Abstract: The goal of this series of lectures is to present an overview of the theory of orbit equivalence, with a particular focus on the probability measure preserving actions of the free groups. I will start by giving the basis of the theory of orbit equivalence and explain the theory of cost. In particular, prove such statements as the induction formula and the computation of the cost of fr...

Purpose: We consider a discrete-time retrial queue where the retrial time follows a general distribution, the server subject to Bernoulli vacation policy and the customer has preemptive resume priority, Bernoulli feedback strategy. The main purpose of this paper is to derive the generating functions of the stationary distribution of the system state, the orbit size and some important performanc...

Results from a series of long-term numerical integrations of orbits centred on that of C/1995O1 (Hale-Bopp) are presented. Initially, 33 orbits taken from various sources were integrated in realistic models of the solar system for various timescales about the present in the range (?3; +2)Myr and analyzed to assess the probability of diierent dynamical outcomes, such as a sungrazing state or the...

We measured room-temperature phonon Raman spectra of changchengite (IrBiS) and compared the experimental wavenumbers to theoretical ones obtained by means \emph{ab initio} density-functional-theory calculations in presence absence spin-orbit coupling effects. Combining two different excitation photon energies all symmetry predicted modes are experimentally observed. The electronic properties Ir...

In this chapter, we study the effects of the acceleration gravity on the sedimentation deposition probability, as well as the aerosol deposition rate on the surface of the Earth and Mars, but also aboard a spacecraft in orbit around Earth and Mars as well for particles with density ρ p = 1,300 kg/m³, diameters d p = 1, 3, 5 μm, and residence times t = 0.0272, 0.2 , respectively. For particles o...

We numerically investigate the migration of dust particles with initial orbits close to those of the numbered asteroids, observed trans-Neptunian objects, and comet Encke. The fraction of silicate asteroidal particles that collided with the Earth during their lifetime varied from 11% for 100 micron particles to 0.008% for 1 micron particles. Almost all asteroidal particles with diameter d >/= 4...

We investigate the occurrence of extreme and rare events, i.e., giant and rare light pulses, in a periodically modulated CO_{2} laser model. Due to nonlinear resonant processes, we show a scenario of interaction between chaotic bands of different orders, which may lead to the formation of extreme and rare events. We identify a crisis line in the modulation parameter space, and we show that, whe...

A gauge-invariant saddle point expansion for the Yang-Mills vacuum transition amplitude on the basis of the squeezed approximation to the vacuum wave functional is outlined. This framework allows the identification of gauge-invariant infrared degrees of freedom which arise as dominant sets of gauge field orbits and provide the principal input for an essentially analytical treatment of soft ampl...

in the linear system ax = b the points x are sometimes constrained to lie in a given subspace s of column space of a. drazin inverse for any singular or nonsingular matrix, exist and is unique. in this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of drazin inverse in solving such systems is investigated. constrained linear system arise ...

An approach for chance constrained programming of large-scale nonlinear dynamic systems is presented. The stochastic property of the uncertainties is explicitly considered in the problem formulation in which some input and state constraints are to be complied with predefined probability levels. The method considers a nonlinear relation between the uncertain input and the constrained variables. ...