Billiards with polynomial mixing rates

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic -- enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.Comment: 39pages, 11 figue

Basic principles of hp Virtual Elements on quasiuniform meshes

In the present paper we initiate the study of $hp$ Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size $h$ and in the polynomial degree $p$ in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included

Deterministic Weak Localization in Periodic Structures

The weak localization is found for perfect periodic structures exhibiting deterministic classical diffusion. In particular, the velocity autocorrelation function develops a universal quantum power law decay at 4 times Ehrenfest time, following the classical stretched-exponential type decay. Such deterministic weak localization is robust against weak enough randomness (e.g., quantum impurities). In the 1D and 2D cases, we argue that at the quantum limit states localized in the Bravis cell are turned into Bloch states by quantum tunnelling.Comment: 5 pages, 2 figure

Beam propagation in a Randomly Inhomogeneous Medium

An integro-differential equation describing the angular distribution of beams is analyzed for a medium with random inhomogeneities. Beams are trapped because inhomogeneities give rise to wave localization at random locations and random times. The expressions obtained for the mean square deviation from the initial direction of beam propagation generalize the "3/2 law".Comment: 4 page

Random billiards with wall temperature and associated Markov chains

By a random billiard we mean a billiard system in which the standard specular reflection rule is replaced with a Markov transition probabilities operator P that, at each collision of the billiard particle with the boundary of the billiard domain, gives the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure (RBM) is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain. RBMs provide simple and explicit mechanical models of particle-surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. We focus on the operator P itself and how it relates to the mechanical/geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we characterize the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics, such as the Maxwell-Boltzmann distribution and the Knudsen cosine law, arise naturally as generalized invariant billiard measures; (2) we obtain some basic functional theoretic properties of P. Under very general conditions, we show that P is a self-adjoint operator of norm 1 on an appropriate Hilbert space. In a simple but illustrative example, we show that P is a compact (Hilbert-Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the features of the microstructure;(3) we explore the latter issue both analytically and numerically in a few representative examples;(4) we present a general algorithm for simulating these Markov chains based on a geometric description of the invariant volumes of classical statistical mechanics

Optimization of the extent of surgical treatment in patients with stage I in cervical cancer

The study included 26 patients with FIGO stage Ia1–Ib1 cervical cancer who underwent fertility-sparing surgery (transabdominaltrachelectomy). To visualize sentinel lymph nodes, lymphoscintigraphy with injection of 99mTc-labelled nanocolloid was performed the day before surgery. Intraoperative identification of sentinel lymph nodes using hand-held gamma probe was carried out to determine the radioactive counts over the draining lymph node basin. The sentinel lymph node detection in cervical cancer patients contributes to the accurate clinical assessment of the pelvic lymph node status, precise staging of the disease and tailoring of surgical treatment to individual patient

Optimization of the extent of surgical treatment in patients with stage I in cervical cancer

The study included 26 patients with FIGO stage Ia1–Ib1 cervical cancer who underwent fertility-sparing surgery (transabdominaltrachelectomy). To visualize sentinel lymph nodes, lymphoscintigraphy with injection of 99mTc-labelled nanocolloid was performed the day before surgery. Intraoperative identification of sentinel lymph nodes using hand-held gamma probe was carried out to determine the radioactive counts over the draining lymph node basin. The sentinel lymph node detection in cervical cancer patients contributes to the accurate clinical assessment of the pelvic lymph node status, precise staging of the disease and tailoring of surgical treatment to individual patient

Characteristics of adverse side effects of corticosteroid therapy in children with nephrotic syndrome and methods of pharmacological correction

The article discusses the issues of the long-term glucocorticosteroid therapy in children with nephrotic syndrome that results in severe adverse side effect

Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets