Impulsive non‐autonomous dynamical systems and impulsive cocycle attractors

In this work, we define the notions of ‘impulsive non‐autonomous dynamical systems’ and ‘impulsive cocycle attractors’. Such notions generalize (we will see that not in the most direct way) the notions of autonomous dynamical systems and impulsive global attractors in the current published literature. We also establish conditions to ensure the existence of an impulsive cocycle attractor for a given impulsive non‐autonomous dynamical system, which are analogous to the continuous case. Moreover, we prove the existence of such attractor for a non‐autonomous 2D Navier–Stokes equation with impulses, using energy estimates. Copyright © 2016 John Wiley & Sons, Ltd.     

Fuzzy processes

In this paper, contributions to fuzzy probability and to differential equations with fuzzy parameters are made.After an introductory section, a review of fuzzy sets and fuzzy algebra is given in Section 2. The main new results of the investigation are contained in Section 3.In Section 3, Zadeh's definition of the probability of a ‘fuzzy event’ the average value of a fuzzy function are extended into the time domain. It is then shown that not only grades of membership, but also probabilistic processes with notions of fuzziness contained, can be defined which obey ordinary, matric, or integro-differential equations. Applications are also given in Section 3.     

The effect of activity-based instruction on conceptual development of seventh grade students in probability

The purpose of this study is to investigate and compare the effects of activity-based and traditional instructions on students’ conceptual development of certain probability concepts. The study was conducted using a pretest–posttest control group design with 80 seventh graders. A developed ‘Conceptual Development Test’ comprising 12 open-ended questions was administered on both groups of students before and after the intervention. The data were analysed using analysis of covariance, with the pretest as covariate. The results revealed that activity-based instruction (ABI) outperformed the traditional counterpart in the development of probability concepts. Furthermore, ABI was found to contribute students’ conceptual development of the concept of ‘Probability of an Event’ the most, whereas to the concept of ‘Sample Space’ the least. As a consequence, it can be deduced that the designed instructional process was effective in the instruction of probability concepts.     

Generalizations of the Goulden–Jackson cluster method

We give several modifications of the Goulden–Jackson cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various ‘weights’ on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at the website for this paper.     

Logic, language games and ludics

Wittgenstein’s language games can be put into a wider service by virtue of elements they share with some contemporary opinions concerning logic and the semantics of computation. I will give two examples: manifestations of language games and their possible variations in logical studies, and their role in some of the recent developments in computer science. It turns out that the current paradigm of computation that Girard termed Ludics bears a striking resemblance to members of language games. Moreover, the kind of interrelations that are emerging could be scrutinised from the viewpoint of logic that virtually necessitates game-theoretic conceptualisations, demonstrating the fact that the meaning of utterances may, in many situations, be understood as Wittgenstein’s language games of ‘showing or telling what one sees’. This provides motivation for the use of games in relation to logic and formal semantics that some commentators have called for. Many of the ideas can be traced to C.S. Peirce, for whom signs were vehicles of strategic communication. The conclusion about Wittgenstein is that the notions of saying and showing converge in his late philosophy.     

The fuzzy decision problem: An approach to the point estimation problem with fuzzy information

This paper is devoted to the extension of the Bayesian method for the point estimation, when the available information is ‘vague’.In the nonfuzzy case, the parametric estimation can be approached as a particularization in the statistical decision problem. This motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem (defined by Tanaka, Okuda and Asia).In this way, concepts in the fuzzy decision problem are first ‘expressed’ in the estimation terminology. Then, on the basis of these concepts, we shall introduce some notions and state some interesting results. Finally, several illustrative examples will be exposed.     

Mathematical modeling and performance analysis of combed yarn production system: Based on few data

The paper describes the availability of combed sliver production system, a part of yarn production plant. The units under study are specialized single purpose machines. Performance analysis of the system is carried out to identify the key factors. The optimum value of ‘r’, where ‘r’ represent the number of repairman to repair the twelve carding machines (r ? 12), is calculated to maximizing the steady state availability of the system. The problem is formulated using probability consideration and supplementary variable technique. Probability considerations at various stages give differential-difference equations, which are solved using Lagrange method to obtain the state probabilities. The numerical analysis carried out helps in increasing the production rate by controlling the factors affecting the system i.e. availability optimization.     

A mathematical model of pricing in a large system of cash bonds

The purpose of this paper is to give a mathematical model to generalize the classical approach of compound interest and to overcome the time structure problem of the interest rates. We introduce a suitable stochastic process called the ‘gauge’ process such that its product with the value of any security is assumed to be a martingale in an appropriate probability space. The framework of this model gives a stochastic actualization formula for the pricing of general securities with options and includes Black and Schole's formula without using arbitrage arguments. Emphasis has been placed on numerical calculation.     

Unique factorization and the fundamental theorem of arithmetic

The fundamental theorem of arithmetic is one of those topics in mathematics that somehow ‘falls through the cracks’ in a student's education. When asked to state this theorem, those few students who are willing to give it a try (most have no idea of its content) will say something like ‘every natural number can be broken down into a product of primes’. The fact that this breakdown always results in the same primes is viewed as ‘obvious’. The purpose of this paper is to illustrate with a number of examples that the ‘Unique Factorization Property’ is a rare property and the fact that the natural numbers possess this property is ‘fundamental’ to our understanding of this number system.     

The reason for analogies between physical theories

An investigation on the reason for analogies between physical theories is carried out. The reason is found in the fact that in every physical theory there are basic physical quantities which are referred to geometrical and chronometrical elements. The analysis is pursued further using the rudiments of algebraic topology, i.e. the notions of ‘chain’ and that of ‘coboundary’. A rational classification scheme for the physical variables and equations of whatever physical theory is then constructed. Some examples of these schemes are given.     

Question selection responding to information on customers from heterogeneous populations to select offers that maximize expected profit

The advent of Internet broking pages allows customers to ‘apply’ to a number of different companies at one time, leading to multiple offers made to a customer. The saturated condition of the personal financial products has led to falling ‘take’ rates. Financial institutions are trying to increase the ‘take’ rates of their personal financial products. Applicants for credit will have to provide information for risk assessment, which can be used to assess the probability of a customer accepting an offer. Interactive channels such as the Internet and telephone allow questions that are asked to depend on previous answers. The questions selected need to provide information to assess the probability of acceptance of a particular variant of financial product. In this paper, we investigate a model to predict the best offer to extend next to a customer based on the response for the questions, as well as the question selection itself.     

Probability Modelling and Optimal Location of a Travelling Salesman

The ‘random part’ of an operations research model may be less satisfactory than the ‘deterministic part’, and it may thus be desirable to design algorithms that require few probability assumptions and make few calls to a suitable ‘probability oracle’. We consider here the problem of locating a service facility on a tree network so as to minimize the expected length of a travelling salesman tour through a random set of demand nodes.     

Remarks on the idealist and empiricist interpretation of frequentism: Robert Leslie Ellis versus John Venn

The goal of this paper is to correct a widespread misconception about the work of Robert Leslie Ellis and John Venn, namely that it can be considered as the ‘British empiricist’ reaction against the traditional theory of probability. It is argued, instead, that there was no unified ‘British school’ of frequentism during the nineteenth century. Where Ellis arrived at frequentism from a metaphysical idealist transformation of probability theory’s mathematical calculations, Venn did so on the basis of an empiricist critique of its ‘inverse application’.     

Functions,equivalence relations,quotient spaces and subsets in fuzzy set theory

In the usual set theory, the power set of a set X is lattice isomorphicto 2^x. The theory of fuzzy sets, or fuzzy set theory, involves generalizing the lattice 2 = {0, 1} to a more general, usually non-Boolean, lattice. A ‘point’ in this lattice need not have Boolean properties. Thus proper generalizations must be based on notions which are defined without explicitly using points. In the light of this we give equivalent definitions of function, relation and quotient set involving the members of a power set but not specifically the singletons.A number of authors [1,2,5] have worked with generalizations of functions and quotient sets which are not strictly point free. It emerges that these are special cases of the ones we produce.     

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In this note we discuss uniform integrability of random variables. In a probability space, we introduce two new notions on uniform integrability of random variables, and prove that they are equivalent to the classic one. In a sublinear expectation space, we give de La Vall\'{e}e Poussin criterion for the uniform integrability of random variables and do some other discussions.     

Games with guaranteed scores and waiting moves

Combinatorial scoring games, with the property ‘extra pass moves for a player will do him no harm’, are characterized. The characterization involves an order embedding of Conway’s normal play games, and we call our class the universe of Guaranteed scoring games. Also, we give a theorem for comparing guaranteed games with scores (numbers) which extends Ettinger’s work on Dicot scoring games.     

Multi-Skilled Workforce Optimisation

This paper describes a problem faced by CS Energy's Swanbank Power Station in the Australian state of Queensland. It involved the personnel scheduling (rostering) of staff with multiple skill levels at the power station. Such a problem can be classified using the six stage construction process proposed by Ernst et al. We assume that the three processes of ‘demand modelling,’ ‘shift starting times’ and ‘task scheduling’ are specified. We are concerned with the essential processes of ‘day off scheduling,’ ‘line of work construction’ and ‘shift assignment to staff’ with requirements to maintain multiple skills. Several other authors have reported results for staff with hierarchical skills while the methods proposed in this paper are for non-hierarchical skill sets. The paper describes a set covering approach to the multi-skilled rostering problem. We propose a number of solution strategies for the set covering approach and give a comparison of the results.     

Probability distribution fitting of schedule overruns in construction projects

The probability of schedule overruns for construction and engineering projects can be ascertained using a ‘best fit’ probability distribution from an empirical distribution. The statistical characteristics of schedule overruns occurring in 276 Australian construction and engineering projects were analysed. Skewness and kurtosis values revealed that schedule overruns are non-Gaussian. Theoretical probability distributions were then fitted to the schedule overrun data; including the Kolmogorov–Smirnov, Anderson–Darling and Chi-Squared non-parametric tests to determine the ‘Goodness of Fit’. A Four Parameter Burr probability function best described the behaviour of schedule overruns, provided the best overall distribution fit and was used to calculate the probability of a schedule overrun being experienced. The statistical characteristics of contract size and schedule overruns were also analysed, and the Wakeby (<AU$1?m and AU$11–50?m), Three Parameter Log-logistic (AU$1–A$10?m) and Beta (AU$51–A$100?m and >AU$101?m) models provided the best distribution fits and were used to calculate schedule overrun probabilities by contract size.     

Some new notions of dependence with applications in optimal allocation problems

Dependence structures of multiple risks play an important role in optimal allocation problems for insurance, quantitative risk management, and finance. However, in many existing studies on these problems, risks or losses are often assumed to be independent or comonotonic or exchangeable. In this paper, we propose several new notions of dependence to model dependent risks and give their characterizations through the probability measures or distributions of the risks or through the expectations of the transformed risks. These characterizations are related to the properties of arrangement increasing functions and the proposed notions of dependence incorporate many typical dependence structures studied in the literature for optimal allocation problems. We also develop the properties of these dependence structures. We illustrate the applications of these notions in the optimal allocation problems of deductibles and policy limits and in capital reserves problems. These applications extend many existing researches to more general dependent risks.