2,229 research outputs found
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 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
and in the polynomial degree 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 -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
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