Universal correlations and power-law tails in financial covariance matrices

We investigate whether quantities such as the global spectral density or individual eigenvalues of financial covariance matrices can be best modelled by standard random matrix theory or rather by its generalisations displaying power-law tails. In order to generate individual eigenvalue distributions a chopping procedure is devised, which produces a statistical ensemble of asset-price covariances from a single instance of financial data sets. Local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets. They are in good agreement with the universal Tracy-Widom distribution and Wigner surmise, respectively. This suggests a strong degree of robustness especially in the low-lying sector of the spectra, most relevant for portfolio selections. Conversely, the global spectral density of a single covariance matrix as well as the average over all unfolded nearest-neighbour spacing distributions deviate from standard Gaussian random matrix predictions. The data are in fair agreement with a recently introduced generalised random matrix model, with correlations showing a power-law decay.     

The first eigenvalue of the Dirac operator on locally reducible Riemannian manifolds

We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest possible eigenvalue is attained.     

Chain formation in a monolayer of dipolar hard spheres under an external field

The phase behavior of a monolayer of dipolar hard spheres under an external field, which makes all dipoles of the monolayer orientate along its direction, is investigated. Using integral equation theory in the reference hypernetted chain (RHNC) approximation we calculate the correlation functions, which are used to obtain the response matrix of grand potential with respect to density fluctuations. The smallest eigenvalue of this response matrix determines the stability of the monolayer. When the smallest ei...     

On the distributions of Laplacian eigenvalues versus node degrees in complex networks

In this paper, the important issue of Laplacian eigenvalue distributions is investigated through theory-guided extensive numerical simulations, for four typical complex network models, namely, the ER random-graph networks, WS and NW small-world networks, and BA scale-free networks. It is found that these four types of complex networks share some common features, particularly similarities between the Laplacian eigenvalue distributions and the node degree distributions.     

Eigenvalue distributions in matrix models for Chern-Simons-matter theories

The eigenvalue distribution is investigated for matrix models related via the localization to Chern-Simons-matter theories. An integral representation of the planar resolvent is used to derive the positions of the branch points of the planar resolvent in the large ?t Hooft coupling limit. Various known exact results on eigenvalue distributions and the expectation value of Wilson loops are reproduced.     

Eigenvalue spectra of random matrices for neural networks

The dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity. In large networks, elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions, making results from random matrix theory highly relevant. Unfortunately, classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory. Therefore, we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances.     

The question of a “best” lower bound in variational calculations

Information on the spectral representation of a trial wave function is used to improve the lower bound to the smallest eigenvalue of a quantum mechanical system, given by the formula of Temple.     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.     

Field Analysis of Pillbox Resonator with Radiation Loss Utilizing Finite-Element Beam Propagation Method Formulation

To deal with the radiation from an axially symmetric pillbox resonator embedded in LiNbO₃ substrate, an eigenvalue equation is derived from the expression of the finite-element beam propagation method in a cylindrical coordinate system. By solving the eigenvalue equation iteratively, the field distributions and the angular phase constants of the pillbox resonator are evaluated. The iterative scheme of solving the eigenvalue equation is an extension of our previous direct solution method which was described for a lossless case. The results obtained by this method are compared with those obtained analytically, and excellent agreement is found.     

Mode theory of plane Fabry-Pérot resonator with inhomogeneous active medium

A new method of the eigenvalue problem solution for a resonator with inhomogeneous active medium is given. The approach is based on Maxwell's wave equation and impedance boundary conditions of resonance-type at open resonator ends. A resonator equipped with mirrors in the form of infinite long strips is studied as an example. A rigorous solutions for the cases of stepped and bounded parabolic active medium profiles are obtained. Transcendental eigenvalue equations are investigated, distributions of field amplitude of active resonator modes are found. Asymptotic behavior of rigorous solutions is investigated. A multilayer approximation method is proposed for the eigenvalue problem solution for a resonator with an arbitrary gradient profile of active medium. The testing of this method was carried out with the rigorous solutions for the bounded parabolic profile.     

三维量子自旋玻璃理论(Ⅱ)——稳定性分析

利用我们提出的三维量子自旋玻璃理论,通过引入一个量子的Goldbart去耦方案,将三维量子自旋玻璃的稳定性Hessian矩阵的本征值问题转化为类伊辛自旋玻璃的相应问题,并求出了所有的本征值。通过分析最小本征值,获得了Almeida-Thouless(AT)下临界线和Gabay-Toulouse(GT)上临界线所满足的方程。最后,数值研究了不同自旋的AT不稳定性并讨论了我们的理论与热场动力学方法、集团展开方法的联系。     

(Al)GaAs/AlGaAs周期光波导中的等幅基模

本文应用周期介质光波导的模场分布函数和特征方程分析了(Al)GaAs/AlGaAs周期光波导中结构参量及介质折射率对模场分布形状的影响,研究了波导中形成等幅基模的条件,讨论了相应参量的优化选择。     

Classical one-dimensional Heisenberg model with an interaction of finite range

It is shown that the thermodynamic quantities and spin correlation functions of the classical Heisenberg model on a linear chain are expressed in terms of the eigenvalue with the smallest absolute value and the corresponding eigenfunction of a homogeneous linear integral equation, where the range of the interaction is assumed to be finite. The magnetization and susceptibility at nonzero external magnetic fields are given as a function of temperature, for the case of the nearest neighbour ferromagnetic and antiferromagnetic interaction. Efforts are paid to determine the properties near zero temperature.     

用科尔-霍普夫变换法求非均匀光波导导模本征方程的严格解

采用科尔-霍普夫(Cole-Hopf)变换法，将渐变折射率波导导模的本征值方程变换为里卡提(Riccati)方程，通过较简洁的数学演算导出导模的传播常量与模场分布的解析解，给出了平方律分布、对称爱泼斯坦(Epstein)分布、爱泼斯坦层的平板波导与平方律分布圆光纤4种折射率分布的计算公式。     

The effect of measurement on the quantum features of a chaotic system

We investigate the quantum dynamics of a periodically kicked nonlinear spin system which exhibits regular and chaotic dynamics in the classical regime. The quantum behaviour is characterised by the evolving eigenvalue distributions for the angular momentum components and the features, including recurrences in the quantum means and the presence of quantum tunneling, are discussed. We employ the evolution operator eigenvalue distribution to prove that coherent quantum tunneling occurs between the fixed points in the regular regions of phase space. Continual quantum measurement is included in the model: the classical dynamics are unchanged but a destruction of coherences occurs in the quantum system. Recurrences in the means are destroyed and quantum tunneling is suppressed by measurement, a manifestation of the quantum Zeno effect.     

外腔锁相二极管激光列阵的超模

 利用耦合理论推导了通过外腔反馈实现锁相运行的激光二极管列阵所满足的本征值方程。在忽略非相邻单元间耦合的串联运行方式下，解析求解了包括端面反射的列阵方程的本征值。求出了各阶超模的近场分布，并得到了考虑单个发光单元影响时的远场分布，具体为：当列阵运行于基超模时，近场各个单元间的光场同相，在远场中心位置出现光强主极大；而在其它高阶模式，近场各个单元间光场不同相，远场光强主极大出现在中心位置两侧；远场总体受到单个发光单元近场分布函数的调制。     

Eigenvalue spectrum analysis for temporal signals of Kerr optical frequency combs based on nonlinear Fourier transform

Based on the nonlinear Schr?dinger equation(NLSE) with damping, detuning, and driving terms describing the evolution of signals in a Kerr microresonator, we apply periodic nonlinear Fourier transform(NFT) to the study of signals during the generation of the Kerr optical frequency combs(OFCs). We find that the signals in different states, including the Turing pattern, the chaos, the single soliton state, and the multi-solitons state, can be distinguished according to different distributions of the eigenvalue spectrum. Specially, the eigenvalue spectrum of the single soliton pulse is composed of a pair of conjugate symmetric discrete eigenvalues and the quasi-continuous eigenvalue spectrum with eye-like structure.Moreover, we have successfully demonstrated that the number of discrete eigenvalue pairs in the eigenvalue spectrum corresponds to the number of solitons formed in a round-trip time inside the Kerr microresonator. This work shows that some characteristics of the time-domain signal can be well reflected in the nonlinear domain.     

初始模型对含噪动态光散射数据正则化反演结果的影响

分别采用最小模型矩阵、最平坦模型矩阵、最光滑模型矩阵作为初始化模型,对加入5种不同水平随机噪声的90nm窄单峰、90nm宽单峰和250nm窄单峰、250nm宽单峰颗粒体系的模拟分布进行了正则化反演,并对反演结果进行比较。结果表明:当噪声水平为0时,正则化初始模型的选择对反演结果没有明显影响。随着噪声水平的增加,采用三种初始化模型反演得到的峰值误差和粒度分布误差都随之变大,但采用最平坦模型和最光滑模型反演得到的峰值和粒度分布误差明显小于采用最小初始模型的反演误差。当噪声水平大于0.01时,选择最平坦初始模型获得的粒度分布结果优于采用最光滑初始模型和最小初始模型获得的结果,而采用最光滑初始模型反演得到的峰值优于最平坦初始模型和最小初始模型的反演峰值。因此,采用正则化算法处理含噪动态光散射数据时,为得到最优的粒度分布信息,宜采用最平坦初始模型,若需要获取最准确的峰值信息,则应选择最光滑初始模型。     

On the joint eigenvalue distribution for the matrix ensembles with non zero mean

Exact distributions are given for the two-dimensional case when the mean of the off-diagonal element is non-zero. The joint eigenvalue distribution for theN dimensional case, derived using the volume element in the space ofN ×N orthogonal matrices, is checked by rederiving the exact results forN=2. The smooth nature of theN-dimensional joint distribution supports the claim of the method of moments that the single eigenvalue distribution is a smooth function of the ratio of mean-to-mean square deviation.