Some remarks on the mathematical aspects of automobile driving

Some aspects which involve the interaction of the human element and of the machine element in driving are discussed. As an example a simple equation is derived for the maximum safe speed of a car on an empty road. The parameters of the equation are partly of physiological nature, partly of mechanical nature. Another example treats in a similar manner the problem of a car passing another car moving in the same direction.     

A note on the mathematical biology of automobile driving

It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.     

Mathematical biology of automobile driving

Continuing a previous study (Bull. Math. Biophysics, 28, 645–654, 1966), the biophysical mechanism of a corrective turn is investigated for the case where the stimulus for the corrective turn is produced not only by the perception of the nearness of an edge of the lane, but also by the rate of approach of the car towards the edge. In that case it is found that the tracking curve of the car may consist of a series of damped sinusoids and safe driving would be possible at any speed if it were not for the endogenous fluctuation in the driver's central nervous system. If the effect of the rate of approach increases sufficiently rapidly as the distance to the edge of the lane decreases, then a stable undamped oscillating tracking curve is possible. The case is also studied where the driver makes a corrective turn in response to a direct perception of the angle between the direction of the lane and the longitudinal axis of the car.     

Contribution to the mathematical biophysics of automobile driving

Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.     

Mathematical biology of automobile driving IV

Previous studies by this author of the mathematical biology of automobile driving have emphasized only the biological aspects, except for such mechanical factors as the size of the car. Otherwise, the ideal case of an inertialess car was considered. In this paper the first step is made toward introducing the effects of the mass of the car and the side-slip of the tires when the direction of driving is even slightly altered and combining these with the previously studied biological aspects. Some tentative comparisons with available data are made. This work was done at the Mental Health Research Institute, University of Michigan, Ann Arbor, Michigan.     

Mathematical biology of automobile driving: V

The role of some inertial properties of the car, studied previously only for the case when the stimulus for the corrective turn is the perception of the angle between the direction of the car and the direction of a straight lane (Bull. Math. Biophysics,32, 71–78, 1970), is generalized to include such stimuli as the nearness to the edge of the lane and the anticipatory effect for a corrective turn, as well as the combination of all three stimuli. Conditions for stability of driving are deduced and discussed. They now depend on both biological parameters and such parameters as the position of the center of gravity of the car, its mass, and the side slip of the tires. This work was done at the Mental Health Research Institute, University of Michigan, Ann Arbor.     

Mathematical biology of automobile driving: III

In a previous paper (Bull. Math. Biophysics,22, 257–262, 1960), an expression for the probability that a car jumps off a road as a function of the speed and the size of the car was derived mostly from geometric and kinematic considerations, introducing only the reaction time as a biological parameter. In subsequent papers (Bull. Math. Biophysics,29, 181–186, 187–188, 1967) a more detailed study was made of the exact shape of the tracking curve of the car which involved several biological parameters of the driver. In the present paper the results of the previous studies are combined, and a more general equation for the probability of jumping off the road is obtained. This probability, as in the earlier study, increases with the speedv, widths _o and lengthl _o of the car, and decreases with widths of the lane. However, this probability also depends on several parameters which characterize the psychobiological constitution of the driver. Unpublished experiments by Ehrlich, which corroborate the general conclusions, are briefly described.     

Further contributions to the mathematical biophysics of automobile driving

The discussions of a previous paper (Bull. Math. Biophysics,21, 299–308, 1959) are generalized by considering that the angular direction error made by the driver, as well as the driver's reaction time are not constant but are randomly distributed. Instead of a critical speed, at which the car will jump off the road, we now find that for every speed there is a probability of the car to jump off the road but that this probability is vanishingly small for sufficiently low speeds, yet increases rapidly for high speeds. Thus a more realistic picture of the process of driving is obtained. When the standard deviation of the distribution functions for the angle and the reaction time are very small, the expression obtained here reduces to the expression obtained previously.     

A contribution to the mathematical biology of automobile driving: II. Passing as a case of psychophysical discrimination

The decision to pass or not to pass in view of an oncoming car is considered as a case of comparative judgment in which it is to be decided whether the time it will take to pass safely is greater or less than the time it will take to collide with the oncoming car. H. D. Landahl's well-known theory of psychophysical discrimination is used, and it is assumed that the “distracting stimuli” considered previously (Rashevsky, 1959,Bull. Math. Biophysics,21, 375–85) tend to increase the standard deviation of Landahl's fluctuation function. Effects of the “distracting stimuli” on the threshold of the neuroelements in Landahl's circuit are also considered. On this basis an expression is derived which gives the probability of a collision accident in passing as a function of the “distracting stimuli.”     

Some remarks on molecular biology

The binding energy of a very long molecular chain, composed of different classes of molecules, depends in general on the order of the molecules. It is shown that under very general conditions there exists for a givenbrutto chemical composition of a chain, a class of chains which is characterized by a total binding energy which is equal to the total binding energy of any other prescribed chain of different composition within the limits of unsharpness of the energy level. This establishes a criterion formapping of a class of configurations of long chain molecules on another class. To the extent that a mapping constitutes a generalized code those results contribute to the theory of molecular codes. Applying to our results the results of a previous paper (1959,Bull. Math. Biophysics,21, 309–326), we arrive at the conclusion that the self-replication of a living molecule may be the property not of a particular structure but of classes of structures.     

Some remarks on topological biology

With reference to several recent papers by the author, it is pointed out that within the principle of biotopological mapping a choice of a primordial graph and of a particular transformation defines a system of abstract biology, similar to systems of abstract geometries. The study of such abstract systems is necessary before one can be found which is isomorphic to the actual biological world. A brief survey of the structure and properties of the system based on the choice of the primordial graph and of the transformationT defined in a previous paper (Bull. Math. Biophysics,16, 317–48, 1954) is made. Two more topological theorems are demonstrated, which, interpreted biologically, lead to the conclusion that the higher an organism, the more adaptable it is. Finally a criticism of that particular system of abstract biology is made, and its inadequacy for the representation of the actual biological phenomena pointed out, and a suggestion is made for a possible point set topological approach to biology.     

Two paradigms of mathematical population biology. Attempting to synthesise

Whatever popular the slogan of nonlinear ecological interactions has been in theory, practical ecology professes the projection matrix paradigm, which is essentially linear, i.e., the linear matrix model paradigm for discrete-structured population dynamics. The dominant eigenvalue lamda1, of the projection matrix L is considered as a growth potential of the population. It provides for a quantitative measure of the fitness at which the species is adapted to the given environment, the measure being adequate and accurate, given the data of"identified individuals" type. The case of"identified individuals with unknown parents" bears uncertainty in the status-specific reproduction rates, which eliminates in a unique way (for a broad class of structures and life cycle graphs) by maximizing lamda1(L) under the constraints ensuing from the data and knowledge of species biology. The paradigm of linearity gives way to nonlinear models when modeled are species interactions, such as competition for shared resources, and where the outcome of interaction depends on the population structure of the competitors. This circumstance dictates a need for the synthesis of two paradigms, which is achieved in nonlinear matrix operators as models of interaction between the species whose populations are discrete-structured.     

Some remarks on the mathematical biophysics of organic assymetry

Zusammenfassung Das Vorhandensein bilateraler Asymmetrie bei Organismen ist an und für sich biophysikalisch verständlich. Schwierigkeiten entstehen jedoch beim Versuch die auffallende Ungleichheit in den Häufigkeiten des Vorkommens von Rechts- und Linksasymmetrie biophysikalisch zu deuten. Es werden zwei Möglichkeiten für solch'eine Deutung besprochen. Die eine stützt sich auf die fundamentale Asymmetrie des elektromagnetischen Feldes, und wird als biologisch weniger geeignet angesehen. Die andere, welche dem Verfasser als wahrscheinlicher erscheint, beruht auf der Asymmetrie der organischen Moleküle.

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Résumé L'existence d'une asymétrie bilaterale en général chez des organismes divers n'offre pas de difficulté particulière du point de vue biophysique. Mais les difficultés surgissent quand on tâche de comprendre l'inégalité très prononcée des cas d'asymétrie droite et gauche. Deux possibilités sont discutées dans ce mémoire. L'une repose sur l'asymétrie fondamentale du champs électromagnétique. Cette explication est considérée comme moins probable. Une autre explication qui nous apparaît plus probable, est fondée sur l'asymétrie moléculaire.

     

Mathematical biology of automobile driving: I. The shape of the tracking curve on an empty straight road

The idea was suggested by the author previously (Bull. Math. Biophysics,22, 257–262, 1962) that the keeping of the car close to the center of the lane is a problem of psychophysical discrimination between two conflicting stimuli, namely a stimulus to turn away from the left, resp. right edge of the lane. This is elaborated in the present paper. The effects of discrimination threshold and of the endogenous fluctuations which result in erroneous judgments are discussed. In order that driving should be possible at all, a relation, derived in this paper, must hold between the threshold of discriminationh, the sensitivity coefficientb of the driver to changes in the distance between the car and the edge of the lane, and the width of the lane. General expressions are derived which characterize the stochastic nature of the tracking curve.     

Mathematical biophysics of automobile driving

In a previous paper (Rashevsky, 1959,Bull. Math. Biophysics,21, 299–308) we derived an approximate expression for the maximal speed of driving in terms of the reaction time of the driver. In the present paper the possible effects of unevennesses of the pavement, of such distracting stimuli as road signs etc. on the reaction time are studied theoretically, using previous developments of the mathematical biophysics of the central nervous system. In this manner expressions are derived which determine the maximal safe speed in terms of road conditions and other distracting stimuli. Effects of those conditions on fatigue are also discussed.     

Some remarks on the mathematical theory of nutrition of fishes

V. S. Ivlev [Experimental Ecology of Nutrition of Fishes, 1955, Moscow (in Russia)] has shown that the food uptake by fishes during a fixed interval of time is an exponential function of the concentration of food. Ivlev's equation is derived here, and it is shown that it can hold only for non-stationary conditions, such as prevailed in Ivlev's experiments. For a stationary state, the rate of food uptake should tend asymptotically to a limiting value as the concentration increases, but the variation is not exponential. Different other aspects of the problem are investigated, and definite new experimental procedures suggested. The implications of Ivlev's findings on the effect of non-uniformity of food distribution upon the rate of food consumption are studied from a mathematical point of view. The conclusion is reached that whereas a fish does not, in the process of eating, move directly to an individual food particle which it perceives, it does move more or less directly to large aggregates of particles, if the latter are distributed nonuniformly.     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

Abstract mathematical molecular biology

The tremendous complexity of even the simplest living unit makes a correct theoretical guess as to its mechanism very difficult. It is therefore suggested that, following the example of the physical sciences, a number of purely abstract cases in molecular biology be studied mathematically at first. Subsequent comparison of the different conclusions of such an abstract theory with available data would enable us to decide which of the conceivable situations are likely to occur in reality. As a first step toward such a study the problem of the minimal size of a living unit is studied. Usually the minimal size is considered to be determined by information-theoretical requirements. It is shown that the minimal size may have a very different origin. It may be determined by the possibility that too small a system, even though performing all necessary biological functions, may not be viable unless it is a member of a large group of other similar systems. This approach is developed both from a metric and from a relational point of view. Some relational characteristics of systems of reactions which possess the elementary metabolic properties of organisms are studied.