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  • Springer  (20,311)
  • Elsevier  (4,376)
  • Oxford University Press  (1,361)
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  • Molecular Diversity Preservation International
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  • 2015-2019
  • 1995-1999
  • 1955-1959  (15,577)
  • 1950-1954  (10,471)
  • 1940-1944
  • 1957  (15,577)
  • 1953  (10,471)
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  • 2015-2019
  • 1995-1999
  • 1955-1959  (15,577)
  • 1950-1954  (10,471)
  • 1940-1944
Year
  • 1
    Electronic Resource
    Electronic Resource
    Random walk with persistence and external bias (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 311-338 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The partial differential equation of the random walk problem with persistence of direction and external bias is derived. By persistence of direction or internal bias we mean that the probability a particle will travel in a given direction need not be the same for all directions, but depends solely upon the particle's previous direction of motion. The external bias arises from an anisotropy of the medium or an external force on the particle. The problem is treated by considering that the net displacement of a particle arises from two factors, namely, that neither the probability of the particle traveling in any direction after turning nor the distance the particle travels in a given direction need be the same for all directions. A modified Fokker-Planck equation is first obtained using the assumptions that the particles have a distribution of travel times and speeds and that the average time of travel between turns need not be zero. The fional equation incopporating the assumption of a persistence of direction and an external bias is then derived. Applications to the study of diffusion and to long-chain polymers are then made.
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    URL: http://dx.doi.org/10.1007/BF02476407
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  • 2
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    Methods of statistical analysis (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 383-383 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476411
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  • 3
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    The transmission of excitation from the membrane to actomyosin (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 385-385 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476412
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  • 4
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    On the spread of information with time and distance (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 367-381 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.
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    URL: http://dx.doi.org/10.1007/BF02476410
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  • 5
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    A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 387-394 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A situation is considered in which a fluid containing a substance flows through a vessel at a constant rate, the substance being permeable to the vessel wall. In the region outside the vessel there is supposed to be rapid mixing in the direction perpendicular to the axis of the vessel but no mixing longitudinally. The solution for the spatial distribution at any time is given for the case of an arbitrary initial distribution along the vessel length in the absence of an input. The solution is also given for the case of a single impulsive input, the concentration being initially zero everywhere.
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    URL: http://dx.doi.org/10.1007/BF02476432
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  • 6
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    A mathematical contribution to the study of orientation of organisms (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 431-476 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Organisms orient themselves to a stimulus by two general methods. One method is by directed orientation (taxis); the other is by undirected locomotory reaction (kinesis). An equation, and the methods for finding the necessary parameters of this equation, is derived for the distribution of organisms within a container, with the following limitations: (1) the organisms have no accommodation, (2) they are always active, and (3) the stimulus changes slowly with position. Necessary modifications of the equation are then derived, so that the last two limitations may be eliminated. The equation cannot be solved excatly because of its complexity; hence an approximation method must be used. This method is discussed, an approximate solution is found, and a time constant for equilibrium to be established is derived. Applications tovarious experiments in the literature are then made with fairly satisfactory results. A new interpretation of the theory of klino-kinesis with accommodation is found upon application of the equations developed to experimental work. Further limitations and uses of these equations are then discussed.
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    URL: http://dx.doi.org/10.1007/BF02476435
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  • 7
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    Structural parameters of communication networks (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 501-507 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Certain parameters are defined which roughly characterize the internal structure of networks. A given network structure uniquely determines the values of the parameters, but the reverse is not true. The parameters therefore define certain classes of networks. One of the parameters, thedispersion D(S) gives an indication of the “compactness” of the internal structure. Addition theorems and inequalities are derived relating the dispersions of sub-systems to the dispersion of the complete structure.
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    URL: http://dx.doi.org/10.1007/BF02476438
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  • 8
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    Determination of diffusion and permeability coefficients in nerve trunks (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 489-500 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical theory is developed which permits the determination of certain parameters of an inhomogenous tissue, such as a nerve trunk without its epineurium. The parameters are the permeability coefficients for entrance into an exit of a substance from the nerve fibers, and the diffusion coefficient of the interstitial material. The experimental data required are the dimensions of the cross-section, the average diameter of the fibers, and the ratio of the cross-sectional are of the fibers to the total cross-section, as well as the time course of the decrease of the fraction of the substance left in the nerve trunk, when the trunk is immersed in a bathing solution containing none of it.
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    URL: http://dx.doi.org/10.1007/BF02476437
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  • 9
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    Enzyme localization as a mechanism of apparent active transport (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 509-522 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model based on enzyme localization is developed which gives rise to an apparent active transport of a metabolite into or out of cells. The model is applied to three simple situations, using Fick's equation and the Rashevsky approximation. It is shown that the apparent efficiency can be made as large as desired if, for constant reaction, the outer cell region is made sufficiently small, or, for autocatalytic reaction, if the metabolite concentration in the outer region is sufficiently small. The physical limitations imposed by this mechanism are developed for all three situations.
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    URL: http://dx.doi.org/10.1007/BF02476439
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  • 10
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    Spread of information through a population with socio-structural bias: I. Assumption of transitivity (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 523-533 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.
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  • 11
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    Spread of information through a population with socio-structural bias: II. Various models with partial transitivity (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 535-546 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The assumption of transitivity treated in part I is modified in various ways to describe an information-diffusion process, in which a certain amount of randomness of contact does occur. In one model a parameter is introduced which is indicative of a tendency to go beyond one's immediate vicinity to spread the information as the vicinity becomes saturated with knowers. In another model the randomness appears in the assumption that new knowers are uniformly distributed among the knowers. Two of the equations thus derived, each with two free parameters are in good agreement with experimental results.
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    URL: http://dx.doi.org/10.1007/BF02476441
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  • 12
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    The transmission of excitation from the membrane to actomyosin (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 1-13 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The impressed field, “Window Field” (WF), due to a half-wave action potential on a muscle fiber, has been calculated on the basis of potential theory. It has been shown that in spite of the small intensity of the field, its integrated action can transfer the energy needed to induce, contraction from the membrane to the interior of the fiber. The energy of polarization has been found to be sufficient to exceed the energy of, thermal agitation on that length of fiber, which can be identified as the length of a sarcomere. The changes of ion concentration, caused by the WF, if calculated on the assumption of the semipermeability of theZ membranes, was found to be equal to the changes necessary to induce contraction of actomyosinin vitro.
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  • 13
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    Nonlinear diffusion in metabolic systems (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 15-21 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Some general properties of the solution of the diffusion equation are deduced for the steady-state, spherically symmetric system. On the basis of these developments some results of N. Rashevsky (Bull. Math. Biophysics,11, 15, 1949) are discussed and the results of a previous investigation (Hearon,Bull. Math. Biophysics,12, 135, 1950b) are extended to more general conditions. In particular these extensions apply to the flow of a soluteagainst its concentration gradient, the nonzero gradient of an inert metabolite, and theaccumulation or exclusion of an inert metabolite in a metabolic system.
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  • 14
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    Comments on the approximate solution of the diffusion equation: I (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 23-31 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The approximation method of N. Rashevsky is discussed and reviewed. It is shown that in addition to theexplicit assumptions and approximations there is involved the assumption that the rate of metabolism is the same at every point in the cell and that theaverage rate of metabolism is different from zero. An expression is given for the error in the approximate method when the rate of metabolism is any function of the concentration. It is also shown that a solution in theform of that obtained by the approximate method is not possible if the generalized laws of diffusion are assumed to apply.
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  • 15
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    The kinetics of the succinoxidase (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 33-42 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Rate equations for the enzymatic oxidation of succinic acid are derived on the assumption that when a single molecule of substrate combines with an enzyme molecule, it can do so with either one or two sites on the enzyme, and that oxidation occurs only in the second case. In addition it is assumed that the product of the reaction, fumaric acid, combines reversibly with the enzyme. With certain enzyme preparations the data fitted such an equation satisfactorily. In others the rate was that of a first-order reaction, but addition of cytochrome changed it to the former type. It was concluded that the transfer of hydrogen to oxygen was a first-order reaction and dominated the whole rate when enzyme preparations were used which had been washed relatively free of cytochrome. When the limiting factor was succino-dehydrogenase the rates followed the new equation. Criteria for recognizing noncompetitive inhibition are given, and inhibition by di-tertiary butyl peroxide was shown to be of this type.
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  • 16
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    The interpretation of exponential dose-effect curves (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 43-47 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper deals with the following question: Which distributions of radiosensitivity in a population can lead to an exponential survival curve? The problem is solved exactly, with statistical fluctuations in dose fully accounted for. It is shown that only an exponential distribution of sensitivities can give rise to an exponential survival curve.
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  • 17
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    An approximation method for the solution of diffusion and related problems (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 49-61 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An approximation method is introduced which enables a number of diffusion-type problems to be solved in an approximate but simple manner. Many cases require only the solution of a simple first-order differential equation. The method is applied to a number of cases in which the exact solutions are available. A comparison shows that the method is quite satisfactory in these cases. The method is applied to diffusion problems with rate of consumption proportional to concentration or to the square of the concentration. In the latter case, the result obtained is essentially the same as that found by H. G. Landau (1950) after elaborate calculations.
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  • 18
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    On spontaneous discharge obtained from a physicochemical model of excitation (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 83-91 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that a slight modification of a model of excitatory phenomena in irritable tissues, which has been treated before, exhibits spontaneous oscillations. The frequency of these oscillations and the time-course of the potential across the model membrane have been determined, together with the dependence of some of their characteristics on some important parameters, particularly (Ca++).
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  • 19
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    Cell division by swelling stresses (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 73-81 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is demonstrated that, if the variations of viscosity throughout a cell are considered, swelling stresses may produce elongation and division. To do this it is necessary to generalize Betti's theorem to cover systems containing viscosity gradients and such a generalization is presented. On the basis of two special assumptions it is shown that most of the results of the diffusion drag theory of cell division may be duplicated by the present theory.
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  • 20
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    Imitative behavior in nonuniformly spatially distributed populations (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 63-71 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory of imitative behavior, developed previously, is applied to the case of two social groups which are separated spatially. If the information of each group as to the behavior of the other is complete, the case reduces to that of a single group. When any information is lacking at all, the two groups are independent. If we have two mutually exclusive behaviorsA andB, all four combinationsAA, AB, BA, andBB are possible. If the mutual information gradually increases from zero, then for a certain value of it, the group which is more informed about the behavior of the other will change to that behavior if it did not already exhibit it. If for constant information the size of the group increases, then above a certain threshold value, the larger group imposes its behavior on the smaller.
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  • 21
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    Note on a theory of membrane permeability (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 103-104 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 22
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    The limiting information capacity of a neuronal link (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 107-107 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 23
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    Heredity in uterine cancer (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 105-106 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 24
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    Some considerations on capillary adjustment to changing metabolism of tissue (1953)
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    Bulletin of mathematical biology 15 (1953), S. 93-101 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Local regulation of blood flow as determined by capillary diameter and the number of open capillaries in a region is considered. The local changes in capillary diameter and in the number of open capillaries are assumed to be due to concentration changes of a diffusible, nonspecified metabolite. This metabolite is produced in the tissue and carried away by the blood stream. Using these assumptions and applying pertinent data on capillaries, deductions are made concerning: (a) the law of blood flow as a function of temperature and capillary radius for the hyperemia of high temperature, (b) high flow as it depends on metabolism during strenuous exercise of muscle, and (c) a first approximation to the time duration of occlusion hyperemia.
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    Some Contributions to the Mathematical Biology of Blood Circulation. Reflections of pressure waves in the arterial system (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 109-109 
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    ISSN: 1522-9602
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    On dominance relations and the structure of animal societies: III The condition for a score structure (1953)
    Springer
    Bulletin of mathematical biology 15 (1953), S. 143-148 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The necessary and sufficient condition is given forn integers to be the score structure of a society with a dominance relation. A proof is also given for a theorem showing that there are members who dominate every other member either directly or indirectly through a single intermediate member.
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    Comments on the approximate solution of the diffusion equation: II (1953)
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    Bulletin of mathematical biology 15 (1953), S. 111-119 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract On the basis of a previous general formulation (Bull. Math. Biophysics,15, 21–29, 1953a) a discussion is given of the error in the approximation method of N. Rashevsky. This error, inherent in the method when the metabolic rate is different at each point in the cell, is discussed in detail and numerical values are presented for two particular cases: the rate proportional to the concentration and the rate a prescribed function of the spatial coordinates. It is shown that the formulation for the first case also applies to several other cases, that the error is negligible provided the rate is sufficiently small, and that the error is fairly sensitive to the cell size. If the rate depends upon the coordinatesalone a small rate is not sufficient to insure a negligible error. The relations between the exact method, the standard approximate method, an earlier approximate method (Physics,7 260, 1936), and a more recent refinement (Bull. Math. Biophysics,10, 201, 1948) of the standard method are discussed.
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    The kinetics of linear systems with special reference to periodic reactions (1953)
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    Bulletin of mathematical biology 15 (1953), S. 121-141 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown on the basis of (1) conservation of mass, (2) positive concentrations, and (3) the principle of detail balancing that periodic reactions cannot occur in a closed system described bylinear differential equations. The matrix,A, of the rate equations must be such that |A|=0,a ij〉0 fori≠j,a ii〈0, andVAV −1=B, whereV is diagonal andB is symmetric. These properties ofA imply that the latent roots are real and non-positive and that neither catalysis nor inhibition can be described bylinear equations. It is further shown that periodic reactions cannot occur in anopen system for which the matrix associated with the chemical reactions has the above properties and in which thesimple law of diffusion is obeyed. The relation of these results to Onsager's reciprocal relations and to previous work on periodic and cyclic chemical reactions is discussed. The utility of certain of these results for the treatment of isotope kinetics is indicated.
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  • 29
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    A note on the flow of blood in capillary tubes (1953)
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    Bulletin of mathematical biology 15 (1953), S. 149-152 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is demonstrated that an explanation of the small radius effect or the so-called sigma phenomenon may be obtained by noting that one of the effects of the presence of suspended particles in a flowing fluid is to increase the velocity of flow near the wall over that existing in the absence of particles. This effect may be considered equivalent to relaxing the boundary conditions at the wall. An expression for the viscosity is compared with data and fit is found to be good.
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    Note on the Donnan equilibrium (1953)
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    Bulletin of mathematical biology 15 (1953), S. 153-159 
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    ISSN: 1522-9602
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    Notes: Abstract The solution for the spatial distribution of ions in a Donnan equilibrium has been given by J. H. Bartlett and R. A. Kromhout (1952). The present note gives an explicit solution for the case in which the length of the region containing the membrane is large; in biological situations this requires only that the length considered should be greater than a few hundred Ångstrom units. The Donnan equilibrium may be considered to be a special case of a situation in which forces other than electrical act upon the ions; in particular, it represents the case in which only one ion is acted upon and the energy difference on the two sides of the membrane is infinite. An expression is given for the difference in energy of theith in terms of the electrical potential and of the ion concentrations. As an illustration, the results are applied to nerve membrane potentials.
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    Some considerations on a carrier mechanism as a model for ion accumulation (1953)
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    Bulletin of mathematical biology 15 (1953), S. 161-165 
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    Notes: Abstract A mechanism is described which accounts for the active transport of Na+ ions through a membrane. It is assumed that at one side of the membrane the ion combines with a carrier ion, the resulting carrier compound then diffuses through the membrane and decomposes at the other side of the membrane. The free diffusion of the ions is also taken into account. The time rate of accumulation of the ion in question at the latter side of the membrane is calculated in terms of the concentrations of the ion at both sides of the membrane.
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    Approximate treatment of the formation of a difference in the concentrations of an ion at the two sides of a membrane (1953)
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    Bulletin of mathematical biology 15 (1953), S. 167-171 
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    Notes: Abstract The recent extension of the approximation method is applied to enable us to arrive at the time course of the concentrations at both sides of a membrane. From the differential equations which govern these, the steady-state solution is obtained in terms of the parameters, which are determined by the thickness of the diffusion layers, the chemical composition and reactions, and the diffusion constant of the membrane.
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    Contribution to the mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population (1953)
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    Bulletin of mathematical biology 15 (1953), S. 173-183 
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    Notes: Abstract An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.
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    Comments on the approximate solution of the diffusion equation: I (1953)
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    Bulletin of mathematical biology 15 (1953), S. 235-235 
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    Contributions to the mathematical biophysics of the cardiovascular system (1953)
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    Bulletin of mathematical biology 15 (1953), S. 185-195 
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    Notes: Abstract The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.
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    On the subnormal variance in the counts of randomly distributed particles: I. Approximate treatment of the three-dimensional case and discussion of experiments (1953)
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    Bulletin of mathematical biology 15 (1953), S. 237-244 
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    Notes: Abstract The question is raised concerning the possible causes of abnormally small standard deviations found in counting samples in which particles are distributed at random (e.g., blood cells, fat globules in milk, etc.). The effect of discarding abnormal samples is discounted inasmuch as small standard deviations occur even when all samples are counted. An approximation method is used to calculate the effect of finite particle size, of known repulsive forces between particles and of convection currents. This calculation shows that neither finite size nor the known repulsive forces are sufficient to account for the observed abnormality of standard deviation, but that convection currents can possibly account for it. The possible presence of long-range repulsive forces cannot, however, be excluded.
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    On the subnormal variance in the counts of randomly distributed particles: II. Exact treatment of the one-dimensional case (1953)
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    Bulletin of mathematical biology 15 (1953), S. 245-250 
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    Notes: Abstract The effect of finite particle size on the standard deviation in sample counts is computed for the one-dimensional case. To a first order of approximation the correction is found to be identifical with that found by H. de Vries (1953) using a general approximation method.
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    Outlines of a new theory of photoreception (1953)
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    Bulletin of mathematical biology 15 (1953), S. 251-260 
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    Notes: Abstract A critical examination of the “classical” theories of photoreception in view of more recent experimental findings yields the result that these theories do not possess the property to describe all the more significant phenomena of photoreception correctly, and to some extent suffer the lack of more general applicability. The basis for a new and presumably more general theory of photoreception based on dynamical aspects is laid out. Emphasis is put on the time course of afferent and efferent excitation in the photoreception model, consisting of a receptor element, an afferent and an efferent neuron of the one-factor Rashevsky-type, and an effector organ.
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    Outline of a mathematical approach to history (1953)
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    Bulletin of mathematical biology 15 (1953), S. 197-234 
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    Notes: Abstract A mathematical model for the development of human society, beginning with the earliest stages of urban cultures, is outlined. In the early stages of history, behavior was characterized largely by adherence to a number of beliefs and prejudices of diffeirent kinds, which were accepted on faith and not subject to critical rational analysis. Due to psychobiological variability a very small number of individuals spontaneously appear at all times who challenge the accepted beliefs and prejudices and do not follow the accepted patterns of social behavior. The effect of these individuals upon the rest of the society, especially upon the younger generation, depends on the facilities with which information spreads in society. In earliest societies, when modern methods of mass communication were unknown, the channels of communication were practically identical with the channels of economic transport. The latter in its turn depended on the nature of the roads, and especially on the presence of waterway, which facilitated transportation. The sizes of the earliest cities and the distances between them were largely determined by relative ease of transporation. Expressions are derived for the average size of the earliest cities and for the average distance between them. The calculated average populations of the earliest cities are of the order of 103; the distance of the order of 102 km. Both are in agreement with some archaeological findings. An expression for the time spaon required for the development from the earliest stages of urban cultures to the present time is derived and shown to depend on the specific shoreline of the country, that is, the length of the shorline divided by the area of the country. It is pointed out that western Europe's specific shoreline, including land bordering both seas and rivers, is ten times as large as the shoreline area of other parts of the world. It is shown that this greater specific shoreline may account quantitatively for the faster social and technological development of western Europe in the last few centuries. The calculated total span of time of development from earliest urban cultures to our days is found to be of the order of magnitude of ten thousand years. It is shown that the model accounts for the existence at the present time of primitive cultures. A number of suggestions is made in regard to other possible applications of mathematics to history.
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    A note on the integral-equation description of metabolizing systems (1953)
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    Bulletin of mathematical biology 15 (1953), S. 269-276 
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    Notes: Abstract The assumptions latent in the derivation of the integral equation of Branson are rendered explicit and discussed. It is shown that the equation is valid only for systems in which the substance disappears according to a linear rate law.
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    A critical investigation of the integral description of metabolizing systems (1953)
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    Bulletin of mathematical biology 15 (1953), S. 261-268 
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    Notes: Abstract It is shown that the validity of Branson's integral description of metabolizing systems is subject to severe limitations. The validity is insured only in cases where the reaction is of first order, or quasi of first order. In all other cases Branson's equation has to be modified to insure general applicability. The consequences of a different definition of the metabolizing functionF have also been investigated. With the new definitionF describes the pure effect of metabolization. It is found that in this case the integral equation is only capable of describing first-order reactions. With a slight modification of the integral equation it is possible to describe metabolites “with age”, which do not have reactions of definite order, but which satisfy the superposition principle.
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    On the statistical distribution of mutant bacteria (1953)
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    Bulletin of mathematical biology 15 (1953), S. 277-292 
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    Notes: Abstract A problem in probability is stated with included the problem of the distribution of bacterial mutants as a special case. This problem is solved exactly but since the resulting expressions are too complicated for practical use, various approximate expressions for the distribution are considered, especially for the bacterial mutation case.
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    Note on the neurobiophysical interpretation of imitative behavior (1953)
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    Bulletin of mathematical biology 15 (1953), S. 293-300 
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    Notes: Abstract Simple reaction and discrimination reaction, under the influence of imitation, are considered for the situation in which the stimulus or the stimuli vary slowly with time. The result is analogous to hysteresis under certain conditions. The calculations are facilitated by the solution of $$x = \int_{ - \infty }^{a + \beta x} {g\left( \xi \right)d} \xi ,$$ g(ξ) being the normal error function. Values ofx(α, β) are given in a table.
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    Blood flow in branching circulatory systems during rest and activity (1953)
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    Bulletin of mathematical biology 15 (1953), S. 301-309 
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    Notes: Abstract On the basis of simple physical considerations the blood flow in a branching circulatory system is studied. The case of two groups of parallel vessels is treated. The vessels of the same group are supposed to be identical. The resistance of each group is determined by the resistance of each vessel in the group and by the number of vessels in the group. From the dependence of the resistance of each vessel on its radius an expression is obtained for the blood flow through each group of vessels in terms of the numbers and sizes of the vessels in each group. The number of open vessels in an organ and the radius of each of those vessels are assumed to depend on the metabolic rate of that organ. The relations so obtained, together with the expression above, are applied to derive the blood flow through an organ as a function of the metabolic rate of that organ. It is indicated that the relations obtained might describe the shifting of blood from one organ to another if the activity of one of them changes. A way is pointed out to treat neural regulation of this phenomenon.
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    An age-dependent stochastic model of population growth (1953)
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    Bulletin of mathematical biology 15 (1953), S. 361-365 
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    Notes: Abstract A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1−σ), 0≤σ〈1, of the organism dividing into two at the end of its lifetime, and probability σ that division will not take place.
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    Some quantitative aspects of history (1953)
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    Bulletin of mathematical biology 15 (1953), S. 339-359 
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    Notes: Abstract In a previous paper, in which a possible mathematical approach to history was outlined, it was shown that urbanization plays an important part in the propagation of new ideas. The rate of such propagation influences the rate of historical developments. The present paper deals in more detail with possible mechanisms of formation of earliest cities. Equations are derived which give the limiting size of such cities and their rate of growth. Of particular importance for the spread of new ideas is the spread of information. The latter largely depends on the fraction of individuals who travel between city and country. Expressions for this quantity are derived. An approach is outlined to the mathematical study of the earliest social classes, which may have been formed as a result of military, religious, or economic stratifications.
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    On the propagation of a disturbance through a viscous liquid flowing in a distensible tube of appreciable mass (1953)
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    Bulletin of mathematical biology 15 (1953), S. 395-409 
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    Notes: Abstract The velocity of propagation of a disturbance wave in a liquid flowing in a distensible tube is computed. The mathematical model is more general than those used in previous analyses: the tube wall properties are realistic; the convective part of the axial inertia forces is taken into account; radial inertia forces of both the fluid and tube wall are present; viscous stresses are present. Four parameters influencing the velocity of propagation are obtained and discussed. Curves are plotted illustrating the effects of the parameters. Contrary to the results of previous analyses, viscous effects are shown to be appreciable in blood flow. It is also shown that radial inertia effects can be important in laboratory set-ups.
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    Theory of the vacuum drying of frozen tissues (1953)
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    Bulletin of mathematical biology 15 (1953), S. 411-429 
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    Notes: Abstract A general theory of the drying of frozen tissue is developed and applied to the measurement of the drying rate of frozen guinea pig liver. It is shown that for a given temperature of the subliming ice crystals the mininum drying time of a piece of guinea pig liver is greater than the minimum sublimation time of a piece of ice of the same size and shape by a factor of the order of one thousand. This fact has many implications in the design of freeze-dry apparatus which will be developed in a following paper.
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    The time course of capillary exchange (1953)
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    Bulletin of mathematical biology 15 (1953), S. 477-488 
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    Notes: Abstract The equations governing the time course of the exchange of substances between the blood in the capillaries and the extracellular space are solved for the case of substances which do not penetrate the cells. The equations given relate the time course of the exchange process to the various tissue and circulation parameters such as the specific capillary wall area, the pore area, the inter-capillary distance, the size of the extra-vascular, extra-cellular space, the diffusion coefficient in this space, and the velocity of blood in the capillaries. Some experimental work on capillary exchange is discussed in relation to the theory and estimates are made of the relative importance of the various tissue and circulation parameters in the exchange of substances in different tissues.
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    The excitation of the heart and its modification under the influence of the chemical mediators and the cardiac nerves: II. A particular case of the one-factor theory (1953)
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    Bulletin of mathematical biology 15 (1953), S. 547-559 
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    Notes: Abstract Previous work (Macey, 1952) in the application of the one-factor theory to the heart is extended. The rate of production of the excitatory state is assumed to be linear. Two possible mechanisms are indicated whereby such a situation might arise. Assumptions are made regarding the mode of action of the chemical mediators on the heart, and an equation is derived relating the heart rate to the frequency of nerve impulses traveling along the cardiac nerves. This result compares favorably with the experimental findings of A. Rosenblueth and F. A. Simeone (1934). Other experimental results are interpreted in terms of the theory.
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    Symposium on Radiobiology. The Basic Spects of Radiation Effects on Living Systems (1953)
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    Bulletin of mathematical biology 15 (1953), S. 561-563 
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    Untersuchungen über die wechselnde Intensität der Hochwildschäden im Felde (1957)
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    European journal of wildlife research 3 (1957), S. 32-39 
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    ISSN: 1439-0574
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary Investigations proved that damages caused in the fields by red deer, fallow deer, and wild boars reach their culmination during certain summer months. Winter damages are caused only by wild boars in the potatoe fields of the year before now sown with rye. The question of the number of nights of damaging following one another could be cleared for the area of investigation. With red and fallow deer likewise a quick decrease of field damages could be observed with the beginning of ruttishness. A noticeable decrease of winter field damages caused by wild boars could not be observed during ruttishness. The opinion stated up to now that wandering fallow deer would not cause concentric field damages did not prove true within the district of observation.
    Abstract: Résumé Les recherches ont prouvé que les dégâts causés dans les champs par les bêtes fauves, les daims er les bêtes noires arrivent à leur point culminant pendant certain mois d'été. Dégâts causés dans les champs hibernales sont uniquement produits par les bêtes noires sur soles de pommes de terre de l'année précédente dont la semaille fut faite avec seigle. La question après le nombre des nuits se succédant pouvait être éclaircie pour la domaine d'observation. Une diminution rapide des dégâts causés dans les champs et produits par les bêtes fauves et les daims pouvait être observé également pour tous deux. Par contre il était impossible de constater une diminution sensible des dégâts causés dans les champs hibernales et produits par les bêtes noires pendant le temps de leur ivresse. L'opinion soutenue jusqu'ici que le daim vagabond ne causera pas de dégâts concentrés dans les champs ne se trouvera pas justifié pour le district d'observation.
    Notes: Zusammenfassung Durch Untersuchungen wurde bewiesen, daß die Feldschäden von Rot-, Dam- und Schwarzwild in bestimmten Sommermonaten ihren Höhepunkt erreichen. Winterfeldschäden werden nur durch Schwarzwild auf den mit Roggen bestellten vorjährigen Kartoffelschlägen verursacht. Die Frage nach der Anzahl der aufeinanderfolgenden Schadensnächte konnte für das Untersuchungsgebiet geklärt werden. Bei Rot- und Damwild wurde übereinstimmend mit Beginn der Brunft ein schnelles Absinken der Feldschäden beobachtet. Ein merkbarer Rückgang der Winterfeldschäden durch Schwarzwild während der Rauschzeit konnte nicht festgestellt werden. Die bisher vertretene Ansicht, daß durch das unstete Damwild keine konzentrierten Feldschäden verursacht werden, erwies sich für das Beobachtungsrevier als nicht zutreffend.
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    Zur Abschußplanung für Rehwild (1957)
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    European journal of wildlife research 3 (1957), S. 154-159 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary Although Mr.Gruschwitz chooses a percentage of increase of 70%, he reaches in his examples only 54%, because he relates the percentage to the does on March 31 st and not on June 1 st. The sexual ratio is not the ratio in numbers of the pubescent animals in summer, but the ratio of all male and female animals. It is necessary to make a difference between this ratio in March and in June. A large sexual ratio has to be reduced essentially to the lack of males, the cause being a much too heavy killing off of males. The insufficient killing of females is of importance, too. Measure and means of each interference with the population has to be decided by the slate pencil. It is not to be recommended to kill half of all the does within one year, because most of the does have young ones and choice is difficult.
    Abstract: Résumé Quoique le pourcentage d'accroissement queM. Gguschwitz avait choisi était 70% il atteint dans ces exemples seulement 54%, puisqu'il se rapporte chez les chevrettes sur le pourcentage du 31 mars et point sur celui du 1 er juin. La proportion entre les sexes n'est point la proportion numérique des pièces en état de puberté pendant l'été; mais la proportion de tous les mâles envers toutes les femelles. Il est donc nécessaire de faire une différence entre la proportion du mois de mars et celle du mois de juin. Une grande proportion entre les sexes doit être attribuée essentiellement à un manque en boucs; la raison en est un abattement en boucs de beaucoup trôp élevé. A part de cela, un abattement insuffisant en chevrettes n'a presque pas d'importance. C'est le crayon d'ardoise qui fixe l'étendue et la façon de chaque empiètement dans l'état. Mais il nést point applicable de faire abattre la moitiè de toutes les chevrettes au cours d'une année, puisque la plupart des chevrettes conduisent et en cela même un abattement sélectif devient de plus difficile.
    Notes: Zusammenfassung ObwohlGruschwitz einen Zuwachsprozent von 70 wählt, kommt er in seinen Beispielen nur auf 54%, weil er den Hundertsatz auf die Geißen vom 31. 3. bezieht und nicht auf die vom 1. 6. Das Geschlechterverhältnis ist nicht das Zahlenverhältnis der geschlechtsreifen Stücke im Sommer zueinander, sondern das Verhältnis aller männlichen zu allen weiblichen Stücken. Nötig ist, dieses Verhältnis im März von dem im Juni zu unterscheiden. Ein weites Geschlechterverhältnis ist im wesentlichen auf Mangel an Böcken zurückzuführen, die Ursache ist ein viel zu hoher Bockabschuß. Ungenügender Rickenabschuß ist daneben kaum von Bedeutung. Das Maß und die Art jedes Eingriffes in den Bestand bestimmt der Rechenstift. Die Hälfte aller Ricken innerhalb eines Jahres abzuschießen, ist nicht angebracht, weil die meisten Ricken führen und der Wahlabschuß erschwert wird.
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    Referate (1957)
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    European journal of wildlife research 3 (1957), S. 172-180 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    Zur Beurteilung der dunklen Ringeltauben im Rheinland (1957)
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    European journal of wildlife research 3 (1957), S. 92-92 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    Referate (1957)
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    European journal of wildlife research 3 (1957), S. 96-100 
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    Untersuchung über das Halten von Rammlern und Häsinnen vor dem Jagdschützen (1957)
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    European journal of wildlife research 3 (1957), S. 122-123 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    Bräuche von Vogelfang und Vogeljagd im südkaspischen Gebiet (1957)
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    European journal of wildlife research 3 (1957), S. 107-114 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary Description of rather a number of methods of catching birds with ring nets, nets, and snares. Dates concerning the catching of quails with throwing sticks, the hunt of snowcock, the catching of ducks by means of pumpkin masks, and hawking. Geese are being shot with shooting weapons out of a rifle pit or after having been rounded in with the field lorry. Rock partridges are approached behind movable screens, pheasant cocks are attracted by the flattering noise of house fowl. — This survey includes also ancient means of trapping birds, possibly no longer in use, and is restricted to the Southern marginal regions of the Caspic Sea (including the Eastern Caucasus), according to literature and to personal observations in 1956. In the beginning questions of the imperilment of birds and the necessity of educational measures are being discussed.
    Abstract: Résumé On donne ici la description de quelques methodes pour attraper les oiseaux, soit à I'aide d'un panneau ou à I'aide des rets ou lacets. En plus sont donnés des indications sur la chasse aux cailles avec javelot, la chasse à courre sur Tetraogallus, la chasse aux canards à I'aide d'une masque faite d'un potiron et la chasse au vol. Les oies sont abattus à I'aide des fusils d'une tranchée ou après les avoir cerné d'une voiture allant en terrain. Les bartavelles sont abordés derrière un paravent portable; les coqs de faisan pipé avec le son ballottant des poules domestique. — Ce précis comprend aussi des methodes anciens pour attraper les oiseaux partiellement éteindus aujourd'hui, et se restreind sur les régions méridionals et marginals de la mer caspienne (les régions caucasiennes orientales inclus), selon la literature et des observations propres en 1956. Au commencement les questions des dangers aux oiseaux et la des mesures pédagogiques son traités.
    Notes: Zusammenfassung Beschreibung einer größeren Zahl von Methoden des Fanges von Vögeln mit Käschern, Netzen und mit Schlingen. Angaben über die Jagd auf Wachteln mit Wurfstöcken, die Hetzjagd auf Königshühner, den Entenfang mit Hilfe der Kürbismaske und die Beizjagd. Mit Schußwaffen werden Gänse vom Schützenloch oder nach Einkreisen mit dem Geländewagen erlegt. Steinhühner werden hinter einem tragbaren Schirm angegangen, Fasanenhähne mit dem Flatterlaut der Haushennen angelockt. — Diese Übersicht bezieht auch alte, vielleicht zum Tiel jetzt erloschene Fangweisen ein und beschränkt sich auf die südlichen Randgebiete des Kaspischen Meeres (einschließlich Ost-Kaukasus), nach der Literatur und nach eigenen Beobachtungen 1956. Eingangs werden die Fragen der Gefährdung des Federwildes und die Dringlichkeit erzieherischer Maßnahmen behandelt.
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    Über Raubwild-, Greifvögel- und Kolkraben-Bekämpfung in der Grafschaft Tecklenburg zur Zeit Friedrich des Großen (1957)
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    European journal of wildlife research 3 (1957), S. 168-172 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    One-sided inequalities for hermitian quadratic forms (1953)
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    Monatshefte für Mathematik 57 (1953), S. 1-5 
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    Zur projektiven Geometrie der Linienelemente fünfter und vierter Ordnung (1953)
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    Monatshefte für Mathematik 57 (1953), S. 19-28 
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    Flächenläufige Bewegungsvorgänge im elliptischen Raum I (1953)
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    Monatshefte für Mathematik 57 (1953), S. 29-43 
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    Über die Lösung der gewöhnlichen Laplaceschen Differentialgleichung durch mehrfache Integrale (1953)
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    Monatshefte für Mathematik 57 (1953), S. 6-18 
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    Allgemeine Gesichtspunkte zur Interpolation und einige durch sie angeregte Untersuchungen (1953)
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    Monatshefte für Mathematik 57 (1953), S. 44-65 
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    Über ein Problem vom Waring-Goldbach'schen Typ (1953)
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    Monatshefte für Mathematik 57 (1953), S. 66-74 
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    Kongruente Verlagerung kollinearer Räume in achsiale Lage (1953)
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    Monatshefte für Mathematik 57 (1953), S. 75-87 
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    Buchbesprechungen (1953)
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    Monatshefte für Mathematik 57 (1953), S. 88-96 
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    Value of quadratic forms III (1953)
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    Monatshefte für Mathematik 57 (1953), S. 97-101 
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    Eine isometrische und singularitätenfreie Einbettung des n-dimensionale hyperbolischen Raumes im Hilbertschen Raum (1953)
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    Monatshefte für Mathematik 57 (1953), S. 102-108 
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    Über die Zerlegung einer Gruppe nach zwei Untergruppen (1953)
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    Monatshefte für Mathematik 57 (1953), S. 109-112 
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    Über ein Problem vom Waring-Goldbach'schen Typ II (1953)
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    Monatshefte für Mathematik 57 (1953), S. 113-116 
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    Flächenläufige Bewegungsvorgänge im elliptischen Raum II (1953)
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    Monatshefte für Mathematik 57 (1953), S. 129-133 
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    Eine topologische Invariante von sechs Linienelementen zweiter Ordnung eines Punktes der Ebene (1953)
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    Monatshefte für Mathematik 57 (1953), S. 117-128 
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    Eine projektive Invariante von drei Linienelementen n-ter Ordnung mit einem gemeinsamen Linienelement (n-1)-ter Ordnung (1953)
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    Monatshefte für Mathematik 57 (1953), S. 134-139 
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    Eine Bemerkung zur projektiven Invariante Jn in der voranstehenden Arbeit von E. Kruppa (1953)
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    Monatshefte für Mathematik 57 (1953), S. 140-141 
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    Die Flachtangenten der ebenen Kurven vierter Ordnung (1953)
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    Monatshefte für Mathematik 57 (1953), S. 142-169 
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    Differenzengeometrie der infinitesimalen Flächenverbiegung (1953)
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    Monatshefte für Mathematik 57 (1953), S. 177-184 
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    Buchbesprechungen (1953)
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    Monatshefte für Mathematik 57 (1953), S. 170-176 
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    Zum Transversalenproblem (1953)
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    Monatshefte für Mathematik 57 (1953), S. 185-198 
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    Zur Geometrie der Drehflächen und ihrer geodätischen Linien (1953)
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    Monatshefte für Mathematik 57 (1953), S. 199-216 
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    Beiträge zur numerischen Integration (1953)
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    Monatshefte für Mathematik 57 (1953), S. 217-245 
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    Über Gitter und Polyeder (1953)
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    Monatshefte für Mathematik 57 (1953), S. 246-254 
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    Buchbesprechungen (1953)
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    Monatshefte für Mathematik 57 (1953), S. 255-264 
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    On the Davenport-Heilbronn theorem (1957)
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    Monatshefte für Mathematik 61 (1957), S. 1-36 
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    Über die homomorphen Bilder des Ringes der ganzen Zahlen und über eine verwandte Ringfamilie (1957)
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    Monatshefte für Mathematik 61 (1957), S. 37-41 
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    Über einen Erweiterungssatz (1957)
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    Monatshefte für Mathematik 61 (1957), S. 51-53 
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    Projektionen projektiver Räume (1957)
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    Monatshefte für Mathematik 61 (1957), S. 54-66 
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    Die Kontinua (Vp)—eine Verschärfung der Cantorschen Mannigfaltigkeiten (1957)
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    Monatshefte für Mathematik 61 (1957), S. 67-76 
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    Reguläre Graphen 3., 4. und 5. Grades mit vorgegebenen abstrakten Automorphismengruppen, Farbenzahlen und Zusammenhängen (1957)
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    Monatshefte für Mathematik 61 (1957), S. 42-50 
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    Die Unmöglichkeit von x6+y6=z6 und x9+y9=z9 in quadratischen Körpern (1957)
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    Monatshefte für Mathematik 61 (1957), S. 147-150 
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    Sur les polynomes orthogonaux dont les dérivés sont orthogononaux (1957)
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    Monatshefte für Mathematik 61 (1957), S. 143-146 
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    Anelli con unica decomponibilità in fattori primi ed un problema di intersezioni complete (1957)
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    Monatshefte für Mathematik 61 (1957), S. 97-142 
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    Buchbesprechungen (1957)
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    Monatshefte für Mathematik 61 (1957), S. 77-96 
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    Eine für die Theorie der relativ-abelschen Körper grundlegende Abelsche Operatorgruppe (1957)
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    Monatshefte für Mathematik 61 (1957), S. 151-156 
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    On representations of the full linear group and some of its subgroups over infinite fields of prime characteristic (1957)
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    Monatshefte für Mathematik 61 (1957), S. 157-160 
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    Einheitsfunktionenspalten (1957)
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    Monatshefte für Mathematik 61 (1957), S. 173-194 
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    Buchbesprechungen (1957)
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    Monatshefte für Mathematik 61 (1957), S. 161-172 
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    Über eine Klasse von $$\mathfrak{M}$$ -Untergruppen (1957)
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    Monatshefte für Mathematik 61 (1957), S. 195-208 
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    Über koaxiale euklidische Schraubungen (1957)
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    Monatshefte für Mathematik 61 (1957), S. 225-245 
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    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01371051
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    Electronic Resource
    Berechnung einiger unendlicher Integrale (1957)
    Springer
    Monatshefte für Mathematik 61 (1957), S. 246-249 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01371052
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