Rate of Convergence of Intermediate Order Statistics

Exact rates are derived for the uniform convergence of the density of intermediate order statistics towards the normal or lognormal density under certain smoothness conditions. Our methods also give the exact rate of convergence in the uniform metric and in the total variation metric.     

A distribution function whose moments are close to those of a normal law

The distance in the uniform metric between two distribution functions one of which is normal is considered in terms of the distances between their corresponding moments (in some order). This result is applied to bounds on the distance in the uniform metric of two distribution functions, the moments of whose convolutions are close to the corresponding moments (in some order) of a normal law, from the set of all normal distribution functions. Three references are given.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 627–633, December, 1967.     

The central limit theorem without the condition of independence

Necessary and sufficient conditions are presented for sums of asymptotically independent random variables to converge to a normal random variable in the sense of total variation distance, uniform metric for characteristic functions. and mean metric of order q. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models. Moscow. Russia. 1996. Part II.     

Stability estimates for decompositions of the composition of Gaussian and Poisson distributions

Stability estimates are obtained for decompositions of the composition of Gaussian and Poisson distribution functions which are sharp in the sense of order in the uniform metric; also obtained are two-sided stability estimates of decompositions of these distribution functions in the Levi metric, which are in some sense close to the sharp ones.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 164–186, 1979.     

The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces

We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.     

Jacobi fields for a nonholonomic distribution

The variational problem with nonholonomic constraints was considered in detail by Bliss. A distribution is a special case of constraints. Horizontal geodesics on a manifold with flat metric and constant tensor of nonholonomity are considered. It is proved that, in the classical adjoint problem, conjugate points appear, which does not involve any loss of optimality. The second variation of the length (or energy) functional of admissible (horizontal) geodesics for a distribution on a smooth manifold is expressed in terms of the distribution curvature tensor.     

A comparison of the eigenvalues of the Dirac and Laplace operators on a two-dimensional torus

We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples. Received: 10 December 1998     

模糊有界变差函数全变差的积分表示与距离导数

定义和讨论了模糊数值函数的距离导数,给出了模糊有界变差函数全变差的积分表示.发现模糊绝对连续函数是几乎处处距离可导的,距离导数的积分等于其原函数的总变差,从而给出了模糊有界变差函数全变差的积分表示.     

On the variation of a metric and its application

Some of the variation formulas of a metric were derived in the literatures by using the local coordinates system, In this paper, We give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method. We establish a relation between the variation of the volume of a metric and that of a submanifold. We find that the latter is a consequence of the former. Finally we give an application of these formulas to the variations of heat invariants. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then （M, g） is either scalar flat or a space form.     

Multi-Valued Mappings of Bounded Generalized Variation

We study the mappings taking real intervals into metric spaces and possessing a bounded generalized variation in the sense of Jordan--Riesz--Orlicz. We establish some embeddings of function spaces, the structure of the mappings, the jumps of the variation, and the Helly selection principle. We show that a compact-valued multi-valued mapping of bounded generalized variation with respect to the Hausdorff metric has a regular selection of bounded generalized variation. We prove the existence of selections preserving the properties of multi-valued mappings that are defined on the direct product of an interval and a topological space, have a bounded generalized variation in the first variable, and are upper semicontinuous in the second variable.     

On moment measures of departure from the normal and exponential laws

For scale mixtures of distributions it is possible to prescribe simple moment measures of distance. In the case of departure from the normal and exponential laws of scale mixtures of the normal and exponential, these distances may be taken as the kurtosis and half the squared coefficient of variation minus one respectively. In this paper these measures of distance are exhibited as bounds on the uniform metric for the distance between distribution functions. The results considerably sharpen earlier results of a similar character in [2].     

Variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation of the Kähler metric. We show as application that the Soliton-Kähler-Ricci flow generated by the Soliton-Ricci flow represents a complex strictly parabolic system of the complex components of the variation of the Kähler metric.     

A language measure for supervisory control

This paper formulates a signed real measure for sublanguages of regular languages based on the principles of automata theory and real analysis. The measure provides total ordering on the controlled behavior of a finite-state automaton plant under different supervisors. Total variation of the measure serves as a metric for the infinite-dimensional vector space of the sublanguages of a regular language over the finite field GF(2). The computational complexity of the language measure is of polynomial order in the number of plant states.     

The Metric Integral of Set-Valued Functions

This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in \({\mathbb {R}}^d\). The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.     

Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.     

信息熵公理及其在决策分析中的应用

针对信息量是消息发生前的不确定性给出一个直观测量信息量公式.为了克服Shannon熵的局限性和分析信息度量本质,借鉴距离空间理论中度量公理定义的思路,通过非负性、对称性、次可加和极大性给出信息熵的公理化新定义.将Shannon熵、直观信息熵和β-熵等不同形式的信息度量统一在同一公理化结构下.应用直观信息熵公式仅采用四则运算进行决策树分析,避免了利用Shannon熵公式的对数运算.     

On the curvature of contact metric manifolds

Some results on Ricci-symmetric contact metric manifolds are obtained. Second order parallel tensors and vector fields keeping curvature tensor invariant are characterized on a class of contact manifolds. Conformally flat contact manifolds are studied assuming certain curvature conditions. Finally some results onk-nullity distribution of contact manifolds are obtained.     

H?lder metric regularity of set-valued maps

This paper is devoted to metric regularity of set-valued maps from a complete metric space to a Banach space. In particular we extend a known characterization of the regularity modulus to maps defined on reflexive spaces. The higher order metric regularity, i.e. an extension of metric regularity to H?lder context, is also investigated using high order variations of set-valued maps and results of similar nature are obtained for conical metric regularity.     

高维空间L1范数约束的一类数据稀疏距离测度学习算法与应用

现有一类分类算法通常采用经典欧氏测度描述样本间相似关系,然而欧氏测度不能较好地反映一些数据集样本的内在分布结构,从而影响这些方法对数据的描述能力.提出一种用于改善一类分类器描述性能的高维空间一类数据距离测度学习算法,与已有距离测度学习算法相比,该算法只需提供目标类数据,通过引入样本先验分布正则化项和L1范数惩罚的距离测度稀疏性约束,能有效解决高维空间小样本情况下的一类数据距离测度学习问题,并通过采用分块协调下降算法高效的解决距离测度学习的优化问题.学习的距离测度能容易的嵌入到一类分类器中,仿真实验结果表明采用学习的距离测度能有效改善一类分类器的描述性能,特别能够改善SVDD的描述能力,从而使得一类分类器具有更强的推广能力.     

Regular Carathéodory-type selectors under no convexity assumptions

We prove the existence of Carathéodory-type selectors (that is, measurable in the first variable and having certain regularity properties like Lipschitz continuity, absolute continuity or bounded variation in the second variable) for multifunctions mapping the product of a measurable space and an interval into compact subsets of a metric space or metric semigroup.