Vol 26, No 4 (2019)
- Year: 2019
- Articles: 11
- URL: https://journals.rcsi.science/1061-9208/issue/view/11362
Article
Motion of a Smooth Foil in a Fluid under the Action of External Periodic Forces. I
Abstract
A plane-parallel motion of a circular foil is considered in a fluid with a nonzero constant circulation under the action of external periodic force and torque. Various integrable cases are treated. Conditions for the existence of resonances of two types are found. In the case of resonances of the first type, the phase trajectory of the system and the trajectory of the foil are unbounded. In the case of resonances of the second type, the foil trajectory is unbounded, while the phase trajectory of the system remains bounded during the motion.
Convergence to Stationary States and Energy Current for Infinite Harmonic Crystals
Abstract
We consider a d-dimensional harmonic crystal, d ⩾ 1, and study the Cauchy problem with random initial data. The distribution μt of the solution at time t ∈ ℝ is studied. We prove the convergence of correlation functions of the measures μt to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in the mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of μt to a limit measure as t → ∞. We apply these results to the case when initially some infinite “parts” of the crystal have Gibbs distributions with different temperatures. In particular, we find stationary states in which there is a constant nonzero energy current flowing through the crystal. We also study the initial boundary value problem for the harmonic crystal in the half-space with zero boundary condition and obtain similar results.
On the Transfer of the Wiener Measure to the Set of Continuous Trajectories in the Heisenberg Group
Abstract
In the paper, problems related to the theory of stochastic processes on nilpotent Lie groups are studied. In particular, a stochastic process on the Heisenberg group H3(ℝ) is considered such that the trajectories of this process, in the stochastic sense, satisfy the horizontality conditions with respect to the standard contact structure on H3(ℝ). The main result claims that the measure defined on the trajectories of this process is completely concentrated on the set C([0, t], H3(ℝ)) of continuous trajectories.
Roe Bimodules as Morphisms of Discrete Metric Spaces
Abstract
For two discrete metric spaces X and Y, we consider metrics on X ⊔ Y compatible with the metrics on X and Y. As morphisms from X to Y, we consider Roe bimodules, i.e., the norm closures of bounded finite propagation operators from l2(X) to l2(Y). We study the corresponding category \(\mathcal{M}\), which is also a 2-category. We show that almost isometries determine morphisms in \(\mathcal{M}\). We also consider the case Y = X, when there is a richer algebraic structure on the set of morphisms of \(\mathcal{M}\): it is a partially ordered semigroup with the neutral element, with involution, and with a lot of idempotents. We also give a condition when a morphism is a C*-algebra.
On the Notions of Hole and Vacuum in Mathematics and in the Humanities
Abstract
The notions of “hole” and “vacuum” in various branches of science are considered. A philosophical generalization of these notions on the basis of examples from physics, history, and linguistics is presented. Some aspects of the further development of these notions in mathematical logic, thermodynamics of nuclear matter, and other branches of science are sketched.
Sobolev Problems with Spherical Mean Conditions and Traces of Quantized Canonical Transformations
Abstract
We consider Sobolev problems (problems for an elliptic operator on a closed manifold with conditions on a closed submanifold) for the case in which these conditions are of nonlocal nature and include weighted spherical means of the unknown function over spheres of a given radius. For such problems, we establish a criterion for the Fredholm property and, in some special cases, obtain index formulas.