B. O. Koopman

The bases of probability

The subject for consideration today forms an aspect of a somewhat venerable branch of mathematical theory; but in essence it is part of a far older department of thought—the ancient science of logic. For it is concerned with a category of propositions of a nature marked by features neither physical nor mathematical, but by their rôle under the aspect of the reason. Their essential characteristic is their involvement of that species of relation between the knower and the known evoked by such terms as probability, likelihood, degree of certainty, as used in the parlance of intuitive thought. It is our threefold task to transcribe this concept into symbols, to formulate its principles, and to study its properties in their inner order and outward application. As prelude to this undertaking it is necessary to set forth certain conventions of logic. Propositions are the elements of symbolic logic, but they may play the rôle of contemplated propositions (statements in quotation marks) or of asserted propositions (statements regarded as true throughout a given manipulation or deduction) ; and it is necessary to take account of this in the notation for the logical constants. We shall employ the symbols for negation ( ^ ) , conjunction or logical product ( • ) , and disjunction or logical sum ( V), and regard them as having no assertive power : they combine contemplated propositions into contemplated propositions and asserted propositions into asserted propositions of the same logical type. Quite other shall be our convention regarding implication ( c ) and equivalence ( = ) : they combine contemplated propositions into asserted propositions, and shall not be used to combine asserted propositions in our present study. If a and b stand for contemplated propositions, the assertion that a is false (true) shall be written a = 0 (# = 1), and the assertion that a implies ô, a c b or a ^ o = 0; it is thus quite different from the contemplated proposition ~(a~b). Finally it is universally asserted that a~a = 0, a \/~a = 1

Air-Terminal Queues under Time-Dependent Conditions

and in fact all the laws of Boolean algebra are regarded as assertions. We shall assume their elements to be fa-

The Optimum Distribution of Effort

The queues formed by aircraft in stacks awaiting landing clearance have usually been treated either by machine simulation, or analytically as stochastic processes with time-independent transition probabilities (possessing stationary solutions). In contrast to such methods, the present paper regards the queue-developing process in question as strongly time-dependent, often with a diurnal (24-hour) periodicity. The formulation and treatment are entirely analytic and make use of machines only to solve the equations for the probabilities, by economical deterministic steps, using the coefficients as given in tabular form. Time-varying Poisson arrivals are assumed, and also an upper limit to queue length. Two laws of servicing are used: Poisson and fixed service time; these extremes are found to lead to numerically close results in the realistic case. This situation contrasts with the much cruder approximation of deterministic flow models. The stochastic equations belong to well studied types of differential or...

A Study of the Logical Basis of Combat Simulation

When a limited amount of effort is available for the performance of two related tasks, the practical of how it is to be divided between them in order to obtain the best over-all result is one which constantly arises in operations research. The object of the present paper is to show how the question can often be put in quantitative form, and then to give it a simple mathematical solution in illustrative cases, interpreting the results in the language of recommended procedures. Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.

An Empirical Formula for Visual Search

The object of this study is the analytic or computer-simulation model, used increasingly in analyses of combat, tactical and weapon-systems evaluation, etc. The thesis is that whenever the method yields quantitative results, it achieves these by setting the computer (or the analytic formulation) in states supposed to correspond to those in the combat, and programming a rule of transitions from state to state—picturing what is occurring in the combat. The transition rule may be deterministic or probabilistic, the latter usually by Monte Carlo selection; but the probabilities are always treated as known. This shows that the combat is assumed (explicitly or implicitly) to act as a stochastic process of the changes of state of a system. The computational procedure used implies further assumptions, which imply, in turn, that it is Markovian (directly, or after a transformation). After bringing out these facts, the necessary conditions for their validity are explored. One is the removal, by conventionalization,...

Necessary and sufficient conditions for Poisson’s distribution

This paper presents an empirically derived formula for computing the threshold contrast C for visually detecting distant objects in search situations. The threshold contrast is the value of contrast that produces detection probability 1/2 on a single glimpse of about 1/3 to 1 second in duration. The formula allows one to compute C as a function of background luminance, target angular area A, and an empirically determined constant K. For small A, the formula reduces to Riccos law, which says that the threshold constrast C, and Area A, are directly proportional for small values of A. The limiting threshold contrast is determined by K. Results of fitting this formula to a set of observational data are shown. The formula provides a useful approximation for determining detection capability for planning visual searches.

New Mathematical Methods in Operations Research

1. Introduction. Poissons law of small numbers is conventionally described as the distribution of the number of successes in a very large number of independent trials of very small individual probability of success. The usual method of deriving Poissons distribution is to set the individual probability of success equal to m/n in the binomial distribution of n repeated trials, and then pass to the limit as n—> °°, m = constant =0: The probability Pn(s) of exactly s successes approaches the Poisson value P(s),

On rejection to infinity and exterior motion in the restricted problem of three bodies

On the somewhat infrequent occasions when a mathematician looks at the practical problems of operations research, his first impression is apt to be that only known and elementary parts of mathematics are needed. But if he dwells with the subject longer, his reaction tends to be quite the opposite: he begins to realize that, when rightly conceived and formulated, these practical problems often turn on mathematical questions deep enough to go beyond existing knowledge and to require for their answer research into new mathematical fields. Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.

General Operation of Search

The cases of the restricted problem of three bodies which have hitherto been considered in detail deal with motion of the particle in the interior of a closed oval of zero velocity, or in the neighborhood of points of equilibrium or periodic orbits. In none of these cases does the question of rejection of the infinitesimal body to infinity play any r6le. It is to the investigation of this phenomenon, and of types of motion in which its possibility is an essential feature, that the present paper is devoted. Attention is restricted to the case of motion in a plane. Chapter I consists of a qualitative study of the individual half orbits which are described by the particle as it goes to or comes from infinity. The methods are altogether elementary. The first salient result is the proof of the existence of such half orbits, with a practical criterion for finding them. The second is the theorem that every half orbit which extends to infinity resembles a hyperbolic or parabolic orbit in this: that it has an asymptote (in the fixed space), and the particle has a limiting velocity. In a special case, resembling the parabolic, this velocity is zero and the asymptote is at infinity (projectively speaking). In Chapter II, we take up the investigation of infinity from the point of view of a singularity of the differential equations of motion. It is shown that by means of a rational substitution this singularity can be reduced to a well known type: a point, at which the right-hand members of a first-order system are analytic, but vanish simultaneously. The power expansions of these functions about the point start with terms of the third degree. When all further terms are omitted, a system is obtained whose solution is wholly known; unfortunately, no method is at hand for extending its properties to the given system. We finish the chapter with a discussion of the totality of half orbits which reach to infinity, regarded as an ensemble.

The Axioms and Algebra of Intuitive Probability

Search, in its broadest sense, is an activity not only of modern man, but of man in his most primitive state and of animals capable of motion directed by senses. Looking for food, prey, shelter, mates, and detecting hostile animals or natural dangers, are operations without which they could not survive. During our evolution we have acquired such a set of instincts involving search in this primitive sense that — as in breathing or walking— we habitually give it little conscious thought. Only with the advance of modern technology have we learned to examine search in the light of science: as an operation having various structural patterns and obeying laws of its own.