State Determination and Sufficiency of Observables

Informational completeness and the possibility of state distinction and determination are among the more important issues of quantum statistics. We use spectral and semispectral (POV) measures to analyse these questions. For a given W ^?-algebra and a family of normal states on it we investigate the relation between sufficiency in Petz’s sense of a W ^?-subalgebra generated by a spectral measure, and the possibility of determination of these states by means of an observable.     

Turbulence on Hyperbolic Plane: The Fate of Inverse Cascade

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as \(\,k^{-5/3}\) .     

Sharp L^{p,\infty }\rightarrow L^q Estimates for the Dyadic-Like Maximal Operators

For each \(1\le q we study the sharp versions of the \(L^{p,\infty }\rightarrow L^q\) estimates for the dyadic maximal operator on \(\mathbb {R}^n\) . Actually, this is done in the more general setting of maximal operators associated with a tree-like structure. The proof rests on a novel combination of the Bellman function technique and optimization arguments.     

$$\,$$ Fu–Kane–Mele invariant as Wess–Zumino action of the sewing matrix

We show that the Fu–Kane–Mele invariant of the 2d time-reversal invariant crystalline insulators is equal to the properly normalized Wess–Zumino action of the so-called sewing-matrix field defined on the Brillouin torus. Applied to 3d, the result permits a direct proof of the known relation between the strong Fu–Kane–Mele invariant and the Chern–Simons action of the non-Abelian Berry connection on the bundle of valence states.     

Refined Second Law of Thermodynamics for Fast Random Processes

We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport and becomes relevant for processes far from quasi-stationary regime. General discussion is illustrated by numerical analysis of the optimal memory erasure protocol for a model for micron-size particle manipulated by optical tweezers.     

Describing and Evaluating an Exemplary Mathematics Elementary Staff Development Project

The purpose of this paper is to describe the model of a mathematics and science staff development cooperative and focus on the evaluation of the mathematics component. The Mathematics and Science Education Cooperative (MSEC) was a comprehensive, long‐range staff development program to improve the teaching and learning of mathematics and science at the elementary school level. The special features of MSEC were (a) it provided year‐round, multiyear involvement, and (b) each year an affective strand was included. Statistically significant student mathematics results from the years 1998–2000 are presented.     

Monte-Carlo analysis of tracer diffusion mechanisms in YBa2Cu3O6+c

A method for estimating, via the Monte-Carlo simulation, the most often realized diffusion mechanisms in 2D ordered structures is presented. Taking as an example the diffusion of oxygen ions in high temperature superconductor we propose several diffusion mechanisms and show to what extent they depend on the temperature and concentration of the diffusing particles. Our results are compared with the ones proposed earlier on the basis of energy arguments. We find also additional trajectories, different from those earlier reported in that system. Received 9 November 1998