On the Asymptotic Behavior of an Ornstein-Uhlenbeck Process with Random Forcing

We study the asymptotic behavior of an inertial tracer particle in a random force field. We show that there exists a probability measure, under which the process describing the velocity and environment seen from the vantage point of the moving particle is stationary and ergodic. This measure is equivalent to the underlying probability for the Eulerian flow. As a consequence of the above we obtain the law of large numbers for the trajectory of the tracer. Moreover, we prove also some decorrelation properties of the velocity of the particle, which lead to the existence of a non-degenerate asymptotic covariance tensor. The research of both authors was supported by the Polish Committee for Scientific Research (KBN) grant No. 2PO3A03123.     

柔性关节机器人全局渐近稳定PID算法控制

"小误差放大,大误差饱和"函数能够提高柔性关节机器人位置的控制品质。利用这一函数,提出了比例-非线性微分加实时重力补偿(P-ND+)位置控制算法。应用Lyapunov稳定性理论和La Salles不变性原理证明了闭环系统的全局渐近稳定性。通过仿真,采用P-ND+算法能使机械臂和关节的稳定时间从1s减小到0.2s左右,结果表明该算法比传统算法响应速度更快,验证了提出的非线性控制(P-ND+)相对于传统的线性(PD+)控制具有更好的控制品质。     

Asymptotic Behavior of Beta-Integers

Beta-integers (“β-integers”) are those numbers which are the counterparts of integers when real numbers are expressed in an irrational base β > 1. In quasicrystalline studies, β-integers supersede the “crystallographic” ordinary integers. When the number β is a Parry number, the corresponding β-integers realize only a finite number of distances between consecutive elements and are in this sense the most comparable to ordinary integers. In this paper, we point out the similarity of β-integers and ordinary integers in the asymptotic sense, in particular for a subclass of Parry numbers – Pisot numbers for which their Parry and minimal polynomial coincide.     

Asymptotic Behavior of a Stationary Silo with Absorbing Walls

We study the nearest neighbors one dimensional uniform q-model of force fluctuations in bead packs,⁽¹⁾ a stochastic model to simulate the stress of granular media in two dimensional silos. The vertical coordinate plays the role of time, and the horizontal coordinate the role of space. The process is a discrete time Markov process with state space ^{1,...,N}. At each layer (time), the weight supported by each grain is a random variable of mean one (its own weight) plus the sum of random fractions of the weights supported by the nearest neighboring grains at the previous layer. The fraction of the weight given to the right neighbor of the successive layer is a uniform random variable in [0, 1] independent of everything. The remaining weight is given to the left neighbor. In the boundaries, a uniform fraction of the weight leans on the wall of the silo. This corresponds to absorbing boundary conditions. For this model we show that there exists a unique invariant measure. The mean weight at site i under the invariant measure is i(N+1–i); we prove that its variance is (i(N+1–i))²+O(N ³) and the covariances between grains ij are of order O(N ³). Moreover, as N, the law under the invariant measure of the weights divided by N ² around site (integer part of) rN, r(0, 1), converges to a product of gamma distributions with parameters 2 and 2(r(1–r))^–1 (sum of two exponentials of mean r(1–r)/2). Liu et al. ⁽²⁾ proved that for a silo with infinitely many weightless grains, any product of gamma distributions with parameters 2 and 2/ with [0, ) are invariant. Our result shows that as the silo grows, the model selects exactly one of these Gamma's at each macroscopic place.     

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp [pA−Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour , where , and ρ<1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for J≠0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.     

Asymptotic Behavior for the Liouville Equations

A new proof of the diffusion approximation for ordinary differential equations is given. It is based on an asymptotic expansion of the solution of the corresponding Liouville partial differential equations. In contrast to previous results obtained for the suspension under Holderian mappings of subshift of finite type or Fourier analysis techniques, our proof relies only on symbolic dynamics.     

The Asymptotic Behavior of the Takhtajan-Zograf Metric

We obtain the asymptotic behavior of the Takhtajan-Zograf metric on the Teichmüller space of punctured Riemann surfaces. The first author is partially supported by JSPS Grant-in-Aid for Exploratory Research 2005-2007. The second author is partially supported by the research grant R-146-000-106-112 from the National University of Singapore and the Ministry of Education.     

Asymptotic Behavior of Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

In this paper, we study the one-dimensional Navier-Stokes equations connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coefficient μ(ρ) is proportional to ρ ^θ with θ > 0, where ρ is the density, we give the asymptotic behavior and the decay rate of the density function ρ(x, t). Furthermore, the behavior of the density function ρ(x, t) near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied. The analysis is based on some new mathematical techniques and some new useful estimates. This fills a final gap on studying Navier-Stokes equations with the viscosity coefficient μ(ρ) dependent on the density ρ.     

Asymptotic Behavior of Solutions in Finite Difference Schemes

Many problems of numerically solving the Schrödinger equation require that we choose asymptotic distances many times greater than the characteristic size of the region of interaction. If the solution to one-dimensional equations can be immediately chosen in a form that preserves unitarity, the invariance of probability (in the form of, e.g., fulfilling an optical theorem) is a real problem for two-dimensional equations. An addition that does not exceed the discretization error and ensures a high degree of unitarity is proposed as a result of studying the properties of a discrete two-dimensional equation.     

On the Asymptotic Behavior of the Hyperbolic Brownian Motion

The main results in this paper concern large and moderate deviations for the radial component of a $n$ -dimensional hyperbolic Brownian motion (for $n\ge 2$ ) on the Poincaré half-space. We also investigate the asymptotic behavior of the hitting probability $P_\eta (T_{\eta _1}^{(n)}<\infty )$ of a ball of radius $\eta _1$ , as the distance $\eta $ of the starting point of the hyperbolic Brownian motion goes to infinity.     

Internal Friction in Glass Fiber- and Carbon Fiber-Reinforced Composites with a T-107 Matrix

Technical Physics - The temperature and amplitude dependences of internal friction in hybrid composites consisting of unidirectional carbon fibers and glass tissues in a T-107 molten epoxy matrix...     

Asymptotic Behavior of Soliton Solutions with a Double Spectral Parameter for Principal Chiral Field

The soliton solutions with a double spectral parameter for the principal chiral field are derived by Darboux transformation. The asymptotic behavior of the solutions as time tends to infinity is obtained and the speeds of the peaks in the asymptotic solutions are not constants.     

Asymptotic Behavior for Rayleigh Problem Based on Kinetic Theory

We investigate the dynamics of the gas bounded by an infinite flat plate which is initially in equilibrium and set at some instant impulsively into uniform motion in its own plane. We use the Boltzmann equation to describe intermolecular collisions and assume the diffuse reflection to describe the interaction of the gas with the boundary. The Mach number of the plate is assumed to be small so that we can linearize the Boltzmann equation as well as the boundary condition. We show that the asymptotic behavior of the gas represents a perturbation to the free molecular gas when the time is much less than the mean free time. On the other hand, if the time is much greater than the mean free time, we show that the gas dynamics is governed by the linearized Navier–Stokes equation with a slip flow on the boundary and establish a boundary layer correction with thickness of the order of the mean free path. We also establish the singularity of velocity distribution function along the particle trajectory near the boundary.     

Asymptotic Behavior of Soliton Solutions with a Double Spectral Parameter for Principal Chiral Field

The soliton solutions with a double spectral parameter for the principal chiral field are derived by Darboux transformation. The asymptotic behavior of the solutions as time tends to infinity is obtained and the speeds of the peaks in the asymptotic solutions are not constants.     

On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent.     

Elastic scattering of a photon by an atom with an open shell

In the nonrelativistic approximation, the absolute values and the forms of the differential cross sections of the anomalous elastic scattering of a linearly polarized x-ray photon by manganese and copper atoms in the energy region of the ionization threshold of the 1s shell are calculated for the wave functions of one-electron states. We take into account the multiparticle effects of relaxation of atomic shells in the field of core vacancies, multiplet splitting, the configuration interaction, and Auger and radiative decays of vacancies, as well as the processes of double excitation/ionization of the ground state of the atom. The calculation results are in good agreement with the synchrotron experiment.