for about 15 min (P < 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 (P < 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.
Variational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel-Kontorova model for a ferromagnetic crystal. For such problems, Aubry-Mather theory establishes the existence of "ground states" or "global minimizers" of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a nontrivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and nonphysical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps
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 ω ∈ R. They are the action minimizers that constitute the AubryMather set. When ω 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.
We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen-Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by Γ-convergence.
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