Franck–Condon Factors and r‐Centroids for the B–, C–, F–, and G–X Band Systems of YF Molecule

The Franck–Condon factors and r‐centroids, which are very closely related to relative transition probabilities, have been evaluated by a more reliable numerical integration procedure for the B¹π–X¹Σ⁺, C¹Σ⁺–X¹Σ⁺, F¹Σ⁺–X¹Σ⁺, and G¹π–X¹Σ⁺ band systems of the YF molecule, using suitable potentials.     

56mm焦距重心定位用光学系统

文章中介绍的光学系统是一个56mm焦距,相对孔径为1/1.3,视场角为11.3°,后截距>10mm,光谱范围为046～0.75μm的六块无胶合透镜组成的镜头.该光学系统只有两种低吸收、物理、化学性能良好的普通光学玻璃材料(Bak6和ZF2)组成,因此可以有高的透过车;全视场范围内各色光重心偏之差小于±1.2μm,象面上80%能量集中在24μm～54μm之间.与一般光学系统相比,它具有各色光重心偏差值小、和轴上、轴外能量分布比较一致的特点.     

Metric Sample Spaces of Continuous Geometric Curves and Estimation of Their Centroids

The metric sample space of Fréechet curves (Fréechet , 1934,1951,1961) is based on a generalization of regular curves that covers continuous curves in full generality. This makes it possible to deal with both smooth and non-smooth, even non-rectifiable geometric curves in statistical analysis. In the present paper this sample space is further extended in two directions that are relevant in practice: to incorporate information on landmark points in the curves and to impose invariance with respect to an arbitrary group of isometric spatial transformations. Properties of the introduced sample spaces of curves are studied, specially those concerning to the generation and representation of random curves by random functions.In order to provide measures of central tendency and dispersion of random curves, centroids and restricted centroids ofrandom curves are defined in a general metric framework, and methods for their consistent estimation are derived.     

Shrunken centroids regularized discriminant analysis as a promising strategy for metabolomics data exploration

Metabolomics datasets generated by modern analytical instruments tend to be increasingly complex. In this study, a recent method named shrunken centroids regularized discriminant analysis (SCRDA) has been introduced and applied in the exploration of metabolomics dataset. It is a supervised method for variable selection, discriminant analysis and biomarker screening. By regularizing the estimate of the within‐class covariance matrix, SCRDA can deal with the singularity issue of linear discriminant analysis. Then a shrinkage estimator is applied to perform variable selection. The method presented is illustrated through the simulated datasets and three complex metabolomics datasets. Commonly used orthogonal partial least squares discriminant analysis and two other similar statistical methods, penalized linear discriminant analysis and nearest shrunken centroids, are used for comparisons. The results illustrate that SCRDA has some desirable abilities in variable selection, classification and prediction. Moreover, the biomarkers identified by SCRDA are further demonstrated to be in accordance with the biochemical research. It has been proved that SCRDA can be applied as a promising strategy in metabolomics. Copyright © 2014 John Wiley & Sons, Ltd.     

Magic of centroids

In this paper, we show some properties of centroids of geometric figures, such as triangles, quadrilaterals and tetrahedra. In particular, we will prove the properties by means of geometric transformations and by introducing extensions of triangles and quadrilaterals, i.e. by adding one, two or three new vertices to the figure. The study of these properties can be used, with profit, in a classroom activity supported by a dynamic geometry system.     

Advances in Studies and Applications of Centroidal Voronoi Tessellations

<正>Centroidal Voronoi tessellations(CVTs) have become a useful tool in many applications ranging from geometric modeling,image and data analysis,and numerical partial differential equations,to problems in physics,astrophysics,chemistry,and biology. In this paper,we briefly review the CVT concept and a few of its generalizations and well-known properties.We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs.Whenever possible,we point out some outstanding issues that still need investigating.     

Lower Dimensional Representation of Text Data Based on Centroids and Least Squares

Dimension reduction in today's vector space based information retrieval system is essential for improving computational efficiency in handling massive amounts of data. A mathematical framework for lower dimensional representation of text data in vector space based information retrieval is proposed using minimization and a matrix rank reduction formula. We illustrate how the commonly used Latent Semantic Indexing based on the Singular Value Decomposition (LSI/SVD) can be derived as a method for dimension reduction from our mathematical framework. Then two new methods for dimension reduction based on the centroids of data clusters are proposed and shown to be more efficient and effective than LSI/SVD when we have a priori information on the cluster structure of the data. Several advantages of the new methods in terms of computational efficiency and data representation in the reduced space, as well as their mathematical properties are discussed.Experimental results are presented to illustrate the effectiveness of our methods on certain classification problems in a reduced dimensional space. The results indicate that for a successful lower dimensional representation of the data, it is important to incorporate a priori knowledge in the dimension reduction algorithms.     

Single particle equivalent transfer monopole sum rules involving general multishell target states

Monopole sum rules for single particle equivalent transfer reactions have been derived for target states where both neutrons and protons occupy the last active shell. The Pauli Principle lends some complexity to the sum rule, which is general in the sense that the inequivalent transfer situation and where the active shell has only neutron occupancy, become its special cases. Calculated values of various energy centroids have reasonable agreement with their measured counterparts.     

Effective two-body forces in light and medium mass nuclei

From the spectroscopic information provided by single particle stripping reactions, we have extracted average effective two-body interaction between nucleons in thep-, sd-, andfp- shells. Using the derived interaction parameters, we have calculated the energy centroids of the level spectra of residual nuclei obtained via single nucleon stripping reactions involving light and medium mass nuclei as targets.     

Philon's line generalized: An optimization problem from geometry

Consider inn-dimensional Euclidean space the intersection of a convex cone and a hyperplane through a given point. The problem is to minimize the (n-1)-volume of this intersection. A geometric interpretation of the first-order optimality condition is given. The special casen=2 is known as a characteristic property of Philon's line.Editorial Note. Because of Professor Wetterling's death on January 21, 1994, this paper was handled by Dr. F. Twilt, Department of Applied Mathematics University of Twente, Enschede, Netherlands.