Evaluations and expectations: A Bayesian model†

This paper focuses on the effect of performance evaluations on expectations about future performances: given a situation where two persons know the evaluations they receive at each step of a series of performances, a process model is proposed to account for changes in their expectations as a function of the given evaluations. The approach consists of looking at the problem as a special case of information processing, and of using Bayes' theorem for the construction of the model. Thus, instead of asking about the effect on a person's expectations of a certain evaluation, we ask how a piece of evidence affects the subjective probability attached to a given hypothesis. A few illustrative runs are presented and discussed.     

A dynamic model for social networks†

A continuous‐time binary‐matrix‐valued Markov chain is used to model the process by which social structure effects individual behavior. The model is developed in the context of sociometric networks of interpersonal affect. By viewing the network as a time‐dependent stochastic process it is possible to construct transition intensity equations for the probability that choices between group members will change. These equations can contain parameters for structural effects. Empirical estimates of the parameters can be interpreted as measures of structural tendencies. Some elementary processes are described and the application of the model to cross‐sectional data is explained in terms of the steady state solution to the process.     

A distance model for sociometric structure†

The conceptual and mathematical framework of a general model for distance within sociometric structure is described. The model characterizes “balance” in terms of the triangle inequality, in which the distance between two people (A and C) should be less than or equal to the sum of the distances to a third person (B), i.e., d (A,C) ≤ d (A,B) + d (B,C). The notion of addition of distances is developed. Different ways of adding distances result in different models of sociometric structure. Two families of models for symmetric graphs are discussed. The general model is extended to asymmetric graphs by generalizing the notion of transitivity. The model's potential for resolving a problem of the transitivity’ model is then discussed. The general model provides a means of examining the relationship between stratification and clustering in the structure of groups.     

A Markov chains model of medical specialty choice†

Data from 118 members of three consecutive cohorts of students at a private Southern medical school are employed to develop a discrete parameter stationary Markov chains model of the medical specialty choice process. This model was then tested on independent subsamples from the same population, generating a series of predicted specialty choices which did not differ from observed choices by more than chance expectations. Suggestions for future research, beyond the present demonstration of the utility of a simple Markovian model of a choice process, are offered.     

A Markov chain model for Asch‐type experiments†

A 4‐state absorbing Markov chain is advanced as a model to represent the response process of subjects in Asch‐type experiments. Both the development of the model and the estimation of its parameters are described. The model is applied to the responses obtained from subjects in an Asch‐type experiment designed to analyze the effects of prior status expectations on social influence.     

Luce's Axiom and occupational prestige: Test of a measurement model†

This paper reports research using a psychological theory, R. D. Luce's (1959) individual choice theory, to provide a ratio‐level measurement basis for an important sociological variable, occupational prestige. It is shown that earlier measurement procedures to scale this sociological variable are essentially arbitrary, although heuristically useful. Luce's Axiom of Choice and his Ranking Postulate are then tested on choice sets of occupations where subjects are asked to pair‐compare or rank the occupations on the basis of their prestige. The Axiom receives qualified support, while the Ranking Postulate is not supported. It is concluded that Luce's choice theory can provide a ratio‐level measurement basis for occupational prestige, although a procedure using rankings must be appropriately structured in order to be an adequate instrument of measurement.     

A psychomathematically‐based teaching method†

Mathematical education is concerned with two questions: what to teach and how to teach. In this article the question of how will be discussed.

We begin with the statement that a necessary, though insufficient, condition for any progress in mathematical education is that it should be based on psychomathematical research, rather than tradition or the imitation of a ‘ good ‘ teacher.

One of the results of psychomathematical research is that a secondary school student's immediate (i.e. direct) memory is very limited. On the other hand, a teaching method is useless if it demands that more items be kept in the immediate (i.e. direct) memory than is possible. Of course, it is not enough merely to say what should not be demanded; we must also explain what should be done. A psychomathematically‐based teaching method (PBTM) has been developed with this aim in view. It suggests the use of external sources of memory in a special form, the details of which are given in this paper. Experience shows that recurrent usage of the external source of memory fixes its contents in the auxiliary (long‐term) memory without any effort.

The method was investigated in a controlled experiment. The difference between the results of the experimental and control groups was significant in most of the cases at the 0‐001 level.

     

Prediction logic: A method for empirical evaluation of formal theory†

This paper proposes an approach to data analysis that assists the investigator in discriminating among specific relations corresponding to alternative scientific predictions about qualitative variates.     

A mathematical criterion for a stable government†

The paper presents a mathematical model of the problem of the politico‐economic stability of a welfare state. The variables characterizing the state of a government are identified. A sufficient criterion which ensures the damping out of the effects arising due to arbitrary perturbations in the variables, is obtained. The model assumes the prevalence of normal conditions, that is, times free from unforeseen calamities such as severe earthquake, outbreak of epidemic on a large scale, invasion by another country, acute famine, etc.

     

The equivalence between ( AB )†= B † A † and other mixed-type reverse-order laws

The standard reverse-order law for the Moore–Penrose inverse of a matrix product is (AB)^??=?B ^? A ^?. The purpose of this article is to give a set of equivalences of this reverse-order law and other mixed-type reverse-order laws for the Moore–Penrose inverse of matrix products.     

A bilinear approach to the pooling problem†

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples     

A comparison theorem and an inequality of redhefferf†

For equations of the form w″+B(e^z)w = 0, where B(ζ) is a rational function which is analytic on 0<|ζ|∞, we determine the regions where the bulk of the zeros of a solution must be located. In the special case of the general Mathieu equation, these results complement earlier results of E. Hille (1924) who considered the special case of real Mathieu equations     

Cascading Knapsack Inequalities: reformulation of a crude oil distribution problem

In this paper we consider the problem in which crude oil is shipped from platforms to terminals using oil tankers at minimum transportation cost. This subproblem, which arises in petroleum supply chain models, can be surprisingly expensive to solve with a straightforward formulation involving inventory balances. We present a reformulation that has a special structure defined in this paper as Cascading Knapsack Inequalities. This is used as the basis for deriving tight reformulations for special cases with a limited number of classes of tankers. Numerical results are presented to demonstrate that significant computational savings can be accomplished with the proposed reformulations.     

A stylized model of ‘Momentum’ processes: a research note

Several phenomena of interest like stock price movements, online auction bidprices or inventory levels, can be stylized as ‘momentum processes’,whereby the very start of activity (‘up’ or ‘down’) cantrigger subsequent activity in that direction. We calculate the transientprobabilities of being in a particular state (eg, stock price or current auctionbid level) in such activities. The calculations can be used to estimate theexpected time to reach a certain price, inventory level or bid. The processescan be modified to show the phenomenon where the lack of activity can have an‘inertia of rest’.     

Primary school geometry: some unusual activities†

The lecture given in Lesotho in November 1984 outlines a series of activities which could enliven primary school geometry. It begins with a brief look at the meaning of geometry and how this is related to children's conceptions of space. It puts forward a curriculum which places a high value on perspective and other questions about children's visual acquisition of information. It continues with drawing activities and with exercises in paper folding. It suggests further ways in which geometry is entwined with geography and asks teachers to determine how their children make connections between the conventional primary school geometry they receive and their individual spatial perceptual structures.

     

The stochastic fluid–fluid model: A stochastic fluid model driven by an uncountable-state process,which is a stochastic fluid model itself

We introduce the Stochastic Fluid–Fluid Model, which offers powerful modeling ability for a wide range of real-life systems of significance. We first derive the infinitesimal generator, with respect to time, of the driving stochastic fluid model. We then use this to derive the infinitesimal generator of a particular Laplace–Stieltjes transform of the model, which is the foundation of our analysis. We develop expressions for the Laplace–Stieltjes transforms of various performance measures for the transient and limiting analysis of the model. This work is the first direct analysis of a stochastic fluid model that is Markovian on a continuous state space.     

A Dynamical model of the atmospheric turbulencefromtheunstablestratificationtothestablestratification

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A kinetic model for coagulation–fragmentation

The aim of this paper is to show an existence theorem for a kinetic model of coagulation–fragmentation with initial data satisfying the natural physical bounds, and assumptions of finite number of particles and finite Lp$L^p$-norm. We use the notion of renormalized solutions introduced by DiPerna and Lions (1989) [3], because of the lack of a priori   estimates. The proof is based on weak-compactness methods in L¹$L^1$, allowed by Lp$L^p$-norms propagation.