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  • Articles  (4,070)
  • 1980-1984
  • 1965-1969  (4,070)
  • 1925-1929
  • 1967  (4,070)
  • Mathematics  (4,070)
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  • 1980-1984
  • 1965-1969  (4,070)
  • 1925-1929
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  • 1
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    Bulletin of mathematical biology 29 (1967), S. 1-16 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A vast number of biologically important processes are based upon bimolecular systems. In these systems intermediate complexes are formed. Bimolecular systems in which no complex-complex interactions occur are called linear systems of complexes. A definition and some characteristic properties of these systems are given here. There may exist a contradiction of Onsager's principle of detailed balancing in these systems; however, no principal differences are found between the steady state behavior of an open system and that of a closed system. It is shown that the steady state behavior of a linear system of complexes of arbitrary complexity has some similarities with the steady state behavior of a simple bimolecular system, e.g., Michaelis-Menten enzymatic reaction. Multiplicity of action of the substances participating in biomolecular processes may produce some qualitative differences in the steady state behavior of the system.
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  • 2
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    Bulletin of mathematical biology 29 (1967), S. 17-32 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A time-dependent DNA histogram is calculated for an irradiated population of cells under the limiting assumption that the cells cannot pass through prophase due to the effects of the radiation. The population is assumed to increase exponentially prior to irradiation, but after irradiation to neither gain nor lose cells. Chromosome-number dispersion is taken into account in the calculation. The qualitative behavior of the calculated and experimental histograms are in reasonable agreement. The quantitative agreement between the two is relatively good at short post-irradiation times but is poor at long post-irradiation times (say, greater than half the doubling time). This suggests that recovery phenomena cannot be neglected at long post-irradiation times.
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  • 3
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    Bulletin of mathematical biology 29 (1967), S. 187-188 
    ISSN: 1522-9602
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    Notes: Abstract 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.
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    Bulletin of mathematical biology 29 (1967), S. 181-186 
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    Notes: Abstract 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.
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  • 5
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    Bulletin of mathematical biology 29 (1967), S. 245-259 
    ISSN: 1522-9602
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    Notes: Abstract The principle of minimal work requires that the conducting airways of the human lung should have a maximum radius for minimal resistance to gas flow. At the same time there is a requirement that the airways should have a minimal volume for economy of space. These two opposing requirements have been investigated mathematically, and a method for calculating the angle of branching which produces minimal volume has been derived. The relationship of the radii of the parent and daughter branches to produce minimal resistance has been similarly defined. By measurement of a bronchial cast from a human lung the extent to which the predicted optimum structure is realized in practice has been shown. The change in structure associated with change of function at the transition from conducting airway to diffusion zone has been demonstrated.
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  • 6
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    Bulletin of mathematical biology 29 (1967), S. 191-206 
    ISSN: 1522-9602
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    Notes: Abstract This paper considers a class of set-theoretical entities, calledn-rank Linnaean structures, which are intended as abstract models of the taxonomic classificatory systems of biology. In the first part, devoted to formalism, finite Linnaean structures are discussed in complete generality; but, in addition, eight distinct subclasses are noted and some of the properties of their elements are explored. In the second part, concerned with applications, it is shown that taxonomic systems may be recast in the form of finite Linnaean structures, and an effort is made to show that some undesirable features of earlier models are avoided without artificiality and without abandoning extensional mathematics.
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  • 7
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    Bulletin of mathematical biology 29 (1967), S. 207-216 
    ISSN: 1522-9602
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    Notes: Abstract Generalizing an idea of M. Richardson (Fundamentals of Mathematics, New York: Macmillan Co., 1958), an APS on a given populationP is a non-empty collection of non-empty subsets ofP such that ifA is in the collection andA⊆B, thenB is in the collection. From a structure of this kind a partial ordering ofP, called therelated bumping order, is derived. The question is raised as to what kinds of partial orderings can be so obtained. For structures determined by voting weights of the members of the population, a complete characterization of all possible bumping orders is obtained.
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  • 8
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    Bulletin of mathematical biology 29 (1967), S. 217-226 
    ISSN: 1522-9602
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    Notes: Abstract The “second method” of Liapunov is used to perform a stability analysis of a mathematical model of the neuron. This analysis is based on the hypothesis that the firing of the neuron coincides with a temporary state of instability of the system, and that the initiation of all-or-none process depends on the magnitude of membrane depolarization and its first time derivative. It is found that the stability (and hence the possibility of a second firing) is restored approximately when the rate of membrane repolarization is at a maximum. This result predicts that the duration of the period of absolute refractoriness in neurons would be about 75 per cent of the spike duration, and thus shorter than the value usually obtained from experimental measurements.
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  • 9
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    Bulletin of mathematical biology 29 (1967), S. 227-232 
    ISSN: 1522-9602
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    Notes: Abstract Some aspects of masking phenomena are considered in terms of the simplest possible model of two-factor neural elements. The effect of a number of variables can be accounted for, but the introduction of an internuncial element results in a masking function which need not be symmetric about zero delay interval. As an illustration, the results for a special case are compared with available data. In general, such a model results in a masking function which depends on the intensity, area, and duration of the stimuli, as well as on the temporal and spatial separation between them.
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    Bulletin of mathematical biology 29 (1967), S. 377-388 
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    Notes: Abstract The general equations are discussed describing two species in competition or in symbiosis or feeding one on the other.
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    Bulletin of mathematical biology 29 (1967), S. 403-404 
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    Bulletin of mathematical biology 29 (1967), S. 389-393 
    ISSN: 1522-9602
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    Notes: Abstract It is shown that the principle of biological epimorphism (Rashevsky,Mathematical Principles in Biology and Their Applications, Springfield, Ill.: Charles Thomas, 1960) is contained in the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967) if an additional postulate not directly connected to mappings is made.
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  • 13
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    Bulletin of mathematical biology 29 (1967), S. 407-407 
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  • 14
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    Bulletin of mathematical biology 29 (1967), S. 409-409 
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    Bulletin of mathematical biology 29 (1967), S. 605-613 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract This paper deals with bimolecular systems in which also complex-complex interactions occur. Because of the complexity of the problem, an approximation in a form of coupled linear systems of complexes (Bull. Math. Biophysics,29, 1–16, 1967) is considered. Two types of couplings, serial and parallel, are studied. In the serial coupling the nonlinear system of complexes has the same behavior as its subsystems. An entity, initial sensitivity, has interesting properties: in serial coupling it is at most equal to the product and in parallel coupling, at most equal to the sum of partial initial sensitivities.
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    Bulletin of mathematical biology 29 (1967), S. 615-623 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Amplification effect in the catalytic bimolecular systems is a consequence of the kinetic characteristic of the catalyst. Two types of the coefficient of amplification are defined. The applicability of these definitions is given by the type of the bimolecular system. In a simple example it is shown that the concept of amplification is meaningful in these systems. Furthermore, two rules, analogous to those for a coupling of amplifiers, are derived for the two basic modes of coupling of catalytic systems. Thus, in biological systems the catalytic reactions may be regarded as biologically effective amplifiers.
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    Bulletin of mathematical biology 29 (1967), S. 583-596 
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    Notes: Abstract It is postulated that cell hydration is governed by adsorption of water on cell proteins in accord with the Bradley adsorption isotherm, and that the action of a solute in the surrounding solution is to lower the vapor pressure of the solution so that cell water adsorption is decreased by moving down the Bradley isotherm. From these concepts, it is derived that cell volume (V) should be related to solute concentration (x) by the equationV=−E log10 x+F whereE andF are constants which are independent of type of solute. For a non-adsorbed solute this agrees well with experimental data. For solutes which are adsorbed by cell proteins, a correction in the above equation may be necessary at higher solute concentrations, which is shown to be compatible with various experimental data. The types of experiments which are generally used to support the osmotic pressure theory of cell hydration agree equally well with the adsorption theory. The virtue of the adsorption theory is that, unlike the osmotic pressure theory of cell swelling, it is compatible with permeability of the cell membrane to solutes, which has been experimentally observed for various solutes.
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    Bulletin of mathematical biology 29 (1967), S. 657-664 
    ISSN: 1522-9602
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    Notes: Abstract Conditions under which a time varying electromagnetic field problem (such as arises in electrophysiology, electrocardiography, etc.) can be reduced to the conventional quasistatic problem are summarized. These conditions are discussed for typical physiological parameters.
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  • 19
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    Bulletin of mathematical biology 29 (1967), S. 711-718 
    ISSN: 1522-9602
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    Notes: Abstract A compartmental lung model with any number of synchronously filling and emptying functional chambers and a common dead space or conducting region is considered. It is shown that the model gives rise to an output, in an open circuit washout determination, which is a weighted sum of exponentials. From estimates of these weights and exponential components, estimates of the model parameters can be recovered. Relations giving the unique correspondence between the output parameters and the model parameters are derived and the existence and uniqueness of solutions established.
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    Bulletin of mathematical biology 29 (1967), S. 677-690 
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    Notes: Abstract A physical model that incorporates all the experimental information on the formation of the visual pigment rhodopsin is presented. The visual pigments consist of a chromophore bound to an appropriate protein. Thus rhodopsin (λm 505 mμ) is formed by a Schiff’s base linkage C19H27CH=NH+-opsin (λm 440 mμ) between 11-cis retinal (λm 380 mμ) and the protein opsin (λm 280 mμ). It is found that there exists a red shift in the spectrum of rhodopsin from the Schiff’s base. The model brings an explanation for this red shift. It is shown that such a shift may be due to a charge transfer process (R. S. Mulliken,J. Am. Chem. Soc.,74, 811–824, 1952) between an electron at the double bond of carbons C11−C12 and an atomic orbital of the sulphur present in cysteine. This provides an explanation of the presence of SH-groups in the protein after the absorption of light. A one-electron approximation is used and the dipole momentμ NV ; hence, the oscillator strengthf of the transitionNV is estimated and compared with the experimentally determined extinction coefficient ∈m by mixing 3.5×10−3 M of 11-cis retinal with 8.3×10−5 M of cysteine at pH ranges 6 through 8. Reasonable agreement is found. Solvent, concentration and temperature dependence are shown also.
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    Bulletin of mathematical biology 29 (1967), S. 841-862 
    ISSN: 1522-9602
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    Notes: Abstract By assigning coordinates to the environmental function space comprising all physical and mental stimuli, mathematical interpretations can be based on such terms as adaptability, and reactivity which relate to individuals interacting with their environment within a society. These psychometric concepts are incorporated into a framework of functional analysis, which permits the optimization of social change by maximizing the satisfaction integral through the use of variational or dynamic programming methods in conjunction with some optimal social policy. The approach provides a mathematical connection between psychology and sociology, and further demonstrates that existing forms of government are simulated by differential equations belonging to the same general class. The synthesis of new classes of functional equations describing social progress is visualized as a legitimate objective for abstract mathematical sociology.
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    Bulletin of mathematical biology 29 (1967), S. 863-877 
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    Notes: Abstract The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.
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    Bulletin of mathematical biology 29 (1967), S. 233-243 
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    Notes: Abstract A neuron is assumed to receive synaptic input of both excitatory and inhibitory natures from a large number of neighboring neurons; it is also assumed that a large number of such impulses are required to raise the neuron’s transmembrane potential to its threshold potential, at which it “fires” or “spikes”. The model is similar to one of Gerstein and Mandelbrot, except that in the absence of input an exponential decay of potential toward a resting level is introduced. Computational methods of determining the spike timeinterval distribution are discussed, along with the inverse problem of estimating the parameters of the system from observed spike time-interval data.
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    Bulletin of mathematical biology 29 (1967), S. 311-318 
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    Notes: Abstract Moens-Korteweg relations are developed for the velocity of wave propagation through an orthotropic elastic tube based on the three dimensional equations of elasticity. Numerical examples are presented for the femoral artery of a dog and several other orthotropic materials. These results are compared with those obtained from the equations of motion for an orthotropic elastic medium.
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    Bulletin of mathematical biology 29 (1967), S. 335-341 
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    Notes: Abstract Up to the present time, the following property of the product component in the reversible one substrate-one intermediate-one product enzymic mechanism has been taken only as anassumption, viz., during the course of the reaction, the time-rate of change of product concentration is never negative and the product concentration never exceeds its equilibrium value. Applying the methods of the geometric theory of ordinary differential equations it is shown that this result follows as a direct deduction from the differential equations governing the mechanism together with the initial conditions. Further, the nature of the equilibrium point as a stable node is established.
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    Bulletin of mathematical biology 29 (1967), S. 319-333 
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    Notes: Abstract An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.
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    Bulletin of mathematical biology 29 (1967), S. 405-405 
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    Bulletin of mathematical biology 29 (1967), S. 395-401 
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    Notes: Abstract In continuation of previous work (Rashevsky,Some Medical Aspects of Mathematical Biology, Springfield, Ill.: Charles C. Thomas, 1964, Chap. 23 and Appendix 14), the study of the effects of the physical parameters of the cells of endocrine glands on the onset of sustained periodical oscillations in the interaction between the anterior pituitary and the thyroid hormones is generalized to include the possible effect of the intercellular fluid and of the degree of vascularization. Some conclusions of the previous study remain valid although some modifications must be made. A decreased relative volume of the intercellular fluid and an increased vascularization favor the conditions for sustained oscillations. The permeability of the cells and the permeability of the capillaries appear explicitly in the expressions which show the conditions for sustained periodicities.
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    Bulletin of mathematical biology 29 (1967), S. 437-449 
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    Notes: Abstract A general model of a large 2m-ploid breeding population, withr loci ands h alleles at the h th locus is considered. It is assumed that the population is bisexual, non-overlapping and breeds by random mating. The genotypic structure of the population is presented as a bilinear form in the gametic output vectors where the genotype distribution is in the matrix form. Using the concept of the segregation distribution, the genotype proportions in the (n+1)st generation are given. An equilibrium condition for random chromosome segregation is obtained in terms of gene frequencies.
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    Bulletin of mathematical biology 29 (1967), S. 541-548 
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    Notes: Abstract The rigidity of the skull and the inertial characteristics and incompressibility of its contents cause the elastic cerebral arteries and veins to act over brief periods of time like rigid tubes of relatively small diameter. Poiseuille's law is applicable to their behavior. The use of this law, in combination with the fact that, during brief intervals, the total volume of the cerebral arteries and veins remains constant, permits derivation of a mathematical expression for the average arterial flow in terms of an average arterial radius. The differentiated equation has five positive roots which represent maxima and minima of the average flow in terms of the average arterial radius. The theoretical results have physiological implications and potential clinical usefulness, which are discussed.
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    Bulletin of mathematical biology 29 (1967), S. 549-563 
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    Notes: Abstract Equations are derived for the flow of an anisotropic fluid in a tube. It is argued that these provide a model for arterial blood flow. Particular attention is paid to the effect of radial differences in hematocrit. Sequels to this paper (Bull. Math. Biophysics,29, 565–574; forthcoming, 1967) will respectively demonstrate possible wall-directed forces on the erythrocyte and enlarge on the physiological consequence of hematocrit variations. The present article develops the basic equations and explores the possible role of anisotropic effects in blood flow.
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    Bulletin of mathematical biology 29 (1967), S. 575-581 
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    Notes: Abstract Concentric circular lines are present on X-ray diffraction patterns of DNA. They cannot be accounted for by the double helix model. It is suggested that they may indicate the presence of double helical side chains.
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    Bulletin of mathematical biology 29 (1967), S. 781-791 
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    Notes: Abstract An extension of an earlier model simulating the effects of light on the drosophila eclosion rhythm is presented. The effects of variable light intensity are described. This allows not only the simulation of certain experiments not covered by the earlier model, but also it permits an extension of the model to other organisms. By changing only its sensitivity to light the model simulates the phase response curves of certain mammals as well as Aschoff’srule.
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    Bulletin of mathematical biology 29 (1967), S. 827-829 
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    Notes: Abstract Previous derivations of the Stevens Power Law relating loudness to intensity assume, in addition to level invariance, that the relation must be analytic or at least differentiable. This last condition is replaced here by the weaker one of requiring only continuity.
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    Bulletin of mathematical biology 29 (1967), S. 33-40 
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    Notes: Abstract A method of analyzing thymidine labeling in a population of cells is formulated. The formulation establishes a unique relation between a specific set of labeling data and a specific set of cells in the population, viz. that set of cells having a particular chromosome number. The analysis employs a cell-state variable, i.e., a quantity which specifies the progress of a cell through its lifecycle. This variable is defined in terms of the nucleo-protein content and configuration of the chromosomes. The relation mentioned above leads immediately to an expression for the number of cells present at a particular time following labeling which have a given amount of label per cell and a given chromosome number.
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    Bulletin of mathematical biology 29 (1967), S. 41-56 
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    Notes: Abstract An equation relating radiation-induced metaphase delay to the dose-rate and duration of irradiation is obtained. The equation is derived from a model specifying the effects of radiation on the normal chromosome coiling process. The basic assumptions of the model are (1) that normal coiling proceeds by contractile protein acting on segements of a viscoelastic chromosomal fiber; (2) that radiation causes cross-linking of adjacent chromosomal fibers which hinders the coiling process.
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    Bulletin of mathematical biology 29 (1967), S. 57-65 
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    Notes: Abstract Normal micturition is controlled primarily by a neural system. Certain physical effects become evident when neural control is destroyed, and the automatic or autonomous bladder phenomena occur. It is shown in this paper that a physical system simulating the alternating periods of continence and voiding of the automatic bladder may comprise only passive elastic components, and that periodic voiding does not per se imply neural control.
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    Bulletin of mathematical biology 29 (1967), S. 91-94 
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    Notes: Abstract A number of inaccuracies in previous papers are pointed out and amended, and some implications of the correct situation are outlined.
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    Bulletin of mathematical biology 29 (1967), S. 67-89 
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    Notes: Abstract We investigate a model of the renal medulla in which active NaCl transport is restricted to the thick ascending limb of Henle's loop. The model contains a vas rectum, a loop of Henle, salt, and water. The model generates interstitial osmolality curves consonant with the known functioning of the kidney in water diuresis. Using data from the white rat and the curves generated by the model, one can predict the permeability of the thin limb of Henle's loop to NaCl and the percentage of total renal blood flow entering the inner medulla. In this model interstitial osmolality at the papilla can be about twice plasma osmolality, so that NaCl transport restricted to the outer medulla can contribute significantly to the work required in producing a hypertonic urine. However, the interstitial osmolality monotonically decreases proceeding from the junction of the outer and inner medulla to the papilla, and the maximum interstitial osmolality in the outer medulla is greater than the maximum interstitial osmolality in the inner medulla. Thus we infer that a source of active transport located in the inner medulla is needed to explain the high osmolalities observed in hydropenia. A sketch of an alternative model, a “lineal multiplication mechanism”, for the renal concentrating process is presented in which active transport in the inner medulla is restricted to active salt transport by the collecting duct. The lineal multiplication mechanism makes no use of counter-current multipliers in the inner medulla.
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    Bulletin of mathematical biology 29 (1967), S. 95-121 
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    Notes: Abstract Starting from the basic flux equation, it is possible to obtain an integral form relating the current componentsI i at an arbitrary pointr 2 to the distribution of mobilities and concentrationsc i, potential forces $$\bar \mu $$ , and chemical productivityp i without any restrictive assumptions such as constant mobilities, constant field, steady state, or electrical neutrality. The equation is $$\begin{gathered} I_i (r_2 ) = G_i (r_2 )\left[ {\Delta \bar \mu _i - \int_{r_1 }^{r_2 } {z_i } FA\left( {p_i - dc_i /dt} \right)\left( {\frac{1}{{G_i (r)}}} \right)dr} \right]; \hfill \\ G_i (r) = 1/\int_{r_1 }^r {\frac{{dr}}{{z_i^2 F^2 c_i u_i }}.} \hfill \\ \end{gathered} $$ On the basis of this equation, it is possible to give a more general and systematic development of the basic equations of electrophysiology which clarifies a number of questions concerning the physical interpretation of and the necessary and sufficient conditions for the applicability of some of the standard equations and gives their proper extensions to more general conditions. It is found that the relation between the current components and chemical reactions present arises in a very natural way via the continuity equation and enables one to discuss the incorporation of the metabolic and active transport parameters by assuming a very general physical condition. On the basis of this general integration technique one may then compare the physical interpretation of the differential conductance, the chord conductance and the integral conductance.
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    Notes: Abstract Previous papers by F. M. Snell (Jour. Theor. Biol.,8, 469–479, 1965) and M. A. Fox and H. D. Landahl (Bull. Math. Biophysics,27, Spec. Issue, 183–190, 1965) have found that the formulation by previous authors for the oxygen flow rates through hemoglobin solution as a function of pressure determined by E. Hemmingsen and P. F. Scholander (Science,132, 1379–1381, 1960) did not give a satisfactory quantitative fit of the curve for constant pressure difference. The suggestion of Fox and Landahl that the Bohr effect involving the shift in acidity accompanying the oxidation of Hb should give rise to voltage and pH differences in oxyhemoglobin transport is examined in more detail. In this paper, the previous expressions for the total oxygen flow rate in terms of the end point concentrations are extended to include the effects of the electrical field. Estimates of the potential difference shows it to be negligible. A derivation of a voltage-pH relation shows that the Nernst relation does not apply and a negligible voltage difference does not preclude a pH shift which is the more probable explanation of the discrepancies observed. Several other predictions suitable for experimental testing are made.
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    Bulletin of mathematical biology 29 (1967), S. 153-174 
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    Notes: Abstract A model for the human eye-movement mechanism is derived. The derivation is based on a literature search directed toward identifying and mathematically describing each component through physiological and anatomical considerations. It is felt that although certain parameter values may not be exactly correct (for the data were taken from a wide variety of animals), we can place a great deal of confidence in the configuration.
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    Bulletin of mathematical biology 29 (1967), S. 175-179 
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    Notes: Abstract The urethra as seen on X-ray films may show alternate regions of constriction and distension. That these regions do not necessarily correspond to high and low tensions in the circumferential muscle sheath is shown by calculated stable configurations under uniform tension.
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    Bulletin of mathematical biology 29 (1967), S. 139-152 
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    Notes: Abstract The discussion as to whether societies are organisms andvice versa has been going on for a long time. The question is meaningless unless a clear definition of the term “organism” is made. Once such a definition is made, the question may be answered by studying whether there exists any relational isomorphism between what the biologist calls an organism and what the sociologist calls society. Such a study should also include animal societies studied by ecologists. Both human and animal societies are sets of individuals together with certain other objects which are the products of their activities. A multicellular organism is a set of cells together with some products of their activities. A cell itself may be regarded as a set of genes together with the products of their activities because every component of the cell is either directly or indirectly the result of the activities of the genes. Thus it is natural to define both biological and social organisms as special kinds of sets. A number of definitions are given in this paper which define what we call here organismic sets. Postulates are introduced which characterize such sets, and a number of conclusions are drawn. It is shown that an organismic set, as defined here, does represent some basic relational aspects of both biological organisms and societies. In particular a clarification and a sharpening of the Postulate of Relational Forces given previously (Bull. Math. Biophysics,28, 283–308, 1966) is presented. It is shown that from the basic definitions and postulates of the theory of organismic sets, it folows that only such elements of those sets will aggregate spontaneously, which are not completely “specialized” in the performance of only one activity. It is further shown that such “non-specialized” elements undergo a process of specialization, and as a result of it their spontaneous aggregation into organismic sets becomes impossible. This throws light on the problem of the origin of life on Earth and the present absence of the appearance of life by spontaneous generation. Some applications to problems of ontogenesis and philogenesis are made. Finally the relation between physics, biology, and sociology is discussed in the light of the theory of organismic sets.
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    Bulletin of mathematical biology 29 (1967), S. 189-190 
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    Bulletin of mathematical biology 29 (1967), S. 261-266 
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    Notes: Abstract Ureteral obstruction has been known to reduce the renal concentration gradients of chloride, sodium, urea, and osmolality. A quantitative analysis of the factors responsible for the decrease in the gradients has been performed. By applying the equation of conservation of matter to data obtained in this laboratory it is concluded that:a. at least, 62.5 per cent of the decrease in concentration gradients is due to drainage by circulatory vessels;b. at most, 25 per cent of the decrease is due to increase in water content of medulla; andc. at most, 12.5 per cent of the decrease is due to diffusion of solutes through the interstitium.
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    Bulletin of mathematical biology 29 (1967), S. 267-289 
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    Notes: Abstract The vector equation for the general motion of a body in an inertial system is used to analyze the accelerations in the semicircular canals of the cat when the head undergoes rotation about a vertical axis only, rotation about the naso-occipital axis only, and both rotations simultaneously. The corresponding mean forces and mean pressures in the endolymph are calculated by means of a closed line integral along each canal circumference. The importance of the area of the semicircular canal and of its orientation in space become evident. One can see through this mathematical analysis that the input pattern received by the labyrinthine system depends on a set of well-specified geometrical and mechanical conditions, which must be precisely evaluated in order to interpret the nystagmic outputs.
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    Bulletin of mathematical biology 29 (1967), S. 291-310 
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    Notes: Abstract This paper describes some analytical models for the system which regulates the daily eclosion rhythm of the drosophila. A general topological model is described which can simulate practically all the known experimental results about the behavior of the system under various light stimuli. From that a more specific model is proposed which can shortly be described as follows: The system contains a basic oscillator whose output is a substances. This is produced in the presence of an enzymer. During part of the cycler is deactivated ands dissipates until it reaches a lower level whenr is reactivated again. Light has the effect of deactivatingr immediately. The substances causes the production of a second substanceq which triggers a series of reactions leading to eclosion when it exceeds a threshold. The main oscillator (s—r) is temperature-compensated, but the production ofq is accelerated in the presence of light or higher temperature.
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    Bulletin of mathematical biology 29 (1967), S. 343-348 
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    Notes: Abstract Some theoretical results obtained in a previous publication (Bull. Math. Biophysics,28: 25–50, 1966) are studied from the numerical point of view. Possible medical interpretations are suggested.
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    Bulletin of mathematical biology 29 (1967), S. 349-361 
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    Notes: Abstract The findings of molecular biology concerning biosynthesis of macromolecules are applied to the deduction of the kinetics of mass and volume growth in individual cells between divisions. The time course of increase of all macromolecules and of the total dry mass is found to be linear, in agreement with the available data; the corresponding volume growth curves are either quadratic, or exponential with a linear asymptote, depending on the relative contributions of metabolism and transport to cell water. A self-limiting mass and volume kinetics is derived by including repression among the other molecular mechanisms.
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    Bulletin of mathematical biology 29 (1967), S. 363-371 
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    Notes: Abstract An error in a previous analysis of a simple model for myelinated nerve propagation is pointed out and rectified. The model previously chosen does not lend itself to analysis because it has a region ofinfinite negative conductivity which leads to an instability in steady propagation. A model which assumes afinite negative conductivity is examined in detail and a more general results is derived.
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    Bulletin of mathematical biology 29 (1967), S. 373-376 
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    Notes: Abstract In a deterministic model for the spread of news in a closed homogeneously mixing population of individuals who never forget (or, of an epidemic without recovery), it is shown that the fractions πi of the population first hearing the news (contracting the disease)i th hand are given by the terms of a truncated Poisson distribution.
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    Bulletin of mathematical biology 29 (1967), S. 465-471 
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    Notes: Abstract A mathematical model of the left ventricle is constructed. This model is a spherical thinwalled chamber with a radius and wall density approximately that of the left ventricle and enclosing a fluid whose pressure is that noted when heart sounds occur. The solution for the natural frequency of vibration is of the same value as that noted physiologically. Substitution of the values for the right ventricle yields similar results. The lowest natural frequencies of the idealized cardiac chamber possessing physical properties similar to a real cardiac chamber are of the same order of magnitude as the lowest significant observed cardiac frequencies. Such observation reinforces the likelihood that cardiac sounds are generated by the vibrating cardiac walls.
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    Bulletin of mathematical biology 29 (1967), S. 411-418 
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    Notes: Abstract From a quantum mechanical standpoint, electron tunneling may occur in a nerve axon because the nerve membrane (60 to 100 angstroms) is thin enough for the tunnel effect and because the external solution and the axoplasma are redox systems which can provide electrons. Calculations and diagrams of the density-of-states are given to predict theN-shaped current-voltage characteristics which have been observed in the studies of squid giant axons, of the reconstituted liquid membranes and of the marine algae.
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    Bulletin of mathematical biology 29 (1967), S. 429-436 
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    Notes: Abstract A theory of growth of a cell which takes up nutrients by diffusion or active transport is discussed. The main conclusion is that the volume should grow at least as fast as the third power of the time. Existing experimental evidence is not a conclusive test of the theory, and further experiments to test it are proposed.
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    Bulletin of mathematical biology 29 (1967), S. 419-428 
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    Notes: Abstract This is an analytic study of mucous flow caused by ciliary motion. The computed flow data may be compared with that already found exprimentally. The effects of mucous density, viscosity and layer depth on flow phenomena are investigated. The effects of cilia diameter, length, spacing and oscillation frequency are determined from the equations governing the flow of the mucous blanket. A pertinent finding of the analysis is that the mucous flow in the airway tubes can satisfy physical constraints only through the assumption of a variable viscosity in the covering mucous blanket. The mucous viscosity must increase considerably from the low value at the cillium layer to a much higher value at the air-mucus interface.
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    Bulletin of mathematical biology 29 (1967), S. 473-483 
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    Notes: Abstract A model of the arterial system is discussed which considers determination of the changes in density of energy dissipation of the pulse with the system entrance being the forcing of blood from the heart to the aorta. The effects of boundary conditions and branching upon dissipation density are analyzed on the basis of the model.
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    Bulletin of mathematical biology 29 (1967), S. 451-464 
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    Notes: Abstract A mathematical blood vessel mode is established of a semi-infinite, fluid filled, cylindrical, elastic tube where a pressure wave is propagated in the fluid. This model has assigned to it physical values of the aorta and is subject to a pressure pulse at the origin of the same configuration and magnitude as that observed in the human. The resulting mathematical solution demonstrates wall displacement in agreement with observed phenomena of (a) a high frequency component analogous to cardiovascular sounds; (b) a low frequency component resembling the pressure pulse analogous to the pulsatile movement of the vessel wall.
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    Bulletin of mathematical biology 29 (1967), S. 499-512 
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    Notes: Abstract We use the dimensional parameters previously derived (Bull. Math. Biophysics,28, 355–362, 1966), the ventricular pressure expressed as a Fourier series, and several additional assumptions to derive expressions for the mechanical parameters of the ventricle: flow, muscle segment length, surface area, transmural force, wall tension and work. The wall of the ventricle is assumed to consist of three layers of muscle. Each of the mechanical parameters is expressed in terms of Fourier series.
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    Bulletin of mathematical biology 29 (1967), S. 485-497 
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    Notes: Abstract Based on conceptions and assumptions concerning the blood oxygenation process, some fundamental quantitative relations for red blood corpuscle oxygenation and blood oxygenation kinetics are presented. A distribution function is introduced expressing the probability density for the occurrence of a red blood cell with a specific oxygen content. By means of a kinetic equation deduced the distribution function is connected with spatial distribution of oxygen pressure and with blood flow rate. For the given initial conditions the kinetic equation is solved for a one-dimensional case, and this solution is applied to a generalized oxygenator in a stationary case. The generalized oxygenator presents a system of through-flow elements in which blood flows and contacts oxygen. Each of the through-flow elements is characterized by length, blood flow rate, probability of red blood corpuscle entry and by a quantity depending on oxygen pressure. Results obtained for the generalized oxygenator are then applied to a disc oxygenator with certain presumptions concerning blood oxygen saturation at the system's output expressed in dependence on geometry and performance conditions. Stress is laid upon the influence of blood flow in the oxygenator, on oxygenation; and two extreme cases are compared—series and parallel types of disc oxygenator.
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    Bulletin of mathematical biology 29 (1967), S. 513-531 
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    Notes: Abstract When material is introduced into the blood-brain-cerebrospinal fluid system, it will distribute throughout the system in a manner dependent upon the geometrical characteristics and the physical and chemical parameters of the system. Since only average values of the concentration of the material are known in each of the parts of the system, and approximation approach is used in analyzing this distribution theoretically. A simplified model is chosen for the system and equations are derived. These equations are then applied to the results of four papers in the literature dealing with the distribution of thiocyanate. A fifth paper dealing with the sulfate extracellular space of brain is also analyzed.
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    Bulletin of mathematical biology 29 (1967), S. 533-540 
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    Notes: Abstract The blood flow through an arterial stenosis can theoretically be increased by gradual expansion of the stenosis. This hypothesis was tested by comparing the flow through abruptly expanding and gradually expanding stenosed plastic conduits. The conduits were tested in non-pulsatile flow with water and bank blood and in pulsatile flow in dog's descending aorta. It was found that: (a) The pressure drop in arterial stenoses is larger than that predicted by non-pulsatile theory. (b) Gradual expansion of stenosis in peripheral arteries will not increase the flow significantly. (c) Gradual expansion of intracardiac stenoses (valvular stenosis, shunts, prosthetic valves) may increase the blood flow markedly.
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    Bulletin of mathematical biology 29 (1967), S. 565-574 
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    Notes: Abstract In a previous paper (Bull. Math. Biophysics,29, 549–563, 1967) the author derived equations to represent the flow of blood in an artery. It was pointed out that these did not completely characterize the system and that an additional hypothesis was required. The hypothesis of minimal energy dissipation had been thought to imply a central tendency on the part of suspended particles (erythrocytes). It is here shown that if the fluid is non-Newtonian this may not be so.
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    Bulletin of mathematical biology 29 (1967), S. 597-603 
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    Notes: Abstract Certain developments in the interrelation of cell population theory and intracellular kinetics, collectively termedcytokinetics, are reviewed and discussed. The results imply the suitability of a set of three general postulates governing the behavior of cells under normal conditions and under conditions of stress.
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    Bulletin of mathematical biology 29 (1967), S. 625-638 
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    Notes: Zusammenfassung Unter Verwendung des Begriffs der Befriedigungsfunktion wird ein mathematisch-biophysikalisches Modell des Wettlaufs zweier Sportler über eine feste Distanz entworfen. Die Durchschnittsgeschwindigkeit, die beide Sportler erreichen, erweist sich als abhängig von den Möglichkeiten der Kommunikation (Austausch und Verarbeitung von Informationen) zwischen ihnen. Das Modell zeigt, daß die Geschwindigkeiten mit steigender Kommunikation abnehmen.
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    Bulletin of mathematical biology 29 (1967), S. 639-642 
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    Notes: Abstract The theory of relations between sets, proposed and outlined in previous publications (Bull. Math. Biophysics,23, 233–235, 1961;28, 117–124, 1966;28, 309–313, 1966), is tentatively expanded and generalized with a view to biological applications.
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    Bulletin of mathematical biology 29 (1967), S. 643-648 
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    Notes: Abstract In continuation of previous studies (Bull. Math. Biophysics,28, 283–308; 655–661, 1966;29, 139–152, 1967) it is shown that the difference between the “metric” aspects of physics and the “relational” aspects of biological and social sciences disappear by accepting the broader definition of “relation”, such as that given in mathematics and logic. A conceptual superstructure then becomes possible from which all three branches of knowledge may be derived, though none of them can be derived from the others.
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    Bulletin of mathematical biology 29 (1967), S. 705-710 
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    Notes: Abstract By utilizing an integral equation formulation of Laplace’s equation it is shown that knowledge of the geometry of the conductivity interface, the values of conductivity, and the potentials and gradients on the conductivity interface are sufficient information to determine the infinite media potentials and infinite media potential gradients. The integral equations may be approximated by finite sums and the results can be implemented practically by using a medium size digital computer.
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    Bulletin of mathematical biology 29 (1967), S. 649-656 
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    Notes: Abstract The Rashevsky Principle of Adequate Design is employed in conjunction with the hypothesis of constant hematocrit throughout the arterial tree to give an estimate of the most likely position for an erythrocyte in an artery. The result accords with the experimental work of Segré and Silberberg (1962).
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    Bulletin of mathematical biology 29 (1967), S. 665-675 
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    Notes: Abstract The present paper is an attempt to outline a possible approach to the study of concrete cellular systems in terms of relational biology as developed by Rashevsky and Rosen. The basic ideas and the formalism of Rosen’s (M,R)-systems, proposed as a model of abstract biological systems, are used in order to represent the cellular protein biosynthesis. A diagram corresponding to the activation of amino acids and synthesis of amino-acyl-transfer RNA, the attachment of t RNA to a specific codon of messenger RNA and peptide bond synthesis with the release of a protein molecule, is constructed. The systemM thus obtained for the synthesis of a proteinp k receives a set of environmental inputs, that is, the twently naturally occurring amino acids and emits a single output, thep k protein. The problem of noncontractibility of inputs in the $$\mathfrak{M}_{p_k }$$ system is then analyzed. In our context, it is found that the noncontractibility is not associated with the whole amino acid setS pk but with an “essential amino acid set” $$\bar S_{p_k }$$ , so that $$\bar S_{p_k } \subseteq S_{p_k }$$ and $$S_{p_k } - \bar S_{p_k }$$ represent the set of amino acids which can be replaced or absent. According to our considerations, the biochemical concept of “essential amino acid” acquires a new significance, that is, what seems “essential” is linked with the ability to form a giventRNA t ∼a i complex in a suitable augmented dependent set essential for the biosynthesis of a functional protein. Eventually the discussion of re-establishability leads to some important biological implications concerning the existence of ambiguous codons and the degeneracy phenomenon in the genetic code, as anecessary biochemical tool involved in adaptive processes.
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    Notes: Abstract Recent experimental evidence has provided increasing support for the hypotheses that 60 to 80 per cent of intracellular Na+ exists in a complexed state, and that intracellular water exists in a semi-organized, non-liquid state having low solubility for Na+. Using these postulates, a previous crude theory of Na+ leakage from the cell based on electronion conduction analogies has been redeveloped in a more complete and detailed fashion, following a non-equilibrium thermodynamic approach. The theory, which is based on the postulate of almost 100 per cent complexing of intracellular Na+, predicts that Na+ leakage from muscle should conform to the Elovich equation, which closely agrees with experiment, despite the fact that experiments indicate that 20 to 40 per cent of muscle Na+ isnot complexed. To resolve this apparent paradox, the leakage of complexed and non-complexed Na+ from muscle was measured by nuclear magnetic resonance (NMR). The non-complexed Na+ leaked much more slowly than the complexed Na+, suggesting that the non-complexed Na+ may be confined within vacuoles surrounded by an activation energy barrier at the vacuolar membrane. This implies that the measured curves of Na+ leakage showing Elovich kinetics are due mostly to leakage of complexed Na+ as the theory requires, and that the leakage of 20 to 40 per cent non-complexed Na+ is mostly delayed until later times.
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    Bulletin of mathematical biology 29 (1967), S. 719-736 
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    Notes: Abstract The signal that drives the respiratory muscles is assumed to be caused by the mutual interaction of two neural networks in the respiratory center. These nets are considered to be spatially homogeneous with all the neurones firing simultaneously so that their dynamical characteristics depend upon the process of temporal summation. A mathematical model of this phenomenon leads to a description of the behavior of the respiratory center consisting of a nonlinear, Volterra-type integral equation equivalent to a second-order, nonlinear autonomous differential equation, which is proved to exhibit, under some conditions, a limit cycle from which stems the basic respiratory drive. It is suggested that the frequency of this drive is controlled by chemical or central influences by varying the central facilitation of the neurones at rest.
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    Bulletin of mathematical biology 29 (1967), S. 753-757 
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    Notes: Abstract Theoretical expressions for the probability of survival of an irradiated cell have been obtained in a simple form. An attempt has been made to calculate the probability that a cell will remain in normal condition or in damaged state at a given instant of time. The important assumption which makes this possible is that once the cell is repaired it will behave like a normal cell for further irradiation.
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    Bulletin of mathematical biology 29 (1967), S. 737-752 
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    Notes: Abstract A one compartment, mechanical model of the human lung-thorax system is presented and mathematically analyzed. The equation relating the thoracic muscular stress to the expired air volume is developed and investigated. Assuming that the pressure drop along the airways is a linear function of air flow rate and that the effective lung-thorax compliance is constant, a form for the muscular stress as a function of time is developed. This is used to predict volume-time and flow-volume curves, which are compared to those measured on a normal individual. It appears that these theoretical results have the essential characteristics of the experimental curves. These results, coupled with the one-to-one correspondence between the parameters of the model and those of the prototype, suggest that this model should have great utility in the study of ventilatory mechanics.
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    Bulletin of mathematical biology 29 (1967), S. 759-765 
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    Notes: Abstract The classical enumeration theorem of Pólya (Acta Math.,68, 145–254, 1937) is applied to a modified version of Harary’s (Pacific J. Math.,8, 743–755, 1958) generating functions for counting bicolored graphs to derive a counting function for the number of balanced signed graphs. Methods for computing these counting polynomial functions are discussed.
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    Bulletin of mathematical biology 29 (1967), S. 793-825 
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    Notes: Abstract A theory of speech perception based on evolutionary adaptive postulates is suggested. The principles are grouped around two categories: (a) the physical properties of the stimulus energy and its distributions in the environment; (b) the tendency of the sensory powers to become optimum in order to aid the survival of the species. The construction of the theory and the experiments performed to test its basic predictions are summarized. The theory is examined rather thoroughly with respect to basic vowel space. A line of generalization for the treatment of consonants is indicated.
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    Bulletin of mathematical biology 29 (1967), S. 767-779 
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    Notes: Abstract A hypothetical neural network is presented to account for errors made by food-deprived albino-rats in a bar-pressing situation (Carlton, 1964). The rats were required to press alternately on two bars in order to activate a device that releases milk to them. When-ever an animal pressed consecutively on the same bar, no milk was released and the animal was scored as having committed an error. In the first part of the paper, a time-independent neural network, part of which is equivalent to the psychophysical discrimination network of H. D. Landahl (1938), is used to interpret the effects on the animals’ performances of the drugs amphetamine, scopolamine and atropine. Suggestions for further experiments are made on the basis of the initial form to the model. In the second part of the paper, certain parameters of the initial form of the model are assumed to be time-dependent and a further generalization occurs through the introduction of the interresponse time distribution. It is shown that, under specified conditions, the second form of the model reduces to the first. The time-dependent form of the model allows certain features to be discussed that could not be discussed in the time-independent form [e.g. P. Dew’s (1961) notion of a possible mode of action for a variety of the behavioral effects of amphetamine]. Furthermore, experiments of an essentially different type from those discussed in the first part can be proposed to aid in the development of a theory for this kind of behavioral situation.
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    Bulletin of mathematical biology 29 (1967), S. 879-880 
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    Bulletin of mathematical biology 29 (1967), S. 831-839 
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    Notes: Abstract The survival probabilities of newly-formed colonies of organisms arising from the branching process formulation of individual reproduction are examined. Six types of 2-parameter discrete offspring distributions common to mathematical ecology are compared with respect to survival of newly formed colonies of offspring. It is found that the survival value of the distribution can be rank ordered in the following descending order: modified Poisson (highest) Neyman A, geometric Poisson, Pólya-Aeppli, negative binomial and the modified geometric. Causal factors for these differences and practical implications of these results are discussed.
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    Monatshefte für Mathematik 71 (1967), S. 1-6 
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    Monatshefte für Mathematik 71 (1967), S. 7-13 
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    Monatshefte für Mathematik 71 (1967), S. 14-23 
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    Monatshefte für Mathematik 71 (1967), S. 24-31 
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    Monatshefte für Mathematik 71 (1967), S. 32-39 
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    Monatshefte für Mathematik 71 (1967), S. 49-54 
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    Monatshefte für Mathematik 71 (1967), S. 40-48 
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    Monatshefte für Mathematik 71 (1967), S. 55-63 
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    Monatshefte für Mathematik 71 (1967), S. 64-79 
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    Monatshefte für Mathematik 71 (1967), S. 80-86 
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    Monatshefte für Mathematik 71 (1967), S. 97-99 
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    Monatshefte für Mathematik 71 (1967), S. 100-113 
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    Monatshefte für Mathematik 71 (1967), S. 87-96 
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    Monatshefte für Mathematik 71 (1967), S. 114-122 
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    Monatshefte für Mathematik 71 (1967), S. 123-142 
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    Monatshefte für Mathematik 71 (1967), S. 143-147 
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    Monatshefte für Mathematik 71 (1967), S. 148-155 
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    Monatshefte für Mathematik 71 (1967), S. 165-179 
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    Monatshefte für Mathematik 71 (1967), S. 156-164 
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    Notes: Zusammenfassung In der vorliegenden Arbeit wird die lineare, inhomogene, gewöhnliche Differentialgleichung 2. Ordnung mit beliebigen regulären Koeffizientenfunktionen mittels Liereihen gelöst. Es werden zwei Wege beschritten, von denen einer auf Rekursionsformeln und der andere auf die iterative Auswertung eines “Störintegrals” führt; ersterer ist programmiertechnisch einfacher, läßt aber keine exakte Fehlerabschätzung zu, während für das Störintegral der Fehler angegeben werden kann.
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    Monatshefte für Mathematik 71 (1967), S. 180-192 
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    Monatshefte für Mathematik 71 (1967), S. 193-202 
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