Quantization of nonintegrable maps

Using Heisenberg's matrix formulation of quantum mechanics, a method is given for quantizing volume-preserving polynomial mappings. The energy levels of the linear map are obtained exactly and those of the cubic, nonintegrable map are obtained approximately and numerically.     

Entropy production in open volume-preserving systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.     

Nilpotent normal form for divergence-free vector fields and volume-preserving maps

We study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue. We prove that when a divergence-free vector field in R³ has nilpotent linearization with maximal Jordan block then, to arbitrary degree, coordinates can be chosen so that the nonlinear terms occur as a single function of two variables in the third component. The analogue for volume-preserving diffeomorphisms gives an optimal normal form in which the truncation of the normal form at any degree gives an exactly volume-preserving map whose inverse is also polynomial with the same degree.     

Local Lyapunov exponents for spatiotemporal chaos

Local Lyapunov exponents are proposed for characterization of perturbations in distributed dynamical systems with chaotic behavior. Their relation to usual and velocity-dependent exponents is discussed. Local Lyapunov exponents are analytically calculated for coupled map lattices using random field approximation. Boundary Lyapunov exponents describing reflection of perturbations at boundaries are also introduced and calculated.     

Merger of coherent structures in time-periodic viscous flows

Inertia-induced changes in transport properties of an incompressible viscous time-periodic flow are studied in terms of the topological properties of volume-preserving maps. In the noninertial limit, the flow admits one constant of motion and thus relates to a so-called one-action map. However, the invariant surfaces corresponding to the constant of motion are topologically equivalent to spheres rather than the common case of tori. This has fundamental ramifications for the effect of inertia and leads to a new kind of response scenario: resonance-induced merger of coherent structures.     

A novel hybrid ES-FE-SEA for mid-frequency prediction of Transmission losses in complex acoustic systems

The widely-used numerical modeling approaches such as the finite element method (FEM) and statistical energy analysis (SEA) often have limited applicability to the transmission loss prediction in mid-frequency range. In this paper, a novel hybrid edge-based smoothed FEM coupled with statistical energy analysis (ES-FE-SEA) method is proposed to further improve the accuracy of “mid-frequency” transmission loss predictions. The application of ES-FEM will “soften” the well-known ‘‘overly-stiff’’ behavior in the standard FEM solution and reduce the inherent numerical dispersion error. While the SEA approach deals with the physical uncertainty in the relatively higher frequency range. The plate of interest is appropriately described by an ES-FEM model, due to its relative robustness to perturbations. Its adjacent reverberation cavities are modeled by employing the SEA approach, because of their high model density. The coupling and interaction between SEA subsystems and the FE subsystem is governed by the “reciprocity relationship” theorem. A standard numerical example for benchmarking is examined and excellent agreement was achieved between the prediction and reference results. The proposed ES-FE-SEA is also verified by various numerical examples. The method is finally applied to the modeling a complicated engineering problem–acoustic fields on both sides of the front windshield in a passenger car.     

Decay of Correlations and Invariance Principles for Dispersing Billiards with Cusps, and Related Planar Billiard Flows

Following recent work of Chernov, Markarian, and Zhang, it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1/n. Since the collisions inside a cusp occur in quick succession, it is reasonable to expect a much faster decay rate in continuous time. In this paper we prove that the flow is rapid mixing: correlations decay faster than any polynomial rate. A consequence is that the flow admits strong statistical properties such as the almost sure invariance principle, even though the billiard map does not. The techniques in this paper yield new results for other standard examples in planar billiards, including Bunimovich flowers and stadia.     

Average exit time for volume-preserving maps

For a volume-preserving map, we show that the exit time averaged over the entry set of a region is given by the ratio of the measure of the accessible subset of the region to that of the entry set. This result is primarily of interest to show two things: First, it gives a simple bound on the algebraic decay exponent of the survival probability. Second, it gives a tool for computing the measure of the accessible set. We use this to compute the measure of the bounded orbits for the Henon quadratic map. (c) 1997 American Institute of Physics.     

On the Volumorphism Group, the First Conjugate Point is Always the Hardest

We find a simple local criterion for the existence of conjugate points on the group of volume-preserving diffeomorphisms of a 3-manifold with the Riemannian metric of ideal fluid mechanics, in terms of an ordinary differential equation along each Lagrangian path. Using this criterion, we prove that the first conjugate point along a geodesic in this group is always pathological: the differential of the exponential map always fails to be Fredholm.Much of this work was completed while the author was a Lecturer at the University of Pennsylvania. The author is grateful for their hospitality.     

Statistical isotropy of the cosmic microwave background

The breakdown of statistical homogeneity and isotropy of cosmic perturbations is a generic feature of ultra-large scale structure of the cosmos, in particular, of non-trivial cosmic topology. The statistical isotropy (SI) of the cosmic microwave background temperature fluctuations (CMB anisotropy) is sensitive to this breakdown on the largest scales comparable to, and even beyond the cosmic horizon. We propose a set of measures,K _l (l = 1, 2,3,...) which for non-zero values indicate and quantify statistical isotropy violations in a CMB map. We numerically compute the predictedK _l spectra for CMB anisotropy in flat torus universe models. Characteristic signatures of different models in theK _l spectrum are noted.     

A decomposed equation for local entropy and entropy production in volume-preserving coarse-grained systems

In this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume-preserving maps. The equation consists of the following terms: unsteady, convection, diffusion, probability-weighted phase space volume expansion rate, nonnegative entropy production, and residuals. The decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse-grained systems on the phase space. When it is applied to two volume-preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by Gilbert et al. [T. Gilbert, J.R. Dorfman, P. Gaspard, Entropy production, fractals, and relaxation to equilibrium, Phys. Rev. Lett. 85 (2000) 1606-1609] and the phenomenological expression both in nonequilibrium steady and unsteady states. The coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site. It follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states.     

Design and evaluation of an abbreviated pixelwise dynamic contrast enhancement analysis protocol for early extracellular volume fraction estimation

IntroductionT1-based method is considered as the gold standard for extracellular volume fraction (ECV) mapping. This technique requires at least a 10 min delay after injection to acquire the post injection T1 map. Quantitative analysis of Dynamic Contrast Enhancement (DCE) images could lead to an earlier estimation of an ECV like parameter (2 min). The purpose of this study was to design a quantitative pixel-wise DCE analysis workflow to assess the feasibility of an early estimation of ECV.MethodsFourteen patients with mitral valve prolapse were included in this study. The MR protocol, performed on a 3 T MR scanner, included MOLLI sequences for T1 maps acquisition and a standard SR-turboFlash sequence for dynamic acquisition. DCE data were acquired for at least 120 s. We implemented a full DCE analysis pipeline with a pre-processing step using an innovative motion correction algorithm (RC-REG algorithm) and a post-processing step using the extended Tofts Model (ECV_ETM). Estimated ECV_ETM maps were compared to standard T1-based ECV maps (ECV_T1) with both a Pearson correlation analysis and a group-wise analysis.ResultsImage and map quality assessment showed systematic improvements using the proposed workflow. Strong correlation was found between ECV_ETM, and ECV_T1 values (r-square = 0.87).ConclusionA DCE analysis workflow based on RC-REG algorithm and ETM analysis can provide good quality parametric maps. Therefore, it is possible to extract ECV values from a 2 min-long DCE acquisition that are strongly correlated with ECV values from the T1 based method.     

Sensitivity of wave field evolution and manifold stability in chaotic systems

The sensitivity of a wave field's evolution to small perturbations is of fundamental interest. For chaotic systems, there are two distinct regimes of either exponential or Gaussian overlap decay in time. We develop a semiclassical approach for understanding both regimes and give a simple expression for the crossover time between the regimes. The wave field's evolution is considerably more stable than the exponential instability of chaotic trajectories seems to suggest. The resolution of this paradox lies in the collective behavior of the appropriate set of trajectories. Results are given for the standard map.     

基于视觉注意的彩色图像检索方法

基于视觉注意模型提取的特征能够反映图像高层语义的新特征,将视觉注意机制引入到图像分析领域能有效地减小语义鸿沟,获得高效的图像检索性能.根据视觉感知的特点,对Itti视觉注意模型进行了改进.采用主分量图表示亮度图,将纹理粗糙度信息融入视觉注意模型,进而提出了一种基于视觉注意空间分布特征的图像检索算法.首先由改进视觉注意模型将图像分解得到38个视觉特征图,然后采用网格平分法提取视觉特征图的空间分布信息,组成特征矢量来多层次地对图像特征进行描述,用于图像检索.实验结果表明,该算法利用基于改进注意力模型方法来提取图像空间分布特征进行图像检索,能获得较高的检索率.     

Extreme statistics of complex random and quantum chaotic states

Complex random states have the statistical properties of the Gaussian and circular unitary ensemble eigenstates of random matrix theory. Even though their components are correlated by the normalization constraint, it is nevertheless possible to derive compact analytic formulas for their extreme values' statistical properties for all dimensionalities. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix theory is conjectured to be applicable to chaotic quantum systems, we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic. The statistical behaviors are consistent with the finite-N formulas.     

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.     

On General Volume-Preserving Mechanical Systems via Cohomology

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold(Μ,ω) via the highest Euler-Lagrange cohomology. It is shown that for every volume-preserving fiow there are some 2-forms that play a similar role to the Hamiltonian in the Hamilton mechanics and the ordinary canonical equations with Hamiltonian H are included as a special case with a 2-form Hω/(n - 1).     

On General Volume-Preserving Mechanical Systems via Cohomology

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M,ω) via the highest Euler-Lagrange cohomology.It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in the Hamilton mechanics and the ordinary canonical equations with Hamiltonian H are included as a special case with a 2-form Hω/(n-1).     

Elastic response of a free half-space to random force excitations applied on the boundary

A response of an elastic half-space to random forces applied normally to the free boundary is studied. This paper is the second part of the study we presented in [I.A. Shalimova, K.K. Sabelfeld, The response of an elastic three-dimensional half-space to random correlated displacement perturbations on the boundary, Physica A 389 (21) (2010) 4436–4449] where the case of random displacements on the boundary was considered. We analyze the white noise excitations in detail, and derive explicitly the mean of the elastic energy, the strain and displacement correlation tensors. Simulation algorithms are constructed both for displacement and strain random fields, which enables us to calculate any desired statistical characteristics.