Inertial manifolds and forms for semilinear parabolic equations subjected to additive white noise

We present a construction of inertial manifolds and forms for semilinear parabolic equations subjected to additive white noise. The existence of inertial manifolds allow us to establish the existence of unique invariant distribution. We also derive the connection between this distribution and inertial forms of the system under consideration.     

POSSIBLE SPACE-TIME MANIFOLDS AND KINEMATICS

In this work, the analysis of the space-time manifolds, their kinematic groups and Lie algebras are made as intuitive as possible. First of all, an analysis ofthe inertial frames shows that the space-time manifold in which there exists a global inertial frame should be a pseudo-sphere so that the kinematic group must be a rotation group. Thus the explicit analytical expressions of such kinematical transformations and the commutation relations among the corresponding generators can be formulated easily. Consequently, the contractions of such manifolds,kinematic groups and Lie algebras can be deduced concretely and intuitively.     

Geometry of inertial manifolds probed via a Lyapunov projection method

A method for determining the dimension and state space geometry of inertial manifolds of dissipative extended dynamical systems is presented. It works by projecting vector differences between reference states and recurrent states onto local linear subspaces spanned by the Lyapunov vectors. A sharp characteristic transition of the projection error occurs as soon as the number of basis vectors is increased beyond the inertial manifold dimension. Since the method can be applied using standard orthogonal Lyapunov vectors, it provides a possible way to also determine experimentally inertial manifolds and their geometric characteristics.     

Active vibration control using an inertial actuator with internal damping

Collocated direct velocity feedback with ideal point force actuators mounted on structures is unconditionally stable and generates active damping. When inertial actuators are used to generate the control force, the system can become unstable even for moderate velocity feedback gains due to an additional -180 degree phase lag introduced by the fundamental axial resonant mode of the inertial actuator. In this study a relative velocity sensor is used to implement an inner velocity feedback loop that generates internal damping in a lightweight, electrodynamic, inertial actuator. Simulation results for a model problem with the actuator mounted on a clamped plate show that, when internal relative velocity feedback is used in addition to a conventional external velocity feedback loop, there is an optimum combination of internal and external velocity feedback gains, which, for a given gain margin, maximizes vibration reduction. These predictions are validated in experiments with a specially built lightweight inertial actuator.     

Influences of subgrid scale dynamics on resolvable scale statistics in large-eddy simulations

Recently, the -expansion and recursive renormalization group (RNG) theories as well as approximate inertial manifolds (AIM) have been exploited as means of systematically modeling subgrid scales in large-eddy simulations (LES). Although these theoretical approaches are rather complicated mathematically, their key approximations can be investigated using direct numerical simulations (DNS). In fact, the differences among these theories can be traced to whether they retain or neglect interactions between the subgrid-subgrid and subgrid-resolvable scales. In this paper, we focus on the influence of these two interactions on the evolution of the resolvable scales in LES: the effect^A which keeps only the interactions between the small and large scales; and, the effect^B which, on the other hand, keeps only the interactions among the subgrid-subgrid scales. The performance of these models is analyzed using the velocity fields of the direct numerical simulations. Specifically, our comparison is based on the analysis of the energy and enstrophy spectra, as well as higher-order statistics of the velocity and velocity derivatives. We found that the energy spectrum and higher-order statistics for the simulations with the effect^A (referred to, hereafter, as model^A) are in very good agreement with the filtered DNS. The comparison between the computations with effect^B (referred to, hereafter, as model^B) and the filtered DNS, however, is not satisfactory. Moreover, the decorrelation between the filtered DNS and model^A is much slower than that of the filtered DNS and model^B. Therefore, we conclude that the model^A, taking into acciunt the interactions between the subgrid and resolvable scales, is a faithful subgrid model for LES for the range of Reynolds numbers considered.     

A general algorithm for improving ILDMs

The method of intrinsic low-dimensional manifolds (ILDMs) has proven to be an efficient tool for the simplification of chemical kinetics. Nevertheless, there are still some open questions with respect to an efficient calculation and implementation of ILDMs. In this paper, we focus on the efficient calculation of ILDMs and present a refinement method for ILDMs. The method is based on an evolution equation of the manifolds towards a steady state solution which then represents the slow manifold. It has the property that it is continuous, differentiable, and in addition, it is an inertial manifold which represents the slow dynamics of the chemical system. In this way, many of the problems associated with the concept of ILDM are overcome.     

惯性质量确实与速度有关

指出了一些文章中关于惯性质量和速度无关的错误根源,阐明了惯性质量和速度有关的理由.同时还指出了经典力学中的"状态"和相对论中的"状态"之差异.     

Influence of Ion Nonlinear Polarization Drift and Warm Ions on Solitary Kinetic Alfven Wave

Considering the effects of ion nonlinear polarization drift and warm ions, we adopt two-fluid model to investigate the character of low-frequency Solitary Kinetic Alfvén Wave (SKAW hereafter) in a magnetic plasma. The results derived in this paper indicate that dip SKAW and hump SKAW both exist in a wide range in magnetosphere (for the pressure parameter β～10^-5～0.01, where β is the ratio of thermal pressure to magnetic pressure, i.e. β=2μ₀nT/B₀²). These two kinds of SKAWs propagate at either Super-Alfvénic velocity or Sub-Alfvénic velocity. In the inertial region β＜＜m_e/m_i, the Sub-Alfvénic velocity dip SKAWs and hump SKAWs both exist; in the transmittal region β～ 2m_e/m_i, dip SKAWs and hump SKAWs propagate at Super-Alfvénic velocity or Sub-Alfvénic velocity; Super-Alfvénic velocity hump SKAWs and Super-Alfvénic and Sub-Alfvénic velocity dip SKAWs are in the kinetic region 1＞＞β＞＞ m_e/m_i. These results are different from previous ones. That indicates that the effects of ion nonlinear polarization drift and warm ions are important and they cannot be neglected. The SKAW has an electric field parallel to the ambient magnetic field, which makes the SKAW take an important role in the acceleration and energization of field-aligned charged particles in magnetic plasmas. And the SKAW is also important for the heating of a local plasma. So it makes a novel physical mechanism of energy transmission possible.     

Inertial Manifolds for a Smoluchowski Equation on the Unit Sphere

The existence of inertial manifolds for a Smoluchowski equation—a nonlinear Fokker-Planck equation on the unit sphere which arises in modeling of colloidal suspensions—is investigated. A nonlinear and nonlocal transformation is used to eliminate the gradient from the nonlinear term.     

Tetraplectic structures, tri-momentum maps, and quaternionic flag manifolds

The purpose of this note is to define tri-momentum maps for certain manifolds with an Sp(1)ⁿ-action. We exhibit many interesting examples of such spaces using quaternions. We show how these maps can be used to reduce such manifolds to ones with fewer symmetries. The images of such maps for quaternionic flag manifolds, which are defined using the Dieudonné determinant, resemble the polytopes from the complex case.     

Subprincipal Symbol for Toeplitz Operators

We establish some subprincipal estimates for Berezin–Toeplitz operators on symplectic compact manifolds. From this, we construct a family of subprincipal symbol maps and we prove that these maps are the only ones satisfying some expected conditions.     

Hölder continuity of two types of bidirectionally coupled generalised synchronisation manifold

This paper studies the existence of Höolder continuity of bidirectionally coupled generalised synchronisation (GS). Based on the slaving principle of synergetics and the modified system approach, it classifies the GS into several types. The existences of two main types of H"older continuous bidirectionally coupled GS inertial manifolds are theoretically analysed and proved by using the Schauder fixed point theorem. Numerical simulations illustrate the theoretical results.     

可能的空-时流形和运动群

对空-时流形、运动群及其李代数作了尽量直观的几何分析与推导。首先,对惯性系的分析指出,利用黎曼几何中的Beltrami定理后可知,存在惯性系的空-时必是伪超球,因而运动群就是旋转群,于是不难推算出运动学变换的具体分析表达式及其生成元间的对易关系式。由此,具体而直观的推出了流形、群、代数的缩并关系。 关键词：     

Velocity and timing of multiple spherically converging shock waves in liquid deuterium

The fuel entropy and required drive energy for an inertial confinement fusion implosion are set by a sequence of shocks that must be precisely timed to achieve ignition. This Letter reports measurements of multiple spherical shock waves in liquid deuterium that facilitate timing inertial confinement fusion shocks to the required precision. These experiments produced the highest shock velocity observed in liquid deuterium (U(s) = 135 km/s at ～2500 GPa) and also the first observation of convergence effects on the shock velocity. Simulations model the shock-timing results well when a nonlocal transport model is used in the coronal plasma.     

Testable scenario for relativity with minimum length

I propose a general class of spacetimes whose structure is governed by observer-independent scales of both velocity (c) and length (Planck length), and I observe that these spacetimes can naturally host a modification of FitzGerald–Lorentz contraction such that lengths which in their inertial rest frame are bigger than a “minimum length” are also bigger than the minimum length in all other inertial frames. With an analysis in leading order in the minimum length, I show that this is the case in a specific illustrative example of postulates for relativity with velocity and length observer-independent scales.     

Calculating effective diffusivities in the limit of vanishing molecular diffusion

In this paper we study the problem of the numerical calculation (by Monte Carlo methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity field, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume-preserving property of the equations of motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non-trivial time-correlation structure and when inertial effects cannot be neglected. The method of modified equations is used to explain failings of Euler-based methods. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.     

An approximate inertial manifold for computing Burgers'' equation

We present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals. Our algorithm is motivated from the dynamical systems point of view. In particular, we adapt the methodology of approximate inertial manifolds to a finite difference scheme. This leads to a differential treatment in which the higher (i.e. unresolved) modes are expressed in terms of the lower modes. As a particular example we derive an approximate inertial manifold for Burgers' equation and develop a numerical algorithm suitable for computing. We perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme. For all values of the parameters (which are the coefficient of viscosity and the cell size), we obtain a decrease in the numerical error by at least a factor of 2.0 with the modified scheme. The decrease in error is substantially greater over large regions of the parameters space.     

Membranes with rotating motors

We study collections of rotatory motors confined to two-dimensional manifolds. These systems show a nontrivial collective behavior since the rotational motion leads to a repulsive hydrodynamic interaction between motors. While for high rotation speed motors might exhibit crystalline order, they form at low speed a disordered phase where diffusion is enhanced by velocity fluctuations. These effects should be experimentally observable for motors driven by external fields and for dipolar biological motors embedded into lipid membranes in a viscoelastic solvent.     

Velocity dependent inertial induction: An extension of Mach’s principle

In this article a model of inertial induction has been presented. According to this model the magnitude of the acceleration dependent inertia force comes out exactly as the product of the acceleration and inertial mass. The model also indicates that even uniform velocity gives rise to inertia force. However, the magnitude of the velocity dependent inertia force is exceedingly small but it causes a cosmological red shift whose order of magnitude is same as that of the observed values.