基于高斯平均的DTI脑模板构建方法

在获取被试的张量数据后通常对其进行多通道线性平均以得到张量模板．但线性平均不仅会忽略张量中的向量信息,还会使灰质和白质的交界处过于平滑,降低模板的分辨率．为了解决以上问题,本文引入了四元数及高斯加权平均来构建高斯扩散张量成像（Diffusion Tensor Imaging,DTI）脑模板．本文首先对55个健康被试的DTI数据进行预处理,使得数据伪影最小化;再通过扩散张量成像工具包（Diffusion Tensor Imaging ToolKit,DTI-TK）将预处理后的数据进行初步空间标准化;然后将张量通过特征分解得到特征向量和特征值;最后,将由特征向量转化的四元数标量和特征值分别进行高斯加权平均得到平均后的特征向量和特征值,并对其进行重建得到张量模板．实验结果表明相比于线性DTI模板,高斯DTI模板在DTED、COH、DVED、OVL、corr FA评估指标上表现更优,而IA指标较差,说明本文提出的高斯DTI模板在整体信息保留方面有所优化,但方向信息有所丢失．     

On a pathology in indefinite metric inner product space

A pathology related to an indefinite metric, which has been pointed out by Ito in connection with construction of a two dimensional quantum field model at a finite cutoff, is mathematically analyzed in a simple model. It is found for a model Hamiltonian with a parameter in an indefinite metric inner product space that eigenvalues with a complete set of eigenvectors changes suddenly from positive integers to negative integers as a parameter crosses a critical value (the Hamiltonian being skew selfadjoint with absolutely continuous spectrum on a pure imaginary axis at the critical value of the parameter), if a fixed (positive definite Hilbert space) topology is used in the completion of the underlying indefinite metric inner product space. However it is also found that if the topology is varied with the parameter of the Hamiltonian in the manner similar to analytic continuation, then the Hamiltonian keeps positive integer eigenvalues with a complete set of eigenvectors.     

一种求解大型稀疏对称矩阵(极端)特征值问题的有效算法

提出一种求解大型稀疏对称矩阵几个最大(最小)特征值和相应特征向量的迭代块DL(即Davidson-Lanczos)算法并且讨论了迭代块DL算法的收敛率     

A new eigensolution of structures via dynamic condensation

Dynamic condensation (or model reduction) is a commonly used algorithm to fast estimate some low eigenvalues and corresponding eigenvectors of structures by reducing the order of the original structural model to a smaller one. This paper proposes a new eigensolution technique via iterated dynamic condensation. The technique retains all the inertia terms associated with the removed degrees of freedom in an iterated form, which generates the reduced mass matrix similar to that obtained in the Guyan reduction method with a frequency-dependent perturbed term. The corresponding eigenvalues and eigenvectors of interest are obtained as those of the Guyan reduction method with perturbations by using an eigensensitivity-based iterative method. The effectiveness and accuracy of the proposed technique are numerically verified by using a steel frame and the GARTEUR structure.     

Products, Coproducts, and Singular Value Decomposition

Products and coproducts may be recognized as morphisms in a monoidal tensor category of vector spaces. To gain invariant data of these morphisms, we can use singular value decomposition which attaches singular values, i.e. generalized eigenvalues, to these maps. We show, for the case of Grassmannand Clifford products, that twist maps significantly alter these data reducing degeneracies. Since non group like coproducts give rise to non classical behavior of the algebra of functions, makeing them noncommutative, we hope to be able to learn more about such geometries. A remarkabe thechnicallity is that the coproduct for positive singular values of eigenvectors in A yields directly corresponding eigenvectors in A⊗ A.     

An improved quantum principal component analysis algorithm based on the quantum singular threshold method

Quantum principal component analysis (qPCA) is a dimensionality reduction algorithm for getting the eigenvectors corresponding to top several eigenvalues of the data matrix and then reconstructing. However, qPCA can only construct the quantum state contains all the eigenvectors and eigenvalues. In this paper, we present an improved quantum principal component analysis (Improved qPCA) algorithm with a fixed threshold. We can reduce the singular value less than the threshold to 0 and obtain a target quantum state which can be used to get an output similar to qPCA after phase estimation. Compared with qPCA, our algorithm has only the target eigenvalues and the probability that we get each target eigenvalue is greater. Furthermore, our algorithm can serve as an additional regularization method and a subroutine for subsequent data processing.     

Rewiring dynamical networks with prescribed degree distribution for enhancing synchronizability

In this paper, we present an algorithm for enhancing synchronizability of dynamical networks with prescribed degree distribution. The algorithm takes an unweighted and undirected network as input and outputs a network with the same node-degree distribution and enhanced synchronization properties. The rewirings are based on the properties of the Laplacian of the connection graph, i.e., the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian. A term proportional to the eigenvectors is adopted to choose potential edges for rewiring, provided that the node-degree distribution is preserved. The algorithm can be implemented on networks of any sizes as long as their eigenvalues and eigenvectors can be calculated with standard algorithms. The effectiveness of the proposed algorithm in enhancing the network synchronizability is revealed by numerical simulation on a number of sample networks including scale-free, Watts-Strogatz, and Erdo?s-Re?nyi graphs. Furthermore, a number of network's structural parameters such as node betweenness centrality, edge betweenness centrality, average path length, clustering coefficient, and degree assortativity are tracked as a function of optimization steps.     

Anharmonic Parameters of Vibrationally Excited Molecules

An algebraic method of perturbation theory for eigenvalues and eigenvectors is used to examine the intramolecular parameters of excited vibrational states. Based on this method, formulas for higher-order coefficients of the matrix of vibrational modes, changes in the Cartesian coordinates of vibrating atoms, and anharmonic elements of the tensor of inertia moment are derived.     

Absence of gluonic components in axial and tensor mesons

A quarkonium–gluonium mixing scheme previously developed to describe the characteristic of the pseudoscalar mesons is applied to axial and tensor mesons. The parameters of the model are determined by fitting the eigenvalues of a mass matrix. The corresponding eigenvectors give the proportion of light quarks, strange quarks and glueball in each meson. However, the predictions of the model for the branching ratios and electromagnetic decays are incompatible with the experimental results. These results suggest the absence of gluonic components in the states of axial and tensor isosinglet mesons analyzed here. Received: 12 November 1999 / Revised version: 27 April 2000 / Published online: 9 August 2000     

The forced oscillator method: eigenvalue analysis and computing linear response functions

A review is given of the forced oscillator method (FOM), an algorithm particularly suitable to treat physical systems described by very large matrices. This scheme enables us to compute spectral densities, eigenvalues and their eigenvectors of both Hermitian and non-Hermitian matrices with high speed and accuracy, in particular when combined with the fast time-evolution method based on the Chebyshev polynomial expansion. In addition, linear response functions can be computed with high speed and accuracy in the context of the FOM. The emphasis will be on the presentation of the efficiency of the FOM for a broad range of applications with their computer source codes for the purpose of wide utility.     

A Relativistic Hidden-Variable Interpretation for the Massive Vector Field Based on Energy-Momentum Flows

This paper is motivated by the desire to formulate a relativistically covariant hidden-variable particle trajectory interpretation of the quantum theory of the vector field that is formulated in such a way as to allow the inclusion of gravity. We present a methodology for calculating the flows of rest energy and a conserved density for the massive vector field using the time-like eigenvectors and eigenvalues of the stress-energy-momentum tensor. Such flows may be used to define particle trajectories which follow the flow. This work extends our previous work which used a similar procedure for the scalar field. The massive, spin-one, complex vector field is discussed in detail and the flows of energy-momentum are illustrated in a simple example of standing waves in a plane.     

On the theory of the linear electro-optical effect in cubical crystals

The linear electro-optical effect in cubical crystals of acentric classes is considered by an approximate covariant method. Simple analytic expressions for the induced optic axes and for the eigenvalues and eigenvectors of the inverse dielectric tensor are obtained for all directions of the electric field in these crystals. The method used enables one to determine the above quantities with a high degree of accuracy. The cases where the electric field rotates in planes orthogonal to the crystallographic directions and in the planes of symmetry of the crystal are considered in detail.     

Testing the self-duality of topological lumps in SU(3) lattice gauge theory

We discuss a simple formula which connects the field-strength tensor to a spectral sum over certain quadratic forms of the eigenvectors of the lattice Dirac operator. We analyze these terms for the near zero modes and find that they give rise to contributions which are essentially either self-dual or anti-self-dual. Modes with larger eigenvalues in the bulk of the spectrum are more dominated by quantum fluctuations and are less (anti-)self-dual. In the high temperature phase of QCD we find considerably reduced (anti-)self-duality for the modes near the edge of the spectral gap.     

General Geometric Optics Formalism in Plasmas

This paper exploits a general approach to geometric optics in inhomogeneous plasmas based on the properties of the local dielectric tensor ?. We express ? in terms of its eigenvalues ?j and eigenvectors ?. Then to zeroth order in the geometric optics approximation the determinant D = ?1?2?3 vanishes and the elements ?j vanish separately in pairs or simultaneously. It is shown that this branching in the dispersion relation changes the formulation of the geometric optics equations. The ray tracing and the transport of the amplitude of the wave in both degenerate and nondegenerate cases is described. The general procedure for transition through a boundary between degenerate and nondegenerate regions, where the rays split into two parts each following a different branch of the dispersion relation is also presented in this paper. We demonstrate our general method in a case, where radiation from a vacuum region enters an inhomogeneous magnetized plasma layer.     

Noise considerations in the determination of diffusion tensor anisotropy

In this study the noise sensitivity of various anisotropy indices has been investigated by Monte-Carlo computer simulations and magnetic resonance imaging (MRI) measurements in a phantom and 5 healthy volunteers. Particularly, we compared the noise performance of indices defined solely in terms of eigenvalues and those based on both the eigenvalues and eigenvectors. It is found that anisotropy indices based on both eigenvalues and eigenvectors are less sensitive to noise, and spatial averaging with neighboring pixels can further reduce the standard deviation. To reduce the partial volume effect caused by the spatial averaging with neighboring voxels, an averaging method in the time domain based on the orientation coherence of eigenvectors in repeated experiments has been proposed.     

Bound-State Calculations for Three Atoms Without Explicit Partial Wave Decomposition

A method to calculate the bound states of three atoms without resorting to an explicit partial wave decomposition is presented. The differential form of the Faddeev equations in the total angular momentum representation is used for this purpose. The method utilizes Cartesian coordinates combined with the tensor trick preconditioning for large linear systems and Arnoldi’s algorithm for eigenanalysis. As an example, we consider the He₃ system in which the interatomic force has a very strong repulsive core requiring the inclusion of a large number of partial waves to achieve convergence. To improve convergence and stability in the Arnoldi’s algorithm, we modify the system of equations by introducing a background potential that removes the large number of unphysical eigenvalues stemming from the hard core. The introduction of such a modifying potential does not affect the physical spectrum of the system. The results obtained by solving the modified, three-dimensional, Faddeev equations compare favorably with other results in the field.     

Multiscale finite element methods for high-contrast problems using local spectral basis functions

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ₁)^1/2, where Λ₁ is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings.     

Modified perturbation method for eigenvalues of structure with interval parameters

In overcoming the drawbacks of traditional interval perturbation method due to the unpredictable effect of ignoring higher order terms,a modified parameter perturbation method is presented to predict the eigenvalue intervals of the uncertain structures with interval parameters.In the proposed method,interval variables are used to quantitatively describe all the uncertain parameters.Different order perturbations in both eigenvalues and eigenvectors are fully considered.By retaining higher order terms,the original dynamic eigenvalue equations are transformed into interval linear equations based on the orthogonality and regularization conditions of eigenvectors.The eigenvalue ranges and corresponding eigenvectors can be approximately predicted by the parameter combinatorial approach.Compared with the Monte Carlo method,two numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm to solve both the real eigenvalue problem and complex eigenvalue problem.     

A Rayleigh–Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices

A procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse Hermitian matrices. The procedure, a modification of an inverse subspace iteration procedure, uses adaptively determined Chebyshev polynomials to approximate the required application of the inverse operator on the subspace. The method is robust, converges with acceptable rapidity, and can easily handle operators with eigenvalues of multiplicity greater than one. Numerical results are shown that demonstrate the utility of the procedure.     

一种基于主分量分析的恒星光谱快速分类法

恒星光谱分类是天体光谱自动识别中的重要组成部分。本文主要介绍一种实用的基于主分量分析(PCA)法对恒星光谱进行快速自动的分类方法。该方法在恒星的主分量空间中对样本点进行投影 ,并利用最近邻分类器进行分类 ,获得与恒星MK分类标准的光谱型基本一致的结果。本文的主要工作有 :(1 )利用PCA方法构造恒星光谱的特征矩阵 ,建构恒星的主分量空间 ;(2 )对恒星光谱进行主分量投影 ,对投影点进行光谱型和光度级的分类器设计 ,利用最近邻法分类 ,最后得出恒星的分类树。该分类法速度快 ,分类准确率较高 ,对目前许多大型光谱巡天计划所获得的大量光谱数据的处理有着重要的意义。