区间值直观模糊集的相似测度和距离测度

对现有的模糊集和直观模糊集的相似测度和距离测度的公理化定义进行分析,并做出改进;然后提出区间值直观模糊集的相似测度和距离测度的公理化定义,并各引入它们的一种计算方法;最后给出区间值直观模糊集的相似测度和距离测度在模式识别中的一个应用实例.     

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.     

数据流的非平稳性度量——纪念李国平院士吴新谋教授诞辰100周年

该文融合遍历论、粗粒化方法和信息论的观点研究数据流的非平稳性度量问题. 引入了数据流的非平稳性度量的概念, 给出了数据流非平稳性度量的有效的近似算法. 数据流的非平稳性度量为该文融合遍历论、粗粒化方法和信息论的观点研究数据流的非平稳性度量问题. 引入了数据流的非平稳性度量的概念, 给出了数据流非平稳性度量的有效的近似算法. 数据流的非平稳性度量为该文融合遍历论、粗粒化方法和信息论的观点研究数据流的非平稳性度量问题. 引入了数据流的非平稳性度量的概念, 给出了数据流非平稳性度量的有效的近似算法. 数据流的非平稳性度量为$0$和$1$之间的实数,平稳性较好的数据流的非平稳性度量较小. 作者将数据流的非平稳性度量应用到模型选择问题中,提出残差序列非平稳性度量最小化的模型选择标准. 作者用数值试验检验了该文提出的数据流非平稳性度量的近似算法, 并检验了其作为模型选择标准的能力.数值试验的结果表明, 非平稳性度量是衡量数据流非平稳程度的一个合理指标, 可以很好地区分趋势平稳数据和差分平稳数据, 区分独立同分布序列、白噪声序列和鞅差序列.     

The Generalized Moment Problem Associated with Correlation Measures

The classical power moment problem can be viewed as a theory of spectral representations of a positive functional on some classical commutative algebra with involution. We generalize this approach to the case in which the algebra is a special commutative algebra of functions on the space of multiple finite configurations. If the above-mentioned functional is generated by a measure on the space of finite ordinary configurations, then this measure is the correlation measure for a measure on the space of infinite configurations. The positiveness of the functional gives conditions for the measure to be a correlation measure.     

How good is lebesgue measure?

So, what is the answer to the question “How good is Lebesgue measure?” In the class of invariant measures, Lebesgue measure seems to be the best candidate to be a canonical measure. In the class of countably additive not necessarily invariant measures, to find a universal measure we have to use a strong additional set-theoretical assumption and this seems to be too high a price. Thus the best improvement of Lebesgue measure seems to be the Banach construction of a finitely additive isometrically invariant extension of Lebesgue measure on the plane and line. However, such a measure does not exist on Rⁿ for n ≤ 3, and to keep the theory of measures uniform for all dimensions we cannot accept the Banach measure on the plane as the best solution to the measure problem. From this discussion it seems clear that there is no reason to depose Lebesgue measure from the place it has in modern mathematics. Lebesgue measure also has a nice topological property called regularity: for every EL and every ɛ > 0, there exists an open set V⊃E and closed set F ⊂ E such that m(V/F) < ɛ. It is not difficult to prove that Lebesgue measure is the richest countably additive measure having this property (see [Ru], Thm. 2.20, p. 50).     

A construction of catalytic super-Brownian motion via collision local time

We give a direct construction of a random measure which is equal in law to the collision local time between a catalytic super-Brownian motion and its catalytic measure. Under a regularity assumption on the catalytic measure, we show that the catalytic super-Brownian motion can be constructed deterministically from this measure.     

三叉树模型下的等价鞅测度

本文研究了三叉树模型下的等价鞅测度刻划问题，得到了三叉树模型的最小熵鞅测度，逆相对熵鞅测度，方差最优鞅测度和极小鞅测度的精确表达式。     

Textbook representations of radian angle measure: The need to build on the quantitative view of angle

Empirical studies of angle, angle measure, and radian angle measure have primarily focused on documenting students' and teachers' conceptions. Representations of angle measure have the potential to inform and build on these conceptions. To gain insight into the concept of radian angle measure from a curricular perspective, I investigated the representations of radian angle measure in eight widely used U.S. high school mathematics textbooks. The data for this study are the 20 figures and the 8 definitions of radian angle measure in the textbooks. Inductive content analysis was used to identify the frequency of observable characteristics in the representations. Textbook figures of radian angle measure most frequently focus on a one-radian angle. Textbook figures also demonstrated radian angle measures in terms of π, in standard position, even when the Cartesian-coordinate plane was not in the figure. The frequently used characteristics of radian representations could inadvertently reinforce a qualitative concept of angle, instead of fostering the quantitative attributes of radian angle measure. Moving forward, analyzing conceptions that are evoked by the frequently used representations of radian angle measure could build a link between evidence from this study and studies of learning.     

The Inequality Between Two Distributions: Applications to the Analysis of Diversity Data

A new measure called diversity difference is proposed for the inequality of a pair of distributions. The diversity difference measure satisfies eight properties of a measure of inequality. This measure is simple to calculate and provides easily interpreted results.Existing inequality measures examine the distribution of a single variable whose data are arranged in a monotonic order. The new measure can employ multiple variables and does not require each to be monotonic but can be used if the data happen to be monotonic. The pair of distributions is useful for organizational diversity data because one of the distributions represents the actual proportions of employees in any class or set of classes and the other distribution is the benchmark or anchoring distribution. Data from the measure can be displayed in diversity difference trees for quick interpretation.The diversity difference measure can be arranged to define a Lorenz curve. An example with three classes (gender, race, and age) is employed to provide examples of the measure, the resulting Lorenz curve, and the disparity ratio.     

New dynamical invariants on hyperbolic manifolds

The rotation measure is an asymptotic dynamical invariant assigned to a typical point of a flow in a fiber bundle over a hyperbolic manifold. The total mass of the rotation measure is the average speed of the orbit and its “direction” is the ergodic invariant probability measure of the hyperbolic geodesic flow which best captures the asymptotic dynamics of the given point. The rotation measure exists almost everywhere and is constant for an ergodic measure of the given flow and so it may be viewed as assigning an ergodic measure of the geodesic flow to one of the given flow. It generalizes the usual notion of homology rotation vector by encoding homotopy information.     

模糊信息度量及推理中蕴涵运算的信息度约束

根据胡固定教授经典集信息量定义给出一种新的模糊糊信息量度量方法,研究了其性质,并与Yager的特征测度进行了比较,表明该潮度是合理的.提出了模糊推理中蕴涵运算的信息度约束,给出了基于合成算法的模糊推理中,模糊蕴涵运算满足该信息度约束的充分条件.     

Self-Conformal Multifractal Measures

A self-conformal measure is a measure invariant under a set of conformal mappings. In this paper we describe the local structure of self-conformal measures. For such a measure we divide its support into sets of fixed local dimension and give a formula for the Hausdorff and packing dimensions of these sets. Moreover, we compute the generalized dimensions of the self-conformal measure.     

A note on the mean correcting martingale measure for geometric Lévy processes

A martingale measure is constructed by using a mean correcting transform for the geometric Lévy processes model. It is shown that this measure is the mean correcting martingale measure if and only if, in the Lévy process, there exists a continuous Gaussian part. Although this measure cannot be equivalent to a physical probability for a pure jump Lévy process, we show that a European call option price under this measure is still arbitrage free.     

Statistical properties of topological Collet-Eckmann maps

We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet-Eckmann”. This condition is weaker than the “Collet-Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem.We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet-Eckmann condition: a rational map satisfies the Topological Collet-Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.     

Operator-decomposability of Gaussian measures on separable Banach spaces

An operator-decomposable Gaussian measure on a separable Banach space can be factorized into a convolution product of a strongly operator-decomposable Gaussian measure and an operator-invariant Gaussian measure (with respect to the same operator). An example for this very factorization is discussed in some detail. In particular it is shown that a strongly operator-decomposable Gaussian measure need not necessarily be supported by the contraction subspace of the operator involved. Finally, the decomposability semigroup of a Gaussian measure turns out to be convex; and the corresponding invariance semigroup belongs to its extreme boundary.     

Vague集上的正规距离与正规相似度

距离与相似度是Vague集上的一对对偶概念,给出了Vague集上加强的距离的公理化定义,引入了正规距离与正规相似度的概念,提出了Vague集上距离的一个一般公式,通过对Vague值的探讨给出了该一般公式成为正规距离的充分条件,同时得到Vague集上的一个新的正规距离公式.最后通过数据分析得出由此正规距离公式生成的正规相似度是合理有效的.     

Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis

A new method is developed for estimating the spectral measure of a multivariate stable probability measure, by representing the measure as a sum of spherical harmonics.     

模糊度定义的不一致性及其理论分析

指出模糊度定义的不一致性，并通过实例进行验证；从理论上分析模糊度定义的不一致性产生的原因，指出当模糊度理论在图像处理中应用时模糊度定义的不一致性可能会产生的影响；给出一个比较合理的模糊度定义。文中提出的问题是在工程应用过程中发现的新问题，所做的工作也只是初步的。     

A decomposition theorem for maxitive measures

A maxitive measure is the analogue of a finitely additive measure or charge, in which the usual addition is replaced by the supremum operation. In contrast to charges, maxitive measures often have a density. We show that maxitive measures can be decomposed as the supremum of a maxitive measure with density, and a residual maxitive measure that is null on compact sets under specific conditions.     

拟相似集的Hausdorff测度的特征

Siegfried GRAF在文献［1］中给出了自相似集上的Hausdorff测度(简称H-测度)的特征.John McLaughlin在文献［2］中引入了拟相似集的概念,K.J.Falconer又在文献［3］中讨论了拟相似集上H-测度和维数的性质.本文研究拟相似集上的H-测度的特征,并得出在一定条件下支撑于其上满足一定条件的测度与H-测度的等价性条件.