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  • Springer  (26,558)
  • American Meteorological Society
  • 1995-1999
  • 1990-1994
  • 1965-1969  (27,018)
  • 1966  (27,018)
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  • 1995-1999
  • 1990-1994
  • 1965-1969  (27,018)
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  • 1
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    Bulletin of mathematical biology 28 (1966), S. 333-345 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper is a sequel to a paper by the author entitled “Restricted Transition Probabilities and Their Applications to Some Problems in the Dynamics of Biological Populations” (Bull. Math. Biophysics, 1966,28, 315–331). The paper is divided into two parts. In part one some aspects of the maximum size attained by the population during a finite time interval are studied for the case the stochastic process underlying the evolution of the population is a birth process. Two interesting by-products emerge from the study presented in part one; namely a combinatorial method of finding solutions to the Kolmogorov differential equations in special cases, and secondly, a set of criteria for the optimum allocation of genotypes in the host population of a host-pathogen system. The optimum allocation of genotypes in the host population is a problem of practical importance in controlling plant pathogens. In part two the theory of restricted transition probabilities developed in the companion paper is applied in finding the distribution of the time to the appearance of the first mutation for the case of a two dimensional birth process. The distribution of the time to the appearance of the first mutation is of importance in understanding the role mutation plays in the evolution of a population, particularly in the pathogen population of a host-pathogen system.
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    Bulletin of mathematical biology 28 (1966), S. 355-362 
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    Notes: Abstract The complex arrangement of the muscle fibers in the ventricular wall and the nonsymmetric contraction and expansion of the ventricle preclude the writing of a differential equation of motion for the ventricle as a whole. We can, however, describe the motion of the ventricle by describing the motion of the dimensional parameters length and diameter; the radius, circumference, cross-sectional area, and volume following naturally from these. The ventricle is assumed to be an ellipsoid of revolution and the dimensional parameters to be periodic functions of time. Each of the parameters is expressed as a Fourier series.
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    Bulletin of mathematical biology 28 (1966), S. 347-354 
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    Notes: Abstract Le modèle de Nelson peut-être considéré comme une approximation du modèle de Hodgkin-Huxley. Moins précis, il est plus maniable. Le modèle de Nelson peut également être considéré comme une généralisation du modèle de Hodgkin-Huxley. En effet, il introduit des liaisons synaptiques localisées ou diffusantes, et un processus de facilitation. Le mécanisme des liaisons synaptiques ne se traduit pas facilement dans le langage de Hodgkin-Huxley. Par contre, le processus de facilitation s'interprète facilement. Nelson's model can be taken as an approximation of Hodgkin-Huxley's model. Its precision is lesser, but it is more usable. Nelson's model can also be taken as a generalization of Hodgkin-Huxley's one; for it introduces localized or diffusing synaptic connexions and a facilitating process. The mechanism of synaptic connexions cannot be easily translated into Hodgkin-Huxley's language. On the contrary, the facilitating process is easily interpreted.
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    Bulletin of mathematical biology 28 (1966), S. 363-370 
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    Notes: Abstract A spherical model for the human left ventricle with two different types of aneurysms, circular and rectangular-square, is proposed and meaningful relations are derived between the parameters of the aneurysms and ventricle. Such ventricular parameters as stroke volume, end-diastolic volume, and end-systolic volume are given normal human values to compute values for end-systolic radius and percentage shortening of muscle for various sized circular and rectangular-square aneurysms.
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    Bulletin of mathematical biology 28 (1966), S. 375-378 
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    Notes: Abstract The Volterra theory of two competing populations is extended to the contemporary social problem of crime control. Domains of stability for the time dependence of the numbers in the criminal and enforcement groups are exposed by a numerical example. Both augmentation and reduction of enforcement can produce a stable system. Average values of the ratio of members in each group show great sensitivity to the control policies adopted by the remaining sector of the total population.
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    Bulletin of mathematical biology 28 (1966), S. 379-390 
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    Notes: Abstract The paper deals with interactions of substances via an enzymatic reaction (Bull. Math. Biophysics,25, 141–154, 1963). The substances are the activators, inhibitors and/or substrates of the reaction. Due to the bimolecularity of the processes in the reaction, the quantitative relation between the steady state amount of complexes and the amounts of the substances assumes a typical form. In multiple enzymatic reactions this form is more complicated, though basically similar. Because the substances may influence the steady state amounts of the complexes in opposite directions, the compensation and blocking effects are the properties of enzymatic reactions. The substances with the same direction of influence may potentiate each other. In the enzymatic reaction here considered, the potentiation is always non-negative.
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    Bulletin of mathematical biology 28 (1966), S. 391-409 
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    Notes: Abstract Growth-rate functions in analytic form have been obtained for cell cultures in which the doubling times follow the Gaussian and Poisson distributions. The growth-rate functions are calculated by using Laplace transforms to solve an integral equation previously presented. Oscillatory solutions result if a substantial fraction of the cells in a culture are synchronized to divide at some particular time. The synchrony and, hence, the oscillatory character of the growth-rate function eventually disappear because of the non-zero variance of the doubling-time distribution. If their variances are sufficiently small, the Gaussian and Poisson doubling-time distributions lead to growth-rate functions that become identical in the limit of large time.
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    Bulletin of mathematical biology 28 (1966), S. 411-416 
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    Notes: Abstract IfN(t) is the expected number of cells in a culture at timet, $$\dot N(t)$$ the corresponding time derivative, andf(t−τ)dt the probability that a cell of aget−τ at timet will divide in the succeeding time intervaldt, then according to Hirsch and Engelberg (this issue) there obtains the integral equation $$\dot N(t) = 2\int_{ - \infty }^t {f(t - \tau )\dot N(\tau )d\tau }$$ for describing the dynamics of the cell population. It is the purpose of this note to give two alternative derivations of this equation, one based on the age density equation of Von Foerster, and the other based on a generalized form of the Harris-Bellman equation describing the first moment of an age dependent, branching process. In addition, a probability model is posed from which the Von Foerster equation and, hence, the Hirsch-Engelberg equation readily follows.
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    Bulletin of mathematical biology 28 (1966), S. 417-432 
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    Notes: Abstract A model of the dissolution kinetics of powdered enamel is developed based on the kinetic rate termq, $$q = K'H - k'C \cdot P_1$$ , whereH=[H +],C=[Ca ++] andP 1=[HPO 4 = ]. The differential equations describing the rate of mineral dissolution (and the linearly related rate of appearance of calcium and phosphate in solution) have been derived and solved for three basic cases: (1) when thepH of the solution and surface area of the enamel are considered constant, (2) when thepH is assumed constant, but the reduction in surface area during dissolution is considered, and (3) when the rise ofpH resulting from the buffering effect of the dissolved enamel is considered along with the change in surface area. Analytical solutions have been obtained for cases (1) and (2), while a numerical solution has been found for case (3). Data from a study on enamel dissolution are presented that agree with the theory of case (3), and it is noted that apH rise as large as 0.5 can occur, as has been shown elsewhere in the literature.
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    Bulletin of mathematical biology 28 (1966), S. 477-481 
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    Notes: Abstract On the basis of Landahl's theory of two-choice learning it is shown that application of punishment for wrong responses, without giving award for correct ones, does not lead to complete learning, no matter how many trials are used. If initially a “wrong response” was learned, then an attempt to inhibit it by punishment alone will in a class of cases lead only to a 50% suppression of that wrong response. Possible connection with the problem of effectiveness of punishment as a deterrent for crime is mentioned.
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    Bulletin of mathematical biology 28 (1966), S. 483-483 
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    Bulletin of mathematical biology 28 (1966), S. 483-483 
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    Bulletin of mathematical biology 28 (1966), S. 485-485 
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    Bulletin of mathematical biology 28 (1966), S. 501-510 
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    Notes: Abstract A set of characteristic parameters is given for electrophoresis accompanied by diffusion, followed by a method of simplification of the transport equation. The concept of electrophoretic similarity is introduced in connection with the presentation of solutions and the final section contains some dimensional considerations of the potential equation.
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    Bulletin of mathematical biology 28 (1966), S. 511-517 
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    Notes: Abstract We show that when we represent (ℓ, ℛ)-systems with fixed genome as automata (sequential machines), we get automata with output-dependent states. This yields a short proof that ((ℓ, ℛ)-systems from a subcategory of automata—and with more homomorphisms than previously exhibited. We show how ((ℓ, ℛ)-systems with variable genetic structure may be represented as automata and use this embedding to set up a larger subcategory of the category of automata. An analogy with dynamical systems is briefly discussed. This paper presents a formal exploration and extension of some of the ideas presented by Rosen (Bull. Math. Biophyss,26, 103–111, 1964;28, 141–148;28 149–151). We refer the reader to these papers, and references cited therein, for a discussion of the relevance of this material to relational biology.
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    Bulletin of mathematical biology 28 (1966), S. 487-500 
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    Notes: Abstract A two-dimensional nonlinear integro-differential equation with time-varying coefficients describing the behavior of the fluttering wing-body systems typical of natural flight mechanisms has been deduced from the Navier-Stokes equation which generalizes local pressure and velocity distributions in the externally oscillating air field. The resulting equation for the wing forces is combined with an analogous expression for the forces of gravitation and acceleration associated with the body. The air acceleration force, not previously considered in bio-physical models of insect and bird flight, is shown to arise from a formal analysis of unsteady or time-varying contributions to the velocity field, while the square form of the conventional steady state aerodynamic forces is derived from the intertial terms in the Navier-Stokes equation with the aid of the approximations of Newtonian impact theory. Previous calculations (Houghton, 1964) have indicated that the contribution to gravitational stability of air acceleration and aerodynamic life are roughly in the ratio of 3:1.
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    Bulletin of mathematical biology 28 (1966), S. 519-536 
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    Notes: Abstract Certain types of cortical electrical events are non-propagated so that the associated electric fields must have standing wave characteristics. However, cortical electric events typically are generated by neurone populations which cannot be activated simultaneously on impulse driving. Hence the sum of the standing wave fields due to asynchronous activation of adjoining regions of cortical neurones must give the appearance of a traveling wave. Analysis of cortical waveforms is further complicated by curvature in cortical surfaces. A model is presented that shows the effects of curvature and time lag in activation on the form of the potential at points in space around a laminar array of elements simulating a population of cortical neurones. The results are compared with waveforms evoked by single-shock stimulation of the prepyriform cortex in cats.
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    Bulletin of mathematical biology 28 (1966), S. 545-554 
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    Notes: Abstract A continuity equation for cell-number density in a population of cells is derived, and a system of equations for eliminating parameters between the general solution and the initial distribution obtained.
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    Bulletin of mathematical biology 28 (1966), S. 537-544 
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    Notes: Abstract Use of an electrical model of the left ventricle of the heart and the arterial system permits analysis of the changes which take place as the capacity of the myocardium for generation of force decreases. The model is simple in structure, and its construction and practical testing would not be difficult. It demonstrates that, as the heart muscle weakens, the peak of intracardiac force occurs later in systole, and the difference between the intracardiac pressure and the aortic pressure in the second half of systole is much greater than for the normal heart. The feedback mechanisms which are proposed to affect myocardial contractility would affect this compensation for cardiac weakening. Indices to categorize the behavior of the normal, compensated though weakened, and decompensated myocardium are proposed.
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    Bulletin of mathematical biology 28 (1966), S. 555-566 
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    Notes: Abstract The frequency distribution in a population of cells of the quantityCD (defined as the amount of some chromosomal substance in a cell which containsC chromosomes) is calculated using the distribution in the population of the amount per chromosome,D, and the distribution of chromosome number,C.
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    Bulletin of mathematical biology 28 (1966), S. 567-574 
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    Notes: Abstract The rate of chromosomal DNA synthesis in an exponentially growing population of cells having chromosome-number dispersion is calculated using DNA histogram data, chromosome-number distribution data, and the assumptions that the synthesis rate is constant and DNA double exactly.
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    Bulletin of mathematical biology 28 (1966), S. 575-584 
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    Notes: Abstract An estimate is made of the rate coefficient for linear DNA synthesis with exact doubling in an exponentially growing population of Ehrlich ascites tumor cells having chromosome-number dispersion. Comparison of calculated and experimental results suggest that the assumptions used in the calculation are tenable, but further experimental evidence is needed to prove this.
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    Bulletin of mathematical biology 28 (1966), S. 655-661 
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    Notes: Abstract The paper develops further some suggestions made previously (Bulletin of Mathematical Biophysics,28, 283–308, 1966) that certain biological phenomena may be more easily interpreted from a “sociological” point of view by considering the organism as a social aggregate of cells and a cell as a social aggregate of genes. In this light the problems of origin of life on earth, of aging, and of parasitism and symbiosis are discussed. The notion of social aggregates of different orders is introduced.
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    Bulletin of mathematical biology 28 (1966), S. 663-663 
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    Notes: Abstract A theoretical and experimental study was made of the mechanical behavior of the cornea. The theoretical analysis included an analytical solution for the symmetrical constraint of a thin, shallow, spherical shell by a rigid indenter. The experimental study investigated the rheology of the cornea with particular emphasis on its compliance with the requirements of the Boltzmann Superposition Principle. Representative results of tests on twenty enucleated hog eyes and two human eyes have been reported. The corneas of the human and hog eyes behaved as linear viscoelastic solids; the human eyes differed from the hog eyes in having a long term creep component. Several eyes were tested at the site of procurement, six to seven minutes after the animal's death, and it was established that creep is not an artifact due to aging or enucleation. The analytical and experimental results were combined to study some instruments used to detect the level of pressure in the eye. The theoretical analysis predicted that a type of elastic instability occurs during the process of flattening a small portion of the cornea; this is discussed with reference to the Goldmann and Mackay-Marg tonometers. The role of corneal creep was considered with reference to the response of the Schiøtz indentation tonometer during the time dependent process known as tonography.
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    Bulletin of mathematical biology 28 (1966), S. 645-654 
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    Notes: Abstract Following previous studies, differential equations are established which determine the variation of the stimulus towards a corrective turn of the steering wheel and its effect on the excitation of the centers in the brain which results in the production of the corrective turn. The equations are derived under the highly oversimplified assumption that all excitation thresholds are so small that they can be neglected. Under these assumptions it is found that the tracking curve of a car is a sinusoid with negative damping, that is, with an ever increasing amplitude. Driving under these assumptions is imposible since the car will always eventually jump off the road. The possible effects of the threshold as well as stimuli towards corrective turns other than the distance from the edge of the lane are very briefly discussed. In spite of the negative results of the paper, its interest lies in the circumstance that with the complication of the model, we find that driving depends not only on the reaction times as the only “purely biological” parameter, but on three other neurobiophysical constants. In a subsequent paper (Rashevsky, 1967) it is shown how the introduction of one or more purely biological parameters of the driver makes a stable driving regime possible.
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    Bulletin of mathematical biology 28 (1966), S. 663-663 
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    Bulletin of mathematical biology 28 (1966), S. 1-10 
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    Notes: Résumé Les premiers étages sensoriels sont étudiés en utilisant notre modèle de neurone et en supposant que les réseaux responsables de la perception sont particulièrement solides, stables, économiques. Nous montrons que les premiers neurones doivent être spontanément périodiques et autorégulés. La nécessité fonctionnelle des premiers étages de la voie visuelle est démontrée. Par analogie, nous étudions la voie auditive.
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    Bulletin of mathematical biology 28 (1966), S. 11-24 
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    Notes: Abstract The problem of the viscous flow of an incompressible Newtonian liquid in a converging tapered tube has been solved in spherical polar coordinates. The method of the solution involves the Stokes' stream function and a technique introduced by Stokes in the study of a sphere oscillating in a fluid. The theory for the flow in a rigid tube includes: (1) the pulsatile flow with both radial and angular velocity components; (2) the steady state flow with both radial and angular velocity components and (3) the very slow steady state flow with only a radial velocity component present. For a tapered elastic tube, the velocity of the propagated pulse wave is determined. The solution given is in terms of the elastic constants of the system and the coordinates for this type of geometry. The pulse velocity is then related to the velocity in an elastic cylindrical tube with the necessary correction terms to account for the tapered tube.
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    Bulletin of mathematical biology 28 (1966), S. 25-50 
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    Notes: Abstract In this paper a class of branching processes applicable to populations reproducing by some asexual means or by a simple selfing system of mating is studied. The paper is divided into three parts. In part one the mathematical model is introduced, part two is a mathematical analysis of the model, and in part three concrete applications and examples are given. Many of the proofs of the theorems in part two are omitted but will appear in a subsequent issue of theBulletin.
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    Bulletin of mathematical biology 28 (1966), S. 51-74 
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    Notes: Abstract The application of the earlier results (Pavlidis, T. 1965. “A New Model for Simple Neural Nets and its Application in the Design of a Neural Oscillator.”Bull. Math. Biophysics,27, No. 2, 215–229) to the design of more complex neural nets is attempted. The following cases are considered: 1. Chains of neurons where it is proven that the frequency of the output pulses does not depend on the value of the input as long as it is above a certain threshold. 2. Groups of neurons with backward inhibition which present an intermittent mode of operation. 3. Neural nets with periodic facilitation which permit time sharing of certain components for different functions. 4. A neural net which can detect the sign of the input even if the main receptor is sensitive only to the absolute value of it, is presented. 5. A velocity estimating neural net which in combination with one of the nets with intermittent response provides a model for the smooth eye tracking movements.
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    Bulletin of mathematical biology 28 (1966), S. 75-90 
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    Notes: Abstract By assigning coordinates to the information space comprising all knowledge, rigorous mathematical interpretations can be placed on such terms as academic ability, memory and creativity such that these psychometric concepts can be incorporated into a framework of functional analysis which then permits the optimization of long-term academic learning processes through the location of the teaching trajectories in information space which will maximize the knowledge accumulated in a generalized educational system composed of a complex of subject-pupil-teacher interactions. The concepts of discrete and continuous information spaces are discussed in connection with subject-subject, subjectpupil and pupil-pupil interactions, and the advantages of using variational versus dynamic programming methods of optimizing alternative educational systems are evaluated.
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    Bulletin of mathematical biology 28 (1966), S. 103-106 
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    Notes: Abstract IfK is a partition of a setK which is partially ordered by the relationR andR is a collection of pairs of sets ofK such that the sets of each pair are related byR in the sense of Rashevsky, thenR is a relation which partially ordersK. Necessary and sufficient conditions thatK be a chain are obtained, and ifK is a chain under these conditions, it is shown thatK is unique.
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    Bulletin of mathematical biology 28 (1966), S. 161-166 
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    Notes: Abstract This paper continues a comparison of the Taylor series and spherical harmonic forms of multipole representations initiated by Yeh (Bull. Math. Biophysics,24, 197–207, 1962). It is shown that while transformations from Taylor series form into spherical harmonic form is always possible, the inverse cannot be accomplished as suggested by Yeh; corrected transformation equations are given. It is also shown that direct measurement of Taylor coefficients, as outlined in Yeh, Martinek, and de Beaumont (Bull. Math. Biophysics,20, 203–216, 1958), is actually not possible. Accordingly, only the spherical harmonic coefficients can be determined by measurement of surface potentials, as in electrocardiography.
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    Bulletin of mathematical biology 28 (1966), S. 181-190 
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    Notes: Abstract This paper is a continuation of a paper, “Some Multi-Dimensional Branching Processes as Motivated by a Class of Problems in Mathematical Genetics I,” by C. J. Mode, which appeared in a previous issue of theBulletin. Its purpose is two-fold; namely to discuss the mathematical existence of the model and to supply the mathematical proofs of some theorems in section two of the paper mentioned above. This paper should be read in conjunction with the previous paper.
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    Bulletin of mathematical biology 28 (1966), S. 191-194 
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    Notes: Abstract Rosen (Bull. Math Biophysics. 1959) has argued that a self-reproducing automaton of the type originally described by von Neumann is impossible because of a logical paradox inherent in its definition. The paradox is resolved by explicitly allowing errors (mutations) in the system and thus introducing evolution. There is no paradox in an automaton, originating from a slightly different ancestor through mutation. The von Neumann model thus becomes realistic and useful for a discussion of biological phenomena.
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    Bulletin of mathematical biology 28 (1966), S. 167-179 
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    Notes: Abstract Previously proposed formulae for the quantitative estimation of bidirectional shunts across ventricular septal defects require determination of the oxygen contents of mixed venous, pulmonary artery, pulmonary venous, and aortic blood. Because these formulae do not take into account the mixing of oxygenated with unoxygenated blood within the ventricles, their use must result in underestimation of shunt flows in each direction. A mathematical model for a ventricular defect is examined, in which it is assumed that mixing of blood occurs in each of six sites in the venae cavae or right atrium, right ventricle, pulmonary artery, left atrium, left ventricle, and aorta. A total of fourteen streams of blood can flow from one to another of these mixing sites. As long as complete mixing occurs in the six specified mixing sites, any degree of mixing or non-mixing of the various streams is permitted. From the equations characterizing the model, formulae are derived in which the shunt flow in each direction is expressed in terms of the oxygen contents in the six mixing sites and the fractions of blood which enter the shunt from either side without prior mixing in a ventricular mixing site. The previously reported formulae, which apply when no ventricular mixing is allowed to occur, lead to theoretical minimum values for the shunt flows in each direction. At the opposite extreme where all the shunting blood is required to mix in a ventricle before entering the shunt, formulae for maximum possible shunt flows are also obtained. The absolute values for the left-to-right and right-to-left shunt flows, which must lie somewhere between the theoretical maximum and minimum values, cannot be computed from blood gas data alone.
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    Bulletin of mathematical biology 28 (1966), S. 195-205 
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    Notes: Abstract Experimental evidence strongly suggests that the contractility of the intact heart in situ, in contrast to that of striated muscle elsewhere in the body, is controlled in a close-cycle system. Thus, the variation of intraventricular pressure during systole follows a complex pattern, whose relative form remains quite constant regardless of the duration of ejection. By use of the single-chambered model of the cardiovascular system, a mathematical representation of a feasible feedback mechanism is developed. The requirement that the feedback system must satisfy mathematical principles eliminates relationships apparently reasonable from a physiological viewpoint. A clinical application which the mathematical development suggests is that early arterial hypertension may arise from an abnormal feedback mechanism with excessively large cardiac output in the initial portion of systole.
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    Bulletin of mathematical biology 28 (1966), S. 207-216 
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    Notes: Abstract Due to the lack of direct X-ray evidence for base pairing being the only mechanism for the formation of double helix in a DNA crystal, an alternative explanation is suggested so that the observed DNA loop becomes essential.
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    Bulletin of mathematical biology 28 (1966), S. 219-233 
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    Bulletin of mathematical biology 28 (1966), S. 217-218 
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    Notes: Abstract Validity of group ring expression of selfed population is shown for cases in which there are differences in recombination probabilities between two sexual sides of a plant.
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    Bulletin of mathematical biology 28 (1966), S. 261-282 
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    Notes: Abstract An integral equation analysis of generaln compartment steady state systems imbedded in static media of arbitrary complexity has been developed. A set of initial entry functions can be found which serve to determine a corresponding set of partitioned initial entry functions. The partitioned functions, in turn, can be used to predict the probabilities and time courses of various transport histories and to determine all steady state rates of flow between measured compartments. The method is quite general, being completely applicable, for example, to closed systems, to cyclic systems and to systems in which relatively rapid (but finite) exchange between compartments occurs.
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    Bulletin of mathematical biology 28 (1966), S. 309-313 
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    Notes: Abstract From the definition of a strong and weakn-ary relation betweenn sets, given in a previous paper (Bulletin of Mathematical Biophysics,27, 477–492), it follows that for a given set ofn sets and givenn-ary relationR between them there can exist only one strong relation, but a large number of weak ones. An expression for the total number of possible weak relations is derived and the notion of the degree of weakness of a relation is introduced and discussed.
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    Notes: Abstract The problem of determining the sequence of a biopolymer from its fragments is stated in mathematical terms. Using concrete properties of a free monoid, certain general classes of biopolymers are shown to be insolvable from fragment data produced by complete digestion where enzymes specific for any possible combination of chemical bonds are employed.
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    Bulletin of mathematical biology 28 (1966), S. 283-308 
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    Notes: Abstract To the extent that all biological phenomena are perceivable only through their physical manifestations, it may be justified to assume that all biological phenomena will be eventually represented in terms of physics; perhaps not of present day physics, but of some “extended” form of it. However, even if this should be correct, it must be kept in mind that representing individual biological phenomena in terms of physics is not the same as deducing from known physical laws the necessity of biological phenomena. Drawing an analogy from pure mathematics, it is possible that while every biological phenomenon may be represented in terms of physics, yet biological statements represent a class of “undecidable” statements within the framework of physics. Such a conjecture is reinforced by the history of physics itself and illustrated on several examples. The 19th century physicists tried in vain todeduce electromagnetic phenomena from mechanical ones. A similar situation may exist in regard to biological and social sciences. Quite generally, the possibility of representing a class B phenomena in terms of class A phenomena does not imply that the phenomena of class B can be deduced from those of class A. The consequences of the above on the relation between physics, biology, and sociology are studied. A tentative postulational formulation of basic biological principles are given and some consequences are discussed. It is pointed out that not only can the study of biological phenomena throw light on some physical phenomena, but that the study of social phenomena may be of value for the understanding of the structures and functions of living organisms. The possibility of a sort of “socionics” is indicated.
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    Bulletin of mathematical biology 28 (1966), S. 371-374 
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    Notes: Abstract It is shown that any (ℳ ℛ) has some component which cannot be re-established after it has been inhibited. If there is only one such component, it must be central, that is, its inhibition stops the whole system. These results hold even when it is not assumed that ℳ is connected.
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    Bulletin of mathematical biology 28 (1966), S. 315-331 
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    Notes: Abstract In this paper a theory of a class of restricted transition probabilities is developed and applied to a problem in the dynamics of biological populations under the assumption that the underlying stochastic process is a continuous time parameter Markov chain with stationary transition probabilities. The paper is divided into three parts. Part one contains sufficient background from the theory of Markov processes to define restricted transition probabilities in a rigorous manner. In addition, some basic concepts in the theory of stochastic processes are interpreted from the biological point of view. Part two is concerned with the problem of finding representations for restricted transition probabilities. Finally, in part three the theory of restricted transition probabilities is applied to the problem of finding and analyzing some properties of the distribution function of the maximum size attained by the population in a finite time interval for a rather wide class of Markov processes. Some other applications of restricted transition probabilities to other problems in the dynamics of biological populations are also suggested. These applications will be discussed more fully in a companion paper.
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    Bulletin of mathematical biology 28 (1966), S. 433-441 
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    Notes: Abstract Equilibrium solubility considerations are presented based on the assumption that equating the kinetic expressionq, developed in part I, to zero can describe the equilibrium or steady state between hydroxyapatite and salt solutions. From this expression is derived Hodge's empirical equilibrium equation,C=KH. Further, a lograithmic transformation of this equation results in an expression that accounts for the equilibrium calcium, phosphorus andpH relation found by Levinskas and Neuman. Finally, it also shows the relation between log (C·P) andpH necessary for typical artificial carious lesions as found by Coolidge, Besic and Jacobs. A discussion of a recent theory of hydroxyapatite solubility of LaMer reveals calculation errors that vitiate his results. It is shown that logK 1 (K 1 is the ratio of the rate constants inq and can serve as a solubility equilibrium constant for hydroxyapatite) varies by only 1.2 units when calculated from three diverse sets of data. This variation is less than that reported by LaMer (when the errors of calculation in that work are corrected) and considerably less than the range of 11 among attempts to calculate a conventionalpK sp , as summarized by Hodge.
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    Bulletin of mathematical biology 28 (1966), S. 465-475 
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    Notes: Abstract In imitative behavior, as studied previously by N. Rashevsky (Mathematical Biology of Sociol Behavior, Chapter XIII, The University of Chicago Press, 1950), the reason for the majority of a society to accept a particular behavior is based on purely voluntary action (band-wagon effect). In the present paper effects of coercion of the majority by a small minority group which poses the means for coercion, are studied. More general types of equations are thus obtained and threshold effects found, which bear a resemblance to some such effects studied previously.
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    Notes: Abstract Part III attempts to develop a diffusion controlled model of caries in the intact enamel employing the kinetic results of the previous two parts. A model of the enamel as a granular bed with a diffusible organic matrix filling the interstices is considered. The basic equations of diffusion and simultaneous reaction are developed under the assumption that all the reactions are so rapid as compared with the diffusion rate, that they are in a quasi-equilibrium state. The resultant system of seven coupled, non-linear parabolic partial differential equations is of such complexity that only numerical solutions could be attempted. Stability restrictions inherent in the problem dictated the use of the DuFort-Frankel numerical solution for parabolic boundary problems. Numerical solutions giving the concentration of all reactants, the rate of mineral loss, and the enamel porosity were obtained for a variety of boundary conditions. It is found that departure from the equilibrium condition expressed in part II is necessary for the occurrence of an attack on the enamel. The rate and pattern of penetration is then determined primarily by the concentrations of undissociated buffer, and salts, together with the rate of diffusion in the surrounding medium. The possibility of a relatively intact surface layer persisting over a demineralized subsurface region due solely to the composition of the demineralizing medium is noted. Remineralization behavior in portions of the carious lesion occurs in the model under certain boundary conditions.
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    Bulletin of mathematical biology 28 (1966), S. 91-102 
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    Notes: Abstract In previous papers (1955–1957) a theory of biological similarity was established, assuming that the limits are the mechanical and the electrodynamical similarity criteria. The range of this theory lies between the coefficient of the time exponent (γ) for mechanical (0.5γ) and electrodynamical (1.0γ) similarities, being the mode 0.93γ. Moreover, for certain functions this restricted theoretical range should be extended to the hydrodynamical similarity criterion (2γ), so that the dimensionless numbers commonly used in Physics (Reynolds, Froude, Weber, etc.) can be included within the total range (0.5–2γ) of biological similarities. From dimensional analysis of physiological, functions it was possible to obtain, by means of dimensional and solution matrices, a group of “nondimensional numbers” by applying Buckingham's Pi-theorem. Nevertheless, only if a single similarity criterion was applied, the residual weight exponent was exactly zero; in all other instances the weight exponent was not zero, due to the existence of a range for biological similarities and to the statistical meaning of exponent (b) of the allometric equations. From the similarity criteria “invariant numbers” can be obtained, by means of which it is possible to establish correlations between numerous morphological and physiological characteristics of a particular system (circulation, respiration, metabolism, etc.).
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    Bulletin of mathematical biology 28 (1966), S. 117-124 
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    Notes: Abstract The notion of relations between sets, defined in a previous publication (Bull. Math. Biophysics,23, 233–235, 1961) is generalized and some biological examples are given. A generalization ton-ary relation is suggested.
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    Bulletin of mathematical biology 28 (1966), S. 107-116 
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    Notes: Abstract A method is introduced for using matrices to represent the organism-graphs of Rashevsky's theory of biotopological mapping. The representation is made in such a way as to reveal the structure of these graphs. Using insight gained from the consideration of the matrix representations, a theorem is proved concerning the primordial origins of organisms and counterexamples are displayed to show the necessity of the hypotheses of this theorem.
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    Bulletin of mathematical biology 28 (1966), S. 137-138 
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    Bulletin of mathematical biology 28 (1966), S. 139-139 
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    Bulletin of mathematical biology 28 (1966), S. 125-135 
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    Notes: Abstract The neurobiophysical model of schizophrenia discussed previously (Bull. Math. Biophysics,26, 167–185, 1964;27, 21–26, 1965) is generalized further, to include catatonic and stuporous states. It is concluded that the development of schizophrenia will proceed through different stages of catatonic and non-catatonic states, depending on parameters which characterize on one hand the general inhibition of the individual, on the other hand what may be called his “stability.” Suggestions for possible clinical verifications of the conclusions are made.
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    Bulletin of mathematical biology 28 (1966), S. 141-148 
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    Notes: Abstract Using the relationship between (M,R) and sequential machines developed in previous work, it is shown that the totality of (M,R) which can be formed over a given categoryA itself forms a category in a natural fashion.
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    Bulletin of mathematical biology 28 (1966), S. 149-151 
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    Notes: Abstract The condition which allows the existence of induced replication maps in (M,R)-systems is shown to place strong restrictions on the “richness” of the category from which these systems can be constructed. This condition also admits of a simple biological interpretation, which can be checked empirically, and which may offer insight into the physical and biological realizations of these abstract systems.
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    Bulletin of mathematical biology 28 (1966), S. 153-160 
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    Notes: Abstract Rosen’s identification of abstract biological systems, called (M,R)-systems, with sequential machines is formally characterized. It is then shown that the determination of environmental alterations of (M,R)-systems from a knowledge of the response sequence and the structure of the system, which we call behavioral reversibility, can be interpreted as information-losslessness of sequential machines. Applying this relationship, necessary conditions for behavioral reversibility are derived. It is further shown that, similar to Rosen’s work on structural reversibility, (M,R)-systems are behaviorally reversible only if the number of physically realizable mappings are restricted.
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    European journal of wildlife research 12 (1966), S. 29-29 
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    European journal of wildlife research 12 (1966), S. 30-30 
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    European journal of wildlife research 12 (1966), S. 31-33 
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    European journal of wildlife research 12 (1966), S. 34-34 
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    European journal of wildlife research 12 (1966), S. 5-11 
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    Description / Table of Contents: Summary This paper discusses the taxonomic significance of hide coloration of the roe deer of Europe and the Near East from several points of view. The coloration as such is certainly no fundamental criterian in the case of environment — parallel color phases („Standortsformen“, ecotypes); but it becomes important as a distinguishing and delimiting character, where isochromatic ecotypes occur, widely separated geographically from one another by other color phases (and thus indicate a divergent evolutionary development). The coloration maintains this preeminent significance a taxonomic character until an exact study of these parallelraces reveals anatomical or other deviations. Most likely, all color characters are directly genetically determined and some observations substantiating this hypothesis are cited. Thus in certain cases of the taxonomic classification of the subspecies group („Formenkreis“) it may be justifiable to use hide color as a decisive criterion the valid subspecies of the prospecies capreolus (the western roe deer) are listed at the close of the paper.
    Abstract: Résumé Les différents aspects de la signification taxonomique de la couleur du pelage du chevreuil (Capreolus capreolus) fait l'objet d'une discussion. En tant que tels, les tons du pelage résultant de conditions de mileu parallèles (variétés stationnelles, écotypes) ne constituent pas un critère taxonomique fondamental; toutefois, ils gardent une signification comme caractères d'identification ou de délimitation géographique dans les cas où des écotypes, présentant un ton du pelage identique, vivent clairement séparés dans l'espace d'autres écotypes et résultent dès lors de précédents évolutifs divergents. La couleur du pelage garde cette signification prioritaire aussi longtemps qu'une analyse de ces races parallèles recouvre des déviations anatomiques ou autres. Les caractères de couleur sont plus que probablement fixés directement dans le matériel héréditaire. Quelques observations sont mentionnées qui s'accordent avec cette hypothèse. Par conséquent, il est justifié, dans la classification taxonomique, de faire usage aussi, dans bien des cas, de la couleur du pelage comme critère décisif de sub-spéciation. A l'issue de cet exposé, l'auteur donne la liste des sous-espèces valables du prospecies capreolus (chevreuil occidental).
    Notes: Zusammenfassung Die taxonomische Bedeutung der Fellfarbe des Rehes wird unter verschiedenen Gesichtspunkten diskutiert. Bei umweltparallelen Farbtönen („Standortformen“, Ökotypen) ist die Färbung als solche zwar grundsätzlich kein Kriterium, sie behält aber als Erkennungs-und Begrenzungszeichen in den Fällen seine Bedeutung, in denen gleichgefärbte Ökotypen geographisch weit voneinander durch andere getrennt leben (und daher eine divergente Entwicklung hinter sich haben). Die Färbung behält diese vorrangige Bedeutung so lange, bis ein genaues Studium dieser Parallelrassen anatomische oder sonstige Abweichungen aufdeckt. — Alle Farbcharaktere sind mit großer Wahrscheinlichkeit unmittelbar erblich verankert. Einige Beobachtungen, die darauf hindeuten, werden angeführt. — Es ist daher berechtigt, bei der taxonomischen Gliederung des Formenkreises auch die Färbung der Decke in manchen Fällen als entscheidendes Kriterium heranzuziehen. — Es werden anschließend die validen Unterarten der Prospeciescapreolus (Westreh) angeführt.
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    European journal of wildlife research 12 (1966), S. 34-35 
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    European journal of wildlife research 12 (1966), S. 35-37 
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    European journal of wildlife research 12 (1966), S. 38-48 
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    European journal of wildlife research 12 (1966), S. 48-48 
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    European journal of wildlife research 12 (1966), S. 16-28 
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    Description / Table of Contents: Summary Grass is an important food-component in rumina of Red Deer (Cervus elaphus L.). On meadows for Red Deer grazing, however, is often so little intensive, that in summer repeated mowing is necessary. This indicates that our knowledge of possible food preferences for certain grass-species, of management of game-meadows and of the optimal size of the meadows is as yet insufficient. By means of small experimental plots in game-pastures, the authors investigated which grass-species were preferred by Red Deer and by what qualities of the grass these preferences are determined. It was found that grasses are accepted only in the period that they are lush. Thus in gamemeadows only grasses should be used that flower late and therefore remain lush for a period of several months. In addition it was found that lushness can be maintained or prolonged by fertilizing with nitrogen twice or three times during the vegetation period. Appreciated species arePhleum pratense, Lolium perenne, Dactylus glomerata, Poa pratensis, Agrostis tenuis, Deschampsia flexuosa (p. 21). Cespitosing in these species proved to be very good already in the year after sowing (p. 22). Qualities like persistance, resistance, production and the like are further investigated in these preferred species. An interaction was found between the palatability of the grass and the grazing-intensity; not only does palatability attract game but also intensive grazing makes the meadow remain lush. The authors hope to establish the optimum size of a game-pasture in relation to the deer population in the same area; that is such a size, that the game keeps the grass cropped short without overgrazing it. Management would benefit from the game-pastures being selfsupporting. We hope to reach final conclusions within two years.
    Abstract: Résumé L'herbe est un constituant important de la panse du cerf (Cervus elaphus L.). Par moments, l'abroutissement des pâtures à cerfs est toutefois tellement peu intense qu'une coupe répétée s'impose au cours de l'été. Ceci indique notre manque de connaissances en ce qui concerne la prédilection du cerf pour certaines graminées, les traitements à appliquer aux pâtures et les dimensions optimales à leur conférer. A l'aide de parcelles d'essai, les auteurs ont étudié les compositions de graminées qui semblent les mieux appétées par le gibier de même que les propriétés qui semblent se trouver à l'origine de ce choix sélectif. A cet égard, il semble que la tendresse de l'herbe constitue un caractère décisif. L'appétence des graminées possédant cette propriété peut être maintenue ou exaltée par des distributions répétées de fumure azotées au cours de la croissance. Les espèces suivantes furent particulièrement appréciées: Herbe à Timothée ou Fléole des prés (Phleum pratense L.), Ray-grass anglais (Lolium perenne L.), Pâturin des prés (Poa pratensis L.), et Agrostis ténue (Agrostis tenuis Sibth.) (cfr. pag. 21). Chez ces différentes sortes la formation de mottes fut largement satisfaisante dès l'année faisant à l'ensemencement (pag. 22). On étudie de façon plus précise les propiétés de certaines races parmi les espèces retenues, telles que la persistance, la résistance, la production etc.... En faisant usage des interactions entre l'appétence et l'intensité de la tonte, les auteurs pensent pouvoir déterminer l'étendue maximale d'une pâture en fonction de la densité locale du gibier. La pratique exige la mise à la disposition d'une pâture à gibier requérant un minimum de travail pour un maximum de profit. Afin d'atteindre ce double but les experiences seront poursuivies pendant un ou deux ans.
    Notes: Zusammenfassung Gras ist ein wichtiger Anteil der Nahrung im Pansen des Rotwildes. Im Gegensatz hierzu werden die Gräser der Wildwiesen oft so wenig abgeäst, daß ein wiederholtes Mähen notwendig ist. Diese Tatsache zeigt, daß unser Wissen über die Bevorzugung bestimmter Grasarten, über die Behandlung der Wildwiesen und über ihre optimale Größe noch mangelhaft ist. Auf Versuchsflächen wurde untersucht, welche Grassorten vom Rotwild bevorzugt werden und welche Eigenschaften des Grases die Vorliebe verursachen. Es zeigte sich, daß die Zartheit des Grases ausschlaggebend ist. Bei Gräsern mit dieser Eigenschaft kann die Schmackhaftigkeit durch wiederholte Stickstoffdüngung während der Vegetationsperiode bewahrt oder erhöht werden. Besonders geschätzt wurden u. a.Phleum pratense L.,Lolium perenne L.,Dactylis glomerata L.,Poa pratensis L.,Agrostis tenuis Sibth. (Tab. S. 21). Die Rasenbildung war bei diesen Arten im Frühjahr nach der Aussaat bereits sehr befriedigend (Tab. S. 22). Bei bestimmten Rassen von geschätzten Grasarten wurden Eigenschaften wie Ausdauer, Widerstandsfähigkeit, Ertrag usw. untersucht. Mit Hilfe der festgestellten Wechselwirkung zwischen Schmackhaftigkeit und Beweidungsintensität meinen die Autoren, die optimale Größe einer Wildwiese für örtliche Wilddichten finden zu können. Es ist der Praxis dienlich, über Wildwiesen verfügen zu können, die bei geringem Arbeitsaufwand reichen Ertrag bringen. Um dieses doppelte Ziel zu erreichen, werden die Versuche noch ein oder zwei Jahre weitergeführt werden.
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    European journal of wildlife research 12 (1966), S. 49-54 
    ISSN: 1439-0574
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary Age determination on teeth ofPhoca vitulina combined with measuring the length of os penis allows age esteemation in this species.
    Abstract: Résumé Des déterminations de l'âge par analyse des canines du phoque marin (Phoca vitulina) ont été exécutées qui combinées à des mensurations de l'os penis, donnent une idée du déroulement de la croissance chez ce pinnipède.
    Notes: Zusammenfassung An den Eckzähnen von Seehunden wurden Altersbestimmungen ausgeführt, die in Kombination mit Messungen am Baculum ein Bild vom Verlauf des Wachstums beim Seehund vermitteln.
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    European journal of wildlife research 12 (1966), S. 85-85 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    European journal of wildlife research 12 (1966), S. 85-87 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    Computing 1 (1966), S. 50-61 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es werden Endliche Automaten betrachtet, deren Übergangsmatrix block-stochastisch ist. Die block-stochastische Struktur definiert eine Äquivalenzbeziehung zwischen Zuständen des Automaten. Die Bedeutung und Auswirkung dieser Relation wird untersucht, und zwar insbesonders in Hinsicht auf die in den einzelnen Zuständen des Automaten angenommenen Sprachen.
    Notes: Summary Finite automata are considered whose transition matrix is blockstochastic. The block-stochastic structure defines an equivalence relation among states of the automata. The implications of this relation are investigated, especially with respect to the languages accepted in the states of the automata.
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    Computing 1 (1966), S. 88-92 
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    Computing 1 (1966), S. 119-126 
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    Description / Table of Contents: Summary This paper gives a contribution to the hypothesis ofLense regarding the movements of the second order zeros of the derivativesI′ ν (z) of Bessel functions for every negativ variable ν.
    Notes: Zusammenfassung Die Arbeit liefert einen Beitrag zur derLense'schen Vermutung über die Bewegung der Doppelnullstellen der AbleitungI′ ν (z) der Besselfunktionen bei veränderlichem negativen ν.
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    Computing 1 (1966), S. 133-145 
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    Description / Table of Contents: Summary In linear programming often it is important whether a given linear programming problem is equivalent to a transportation problem. In this case, the stepping-stone method could be taken for solving the problem, instead of the simplex method, which requires more storage capacity and computing time.—To decide this question a so-called simplex matrix is used, which results from the given linear programming problem treated by the simplex method. By help of two necessary conditions as well as a necessary and sufficient condition it can be concluded whether the linear programming problem belonging to that simplex matrix is equivalent to a transportation problem or not.—The practical handling of the developed algorithm is shown by an example.
    Notes: Zusammenfassung In der linearen Planungsrechnung interessiert oft, ob ein gegebenes lineares Optimierungsproblem sogar ein Transportproblem ist. Dann könnte man nämlich zur Lösung des Problems statt der Simplexmethode die Stepping-Stone-Methode anwenden, die weniger Speicherplatz und Rechenzeit erfordert.—Zur Klärung dieser Frage geht man von einer sogenannten Simplexmatrix aus, die aus dem mit der Simplexmethode behandelten linearen Optimierungsproblem entstanden ist. Mit Hilfe von zwei notwendigen Bedingungen sowie einer notwendigen und hinreichenden Bedingung läßt sich dann entscheiden, ob das zu jener Simplexmatrix gehörige lineare Optimierungsproblem ein Transportproblem ist oder nicht.—Die praktische Handhabung des entwickelten Verfahrens wird an einem Beispiel gezeigt.
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    Computing 1 (1966), S. 197-213 
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    Topics: Computer Science
    Description / Table of Contents: Summary Besides the electronic digital computers electronic analog computers and their generalisations to iterativ-analog computers or analog computers controlled by digital computers and finally the connection of analog and digital computers in hybrid computers are of increasing importance especially for those mathematical problems, which do not need the high accuracy of digital computers, whereas the typical possibilities of the analog computers allow a quick survey of the features of the solutions. From these applications we choose a few problems of operations research.
    Notes: Zusammenfassung Neben den elektronischen Digitalrechenanlagen gewinnt heute der elektronische Analogrechner und seine Verallgemeinerung zum iterativen Analogrechner, sowie der durch einen Digitalrechner gesteuerte Analogrechner und schließlich die Verbindung von Analog- und Digitalrechner im Hybridrechner zunehmende Bedeutung. Das gilt vor allem für solche mathematische Problemstellungen, bei denen die große Genauigkeit des Digitalrechners nicht erforderlich ist, andererseits aber die typischen Eigenschaften des Analogrechners einen raschen Überblick über die Eigenschaften der Lösungen ermöglichen. Aus der Fülle derartiger Anwendungen seien im folgenden Probleme der Unternehmungsforschung herausgegriffen.
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    Computing 1 (1966), S. 282-282 
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    Computing 1 (1966), S. 273-280 
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    Description / Table of Contents: Summary There exists a general principle for finite matrices, according to which to every matrix norm corresponds an inclusion theorem for eigenvalues. If the norm is the row-sum norm, we have theGershgorin theorem. Another inclusion theorem is used for obtaining bounds for the deviations of the eigenvalues from the diagonal elements, involving the order of the matrix, the maximal modulus of the off-diagonal elements and the distances of the diagonal elements. These bounds yield estimates for theJacobi method for determination of eigenvalues of symmetric matrices.
    Notes: Zusammenfassung Für endliche Matrizen wird ein allgemeines Prinzip gezeigt, das jeder Matrixnorm einen Einschließungssatz für eigenwerte zuordnet. Für die Norm der maximalen Zeilenbetragssumme ergibt sich speziell der Satz vonGerschcorin. Ein anderer Einschließungssatz wird dazu benutzt, für die Abweichungen der Eigenwerte von den Diagonalelementen Schranken aufzustellen, die von der Ordnung der Matrix, dem Maximalbetrag der Nichtdiagonalelemente und den Abständen der Diagonalelemente abhängen. Diese Schranken liefern Fehlerabschätzungen für dasJacobi-Verfahren zur Bestimmung der Eigenwerte symmetrischer Matrizen.
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    Computing 1 (1966), S. 327-340 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung In dieser Arbeit wird die algebraische Struktur von abstrakten Automaten untersucht. Als Hilfsmittel dazu dient die Automorphismengruppe von Automaten, das ist die Gruppe aller Zustandspermutationen, bei welchen die Übergangsfunktion erhalten bleibt. Die Zustandsmenge eines Automaten wird in Äquivalenzklassen von “eng verbundenen” (strongly connected) Teilmengen zerlegt. In der Menge dieser Äquivalenzklassen erklären wir eine Teilordnung, deren minimale Elemente “Quellklassen” (source-classes) genannt werden. Wenn nur eine Quellklasse existiert, dann heißt der Automat zyklisch (Oehmke [3]); wenn jeder Automorphismus alle eng verbundenen Äquivalenzklassen auf sich selbst abbildet, dann nennen wir den Automaten normal. In den bisherigen Arbeiten wurden fast ausschließlich nur eng verbundene Automaten behandelt. Hier hingegen erstrecken sich die Untersuchungen auf zyklische und normale Automaten. Dabei werden auch einige Resultate vonA. Fleck [1] über eng verbundene Automaten verallgemeinert. Gelegentlich beschränken wir uns auf abelsche Automaten.
    Notes: Summary In this paper the structure of automata is investigated using the concept of the automorphism group. The investigations about strongly connected automata are extended to cyclic (Oehmke) and normal automata. The set of states is divided into equivalence classes of strongly connected subsets (SCEC). In the set of all SCEC we explain a partial ordering whose minimal elements are called sourceclasses. If there is only one source-classe, the automaton is called cyclic. If each automorphism maps every SCEC onto itself, then the automaton is said to be normal. We generalize some results ofA. Fleck [1]. In some cases we restrict ourselves to Abelian automata.
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    Computing 1 (1966), S. 358-367 
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    Computing 1 (1966), S. 233-255 
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    Description / Table of Contents: Summary As in many branches of science, there has been a change of inner structure taking place in Applied Mathematics in recent times, with a marked turn towards abstraction, exemplified by the wide use of methods of functional analysis. In this summarizing lecture, an attempt is made to demonstrate the useful application of abstract notions in Numerical Mathematics, especially by means of the concept of partial order. This concept is used, for instance, in connection withRiesz' partially ordered Banachspaces, pseudometric spaces, intervals, monotone and positive operators, linear and nonlinear optimization. Applications of monotone operators are given for benting of beams, boundary value problems for linear and nonlinear elliptic differential equations, extrapolation for initial value problems and eigenvalue problems. Emphasis is laid on the interrelation between various problem areas, e.g. between boundary value problems and optimization problems.
    Notes: Zusammenfassung In vielen Wissenschaften, so auch in der Angewandten Mathematik, hat sich in letzter Zeit ein innerer Strukturwandel, insbesondere eine starke Wendung zum Abstrakten hin, vollzogen, welche sich z. B. in der ausgiebigen Verwendung funktionalanalytischer Methoden äußert. In diesem zusammenfassenden Vortrag wird versucht, am Beispiel der Halbordnung die Verwendung und den Nutzen abstrakter Begriffe in der Numerischen Mathematik zu zeigen. Die Halbordnung wird u. a. benutzt bei den Begriffen:Rieszscher Halbordnungs-Banachraum, pseudometrischer Raum, Intervall, monotoner Operator, positiver Operator, lineare und nichtlineare Optimierung. Anwendungen der monotonen Operatoren werden beschrieben bei der Biegegleichung für Träger, bei Randwertaufgaben linearer und nichtlinearer elliptischer Differentialgleichungen, bei der Extrapolation für Anfangswertaufgaben und bei Eigenwertaufgaben. Es werden die Zusammenhänge zwischen verschiedenen Gebieten betont, z. B. zwischen Randwertaufgaben und Optimierungsaufgaben.
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    Computing 1 (1966), S. 309-315 
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    Description / Table of Contents: Summary In this paper a universal recurrence formular is given for the calculation of theLegendre-Polynomials an their derivates. The first values for the beginning of the recurrence are calculated. Then there are given some simple formulas for the values of functions with the arguments 0 and 1 at the ends of the considered interval. At last the functional equations are generalized.
    Notes: Zusammenfassung Der vorliegende Beitrag gibt eine universelle Rekursionsformel zur Berechnung derLegendre-Polynome und ihrer Ableitungen an. Die Ausgangswerte der Rekursion werden berechnet. Ferner werden einfache Ausdrücke zur Berechnung der Funktionen an den Intervallenden hergeleitet. In einem letzten Abschnitt werden die bekannten Differentialbeziehungen derLegendre-Polynome verallgemeinert und die Differentialgleichung der (m-1) ten Ableitung des Polynomesn-ten Grades angegeben.
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    Monatshefte für Mathematik 70 (1966), S. 1-7 
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    Monatshefte für Mathematik 70 (1966), S. 15-32 
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    Monatshefte für Mathematik 70 (1966), S. 8-14 
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    Notes: Abstract In this paper giving the definitions of the various generalised Bessel coefficients and of some of their associated functions, Addition Theorems and their extension corresponding to Neumann's generalisation have been studied.
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    Monatshefte für Mathematik 70 (1966), S. 33-38 
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    Monatshefte für Mathematik 70 (1966), S. 39-46 
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    Monatshefte für Mathematik 70 (1966), S. 47-57 
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    Monatshefte für Mathematik 70 (1966), S. 74-80 
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    Monatshefte für Mathematik 70 (1966), S. 64-73 
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    Monatshefte für Mathematik 70 (1966), S. 58-63 
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    Monatshefte für Mathematik 70 (1966), S. 81-96 
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    Monatshefte für Mathematik 70 (1966), S. 97-105 
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    Monatshefte für Mathematik 70 (1966), S. 106-110 
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    Monatshefte für Mathematik 70 (1966), S. 111-117 
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    Monatshefte für Mathematik 70 (1966), S. 118-126 
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    Monatshefte für Mathematik 70 (1966), S. 127-133 
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