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  • Springer  (65,535)
  • 2020-2024
  • 1990-1994
  • 1970-1974  (65,535)
  • 1971  (33,802)
  • 1970  (31,733)
  • 1
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    Electronic Resource
    Contributions to the theory of organismic sets, II: Aggressiveness and submissiveness (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 49-54 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In the theory of organismic sets (Bull. Math. Biophysics,31, 159–198, 1969) we considered organisms as sets endowed with certain “activities,” the latter’s resulting in a set of “products.” Those products may be of a material nature, like a hormone secreted by a cell, or of a non-material nature, like a feeling or an attitude. In the present paper aggressiveness and submissiveness are considered as such non-material products of the activities of the brain cells. A general description of aggressiveness and submissiveness is given in terms of organismic sets. Cycles in “peck order” are thus naturally explained.
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    URL: http://dx.doi.org/10.1007/BF02476664
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  • 2
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    Contribution to the theory of organismic sets, III (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 55-66 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In line with previous studies on organismic sets, the division of all organismic sets intogeneral autotrophic and heterotrophic is introduced. The first produce their food themselves from some external source of energy, which in general may be an energy of any kind. The others use other organismic sets as the source of their food and energy. On earth we know only one kind of generalgeneral autotrophic organismic sets, namely, the autotrophic plants which use solar radiation as their source of energy and for production of their own food. It is shown why autotrophic animals do not exist on earth except as microorganisms like, e.g.,Euglena. A rigorous proof of the previously derived theorem that in an organismic set of ordern〉1 no element can be completely specialized is given. It requires the introduction of new postulates. Finally, in considering the organic world as a whole, the notion of organismic sets ofmixed order is introduced.
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    URL: http://dx.doi.org/10.1007/BF02476665
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  • 3
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    “A stochastic solution of the 3-body problem in chemical kinetics. The termolecular stochastic process I” (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 67-81 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It appears to be axiomatic that termolecular and higher order reactions occur relatively rarely. The basis for this judgment seems to lie in the supposition that successful 3-Body collisions of 3 interactive species of molecules cannot occur frequently enought to account for chemical or biochemical transformation. In order to provide a more complete mathematical framework than now exists for examining this hypothesis the probability of effective termolecular “δ-collisions” as a function of time is derived. This amounts to adding to the class of reactions for which stochastic models are now available the termolecular reaction. In common with the unimolecular and bimolecular cases this process is seen to satisfy the criterion of consistency-in-the-mean with respect to deterministic formulations. It is planned next to use the termolecular process and the lower order processes in computer-assistedin numero experimental studies aimed at comparing alternative mechanisms of reaction.
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  • 4
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    Maximum likelihood estimation for stochastic first order reactions (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 83-96 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Small sample properties of the maximum likelihood estimator for the rate constant of a stochastic first order reaction are investigated. The approximate bias and variance of the maximum likelihood estimator are derived and tabulated. If observations of the system are made at timesiτ,i=1, 2, ...,N; τ〉0, the observational spacing τ which minimizes the approximate variance of the maximum likelihood estimator is found. The non-applicability of large sample theory to confidence interval derivation is demonstrated by examination of the relative likelihood. Bartlett’s method is employed to derive approximate confidence limits, and is illustrated by using simulated kinetic runs.
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    URL: http://dx.doi.org/10.1007/BF02476667
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  • 5
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    Organismic supercategories and qualitative dynamics of systems (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 339-354 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The representation of biological systems by means of organismic supercategories, developed in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–71;32, 539–561), is further discussed. The different approaches to relational biology, developed by Rashevsky, Rosen and by Băianu and Marinescu, are compared with Qualitative Dynamics of Systems which was initiated by Henri Poincaré (1881). On the basis of this comparison some concrete result concerning dynamics of genetic system, development, fertilization, regeneration, analogies, and oncogenesis are derived.
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    URL: http://dx.doi.org/10.1007/BF02476778
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  • 6
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    Some realizations of (M, R)-systems and their interpretation (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 303-319 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Some years ago (Rosen 1958a, b; 1959) we described a class of metaphorical, relational paradigms for cellular activity which we termed (M, R)-systems. A sizable amount of subsequent work, to be itemized below, has been devoted to an exploration of some of the properties of these systems. The main purpose of the present paper is to put this class of paradigms into a general system-theoretic perspective, with a particular view to appraising the relation between the type of system description embodied in the (M, R)-system and other kinds of physical and mathematical descriptions of cellular systems. Thus, the principal aim is to establish the relationships and connections between the global relational formalism embodied in the (M, R)-systems and the empirical descriptions which still represent the bulk of our biological knowledge.
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    URL: http://dx.doi.org/10.1007/BF02476776
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  • 7
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    Relational forces and structures produced by them (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 321-338 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract After giving a brief review of the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967;31, 159–198, 1969), in which the concept of relational forces, introduced earlier (Bull. Math. Biophysics,28, 283–308, 1966a) plays a fundamental role, the author discusses examples of possible different structures produced by relational forces. For biological organisms the different structures found theoretically are in general agreement with observation. For societies, which are also organismic sets as discussed in the above references, the structures can be described only in an abstract space, the nature of which is discussed. Different isomorphisms between anatomical structures, as described in ordinary Euclidean space, and the sociological structures described in an abstract space are noted, as should be expected from the theory of organismic sets.
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  • 8
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    Probabilistic and multiple regression modeling of the biologist’s decision processes and its implications for medical diagnosis (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 439-449 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Current psychological research into the inference (diagnostic) process is briefly reviewed, using as a vehicle an investigation of the prediction of the probability of success of hypothetical applicants to a graduate program in biology. Brunswik’s lens model and multiple regression analysis are used, as is a Bayesian approach. Four judges’ (biologists’) predictions are analyzed. Some general conclusions about inference, drawn from the current data in psychology, are presented.
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  • 9
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    Anin numero study of glucose-insulin interaction (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 451-462 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model has been developed to simulate the glucose-insulin interaction following a glucose load such as occurs in an IVGTT. This model differs from earlier models in that the insulin response to glucose loading is a recurring all or none threshold response. The model has been simulated on a digital computer using the digital analog simulation language CSMP.
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    URL: http://dx.doi.org/10.1007/BF02476786
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  • 10
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    Wave propagation in bone media (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 463-479 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The composite nature of bone dictates the use of a model for bone which is transversely isotropic. We solve the associated sets of partial differential equations governing the dynamic elastic behavoor of a two-layered cylindrical-shaped bone. The solution is analyzed for long, short, and intermediate length waves. The special case of compact bone is treated for long and short wave lengths and a numerical example is worked out to determine the wave speeds (for short wave lengths) given a set of elastic constants, determined by ultrasonic methods, and the bone density, wave frequency, and radius.
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  • 11
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    Fourth International Biophysics Congress (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 481-481 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 12
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    Stochastic models for an enzyme reaction in an open linear system (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 97-115 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A stochastic model is developed for an enzyme reaction in an open linear system. The proposed model assumes that the open system maintains the concentration of substrate and inhibitor at constant levels and that the product molecules are removed from the system by a first order reaction. Stochastic models for several enzyme reactions occurring in this open system are shown to correspond to special cases of theGI/M/∞ queue. Takács’ (1958) results for this queueing system are used to obtain the stochastic properties of the enzyme systems. A specific model we studied assumed completely competitive inhibition in an open system. The stationary distribution for the number of product molecules in the system is obtained. The enzyme reaction which incorporated the “intermediate chain hypothesis” can also be investigated by the queueing theory approach. It is shown that for this open system, if the model which incorporated the intermediate chain hypothesis has the same deterministic properties as the Michaelis-Menten model, then the latter has greater stochastic variation than the former.
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  • 13
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    International Conference on Stochastic Point Processes (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 153-153 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 14
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    An integral method for the analysis of blood flow (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 117-128 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.
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  • 15
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    Methode d’objectivation d’organisations temporelles privilégiées d’activités neuronales unitaires (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 129-151 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Résumé Le but de ce travail est la mise en évidence d’éventuels “patterns” temporels privilégiés de potentiels d’action neuronaux masqués par la superposition d’une activité aléatoire. Dans la première partie, on propose un modèle susceptible de rendre compte de cette activité aléatoire. Dans la seconde, on expose une méthode d’extraction des patterns privilégiés, compatible avec les paramètres du modèle neuronal proposé. Son algorithme fait notamment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, ment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, être appliquée à l’étude de séries temporelles d’événements dans des domaines très divers.
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    URL: http://dx.doi.org/10.1007/BF02476670
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  • 16
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    International Symposium on Mathematical Biology (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 155-156 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 17
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    Molecular set theory. III: The wide sense kinetics of molecular sets (1971)
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    Bulletin of mathematical biology 33 (1971), S. 355-372 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An effort is made to begin widening the scope of kinetics by merging the concepts and point of view of molecular set theory with the stochastic approach to kinetics, beginning with the simplest unimolecular molecular set transformation. In this spirit the new concept ofmolecular set variable is introduced as the basic unit of kinetics as opposed to simply the traditionalconcentration (or cardinality) unit, connoting that the composition as well as the size of a molecular set are significant dynamic features of a system. The changes in state (or “value”) of the molecular set variable are characterized by a Markovian stochastic process and the relationship between this process and the corresponding unimolecular process for the concentration variable introduced earlier is discussed. The possible role of molecular set theory in terms of the underlying biomathematical structure of relational biology is also considered.
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  • 18
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    Contribution a l’etude de la forme des plantes: discussion d’un modele de ramification (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 387-401 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The problem of the forms of plants and models of branching are discussed using experimental data on the mistletoe. The number of branches by division, the distribution of divisions with regard to the number of branches per division and to the level of division, the geometrical characters of branches according to the level of division and the host, the stability of model are studied. One gives an interpretation of the model of branching as a model of growth.
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  • 19
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    A quantum model for controlled enzymic reactions (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 373-386 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A quantum model for the general enzymic reaction,E+S ⇌ ES → P, is presented, starting with the assumptions that any chemical substanceS, which may be a substrate for a particularE (S)-enzyme is a microphysical system and any enzymeE-molecule, capable of interacting with anS-substrate is a “measuring system” which will “measure” one or more of theS-observables. According to the above assumptions a stochastic model of the reaction is constructed and a computer simulation of the steady state performed. The results thus obtained predicted fluctuations in the enzymic reaction rate, function of the substrate “perturbation”. On an experimental basis it is demonstrated that the irradiation of an enzymic substrate with low energies results in the inducement of a dose-dependent oscillatory behavior in the corresponding enzymic reaction rate. In the reaction type, the oscillations thus induced in theE-activity by the corresponding substrates are out-of-phase, realizing a biochemical discriminating net. Likewise, in an $$S_1 \xleftarrow{{E_1 }}S\xrightarrow{{E_2 }}S_2$$ reaction type, the oscillations induced by the irradiatedS-substrate in the activities of the respective enzyme, realize a biochemical switching net.
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  • 20
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    Basic studies on metabolic steady state. Incompletely mixed systems (1971)
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    Bulletin of mathematical biology 33 (1971), S. 403-412 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper is the second of a pair dealing with some mathematical properties of metabolic steady state. An investigator wishing to compute the rate of appearance and/or disappearance of a metabolite in steady state within an intact biological system will usually appeal to a method involving radioactive tracers. It is shown that while the investigator’s choice of the mode of tracer administration (constant infusion or single injection) is largely arbitrary, the mathematical interpretation of the results may depend upon the presence or absence of gradients in certain of the variables of the system. The latter will be the case if the system is sampled at a point within the distribution space of the metabolite which is not a source point but is otherwise arbitrary. In order to deduce a formula which gives the required rate, he must have knowledge of the gradient of concentration of the traced substance, and sometimes of the gradient of specific activity.
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  • 21
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    Some mathematical aspects of the medical diagnostic process. I: a general mathematical model (1971)
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    Bulletin of mathematical biology 33 (1971), S. 413-424 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Making a medical diagnosis consists of correlating knownpatterns of disease with the various classes of clinical data elicited from the history, physical examination, and batteries of tests relative to the diagnostic dynamics symbolized by atree branching into the various possible diagnostic decisions. In this paper a relational mathematical model of the reasoning aspects of the conventional medical diagnostic process is suggested as a way of extracting a general, formal concept of medical diagnosis. Computer implementation of the model is discussed briefly.
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  • 22
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    Forecasting and chronic illness (1971)
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    Bulletin of mathematical biology 33 (1971), S. 425-437 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Biomedical data in the form of series of observations made on a single process at regular intervals constitute a discrete time series and are eligible for time series methods of analysis. The models yielded by this analysis provide the framework within which exponential smoothing methods may operate on the data to provide recurrent forecasts of future states of the process. Because the forecasts may be made on an individual basis and are sensitive to the past behavior of the individual process, the methods are presented as being potentially of great utility in the management of chronic and progressive illnesses. When incorporated into automated testing and diagnostic systems, the forecasting method will provide the capability of making prognoses for large numbers of individuals, quickly, routinely and reproducibly.
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  • 23
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    The influence of noninstantaneous mixing on the measurement of fluxes of solute and water across a biological membrane (1970)
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    Bulletin of mathematical biology 32 (1970), S. 1-24 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The usual method of tracer analysis for calculating the flow across a biological membrance is based on the assumption that the compartments on either side are well-stirred. Thus, the validity of the rate of flow determination is questionable for cases where the distribution of tracer is not homogeneous. In this study, a mathematical model is developed for the purpose of estimating the effect of slow mixing on the calculation of the flow rate. The model is applied to the measurement of the rate of flow of aqueous humor through the living eye by use of a fluorescent dye as a tracer. A transit time of several minutes for the passage of fluorescein through the posterior chamber and an extended period of nonuniform distribution of fluorescein in the anterior chamber was observed. The effect of slow mixing on the calculated flow rate is compared to rates derived from equations based on the assumption of rapid mixing. Aqueous flow rates determined by the two methods were found to agree to within ≈20%.
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  • 24
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    The dispersion of indicator in the cardio-pulmonary system (1970)
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    Bulletin of mathematical biology 32 (1970), S. 25-43 
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    Notes: Abstract This paper presents a three-parameter model of the mechanism of dispersion of an indicator in the cardio-pulmonary system, based on the postulates that this dispersion can be described by the one-dimensional diffusion equation and that dispersion continues past the sampling site. The model is tested using indicator dilution curves obtained from dogs, and the coefficient of diffusion is thus measured. It is found that this coefficient increases in magnitude non-linearly with increasing blood speed.
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  • 25
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    The effect of distension of the left ventricle of the heart on the length of the individual myocardial fibers (1970)
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    Bulletin of mathematical biology 32 (1970), S. 45-58 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract Based on the ellipsoid model of the left ventricle and the helicoidal course of the left ventricular myocardial fibers, a theory has been developed for calculating the length of the individual myocardial fibers. Numerical solutions of the final equation show that when the left ventricle is distended, the increase in length of the myocardial fibers is not uniform throughout the thickness of the myocardial wall. It was shown that with increasing dimensions of the left ventricle, the distension of the myocardial fibers becomes smaller as one advances from the endocardium to the middle layer of fibers, whereas it increases as one advances from the middle layer to the epicardial layer. The mechanism by which this effect is brought about as well as its physiological implications are discussed.
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  • 26
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    The gas washout determination under a symmetry assumption (1970)
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    Bulletin of mathematical biology 32 (1970), S. 59-63 
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    Topics: Biology , Mathematics
    Notes: Abstract In an open circuit gas washout determination the output of test substance is shown to be of the form ∑ i=0 ∞ A i λ i k on thekth expiration whereA i 〉0,i=1,2,... and 1 〉 λ1 ≥ λ2...≥ 0 provided the transition from inspiration to expiration has certain symmetry properties with the transition from expiration to inspiration. In general, no direct physical interpretation such as volume for theA t ’s or fraction of retained gas on expiration for the λ i is justified.
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  • 27
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    Mathematical biology of automobile driving IV (1970)
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    Bulletin of mathematical biology 32 (1970), S. 71-78 
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    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract 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.
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    Relaxation methods for the determination of drug receptor affinities (1970)
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    Bulletin of mathematical biology 32 (1970), S. 65-69 
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    ISSN: 1522-9602
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    Notes: Abstract According to the occupation theory of drug receptor interaction, the response is a functionf of the number of receptors occupied by drug molecules. Considerable controversy exists regarding assumptions about this function. Without knowledge of the nature of the function, it is not possible to determine directly the rate constants, and hence the affinity constant, in the reaction between the receptor and an agonist drug. Instead, indirect determinations involving the use of antagonists have been employed, limiting the determination of affinity to those agents for which specific antagonists exist. The present paper discusses a method for the direct determination of affinity of an agonist drug. It is a “relaxation method,” i.e., the equilibrium is perturbed and the kinetics of the restoration process are studied. Assuming only thatf is non-decreasing and approximately linear over a limited domain of concentrations, it is shown that the change in response obeys first order kinetics, permitting a determination of the rate constants from the time course of the restoration process.
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    A remark on the course of development of organismic sets (1970)
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    Bulletin of mathematical biology 32 (1970), S. 79-81 
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    Notes: Abstract The basic postulate la, which governs the development of organismic sets, introduced previously (Bulletin of Mathematical Biophysics,31, 159–198, 1969), is generalized so as to contain also the rates of changes of the number and variety of differentQ-relations which determine an organismic set. It is thus brought closer to the Lagrangian principle in physics. It is pointed out that the postulate also provides a criterion of stability of an organismic set.
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    In-vivo estimations of the nonlinear pressure-volume relationship of the aorta and instantaneous left ventricular volume (1970)
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    Bulletin of mathematical biology 32 (1970), S. 155-172 
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    Notes: Abstract A method is presented in this paper for the in-vivo estimation of the nonlinear pressure-volume relationship of the human aorta. The method is based on nonlinear elastic reservoir theory and utilizes clinical data that can be obtained with a high degree of accuracy, namely stroke volume, end diastolic ventricular volume and aortic pressure trace data. The computational procedure is described and then carried out for six cardiac patients. A method for the estimation of instantaneous left ventricular volume during the ejection period based on the considered nonlinear elastic reservoir theory is also presented. The method is applied for the six cardiac patients cited and the results compared with those obtained for the same subjects by a method of estimation based on linear elastic reservoir theory described in a previous paper by the author (1969).
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    The application of the minimal energy hypothesis to a casson fluid (1970)
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    Bulletin of mathematical biology 32 (1970), S. 249-262 
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    Notes: Abstract Deakin (1967b) suggested that flow of blood might obey a law of minimal energy dissipation. The present paper presents a simpler derivation of Deakin’s equations pointing out several previously unrecognized features. It is shown that these equations are unlikely to be applicable. In particular, the solution obtained by Deakin and Jones (1968) does not yield a true minimum for energy dissipation. The solution for which energy dissipation is actually minimized is shown to possess features which render it unlikely to apply to a real flow.
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    Some remarks on the Kedem-Katchalsky equations for non-electrolytes (1970)
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    Bulletin of mathematical biology 32 (1970), S. 237-247 
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    Notes: Abstract The Kedem-Katchalsky equation for the flow of a non-electrolyte through a homogeneous membrane is shown to be a first order expansion of an exact integral of the Spiegler-Bearman-Kirkwood frictional equations under the assumption that the partial frictional coefficients, ζ ij , are concentration independent. The equations are solved in terms of volume flow; there are no water-to-volume flow correction terms for the permeability, ω, or the reflection coefficient, σ. The precision of the expansion depends upon the magnitude of the water flow. The frictional coefficientsf sm andf sw are given as functions of the experimentally determined parameters ω and σ; the frictions, are shown to be independent ofL p .
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    On the average cellular volume in synchronized cell populations (1970)
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    Bulletin of mathematical biology 32 (1970), S. 459-473 
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    Notes: Abstract As was done by Sinclair and Ross (1969(, we consider a cellular population that consists initially (at time zero) ofN 0 newborn cells, all with the same volumev o. It is assumed that the occurrence of cell division is determined only by a cell’s age, and not by its volume. The frequency function of interdivision times, τ, is denoted byf(τ). If cell death is negligible, the expected number of cells,N(t), will increase according to the laws of a simple age-dependent branching process. The expression forN(t) is obtained as a sum over all generations; thevth term of this sum, in turn, is a multiple convolution integral, reflecting the life history ofvth generation cells (i.e., the lengths of thev successive interdivision periods plus the age of the cell at timet). Assuming that cell volume is a given function of cell age, e.g., linear or exponential, and that cellular volume is exactly halved at each division, it is possible to calculate the volume of a cell with a given life history, and thus the average cellular volume of the whole population as a function of time. If at time zero the volumes differ from cell to cell, the final equation must be modified by averaging over initial volumes. In the case of linear volume increase with age, a very simple asymptotic expression is found for the average cellular volume ast→∞. The case of exponential volume increase with age also leads to a simple asymptotic formula, but the resulting volume distribution is unstable. The mean cellular volume at birth and the second moment of the volume distribution can be calculated in a similar manner.
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    Dynamic analysis, with feedback characterization, of hybrid human musculoskeletal frameworks by the linegraph-flowgraph procedure (1970)
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    Bulletin of mathematical biology 32 (1970), S. 499-520 
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    Notes: Abstract The dynamic response of human musculo-skeletal framework is treated by (i) idealization of the musculo-skeletal framework as hybrid structural networks possessing feedback characteristics and then (ii) employing linegraph-flowgraph procedures for the feedback characterization of the hybrid structural networks. Topological procedures are used in which a “tree” of a network furnishes the skeleton upon which the “linkage” (muscle representing) members provide interaction. Feedback characterization (representing the sensitivity of the skeletal members to the tensile forces) is defined, between the internal “linkage” and “tree” members, by means of the flowgraph. Mikusinski operational calculus is used to facilitate representation of inertia effects by dynamic feedback characterization, with inclusion of initial conditions.
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    Organismic supercategores: II. On multistable systems (1970)
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    Bulletin of mathematical biology 32 (1970), S. 539-561 
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    Notes: Abstract The representation of biological systems in terms of organismic supercategories, introduced in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–70) is further discussed. To state more clearly this representation some new definitions are introduced. Also, some necessary changes in axiomatics are made. The conclusion is reached that any organismic supercategory has at least one superpushout, and this expresses the fact that biological systems are multistable. This way a connection between some results of Rashevsky’s theory of organismic sets and our results becomes obvious.
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    On the mass distribution model for microbial cell populations (1970)
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    Bulletin of mathematical biology 32 (1970), S. 521-537 
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    Notes: Abstract The full implications of a statistical model for growth of a microbial cell population using cell mass as the index of physiological state have been examined by solving the partial differential integral equations resulting from the model. Calculations reveal that a lag phase is predicted during the initial stages of batch growth although no specific cellular mechanism for the phenomenon of lag had been incorporated into the model. The model predicts several situations of batch and continuous growth in which the population density and biomass concentration show opposing trends due to significant variation in the cell mass distribution with time.
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    Traitement des informations chez les etres vivants: Systemes a services par essais successifs (1970)
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    Bulletin of mathematical biology 32 (1970), S. 83-148 
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    Notes: Abstract Information treatment, especially in a natural medium, may be done by a random system: given an input, possible solutions are randomly selected and successively tested until the right one is found and then emitted. This paper is a mathematical study of this type of system and its use as a model of some natural (biological) systems. The theoretical development is illustrated by several examples of possible applications to biological cases. We study, under different assumptions, the distribution of the number of trials needed to treat the input, the distribution of the time for this treatment, and the distribution of the average amount of information treated in a trial or in a unit of time. We consider the case of error, and study some optimal conditions of operation and the occurrence of memory influencing the treatment of an input according to previously processed inputs.
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    Energy of nerve impulse (1970)
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    Bulletin of mathematical biology 32 (1970), S. 151-153 
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    Comments concerning P. C. Voukydis’ “Physiological significance of the geometrical shape of the left ventricle: Course and curvature of the individual myocardial fibers” (1970)
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    Bulletin of mathematical biology 32 (1970), S. 149-149 
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    Mathematical biology of automobile driving: V (1970)
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    Bulletin of mathematical biology 32 (1970), S. 173-178 
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    Notes: Abstract 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.
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    Definition d’une distance entre deux mobiles se deplacant de facon aleatoire sur une droite (1970)
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    Bulletin of mathematical biology 32 (1970), S. 179-195 
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    Description / Table of Contents: Résumé Pour le biologiste, la notion dedistance entre deux ou plusieurs éléments d’un ensemble est très utile car elle sert à mesurer laproximité, laparenté, laressemblance qui existe entre ces éléments ou parties selon le caractère auquel on s’intéresse (position géographique, situation chronologique, aptitudes, phénotypes, composition chimique etc...).
    Notes: Abstract Adistance between two mobiles performing a random walk in one dimension is defined. At a given time this distance is directly related to theprobability of encounter for the mobiles. This definition is used when the motion of the mobiles is a Wiener-Levy process, first in the case of an unrestricted random walk, then if a reflecting or absorbing barrier is introduced.
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    Effects of anisotropy and nonhomogeneity on left ventricular stresses in the intact heart (1970)
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    Bulletin of mathematical biology 32 (1970), S. 197-213 
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    Notes: Abstract The qualitative effects of anisotropy and nonhomogeneity are considered in the evaluation of left ventricular stresses in the intact heart. Maximum stresses and their location are significantly dependent on the nonhomogeneity factors and to a lesser degree on anisotropy of the ventricular wall material. If the circumferential elastic modulus is assumed to vary in a parabolic manner through the wall thickness, maximum stresses occur within the endocardial layers, a result in qualitative agreement with experimental studies.
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    A correction for the effect of the dead space in pulmonary gas washout tests (1970)
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    Bulletin of mathematical biology 32 (1970), S. 215-218 
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    Notes: Abstract Under the assumption that there is complete mixing in the dead space a series of equations are given which give the values for a washout test of the functional component of the pulmonary bellows (without a dead space) in terms of the washout values of the bellows with a dead space.
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    Respiration of tissue as a medium of heterogeneous permeability (1970)
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    Bulletin of mathematical biology 32 (1970), S. 219-235 
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    Notes: Abstract Expressions are derived for the overall oxygen consumption and the O2 penetration depth into a tissue section in terms of the basic parameters, of the system under steady-state conditions. The approach differs from many previous analyses in so far as the oxygen molecules are regarded as reaching their sites of chemical assimilation by diffusion through extracellular fluid followed by bulk diffusion into irregular cells of significantly lower permeability. This “two-phase” model would seem to be compatible with the major experimental features of steady-state respiration, and gives a ratio of cellular to extracellular diffusion coefficients of the same low order as that found for inert gases under transient conditions. The greater oxygen penetration predicted by this model is discussed in relation to the survival of ischemic tissue and is shown to be consistent with data for myocardial infarction.
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    Analytical solutions for the specific activities of a traced substance within the component parts of a simulated kidney-ureter system (1970)
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    Bulletin of mathematical biology 32 (1970), S. 263-278 
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    Notes: Abstract Two models for a kidney-ureter system are considered: one model of one vessel in which a traced substance, undergoing exchange between the vessel and an external compartment, is emptying into the ureter; the second model of two approximately parallel, identical vessels in which a traced substance, undergoing exchange between each vessel and an external compartment, is emptying into the ureter. A single impulsive input of label into a vessel is assumed. For mathematical simplicity, the major conditions imposed on each system are: (1) rapid mixing transverse to a vessel axis and no mixing longitudinal to a vessel axis within the plasma; (2) small variation of the specific activity within the plasma in the longitudinal direction to a vessel axis; (3) constant flow rate of urine into the ureter and (4) constant exchange coefficients, tubule flow velocity and traced substance concentrations within individual compartments.
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    The motion of the left ventricle—III (1970)
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    Bulletin of mathematical biology 32 (1970), S. 279-291 
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    Notes: Abstract We use the concept of a layered wall, where each separate layer is to be homogeneous, isotropic, and incompressible, to derive stress-strain relations for the middle layer muscle ring at the transverse midsection of the left ventricle; a convenient method of formulation is that based on the elastic potential function. The hoop or circumferential stress in all three layers is found using dimensional and mechanical parameters derived earlier. The various parameters are expressed as Fourier series so that their behavior over a complete ventricular cycle is known analytically. The cases of simple elongation and what we termcurvilinear simple elongation are considered for the middle layer muscle ring strain, and the resulting stress-strain relations are derived. The results are compared with an incompressible rubber-like material known as a Mooney material.
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    Transient thermal behavior in biological systems (1970)
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    Bulletin of mathematical biology 32 (1970), S. 303-314 
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    Notes: Abstract The Peaceman-Rachford finite difference method is applied to cylindrically symmetric, transient heat conduction problems in biological media. Inhomogeneous media and internal sources which vary in both space and time are permitted. Boundary conditions are satisfied without sacrificing high local resolution by means of an exponentially stretched grid. Computation time on a Philco 2000/210 computer is approximately 5 msec per grid point per time step.
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    International Symposium on Mathematical Biology (1970)
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    Bulletin of mathematical biology 32 (1970), S. 601-601 
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    Physiological significance of the geometrical shape of the left ventricle: Course and curvature of the individual myocardial fibers (1970)
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    Bulletin of mathematical biology 32 (1970), S. 599-600 
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    A concentric layer model for estimating the energy expenditure of the left ventricle (1970)
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    Bulletin of mathematical biology 32 (1970), S. 581-598 
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    Notes: Abstract The energy cost of the left ventricle is quantitatively analyzed on the basis of the following assumptions: (1) The left ventricle is assumed to be an isotropic, homogeneous elastic, thick, spherical shell. (2) The ventricular wall is made up of a finite number of thin concentric shells. (3) The energetics of the left ventricle is in accordance with the second law of thermodynamics. An expression for the work done during ventricular contraction is derived according to the definition of physical work. The energy liberation during isovolumic contraction is formulated parallel to the concepts of heat production in skeletal muscle during isometric contraction. This expression gives the total work done per stroke in terms of mean systolic pressure, end diastolic volume, stroke volume and wall thickness during diastolic phase.
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    Ernesto Trucco (1970)
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    Bulletin of mathematical biology 32 (1970), S. i 
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    Chorioretinal temperature increases from solar observation (1971)
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    Bulletin of mathematical biology 33 (1971), S. 1-17 
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    Notes: Abstract Chorioretinal temperature increases produced by solar observations are computed digitally. The spectral characteristics of solar radiation and the spectral transmittances of the atmosphere, atmospheric water vapor and ocular media are considered. Image spread is employed in all calculations. The calculations are executed for a variety of observation angles, and the effects of pupil diameter, fixed filters and optical instruments are predicted. Temperature increases are also computed for solar eclipse observations by the appropriate superposition of unobscured solar disk solutions.
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    The kinetics of diffusion between a spherical cell and a surrounding medium with different diffusion properties (1971)
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    Bulletin of mathematical biology 33 (1971), S. 19-26 
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    Notes: Abstract The diffusion equation according to Fick’s law is solved for a spherical cell, surrounded by an infinite medium with different diffusion properties. The method of Laplace transform is used to obtain the formal solution, however, no inversion can be found for all times and an expansion suitable for small times is performed. The final expression found is expanded further to be more suited for the determination of the diffusion coefficient from an experimental curve. Application to a biological problem is dissussed.
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    A mathematical model for first degree block and the Wenckebach phenomenon (1971)
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    Bulletin of mathematical biology 33 (1971), S. 27-38 
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    Notes: Abstract A simple mathematical model of first and second degree heart block is developed on the assumption that eachR wave, with its associated ventricular contraction, results in the release of a substance which raises the excitation threshold of the conducting tissue lying between theS-A andA-V nodes. The conduction pathway is represented by a chain of excitatory elements, the first member of which is acted upon by a regularly occurringP wave. The rate constant of the removal of this inhibitory substance, i.e., recovery rate, is the only constant necessary to be varied in order to pass from normal to first degree block to second degree. In this latter case one can predict the progressive increase inP-R interval until anR wave is missed, as occurs in the Wenckebach phenomenon. The model indicates that an increasedP wave frequency which would have little effect on theP-R inerval for a normal individual with rapid recovery, could produce a Wenckebach pattern in an individual with even a mild first degree block.
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    Temperature dependence of the Elovich equation derived from activated electron or ion transport across a biological interface, with an application to muscle spindle adaptation (1971)
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    Bulletin of mathematical biology 33 (1971), S. 39-47 
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    Notes: Abstract The temperature dependence of the Elovich equation, which is−dx/dt=m exp (nx) (wherex is concentration of substrate or ion,t is time, andm andn are coefficients dependent upon temperature), has been derived from the hypothesis of electron or ion conduction across an activation energy barrier at the surface of a biological particle or membrane, driven by a difference in redox or ion potentials. Using the additional hypothesis of reversible, temperature-dependent, inactivation of sites of reduction or complexing, the theory predicts that both coefficientsm andn have linear Arrhenius plots, in agreement with experimental data for gas adsorption on inorganic solid surfaces, and with two studies of muscle spindle adaptation.
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    Some recent results inn-person game theory (1971)
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    Mathematical programming 1 (1971), S. 58-67 
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    Notes: Abstract The discussion will center mainly on some work on two solution concepts: the core for gaines without side payments and the nucleolus for games with side payments (characteristic funtion games). The core has become an important equilibrium concept in mathematical economics. The nucleolus is related to the theory of bargaining sets.
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    Publications (1971)
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    Mathematical programming 1 (1971), S. 117-120 
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    The Mathematical Programming Society (1971)
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    Mathematical programming 1 (1971), S. 123-125 
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    Hybrid programs: Linear and least-distance (1971)
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    Mathematical programming 1 (1971), S. 153-167 
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    Notes: Abstract We consider a quadratic program equivalent to the general problem of minimizing a convex quadratic function of many variables subject to linear inequality constraints. In previous work [12], one of us presented an algorithm related to such problems, and classified them under “combinatorial equivalence”. The classification contained linear programs at one end (where the function has zero quadratic part) and least-distance programs at the other. A least-distance program is a problem of finding a point of a convex polyhedron which is at least distance from a given point; such programs have been studied by one of us [15, 17] and were shown to correspond to that case of the general problem where the function has positive definite quadratic part. We now extend the work on least-distance programs to those programs intermediate between linear and least-distance (called “essentially bisymmetric” in [12]), and show that such programs are really “hybrids”, with traits inherited from both parent programs: linear and leastdistance.
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    An “out-of-kilter” algorithm for solving minimum cost potential problems (1971)
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    Mathematical programming 1 (1971), S. 275-290 
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    Notes: Abstract An algorithm is given for solving the optimum potential problem, which is the dual of the classical “out-of-kilter” algorithm for flow problems. Moreover, a new proof of finiteness is provided, which holds even for non-rational data; it applies to all the algorithms of network theory which include a labeling process.
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    On the basic theorem of complementarity (1971)
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    Mathematical programming 1 (1971), S. 68-75 
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    Notes: Abstract Using a fixed point theorem of Browder, the basic existence theorem of Lemke in linear complementarity theory is generalized to the nonlinear case.
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    An example of an ill-conditioned nlp problem (1971)
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    Mathematical programming 1 (1971), S. 113-116 
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    Matroids and the greedy algorithm (1971)
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    Mathematical programming 1 (1971), S. 127-136 
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    Notes: Abstract Linear-algebra rank is the solution to an especially tractable optimization problem. This tractability is viewed abstractly, and extended to certain more general optimization problems which are linear programs relative to certain derived polyhedra.
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    Penalty function versus non-penalty function methods for constrained nonlinear programming problems (1971)
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    Mathematical programming 1 (1971), S. 217-238 
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    Notes: Abstract The relative merits of using sequential unconstrained methods for solving: minimizef(x) subject tog i (x) ⩾ 0, i = 1, ⋯, m, h j (x) = 0, j = 1, ⋯, p versus methods which handle the constraints directly are explored. Nonlinearly constrained problems are emphasized. Both classes of methods are analyzed as to parameter selection requirements, convergence to first and second-order Kuhn-Tucker Points, rate of convergence, matrix conditioning problems and computations required.
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    Discrete optimization on a multivariable boolean lattice (1971)
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    Mathematical programming 1 (1971), S. 301-306 
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    Notes: Abstract Consider the problem of finding the minimum value of a scalar objective function whose arguments are theN components of 2 N vector elements partially ordered as a Boolean lattice. If the function is strictly decreasing along any shortest path from the minimum point to its logical complement, then the minimum can be located precisely after sequential measurement of the objective function atN + 1 points. This result suggests a new line of research on discrete optimization problems.
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    Referees 1971 (1971)
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    Mathematical programming 1 (1971), S. 376-376 
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    Die Distributivität in der Intervallarithmetik (1970)
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    Computing 5 (1970), S. 6-16 
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    Description / Table of Contents: Summary This paper deals with a characterization of special cases when the distributive law is satisfied in interval arithmetic, a question first formulated byMoore [1]. To find a solution, the basic problem is subdivided into eight cases; in three of them distributivity ever holds, while in other three cases the law is failing. The conditions concerning the remaining cases are combined to a general characterization.
    Notes: Zusammenfassung Diese Arbeit behandelt die vonMoore [1] aufgeworfene Frage nach einer Charakterisierung der Sonderfälle, für die das distributive Gesetz in der Intervallarithmetik Gültigkeit hat. Das Ausgangsproblem wird in acht Fälle unterteilt, die einzeln behandelt werden; in dreien gilt das Gesetz immer, in weiteren dreien versagt es. Die Bedingungen, die sich für die beiden restlichen Fälle angeben lassen, werden zu einer allgemeinen Charakterisierung zusammengefaßt.
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    Über einen Automaten mit Pufferspeicherung (1970)
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    Computing 5 (1970), S. 57-70 
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    Description / Table of Contents: Summary A class of automata with tape has been defined and considered as recognition devices. Our automaton can use its working tape only as a buffer, i. e. in a “first-in-first-out” manner. Some modifications of the general model are considered, especially the deterministic, real-time version. The position in theChomsky hierarchy of the class of languages accepted by such automata and some operations with these languages have been investigated. Some results on decidability questions are cited.
    Notes: Zusammenfassung Es wird ein Automat mit Speicherband eingeführt und in seiner Eigenschaft als erkennender Automat untersucht. Er kann sein Arbeitsband nur als Puffer benutzen, d. h. was zuerst eingelesen wird, wird auch zuerst wieder ausgelesen. Es werden verschiedene Modifikationen des Grundmodells, insbesondere aber die deterministische, in Realzeit arbeitende Version betrachtet, sowohl hinsichtlich der Lage der durch den Automatentyp definierten Sprachklasse in derChomsky-Hierarchie, als auch das Verhalten dieser Sprachen gegenüber gewissen Operationen. Einige Resultate über Entscheidbarkeitsfragen werden zitiert.
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    Buchbesprechungen (1970)
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    Computing 5 (1970), S. 89-96 
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    Zur Einschließung der Eigenwerte vollstetiger positiver Operatoren in separablen Hilbert-Räumen (1970)
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    Computing 5 (1970), S. 17-26 
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    Description / Table of Contents: Summary In the first part of this paper a method is given, which permits enclosing the first eigenvalue of positive completely continuous Operators in a separableHilbert space. The interval of enclosing, constructed by the orthogonal-invariants does not need arbitrary initial elements but needs knowledge of variety of the first eigenvalue. For numerical computation of the variety a method is proposed which supposes knowledge of estimation of the quotient of the first and second eigenvalue. The problem of enclosing higher order eigenvalues is treated in the second part of this paper.
    Notes: Zusammenfassung Im ersten Teil dieser Arbeit soll ein Verfahren angegeben werden, das es gestattet, den ersten Eigenwert eines positiven, vollstetigen Operators in einem separablenHilbert-Raum einzuschließen. Das mit Hilfe der Orthogonalinvarianten konstruierte Einschließungsintervall bedarf keinerlei willkürlicher Anfangselemente, benötigt aber die Kenntnis der Vielfachheit des ersten Eigenwertes. Zur numerischen Berechnung der Vielfachheit wird ein Verfahren vorgeschlagen, das die Kenntnis einer Abschätzung des Quotienten aus dem ersten und zweiten Eigenwert voraussetzt. Die Einschließung der höheren Eigenwerte wird im zweiten Teil der vorliegenden Arbeit behandelt.
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    Algorithmus 12 Ein ableitungsfreies Verfahren zur Lösung endlich-dimensionaler nichtlinearer Gleichungssysteme (1970)
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    Computing 5 (1970), S. 82-88 
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    Zur Einschließung der Eigenwerte vollstetiger positiver Operatoren in separablen Hilbert-Räumen (1970)
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    Computing 5 (1970), S. 119-127 
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    Description / Table of Contents: Summary The construction of intervals of enclosing for higher order eigenvalues of positiveHilbert-Schmidt operators in a separableHilbert space is reduced to the problem of the interval of enclosing for the first order eigenvalue of a positiveHilbert-Schmidt operator.
    Notes: Zusammenfassung Die Konstruktion von Einschließungsintervallen für die höheren Eigenwerte positiverHilbert-Schmidtscher Operatoren in separablenHilberträumen wird auf die Aufgabe zurückgeführt, ein Einschließungsintervall für den ersten Eigenwert eines positivenHilbert-Schmidtschen Operators anzugeben.
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    Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle (1970)
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    Computing 5 (1970), S. 184-184 
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    Zur Mehrdeutigkeit bei der Approximation in der L p -Norm mit Hilfe rationaler Funktionen (1970)
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    Computing 5 (1970), S. 349-355 
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    Description / Table of Contents: Summary In 1967Cheney andGoldstein presented a paper on mean square approximation by generalized rational functions. The problem of uniqueness of the best approximation remained unsolved. In this paper it is shown, that there may be several different rational functions which are bestL p -approximation (1≤p〈∞) of a given function.
    Notes: Zusammenfassung In einer Arbeit vonCheney undGoldstein aus dem Jahre 1967 wurden die Probleme untersucht, die sich im Zusammenhang mit der rationalen Approximation einer gegebenen stetigen Funktion in derL 2-Norm stellen. In der vorliegenden Arbeit soll unter anderem die vonCheney undGoldstein offengelassene Frage der Eindeutigkeit der besten rationalen Approximation untersucht werden.
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    Die Frequenzkennlinien (1970)
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    Computing 5 (1970), S. 394-394 
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    Domains of uncertainty for perturbed operator equations (1970)
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    Computing 5 (1970), S. 312-323 
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    V. Internationaler Kongreß über Anwendungen der Mathematik in den Ingenieurwissenschaften in Weimar (V. IKM) (1970)
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    Computing 5 (1970), S. 332-332 
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    Ableitungsfreie Verfahren ohne Auflösung linearer Gleichungen (1970)
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    Computing 5 (1970), S. 71-81 
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    Description / Table of Contents: Summary Iteration methods are given for the solution of nonlinear equations in normed spaces requiring functionvalues and first-order divided differences. The latter may be replaced by derivations. The linear equations of each iteration step are solved only approximatively but this does not diminish the order of convergence.
    Notes: Zusammenfassung Zur Lösung von nichtlinearen Gleichungen in normierten Räumen werden Iterationsverfahren angegeben, die Funktionswerte und Steigungen erster Ordnung benutzen. Es ist möglich, die Steigungen durch Ableitungen zu ersetzen. Die linearen Gleichungen, die in jedem Iterationsschritt auftreten, werden nur näherungsweise gelöst. Dabei ist es wichtig, daß hierdurch keine Verringerung der Konvergenzgeschwindigkeit eintritt.
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    Mechanisierung von Dreieckskonstruktionen (1970)
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    Computing 5 (1970), S. 128-135 
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    Description / Table of Contents: Summary The intelligent behavior of a highschool student, who constructs triangles from three given pieces shows that he generates auxiliary points of a tree-like structure by means of geometric loci. The formalization of this leads to a system of points, geometric loci and rules. In each case the mechine is able to decide whether and how the triangle can be contructed and this leads to the decision-procedure we called the „Dreiecksalgorithmus”.
    Notes: Zusammenfassung Analysiert man das Verhalten eines intelligenten Schülers bei der Konstruktion eines Dreiecks aus drei gegebenen Stücken, so erkennt man, daß er mittels geometrischer Örter Hilfspunkte eines baumartigen Gebildes erzeugt. Eine Formalisierung dessen führt auf Listen von Punkten, geometrischen Örtern und auf ein Regelsystem. Die Maschine kann damit in jedem Fall entscheiden, ob und wie das Dreieck aus den gegebenen Stücken konstruierbar ist, und das ist das Entscheidungsverfahren, das wir „Dreiecksalgorithmus” genannt haben.
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    Topological invariants of matrices (1970)
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    Computing 6 (1970), S. 97-103 
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    Description / Table of Contents: Zusammenfassung Es wird eine Methode zur Definition und Charakterisierung der Zusammenhangs- und Repetitionsstruktur von Matrizen vorgeschlagen.—Im Anschluß an die Definition des “Vokabulars” der einzelnen Zeilen (Spalten) werden die numerischen Invarianten (Betti-Zahlen) eines simplizialen Komplexes bestimmt, der als der sogenannte “Nerv” des Systems der Zeilen-(Spalten-)Vokabulare erhalten wird. Die Berechnung erfolgt über die Inzidenzmatrizen des Nervs. Einige Hinweise auf Anwendungsmöglichkeiten und ein elementares Beispiel beschließen die Note.
    Notes: Summary A method is proposed for characterising the connectivity- and repetitivity-structure of matrices. After defining the “vocabularies” of the single rows (columns) we calculate the numerical invariants (Betti-numbers) of a simplicial complex, the “nerve” of the system of vocabularies, via incidencematrices.
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    Über eine Klasse komplexer Intervallgleichungssysteme mit iterationsfähiger Gestalt (1970)
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    Computing 6 (1970), S. 104-106 
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    Description / Table of Contents: Summary In two previous papers [3] and [4] were deduced some statements for certain interval equations over the real number field concerning the existence and uniqueness of the solution and the convergence of iteration methods for the evaluation of the solution. This paper shows, how these statements and their proofs can be extended to analogous interval equations over the complex number field.
    Notes: Zusammenfassung In früheren Arbeiten [3] und [4] wurden für gewisse Intervallgleichungssysteme über dem Körper der reellen Zahlen Kriterien für die Existenz und Eindeutigkeit einer Lösung sowie die Konvergenz von Iterationsverfahren zur Bestimmung der Lösung bewiesen. Die vorliegende Arbeit zeigt, wie diese Kriterien und ihre Beweise zu verallgemeinern sind, damit sie auch für entsprechende Intervallgleichungssysteme über den komplexen Zahlen gelten.
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    Ein kontinuierliches Modell für Kraftfahrzeugströmungen (1970)
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    Computing 6 (1970), S. 107-120 
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    Description / Table of Contents: Summary Mathematical description of traffic flow has been tried repeatedly. Two types of theories are to be distinguished: Discrete traffic flow theories start from a system of differential equations for functionsx n (t) describing the routes of single cars. Continuous flow models require solution of a partial differential equation for the velocityv (x, t) and determination of trajectoriesx (t; c) of the velocity field thus obtained. In this paper these integrations for general initial and boundary conditions of a continuous flow model are performed. Treatment of a special case is added, yielding a flexible graphic method for solution of traffic flow problems. As an example of application, a frequently occurring traffic situation is analysed with this theory.
    Notes: Zusammenfassung Die Beschreibung des Kraftfahrzeugverkehrs in mathematischer Form ist mehrfach versucht worden. Es lassen sich zwei Typen von Ansätzen unterscheiden: Die diskreten Fahrzeugfolgetheorien gehen von einem Differentialgleichungssystem für die Bahnkurvenx n (t) der einzelnen Fahrzeuge aus. Die kontinuierlichen Strömungsmodelle verlangen die Lösung einer partiellen Differentialgleichung für die Strömungsgeschwindigkeitv (x, t) sowie die Ermittlung der Trajektorienx (t; c) im so entstandenen Geschwindigkeitsfeld, die als Bahnkurven der Fahrzeuge interpretiert werden. In der vorliegenden Arbeit werden für ein kontinuierliches Modell die genannten Integrationen bei allgeneinen Anfangs- und Randbedingungen geschlossen durchgeführt. Daran schließt sich die Behandlung eines Spezialfalles an, aus dem sich ein flexibles graphisches Verfahren zur Lösung von Kraftfahrzeugströmungsproblemen gewinnen läßt. Als Beispiel einer Anwendung wird ein im Verkehr häufig auftretender Effekt mit Hilfe dieser Theorie graphisch untersucht.
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    Investigation of a certain stray process applied to the optimum synthesis problems (1970)
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    Computing 6 (1970), S. 139-160 
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    Description / Table of Contents: Zusammenfassung Es wird ein neues Stichprobenverfahren vorgeschlagen, das zur Untersuchung von zulässigen Gebieten in der Optimalsynthese dient. Die Methode ist gekennzeichnet durch einen größeren Wirkungsgrad der Stichproben und bevorzugt die Grenze und ihrer Nachbarschaft gegenüber dem Inneren des zulässigen Gebietes. Der Algorithmus des Irren-Prozesses und die Untersuchungsresultate werden beschrieben. Die Anwendung wird durch Beispiele illustriert. Die Resultate bei der Anwendung verschiedener Methoden werden verglichen. Die verwendeten Rechenautomaten waren GIER und ZAM-2 Computer.
    Notes: Summary A new method is proposed for the random checking of the region of admissible solutions of synthesis problems by means of digital computers. This method is characterized by increasing efficiency of random checking and by preference of the boundary and its neighbourhood to the interior of the region. The algorithm of the stray process is described. The paper contains also the test results of the influence of various parameters on the process. The application of the method is illustrated by examples. The results obtained by different methods are confronted. The computers used were the GIER and ZAM-2 computers.
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    Über die aus monoton zerlegbaren Operatoren gebildeten Iterationsverfahren (1970)
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    Computing 6 (1970), S. 161-172 
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    Description / Table of Contents: Summary By means of monoton decomposable operators,Collatz, Albrecht andSchröder have considered a variety of iterative methods, by which they get inclusion sets for the solution of system of simultaneous linear or nonlinear equations for example. It is shown, that these methods are special cases of basic iterative methods for equations with interval coefficients. In the concluding part of this paper a general convergence theorem in aRieszian-space is given.
    Notes: Zusammenfassung Vor allem vonCollatz, Schröder undAlbrecht wurden mit Hilfe des Begriffes des monotonen Operators verschiedene Iterationsverfahren betrachtet, welche Einschließungsmengen zum Beispiel für die Lösung eines linearen oder nichtlinearen Gleichungssystems liefern. Im folgenden wird der Zusammenhang zwischen diesen Verfahren und der Intervallrechnung hergestellt. Es zeigt sich, daß diese Verfahren als Spezialfälle in den auf Intervallbasis gebildeten Iterationsverfahren enthalten sind. Im abschließenden Teil der Arbeit wird ein allgemeiner Konvergenzsatz in einemRieszschen Raum angegeben.
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    A finite difference method for analyzing the compression of poro-viscoelastic media (1970)
    Springer
    Computing 6 (1970), S. 342-348 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Das Zunehmen der Kompression von wassergesättigten, porösen Medien läßt sich aus der Kontinuitätsgleichung, demDarcyschen Gesetz und einer passenden Druck-Ausdehnungsbeziehung herleiten. Theorien einer sekundären Verdichtung nehmen an, daß die tatsächliche Druck-Ausdehnungsbeziehung zeitabhängig ist. Eine Form dieser Charakterisierungsart ist ein System von linearen, viscoelastischen Modellen. Das ausgewählte Model besteht aus einem elastischen Element in Serie mit einer beliebigen Anzahl vonKelvin-Einheiten. Formuliert wird dieses System als Differential-Integral-Gleichung. Es wird ein endliches Differenzenschema entwickelt, in dem die beherrschende gewöhnliche Differential-Integralgleichung in ein Gleichungssystem übergeführt wird, das aus WärmeleitungsundEuler-Gleichungen besteht. Ein Stabilitätstheorem wird bewiesen.
    Notes: Summary The progress of compression of a water saturated porous medium is derivable from the equation of continuity,Darcy's law, and an appropriate effective stressdilation relationship. Theories of secondary consolidation assume that the effective stress-dilatation relationship is time-dependent. One form of this type of characterization is a system of linear viscoelastic models. The system chosen consists of an elastic element in series with an arbitrary number ofKelvin units. The formulation of this system is a differential-integral equation. The integral portions of the equation are a series of convolution integrals. A finite difference scheme is developed in which the single governing differential-integral equation is broken up into a system of equations of the heat conduction andEuler types. A stability theorem is proved.
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    Sur une propriété de transfert concernant les boules de Rn (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 1-5 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01298295
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    Modules over commutative Banach algebras (1970)
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    Monatshefte für Mathematik 74 (1970), S. 6-14 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    Über Treffanzahlen in Figurengittern (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 21-29 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01298298
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    Über Wahrscheinlichkeiten räumlicher Bündelungserscheinungen (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 30-40 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01298299
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    Anwendung der Störungstheorie auf den Vertauschungssatz der Liereihen (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 15-20 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01298297
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    Nullstellen bei Lösungen komplexer linearer Differentialgleichungen 2. Ordnung (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 41-49 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01298300
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    Algebraische Gleichungen über universalen Algebren (1970)
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    Monatshefte für Mathematik 74 (1970), S. 50-55 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01298301
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    Über Unterbäume eines Baumes (1970)
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    Monatshefte für Mathematik 74 (1970), S. 56-62 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01298302
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    Zur Linearisierung analytischer Differentialgleichungssysteme in der Nähe einer Gleichgewichtslage (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 63-69 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    Trennung und Zusammenhang (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 70-87 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    Der Vierscheitelsatz in Relativ- und Minkowski-Geometrie (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 97-107 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01299032
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    Buchbesprechungen (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 88-96 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    Ein metrischer Satz in der Theorie der C-Gleichverteilung (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 108-118 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01299033
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    Die Doppeltangenten und Haupttangenten des Torus (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 119-137 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01299034
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    Fields of planar bodies tangent to spheres (1970)
    Springer
    Monatshefte für Mathematik 74 (1970), S. 145-149 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01299036
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