Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

On tightness and stopping times

This paper studies conditions of tightness for sequences of processes, which conditions are mostly based on the use of ‘dominating’ increasing processes. The results obtained follow in directions initiated by Aldous and Rebolledo and are particularly well-suited for studying sequences of semimartingales. Also obtained are results that extend sufficient conditions of Aldous's type to processes that are not quasi-left-continuous.     

Deterministic macroeconomic dynamic equations from a stochastic microeconomic description: The role of time scales

In the present paper, we develop the ‘stochastic’ microeconomic approach to obtained a deterministic macrodynamic equation. Three time scales have to be introduced. To be valid, the dynamic equation has to be written on the meso-scopic time scale. ‘Discrete’ and ‘continuous’ time versions are given. On the other hand, the comparison between the ‘stochastic’ and the ‘average’ (all the agents have the same characteristics, the ‘average’ ones) approaches is presented. Both methods give the same form for the dynamic equation, nevertheless the ‘average’ approach gives conditions of validity which are not strict enough. An advantage of the ‘stochastic’ approach is to inforce this point and to propose a better framework for writing down deterministic macrodynamic equations and for making further developments if necessary.     

ON PRIME RINGS AND THE RADICALS ASSOCIATED WITH THEIR DEGREES OF PRIMENESS

Abstract

Prime ringsMaybe classified by the sizes of the sets that ‘insulate’ their elements from annihilation. For a cardinal m > 0, the class [Pbar]_r,(m) of all rings that are right prime of ‘bound at most m’ is studied, with particular reference to its closure under constructions such as matrix rings, semigoup rings, orders and extensions. The classes [Pbar]_r,(m) are special in the sense of radical theory for each m > 0. The attendant upper radicals υ[Pbar]_r,(m) are right (and not left) strong; their compatibility with certain ring constructions is examined. In the lattice of radicals (where they form a strictly descending chain), their positions are described, relative to various familiar radicals.     

中止规则的平均延迟时间及其应用

本文以平均延迟时间为度量，对适用于连续抽样方案的四种中止规则，即规则［Ｓ］，［Ｒ］，［Ｎ，ｃ］及［Ｒ，ｄ］的中止“速度”进行了比较。结果表明：［Ｒ］优于［Ｓ］，而［Ｎ，ｃ］与［Ｒ，ｄ］均优于［Ｒ］。这些结论及方法可被用来适当地选择中止规则，以提高连续型生产的质量控制水平     

Measuring Need to Spend on Education

As part of a legislative package designed to reform the finance of local government in England and Wales, the Government has proposed simplifying the measures of ‘need to spend’, which will continue to be a key determinant of a local authority's grant income. In this paper we examine the construction of a measure of ‘need to spend’ on education in some detail, and argue that ‘fairness’ would be better served by slightly increasing the degree of complexity of the current measure rather than attempting to follow the path to simplification advocated by the Government. The impact on local authorities of adopting the more complex measure of ‘need to spend’ which we propose is described.     

Asymptotic independence of ‘pure head’ stopping times

The asymptotic independence and exponentiality of ‘pure head run’ stopping times will be proved.     

A footnote to The crisis in contemporary mathematics

We examine the preparation and context of the paper “The Crisis in Contemporary Mathematics” by Errett Bishop, published 1975 in Historia Mathematica. Bishop tried to moderate the differences between Hilbert and Brouwer with respect to the interpretation of logical connectives and quantifiers. He also commented on Robinson's Non-standard Analysis, fearing that it might lead to what he referred to as ‘a debasement of meaning.’ The ‘debasement’ comment can already be found in a draft version of Bishop's lecture, but not in the audio file of the actual lecture of 1974. We elucidate the context of the ‘debasement’ comment and its relation to Bishop's position vis-a-vis the Law of Excluded Middle.     

Harnack and super poincaré inequalities for generalized Cox-Ingersoll-Ross model

Abstract

In this paper, Wang’s Harnack inequalities and super Poincaré inequality for generalized Cox-Ingersoll-Ross model are obtained. Since the noise is degenerate, the intrinsic metric has been introduced to construct the coupling by change of measure. By using isoperimetric constant, some optimal estimate of the rate function in the super Poincaré inequality for the associated Dirichlet form is also obtained.     

Approximating minimum-length-sequence metrics: a cautionary note

In numerical taxonomy one may wish to measure the dissimilarity of classifications S and T by computing the distance between them with an appropriate metric. A minimum-length-sequence (MLS) metric requires that the user identify a set X of meaningful transformations of classifications; the MLS metric μ_x is then defined by requiring that μ_x (S,T) be the length of a shortest sequence of transformations from X that carries S into T.For a given application it may be relatively easy to identify an appropriate set X of transformations, but it may be difficult or impossible to design an efficient algorithm to compute μ_x. In this case it is natural to restrict the definition to obtain an approximation ? to the original metric μ_x such that ? has an efficient algorithm for its computation. This restriction process must be performed carefully lest the approximation fail to satisfy the metric properties. We present a general result about this problem and apply it in two ways. First we prove that a published ‘metric’ on partitions of a set in fact violates the triangle inequality and so is merely a semimetric. Then we clarify the relationship between the nearest neighbor interchange metric on labeled binary trees and the closest partition distance measure proposed by Waterman and Smith (1978).     

Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion

We study the ergodicity of stochastic reaction–diffusion equation driven by subordinate Brownian motion. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution’s law. These properties imply that this stochastic system admits a unique invariant measure according to Doob’s and Krylov–Bogolyubov’s theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by $\alpha$-stable type noises do not satisfy Freidlin–Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker–Varadhan type large deviation as time tends to infinity.     

The impact of instructor pedagogy on college calculus students’ attitude toward mathematics

College calculus teaches students important mathematical concepts and skills. The course also has a substantial impact on students’ attitude toward mathematics, affecting their career aspirations and desires to take more mathematics. This national US study of 3103 students at 123 colleges and universities tracks changes in students’ attitudes toward mathematics during a ‘mainstream’ calculus course while controlling for student backgrounds. The attitude measure combines students’ self-ratings of their mathematics confidence, interest in, and enjoyment of mathematics. Three major kinds of instructor pedagogy, identified through the factor analysis of 61 student-reported variables, are investigated for impact on student attitude as follows: (1) instructors who employ generally accepted ‘good teaching’ practices (e.g. clarity in presentation and answering questions, useful homework, fair exams, help outside of class) are found to have the most positive impact, particularly with students who began with a weaker initial attitude. (2) Use of educational ‘technology’ (e.g. graphing calculators, for demonstrations, in homework), on average, is found to have no impact on attitudes, except when used by graduate student instructors, which negatively affects students’ attitudes towards mathematics. (3) ‘Ambitious teaching’ (e.g. group work, word problems, ‘flipped’ reading, student explanations of thinking) has a small negative impact on student attitudes, while being a relatively more constructive influence only on students who already enjoyed a positive attitude toward mathematics and in classrooms with a large number of students. This study provides support for efforts to improve calculus teaching through the training of faculty and graduate students to use traditional ‘good teaching’ practices through professional development workshops and courses. As currently implemented, technology and ambitious pedagogical practices, while no doubt effective in certain classrooms, do not appear to have a reliable, positive impact on student attitudes toward mathematics.     

Ordered Involutive Operator Spaces

This is a companion to recent papers of the authors; here we construct the ‘noncommutative Shilov boundary’ of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider ‘maximal’ and ‘minimal’ unitizations of such ordered operator spaces.     

Axioms for a fuzzy measure of inequality

It has been argued that the concept of inequality is inherently imprecise. A difficulty with standard inequality measures is that they generally make no allowances for this, and when they do, it is by dropping the ‘completeness’ axiom in ranking social states (e.g. the Lorenz criterion). It is argued here that the erring axiom is not ‘completeness’ but ‘exactness’ which, being implicit, tends to escape notice. A fuzzy measure of inequality, along with a set of necessary and sufficient axioms, is established. The new measure has several attractive properttes: It allows for tentative judgements and doubts. It is easy to interpret and compute, and the Gini ranking turns out to be a nearest exact approximation of it.     

Discrete-time trawl processes

We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent ‘seed’ processes on shrinking intervals (‘trawl heights’). Related trawl processes in continuous time were studied in Barndorff-Nielsen et al. (2011, 2014).In the case when the trawl function decays as a power function of the lag with exponent $1<\alpha <2$, the trawl process exhibits long memory and its covariance function is non-summable. We show that under general conditions on generic seed process, the normalized partial sums of such trawl process may tend either to a fractional Brownian motion or to an $\alpha$-stable Lévy process. Moreover if the trawl function admits a faster decay rate, then the classical Donsker’s invariance principle holds true.     

Liquidity Risk with Coherent Risk Measures

This paper concerns questions related to the regulation of liquidity risk, and proposes a definition of an acceptable portfolio. Because the concern is with risk management, the paper considers processes under the physical (rather than the martingale) measure. Basically, a portfolio is ‘acceptable’ provided there is a trading strategy (satisfying some limitations on market liquidity) which, at some fixed date in the future, produces a cash‐only position, (possibly) having positive future cash flows, which is required to satisfy a ‘convex risk measure constraint’.     

Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

First order Feynman–Kac formula

We study the parabolic integral kernel for the weighted Laplacian with a potential. For manifolds with a pole we deduce formulas and estimates for the derivatives of the Feynman–Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge.     

Structural Electricity Models and Asymptotically Normal Estimators to Quantify Parameter Risk

ABSTRACT

We estimate a structural electricity (multi-commodity) model based on historical spot and futures data (fuels and power prices, respectively) and quantify the inherent parameter risk using an average value at risk approach (‘expected shortfall’). The mathematical proofs use the theory of asymptotic statistics to derive a parameter risk measure. We use far in-the-money options to derive a confidence level and use it as a prudent present value adjustment when pricing a virtual power plant. Finally, we conduct a present value benchmarking to compare the approach of temperature-driven demand (based on load data) to an ‘implied demand approach’ (demand implied from observable power futures prices). We observe that the implied demand approach can easily capture observed electricity price volatility whereas the estimation against observable load data will lead to a gap, because – amongst others – the interplay of demand and supply is not captured in the data (i.e., unexpected mismatches).     

Smoothness of the intensity measure density for interacting branching diffusions with immigrations

We consider the invariant measure for finite systems of interacting branching diffusions with immigrations. We use Malliavin calculus in order to show that the intensity measure of the invariant measure admits a density which is continuous, one times partially differentiable and bounded provided the immigration measure is absolute continuous.