Spectral properties of a limit-periodic Schr\"odinger operator in dimension two

We study Schr\"{o}dinger operator $H=-\Delta+V(x)$ in dimension two, $V(x)$ being a limit-periodic potential. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle \vec k,\vec x\rangle }$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.Comment: 89 pages, 6 figure

Gevrey-Type Resolvent Estimates at the Threshold for a Class of Non-Selfadjoint Schrödinger Operators

In this article, we show that under some coercive assumption on the complex-valued potential V(x), the derivatives of the resolvent of the non-selfadjoint Schröinger operator H = −∆ + V(x) satisfy some Gevrey estimates at the threshold zero. As applications, we establish subexponential time-decay estimates of local energies for the semigroup e−tH, t > 0. We also show that for a class of Witten Laplacians for which zero is an eigenvalue embedded in the continuous spectrum, the solutions to the heat equation converges subexponentially to the steady solution

On drivers’ reasoning about traffic signs: The case of qualitative location

This article explores the most appropriate arrangement (vertical, horizontal) and the frame of reference adopted by drivers (intrinsic, relative) as determinants of the comprehension of new traffic messages (e.g., congestion before arriving to Milan). Two specific cases for location (event-before-city, event-after-city) were tested following two layouts: H (horizontal, left-right) and V (vertical, bottom-up). Four comprehension tests carried out between 2006 and 2013 with 10, 099 drivers in four countries (Italy, Netherlands, Spain, Sweden) were analyzed in a 2 (case: Before vs. After) x 2 (disposition: H, V) x 4 (Country) between-subject design. The comprehension of the V variants (78.1%) exceeded the comprehension of the H variants (54.1%) in all the countries in the "before" case. In no country did the V or H variants come close to functional understanding in the "after" case. The results provided evidence of the preferred model and relative frame of reference as determinants of message understanding. General Audience Summary A controversial aspect that arises from the use of different traffic signaling devices is that drivers often have to understand messages they are seeing for the very first time. This article analyzes the results of a series of empirical studies carried out with the aim of internationalizing variable message signs (VMS) by substituting keywords (e.g., prepositions) for abstract graphic signs (e.g., an arrow). Faced with novel elements in a traffic message about which drivers must conclude something in real time, they have no choice but to reason. This article explores the most appropriate arrangement (vertical, horizontal) and the frame of reference adopted by drivers (intrinsic, relative) as determinants of the comprehension of novel and complex VMS (e.g., congestion before arriving to Milan). Our study focuses on the design variants tested to inform drivers about two cases for location (event-before-city and event-after-city), following two basic layouts: H (horizontal, left-right) and V (vertical, bottom-up). Four comprehension tests carried out between 2006 and 2013 with 10, 099 drivers in four countries (Italy, Netherlands, Spain, Sweden) were analyzed in a 2 (case: Before vs. After) x 2 (disposition: H, V) x 4 (Country) between-subject design. The comprehension of the V variants (78.1%) exceeded the comprehension of the H variants (54.1%) in all the countries in the "before" case. However, in no country did the V or H variants come close to functional understanding in the "after" case. The results provided evidence of the preferred model and relative frame of reference as determinants of message understanding. Although it is not realistic to expect national or international drivers to memorize all possible traffic messages, it is feasible to understand how their prior knowledge and preferences modulate their conclusions to design more functional traffic messages

Separation of variables for quantum integrable models related to Uq(sl^N) U_q(\hat{sl}_N)

In this paper we construct separated variables for quantum integrable models related to the algebra $U_q(\hat{sl}_N)$. This generalizes the results by Sklyanin for $N=2,3$.Comment: 12 pages, Latex, AMS font

Global Well-posedness of the 3D Primitive Equations With Partial Vertical Turbulence Mixing Heat Diffusion

The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data

A simple derivation of BV bounds for inhomogeneous relaxation systems

We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. They are the most standard and simplest way to prove compactness and convergence. We provide a novel and simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, but closely related, notion. We give two examples motivated by renal flows which consist of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general