Comparison of the probabilistic approximate classification and the fuzzy set model

It is shown that the generalized notion (probabilistic approximate classification) of rough sets can be conveniently described by the concept of fuzzy sets. A discussion of the proper choice of the definition for the membership function of the intersection (union) of fuzzy sets is also presented. However, from the point of view of the probabilistic approximation space, it is argued that there does not exist a universal definition for the fuzzy intersection (union) operation.     

模糊随机变量及其概率分布

本文在模糊σ－代数及模糊数的极大全序子集之上定义了模糊随机变量，进而首次定义了客观实用的模糊随机变量的概率分布函数，并讨论了数学期望及方差等数字特征。     

中国现代数学家寿命分析

根据 2 1 2位中国现代数学家 (1 1 7位逝世 )的生存资料进行分析 ,得到如下结果 . 62位院士的期望寿命为 84.68岁 ,标准误差为 1 .96岁 ;1 5 0位非院士数学家的期望寿命为 79.2 6岁 ,标准误差为 1 .1 3岁 .院士和非院士数学家的寿命差异有显著性意义 (P =0 .0 5 ) .分别给出了院士和非院士数学家两个群体的寿命表 .结论 :中国现代数学家属于长寿之列 .脑部疾病、心脏疾病和癌症为数学家的主要死因 .     

The Kolmogorov–Čentsov theorem and Brownian motion in vector lattices

The well known Kolmogorov–?entsov theorem is proved in a Dedekind complete vector lattice (Riesz space) with weak order unit on which a strictly positive conditional expectation is defined. It gives conditions that guarantee the Hölder-continuity of a stochastic process in the space. We discuss the notion of independence of projections and elements in the vector lattice and use this together with the Kolmogorov–?entsov theorem to give an abstract definition of Brownian motion in a vector lattice. This definition captures the fact that the increments in a Brownian motion are normally distributed and that the paths are continuous.     

“Are the genre and the Geschlecht one and the same number?” An inquiry into Alfred Clebsch's Geschlecht

This article is aimed at throwing new light on the history of the notion of genus, whose paternity is usually attributed to Bernhard Riemann while its original name Geschlecht is often credited to Alfred Clebsch. By comparing the approaches of the two mathematicians, we show that Clebsch's act of naming was rooted in a projective geometric reinterpretation of Riemann's research, and that his Geschlecht was actually a different notion than that of Riemann. We also prove that until the beginning of the 1880s, mathematicians clearly distinguished between the notions of Clebsch and Riemann, the former being mainly associated with algebraic curves, and the latter with surfaces and Riemann surfaces. In the concluding remarks, we discuss the historiographic issues raised by the use of phrases like “the genus of a Riemann surface”—which began to appear in some works of Felix Klein at the very end of the 1870s—to describe Riemann's original research.     

Optimally Dense Packings of Hyperbolic Space

In previous work, a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an ergodic theorem of Nevo to provide an appropriate definition of those packings to be considered optimally dense. Examples are given to illustrate various aspects of the density problem, in particular the shift in emphasis from the analysis of individual packings to spaces of packings.     

Belief and probability: A general theory of probability cores

This paper considers varieties of probabilism capable of distilling paradox-free qualitative doxastic notions (e.g., full belief, expectation, and plain belief) from a notion of probability taken as a primitive. We show that core systems, collections of nested propositions expressible in the underlying algebra, can play a crucial role in these derivations. We demonstrate how the notion of a probability core can be naturally generalized to high probability, giving rise to what we call a high probability core, a notion that when formulated in terms of classical monadic probability coincides with the notion of stability proposed by Hannes Leitgeb [32]. Our work continues by one of us in collaboration with Rohit Parikh [7]. In turn, the latter work was inspired by the seminal work of Bas van Fraassen [46]. We argue that the adoption of dyadic probability as a primitive (as articulated by van Fraassen [46]) admits a smoother connection with the standard theory of probability cores as well as a better model in which to situate doxastic notions like full belief. We also illustrate how the basic structure underlying a system of cores naturally leads to alternative probabilistic acceptance rules, like the so-called ratio rule initially proposed by Isaac Levi [34].Core systems in their various guises are ubiquitous in many areas of formal epistemology (e.g., belief revision, the semantics of conditionals, modal logic, etc.). We argue that core systems can also play a natural and important role in Bayesian epistemology and decision theory. In fact, the final part of the article shows that probabilistic core systems are naturally derivable from basic decision-theoretic axioms which incorporate only qualitative aspects of core systems; that the qualitative aspects of core systems alone can be naturally integrated in the articulation of coherence of primitive conditional probability; and that the guiding idea behind the primary qualitative features of a core system gives rise to the formulation of lexicographic decision rules.     

GENERALIZED RIEMANN INTEGRAL ON FRACTAL SETS

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.     

D-boundedness andD-compactness in finite dimensional probabilistic normed spaces

In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion ofD-compactness and D-boundedness in probabilistic normed spaces.     

算子概率范数与共鸣定理

提出概率赋范线性空间上集合有界性的简化定义,利用算子概率范数概念。进一步研究概率赋范线性空间上的线性算子理论,并在算子概率赋范空间上,建立了概率有界、概率半有界、非概率无界意义下的共鸣定理。     

条件期望的两种定义及其等价性探讨

讨论随机变最在给定子σ代数下条件期望的定义,利用投影定理这一数学工具给出条件期望的几何定义,并通过对它与现今各种概率论基础或随机过程教材中常见的公理化定义相互等价性的证明,揭示了条件期望这一概念的内涵.     

A stopping game in a stochastic and fuzzy environment

A zero-sum stopping game for a sequence of fuzzy-valued random variables is discussed. The fuzzy random variables are estimated by probabilistic expectation and fuzzy expectation. A saddle point is given for the stopping game.     

Limits of Functions as They Developed Through Time and as Students Learn Them Today

This article compares first-year university students' development of the concept of limits to mathematicians' historical development of the concept. The aim was to find out if students perceive the notion as mathematicians of the past did, as understandings of the concept evolved. The results imply that there are some similarities—for example, the struggle with rigor and attainability. Knowledge of such critical areas can be used to improve students' opportunities of learning limits of functions. Some teaching aspects related to the study are also discussed.     

A general framework for defining and extending actual causation using CP-logic

A central problem in the field of causal modelling is to provide a suitable definition of actual causation, i.e., to define when one specific event caused another. Although current research contains many different definitions, it is pervaded with ambiguities and confusion. Our research has two main goals. First, we wish to provide a clear way to compare competing definitions, and improve upon them so that they can be applied to a more diverse range of instances, including non-deterministic ones. To achieve this we provide a general, abstract definition of actual causation, formulated in the context of the expressive language of CP-logic (Causal Probabilistic logic). We will then show that three recent definitions by Ned Hall (originally formulated for structural models) and a definition of our own (formulated for CP-logic directly) can be viewed and directly compared as instantiations of this abstract definition, which also allows them to deal with a broader range of examples. Second, our framework allows for improving on definitions of actual causation in another way, by incorporating the influence of normality. A recent paper by Halpern and Hitchcock draws on empirical research regarding people's causal judgements, to suggest a graded and context-sensitive notion of actual causation. We rephrase their approach into the probabilistic setting of our abstract definition, allowing us to improve it.     

Injectivity of directed complete poset acts

Action of a monoid on a set have always been of interest for mathematicians and computer scientists. On the other hand, Domain Theory, which studies directed complete partially ordered sets, was introduced by Scott in 1970 when he explored the possibility of using ordered topological spaces to give meaning first to typed and then untyped λ-calculi.

In this paper, combining the above two notions, we consider actions of a monoid on domains and study the algebraic notion of injectivity with respect to monomorphisms and embeddings in the category so obtained.     

关于概率内积空间*

本文给出概率内积空间以一新的定义。借助于这一定义,在本文中建立了概率内积空间的Schwarz不等式、收敛性定理及正交性的概念。并讨论了概率内积空间与概率赋范空间的关系等。     

Some Properties of Gromov–Hausdorff Distances

The Gromov?CHausdorff distance between metric spaces appears to be a useful tool for modeling some object matching procedures. Since its conception it has been mainly used by pure mathematicians who are interested in the topology generated by this distance, and quantitative consequences of the definition are not very common. As a result, only few lower bounds for the distance are known, and the stability of many metric invariants is not understood. This paper aims at clarifying some of these points by proving several results dealing with explicit lower bounds for the Gromov?CHausdorff distance which involve different standard metric invariants. We also study a modified version of the Gromov?CHausdorff distance which is motivated by practical applications and both prove a structural theorem for it and study its topological equivalence to the usual notion. This structural theorem provides a decomposition of the modified Gromov?CHausdorff distance as the supremum over a family of pseudo-metrics, each of which involves the comparison of certain discrete analogues of curvature. This modified version relates the standard Gromov?CHausdorff distance to the work of Boutin and Kemper, and Olver.     

Approximation Theoretic Aspects of Probabilistic Representations for Bi-continuous C-semigroups

By means of Riemann-Stieltjes stochastic process, moment-generating functions and operator-valued mathematical expectation, the problem of probabilistic approximation for bi-continuous C-semigroups is studied and the general probabilistic approximation of exponential formulas and the generation theorems are given.     

Convex fuzzy random variables

A theory of fuzzy random variables is developed that applies to situations involving both randomness and fuzziness. The use of membership functions that are quasi-concave play an important role in the theory. The expectation of a fuzzy random variable is a fuzzy variable (fuzzy set). The usual linearity properties of probabilistic expectation carry over to fuzzy random variables. A special case of a fuzzy Law of Large Number is proven.     

Reviewers for Volume 10

This article compares first-year university students' development of the concept of limits to mathematicians' historical development of the concept. The aim was to find out if students perceive the notion as mathematicians of the past did, as understandings of the concept evolved. The results imply that there are some similarities—for example, the struggle with rigor and attainability. Knowledge of such critical areas can be used to improve students' opportunities of learning limits of functions. Some teaching aspects related to the study are also discussed.