Correlated fermions on a lattice in high dimensions

The limit of infinite dimension of the Hubbard model was recently introduced by Metzner and Vollhardt as a new type of model with interesting implications. In the present paper the same limit is applied to a family of lattice fermion models with generalized kinetic and potential energies. A simple method is introduced for investigating the important issue of handling momentum conservation at the interaction vertices. It is shown that the irrelevance of momentum conservation found by Metzner and Vollhardt applies more generally than their Gaussian density of states. Interactions between particles on different sites are shown to simplify to their Hartree substitute. This leaves on-site interactions as the only dynamical interactions in the limit of infinite dimension.     

Gravitational radiation in asymptotic de Sitter space

A solution of the gravitational field equations is found by using an axially symmetric metric which is asymptotically a de Sitter space metric. We use the general approach of Bondi, van der Burg, and Metzner as applied to the asymptotic flat-space case and search for the necessary conditions for gravitational radiation in asymptotic de Sitter space. We find that the character of the gravitational radiation, if it exists at all, is considerably different from that obtained in the case of asymptotic flat space.     

Numerical solution of an extended White–Metzner model for eccentric Taylor–Couette flow

In this study, we have developed a new numerical approach to solve differential-type viscoelastic fluid models for a commonly used benchmark problem, namely, the steady Taylor—Couette flow between eccentric cylinders. The proposed numerical approach is special in that the nonlinear system of discretized algebraic flow equations is solved iteratively using a Newton–Krylov method along with an inverse-based incomplete lower-upper preconditioner. The numerical approach has been validated by solving the benchmark problem for the upper-convected Maxwell model at a large Deborah number. Excellent agreement with the numerical data reported in the literature has been found. In addition, a parameter study was performed for an extended White–Metzner model. A large eccentricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. We detected several interesting phenomena caused by the large eccentricity ratio of the cylinder system and by the viscoelastic nature of the fluid. Encouraged by the results of this study, we intend to investigate other polymeric fluids having a more complex microstructure in an eccentric annular flow field.     

Energy radiation by an arbitrary axisymme tricsystem

The Bondi/Van der Burg/Metzner solution of the Einstein field equations for an arbitrary axisymmetric system is examined. Expressions for the energy and momentum of the system are obtained by tetrad formalism.     

A unified approach to attractor reconstruction

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.     

Effective dimensionality approach to the RKKY interaction in multilayer systems

We consider RKKY interaction in a quasi 2D system with nonparabolic dispersion. In our paper we calculate the RKKY range function assuming the in-layer confinement via effective dimensionality approach. We show, that indirect magnetic exchange in our system can be modelled by the effective spectral dimension which equals one.     

Statistical Properties of Maps¶with Indifferent Periodic Points

In this note we present an axiomatic approach to the decay of correlations for maps of arbitrary dimension with indifferent periodic points. As applications, we apply our results to the well-known Manneville–Pomeau equation and the inhomogeneous diophantine approximation algorithm. Received: 5 November 1999 / Accepted: 11 October 2000     

Scattering Problem for Local Perturbations\newline of the Free Quantum Gas

A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern–Simons Higgs model and have been studied recently by various authors. Received: 30 September 1997 / Accepted: 21 July 1998     

A variational treatment of the periodic Anderson model with antiferromagnetic order

We discuss a variational ground state wave function for the symmetric periodic Anderson model with commensurate spin density order. The energy of this ansatz is evaluated in closed form. Our approach generalizes a variational treatment proposed recently by Strack and Vollhardt and results in significantly lower energy. Contrary to the Gutzwiller ansatz the wave function recovers the Schrieffer-Wolff limit for large Coulomb repulsion. We clarify the relation of our approach to unrestricted Hartree-Fock and present a comparison with existing quantum Monte Carlo calculations for one dimension.     

A new proof of the local regularity of the eta invariant of a Dirac operator

In this paper we use the approach of our earlier proof of the local index theorem to give a new proof of Bismut–Freed’s result on the local regularity of the eta invariant of a Dirac operator in odd dimension.     

Ergodic Properties of the Non-Markovian Langevin Equation

We discuss the dissipative dynamics of a classical particle coupled to an infinitely extended heat reservoir. We announce a number of results concerning the ergodic properties of this model. The novelty of our approach is that it extends beyond Markovian dynamics to the case where the Langevin equation is driven by colored noise. Our method works in arbitrary space dimension, and for fully nonlinear systems.     

The size effects of a rigid reflector in the scattering of transient ultrasonic field

A model is proposed to assess the transient ultrasonic field radiated by a planar transducer and scattered by a target with a known dimension and geometry. The approach is based on the hypothesis of linear acoustics for a perfectly rigid reflector immersed in an isotropic, homogeneous and lossless fluid. The detected acoustic pressure is explained in terms of the plane and edge waves. In order to determine the boundary effects on the detected pressure, targets of different sizes were used in our simulations. An experimental verification of the proposed model is presented for the case of circular planar targets with different radii made of duralumin and immersed in distilled water. In general, the theoretically predicted results are in good agreement with the experimentally measured results.     

Quantifying the architectural complexity of microscopic images of histology specimens

Tumour grade (a measure of the degree of cellular differentiation of malignant neoplasm) is an important prognostic factor in many types of cancer. In general, poorly differentiated tumours are characterized by a higher degree of architectural irregularity and complexity of histological structures. Fractal dimension is a useful parameter for characterizing complex irregular structures. However, one of the difficulties of estimating the fractal dimension from microscopic images is the segmentation of pathologically relevant structures for analysis. A commonly used technique to segment structures of interest is to apply a pixel intensity threshold to convert the original image to binary and extract pixel outline structures from the binary representation. The difficulty with this approach is that the value of the threshold required to segment the histological structures is highly dependent on the staining technique chosen and imaging conditions (i.e., illumination time, intensity, and uniformity) of the microscopic system. In this work, we present a method for finding the optimal intensity threshold by maximizing the corresponding fractal dimension. This method results in the segmentation of histological structures and the estimation of their fractal dimension (independent of imaging conditions). We applied our technique to 164 prostate histology sections from 82 prostate core biopsy specimens (two serial sections from each of the 63 benign prostate tissues and 19 high grade prostate carcinoma). We stained one of the serial sections with conventional hemotoxylin and eosin (H&E) and the other with pan-keratin, and found that the difference in mean fractal dimension between the two groups was statistically significant (p < 0.0001) for both stains. However, using receiver operating characteristics (ROC) analysis, we conclude that our fractal dimension method applied to the images of pan-keratin stained sections provides greater classification performance (benign versus high grade) than with those stained with H&E when compared to the original histological diagnosis. The sensitivity and specificity achieved with the pan-keratin images were 89.5% and 90.5%, respectively.     

Role of the 2D surface state continuum and projected band gap in charge transfer in front of a Cu(111) surface

Electron capture by Li+ and H projectiles in grazing scattering from Cu(111) and Cu(110) surfaces is studied experimentally and theoretically. Whereas data for Cu(110) can be described by established theoretical methods treating resonant charge transfer with a free-electron metal, data for Cu(111) show pronounced deviations from this approach. We interpret our observations by the effect of the projected L-band gap of the Cu(111) surface. In particular, the quantum states of reduced dimension (2D surface state continuum) play a dominant role in electron transfer.     

The equality of fractal dimension and uncertainty dimension for certain dynamical systems

[MGOY] introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system. In [MGOY] and [P] it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.Research in part supported by AFOSR and by the Department of Energy (Scientific Computing Staff Office of Energy Research)     

A new approach to estimating the signal dimension of concatenated resting-state functional MRI data sets

Estimating the effective signal dimension of resting-state functional MRI (fMRI) data sets (i.e., selecting an appropriate number of signal components) is essential for data-driven analysis. However, current methods are prone to overestimate the dimensions, especially for concatenated group data sets. This work aims to develop improved dimension estimation methods for group fMRI data generated by data reduction and grouping procedure at multiple levels. We proposed a “noise-blurring” approach to suppress intragroup signal variations and to correct spectral alterations caused by the data reduction, which should be responsible for the group dimension overestimation. This technique was evaluated on both simulated group data sets and in vivo resting-state fMRI data sets acquired from 14 normal human subjects during five different scan sessions. Reduction and grouping procedures were repeated at three levels in either “scan–session–subject” or “scan–subject–session” order. Compared with traditional estimation methods, our approach exhibits a stronger immunity against intragroup signal variation, less sensitivity to group size and a better agreement on the dimensions at the third level between the two grouping orders.     

Resolving statistical uncertainty in correlation dimension estimation

In this paper we propose a novel method for obtaining standard errors and confidence intervals for the correlation dimension estimated on an observed chaotic time series. This method is based on the U-Statistics theory and an ingenious combination of the moving block and parametric bootstrap procedures. We test the method on the basis of computer simulations for both clean and noisy series. We show that the distribution of the correlation dimension estimate obtained by our method agrees very well with the "true" distribution obtained by the Monte Carlo simulation. One of the main advantage of our method is the ability to estimate the distribution (and hence, the standard error) of the correlation dimension estimate using only one observed time series.     

Universal Scaling Behavior of Directed Percolation Around the Upper Critical Dimension

In this work we consider the steady state scaling behavior of directed percolation around the upper critical dimension. In particular we determine numerically the order parameter, its fluctuations as well as the susceptibility as a function of the control parameter and the conjugated field. Additionally to the universal scaling functions, several universal amplitude combinations are considered. We compare our results with those of a renormalization group approach.     

A perfectly matched layer approach to the nonlinear Schrödinger wave equations

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrödinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrödinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrödinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrödinger equations and some Schrödinger-coupled systems in each spatial dimension.     

混沌时序相空间重构参数确定的信息论方法

根据信息论基本原理，研究了混沌时间序列相空间重构参数延迟时间和嵌入维数的选取.提出了用符号分析的方法计算互信息函数，确定出延迟时间，在此基础上，提出了一种估计嵌入维数的信息论方法，即根据重构向量条件熵随向量维数的变化关系来确定嵌入维数，通过对几种典型混沌动力学系统的数值验证，结果表明该方法能够确定出合适的相空间重构嵌入维数. 关键词： 混沌 相空间重构 互信息 条件熵 符号分析