Kohn-Sham Density Matrix and the Kernel Energy MethodSCI北大核心CSCD

The kernel energy method(KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries.KEM breaks a molecule into smaller subsets,called kernels,for the purposes of calculation.The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy.A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals.In this study,the kernel expansion for the density matrix is examined in the context of density functional theory(DFT) Kohn-Sham(KS) calculations.A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined,and is then converted into a normalizedprojector by using the Clinton algorithm.Such normalized projectors are factorizable into linear combination of atomic orbitals(LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis.Both straightforward KEM energies and energies from a normalized,idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion.Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules.In the case of the proof-of-concept system,calculations were performed using the STO-3 G and6-31 G(d,p) bases over a range of atomic separations,some very far from equilibrium.The water cluster was calculated in the 6-31 G(d,p) basis at an equilibrium geometry.The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases.In the case of the water cluster,the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result.The KS density matrices of this study are applicable to quantum crystallography.     

New permutation statistics: Variation and a variant

For each permutation π we introduce the variation statistic of π, as the total number of elements on the right between each two adjacent elements of π. We modify this new statistic to get a slightly different variant, which behaves more closely like Mahonian statistics such as maj. In this paper we find an explicit formula for the generating function for the number of permutations of length n according to the variation statistic, and for that according to the modified version.     

Chordal Komatu–Loewner equation for a family of continuously growing hulls

In this paper, we discuss the chordal Komatu–Loewner equation on standard slit domains in a manner applicable not just to a simple curve but also a family of continuously growing hulls. Especially a conformally invariant characterization of the Komatu–Loewner evolution is obtained. As an application, we prove a sort of conformal invariance, or locality, of the stochastic Komatu–Loewner evolution ${\mathrm{SKLE}}_{\sqrt{6},?b_{\mathrm{BMD}}}$ in a fully general setting, which solves an open problem posed by Chen et al. (2017).     

On the kernel space for an L0‐linear function

A random normed module is a random generalization of an ordinary normed space, and it is the randomization that makes a random normed module possess rich stratification structures. On the basis of these stratification structures, this paper shows that either the kernel space N(f) for an L⁰‐linear function f from a random normed module S to the algebra is a closed submodule or N(f) on some specifical stratification is a dense proper submodule of S, which generalizes the classical case. In the meantime, a characterization for the kernel space N(f) to be closed is also given.     

Peano's kernel theorem for vector-valued functions and some applications

We extend the well-known Peano Kernel Theorem to a class of linear operators L : Cⁿ⁺¹([a,b];X}→ X, X being a Branch space, which vanish on abstract polynomials of degree ≤ n. We then recover, in the abstract setting, classical estimates of remainders in polynomials interpolation and quadrature formulas. Finally, we present an application to the error analysis of the trapezoidal time discretization scheme for parabolic evolution equations.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

A fixed-center spherical separation algorithm with kernel transformations for classification problems

We consider a special case of the optimal separation, via a sphere, of two discrete point sets in a finite dimensional Euclidean space. In fact we assume that the center of the sphere is fixed. In this case the problem reduces to the minimization of a convex and nonsmooth function of just one variable, which can be solved by means of an “ad hoc” method in O(p log p) time, where p is the dataset size. The approach is suitable for use in connection with kernel transformations of the type adopted in the support vector machine (SVM) approach. Despite of its simplicity the method has provided interesting results on several standard test problems drawn from the binary classification literature. This research has been partially supported by the Italian “Ministero dell’Istruzione, dell’Università e della Ricerca Scientifica”, under PRIN project Numerical Methods for Global Optimization and for some classes of Nonsmooth Optimization Problems (2005017083.002).     

On the Kernel Rule for Function Classification

Let X be a random variable taking values in a function space , and let Y be a discrete random label with values 0 and 1. We investigate asymptotic properties of the moving window classification rule based on independent copies of the pair (X,Y). Contrary to the finite dimensional case, it is shown that the moving window classifier is not universally consistent in the sense that its probability of error may not converge to the Bayes risk for some distributions of (X,Y). Sufficient conditions both on the space and the distribution of X are then given to ensure consistency.     

A unified framework for harmonic analysis of functions on directed graphs and changing data

We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner.     

高光谱图像与稀疏核典型相关分析冷鲜羊肉新鲜度无损检测

羊肉新鲜度受多种因素影响，通常由多个指标来综合评价，常规试验操作复杂不适合在线检测。高光谱成像数据能够反映羊肉新鲜度变化过程中多种成分的变化信息，但是光谱特征提取与评价模型的建立对最终结果影响较大。为了研究高光谱成像与多指标的快速检测羊肉新鲜度的可行性，提出一种稀疏核典型相关分析方法，借助实验室测定的多个标准值，研究多指标的羊肉新鲜度无损检测。采集了70个代表各级新鲜程度的羊肉样本400~1 000 nm高光谱图像，采用实验室方法测定了挥发性盐基氮(TVB-N)和菌落总数(TAC)标准值，选择感兴趣区域(ROIs)提取代表性光谱图像，利用所提出的特征提取方法提取光谱特征信息，并按照3:1划分校正集和预测集，利用三层神经网络进行分类识别试验。结果表明，新鲜度等级分类总体精度(OA)为0.939 3，Kappa系数为0.906 0，均方根误差(RMSEC)为0.297。研究表明，所提出的多指标光谱特征提取方法可用于快速无损检测羊肉新鲜程度，为采用高光谱成像综合多个新鲜度检测指标，改善由于单一检测指标造成评价模型的适用性和鲁棒性提供了基础。