Convergence in measure with respect to a matrix-valued measure and some matrix completion problems

For a matrix-valued measure M we introduce a notion of convergence in measure M, which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of M. Let S be a subset of the set of matrices of given size. It is easy to see that the set of S-valued measurable functions is closed under convergence in measure with respect to a matrix-valued measure if and only if S is a ρ-closed set, i.e. if and only if SP is closed for any orthoprojector P. We discuss the behaviour of ρ-closed sets under operations of linear algebra and the ρ-closedness of particular classes of matrices.     

Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

In this paper, we show that every infinite dimensional Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure. Therefore, it resolves a long-standing question.     

相伴于随机自相似分形的随机测度的强测度的收敛性

构造了随机自相似分形及其上的记忆函数,并得出了有关结论,在此基础上,我们可以定义一个随机概率测度dΦn(τ)=Kn(τ)dτ,Φn(τ)弱收敛于Φ,进一步可得到强测度序列Ψn(·)=EΦn(·),则{Ψn}弱收敛于Ψ=EΦ.     

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.     

A possibility measure is not a fuzzy measure

In this note we show that a possibility measure is not a particular type of fuzzy measure, except in trivial cases.     

A modification of the certainty measure to handle subnormal distributions

We suggest a modification of the definition of certainty associated with a possibility distribution so that we are assured that the certainty measure is always less than or equal to the possibility measure even in the face of subnormal distributions. We note that when the possibility distribution is normal this new definition reduces to the original definition of certainty. We use this new measure along with the definition of possibility measure to obtain measures of belief and plausibility in the fact of fuzzy information in the Mathematical Theory of Evidence.     

A network efficiency measure with application to critical infrastructure networks

In this paper, we demonstrate how a new network performance/efficiency measure, which captures demands, flows, costs, and behavior on networks, can be used to assess the importance of network components and their rankings. We provide new results regarding the measure, which we refer to as the Nagurney–Qiang measure, or, simply, the N–Q measure, and a previously proposed one, which did not explicitly consider demands and flows. We apply both measures to such critical infrastructure networks as transportation networks and the Internet and further explore the new measure through an application to an electric power generation and distribution network in the form of a supply chain. The Nagurney and Qiang network performance/efficiency measure that captures flows and behavior can identify which network components, that is, nodes and links, have the greatest impact in terms of their removal and, hence, are important from both vulnerability as well as security standpoints.     

A combinatorial problem related to Mahler's measure

We give a generalization of a result of Myerson on the asymptoticbehavior of norms of certain Gaussian periods. The proof exploitsproperties of the Mahler measure of a trinomial.     

Quantum measure theory

We first present some basic properties of a quantum measure space. Compatibility of sets with respect to a quantum measure is studied and the center of a quantum measure space is characterized. We characterize quantum measures in terms of signed product measures. A generalization called a super-quantum measure space is introduced. Of a more speculative nature, we show that quantum measures may be useful for computing and predicting elementary particle masses.     

Comparing the minimal Hellinger martingale measure of order q to the q-optimal martingale measure

This paper investigates the relationship between the minimal Hellinger martingale measure of order q$q$ (MHM measure hereafter) and the q$q$-optimal martingale measure for any q≠1$q\ne 1$. First, we provide more results for the MHM measure; in particular we establish its complete characterization in two manners. Then we derive two equivalent conditions for both martingale measures to coincide. These conditions are in particular fulfilled in the case of markets driven by Lévy processes. Finally, we analyze the MHM measure as well as its relationship to the q$q$-optimal martingale measure for the case of a discrete-time market model.     

有界闭凸值测度的集值积分

引入实值函数关于有界闭凸值测度的集值积分,并讨论了集值积分的收敛定理,证明了当集值测度为有界闭凸集值的有界变差集值测度时,关于弱紧凸集值测度的积分性质对有界闭凸集值测度仍然保持.推广了实值函数关于弱紧凸值测度的积分.     

Contribution to application of energy measure of fuzzy sets

The works of De Luca & Termini continued by, for example, Knopfmacher, Loo and Gottwald, are the most important on the topic of determination of measures of fuzzy sets. The matter is to evaluate how fuzzy a fuzzy set is. There are two general concepts of measures of fuzzy set, i.e. entropy and energy measures.We show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations.Applications of the use of energy measure discussed in detail include decision making, fuzzy process control and prediction in fuzzy systems.     

A counter-example to Dye's Theorem for all non-separable measure algebras

Summary In 1959, H. Dye showed that any two ergodic, measure-preserving automorphisms of a Lebesgue measure algebra were weakly equivalent. In this paper, we study weak equivalence, for ergodic measure-preserving automorphisms on non-separable measure algebras. It is shown that, in general, Dye's Theorem does not hold, and in particular, it holds only on separable, i.e. Lebesgue, measure algebras.     

Norms extremal with respect to the Mahler measure

In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmer?s conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures.     

Fuzzy integral on credibility measure

Fuzzy measure and credibility measure are two dissimilar concepts, and fuzzy integral based on fuzzy measure has been researched. This paper focuses on introduction of fuzzy integral on credibility measure and discusses its properties, since it has been not studied yet so far.     

Self-similar sets of zero Hausdorff measure and positive packing measure

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ ₀ ⁸ a _n 4⁻ⁿ a _n 4⁻ⁿ with digits witha _n ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. Research of Y. Peres was partially supported by NSF grant #DMS-9803597. Research of K. Simon was supported in part by the OTKA foundation grant F019099. Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.     

Statistical convergence and measure convergence generated by a single statistical measure

The purpose of this paper is to discuss those kinds of statistical convergence,in terms of filter F,or ideal L-convergence,which are equivalent to measure convergence defined by a single statistical measure.We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l_∞/l_∞(I_μ).We also show that the usual sequential convergence is not equivalent to a single measure convergence.     

A new dependence measure for importance analysis: Application to an environmental model

This paper presents a new dependence measure for importance analysis based on multivariate probability integral transformation (MPIT), which can assess the effect of an individual input, or a group of inputs on the whole uncertainty of model output. The mathematical properties of the new measure are derived and discussed. The nonparametric method for estimating the new measure is presented. The effectiveness of the new measure is compared with the well-known delta and extended delta indices, respectively, through a linear example, a risk assessment model and the Level E model. Results show that the proposed index can produce the same importance rankings as the delta and extended delta indices in these three examples. Yet the computation of the proposed measure is quite tractable due to the univariate nature of MPIT. The results also show that the established estimation method can provide robust estimate for the new measure in a quite efficient manner.     

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).