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  • Springer  (48,942)
  • PANGAEA
  • 1980-1984
  • 1970-1974  (31,974)
  • 1960-1964  (17,220)
  • 1970  (31,974)
  • 1961  (17,220)
Collection
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Publisher
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  • 1980-1984
  • 1970-1974  (31,974)
  • 1960-1964  (17,220)
Year
  • 1
    Electronic Resource
    Electronic Resource
    An hypothesis of freese and the DNA-protein coding problem (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 305-318 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Freese’s Hypothesis states that a single specific alteration in the sequence of nucleotides of an information-bearing DNA molecule results in a specific mutational effect. Within the framework of the DNA-protein coding problem developed elsewhere, and assuming the quasi-ergodicity of the general coding process, it is shown that Freese’s Hypothesis allows us to derive expressions for the length of the smallest mutable DNA molecule and to obtain a bound for the maximal number of allelic molecules of fixed length. To illustrate these ideas, calculations are carried out on appropriate data from bacternophage and man, and the results are shown to differ by a factor of 10 (modulo the rather crude approximations used). It is further shown that, if ρ(N) and ϱ(N) are respectively the number of information-bearing words of lengthN in a given code and the number of words of lengthN, then the number lim ρ(N)/ϱ(N) depends sensitively on the parameter ∈ which specifiesN→∞ the given code. The implications of this result for the spontaneous aggregation of a sufficient number of information-bearing words to characterize an organism are discussed.
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    URL: http://dx.doi.org/10.1007/BF02476743
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  • 2
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    Post-and predoctoral fellowships in mathematical biology (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 319-319 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476744
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  • 3
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    The perfect learner (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 321-335 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract As a “base line” of memorization performance, the behavior of a “perfect learner” is considered. He is characterized by a perfect memory and by the ability to choose the best search procedure in problems where the correct response from a given repertoire is to be found to each of several stimuli under the condition of “right” and “wroing” promptings by the experimenter. Expected learning curves are derived for the case of disjoint response repertoires associated with the stimuli under cyclic and random presentation of the stimuli and for the case of a single response repertoire (a one-to-one matching problem) under cyclic presentation.
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    URL: http://dx.doi.org/10.1007/BF02476490
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  • 4
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    A method for the computation of aortic distensibility in the living human patient and its use for the determination of the aortic effects of aging, drugs, and exercise (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 337-353 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Detailed equations are given for the computation of aortic distensibility in the intact living human patient from measurements of systolic and diastolic arterial pressures, heart rate and cardiac output. From these equations, the aortic characteristics of a large series of normal men of different ages were computed. Comparing these results with measurements on excised aortas, a more pronounced trend toward increasing aortic stiffness with increasing age is evident in living than in dead aortas. Nor-epinephrine and exercise apparently cause the living aortas to constrict but to become more distensible. The same change occurs after 30 minutes of high spinal anesthesia. The ganglionic blocking agents hexamethonium, pentamethonium, and tetraethylammonium usually cause the living aorta to become stiffer, presumably due to dilatation. The aortas of patients with pulmonary disease usually react in different fashion to exercise or tetraethylammonium. The increased aortic distensibility that occurs with the hypertension induced by nor-epinephrine or exercise acts as a compensatory mechanism by decreasing systolic pressure.
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    URL: http://dx.doi.org/10.1007/BF02476491
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  • 5
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    Dimensional analysis in mathematical biology I. General discussion (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 355-376 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.
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    URL: http://dx.doi.org/10.1007/BF02476492
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  • 6
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    A minimal principle in biomechanics (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 377-391 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Expenditure of energy under several simultaneous forms (mechanical, chemical, etc.) is associated with all muscular activity. The energy is directly related to what is commonly called exertion or effort. This paper defines “muscular effort” quantitatively in terms of some of the elements of the dynamics of the human (and animal) body. It postulates that in all likelihood the individual will, consciously or otherwise, determine his motion (or his posture, if at rest) in such a manner as to reduce his total muscular effort to a minimum consistent with imposed conditions, or “constraints”. The principle, formulated in mathematical terms, is sufficient to ascribe to the moments at all body joints—a matter generally of free choice on the part of the individual—their most likely magnitudes. It therefore renders the equations of human (and animal) motion determinate within this context. The paper also describes briefly an iteration method for the solution of these equations, once they have been made determinate. A simple illustrative application of the principle is included.
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    URL: http://dx.doi.org/10.1007/BF02476493
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  • 7
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    On the role of chemical systems in the microphysical aspects of primary genetic mechanisms (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 393-403 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is pointed out that two fundamentally different views of primary genetic processes occur in the literature which are frequently confused. The first is a true communication-theoretic view, which regards the genetic apparatus as containing a real information-source and a transducer which converts that information to useful form. The second view is generally expressed as a template scheme based on the Watson-Crick model; it is shown that in this model there is actually no such thing as genetic information in a communication-theoretic sense. Both views are then discussed on the basis of microphysical principles developed in previous work of the author (Bull. Math. Biophysics,22, 227–255, 1960) in an attempt to find which approach is in closer accord with the biological facts. It is shown that, if the communication-theoretic view is correct, then the information-bearing object must act as a “catalyst,” but it is pointed out that the type of catalysis involved must be of a fundamentally different nature than that occurring in familiar enzyme-catalyzed reactions. On the basis of general considerations of irreversible changes in microphysical measuring systems, it is shown that any type of template must suffer a gradual and irreversible denaturation, which seems to make it unlikely that a template could play a primary role in fundamental genetic processes.
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    URL: http://dx.doi.org/10.1007/BF02476494
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  • 8
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    On imitative behavior. II. Time course of change from one behavior to another (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 405-411 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory developed in previous papers and based on distribution curves of definite form is generalized to any form of unimodel distributions. The time course of the change from one behavior to another is discussed and a general theorem about the time course is established.
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    URL: http://dx.doi.org/10.1007/BF02476495
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  • 9
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    Abstract mathematical molecular biology (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 417-417 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476497
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  • 10
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    Mathematical studies of the interaction of respiratory gases with whole blood II. CO2 absorption (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 1-14 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Some progress has been made on the problem of the interaction of respiratory gases with whole blood. A working mathematical model for the O2−CO2 interaction phenomena has been developed from mathematical studies of the data. The Edsall-Wyman (1958) model for CO2 absorption is improved upon in this paper by consolidating it with the O2 absorption model developed in paper I of this set (Bernard, S. R.,Bull. Math. Biophysics,22, 391–415, 1960). This improved model assumed the effect of O2 on CO2 absorption is mediated through the electrical charge possessed by the hemoglobin molecule,i.e., O2 molecules bound to hemoglobin displace protons from the hemoglobin thereby increasing the negative charge on the hemoglobin and at the same time increasing the acidity of the solution. The model is tested against the data.
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    URL: http://dx.doi.org/10.1007/BF02476569
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  • 11
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    On the mutual influence of individuals in a society (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 15-18 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract On the basis of previously proposed mathematical models of social behavior, the present note investigates the possibility of the control of behavior remaining permanently in the hands of one class, if this class possesses sufficient means for influencing mass behavior. The conclusion is reached that, with the assumptions made, if the behavior imposed by the controlling class leads to sufficiently strong dissatisfaction, the control will pass to another class, no matter how strong the controlling power of the first.
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    URL: http://dx.doi.org/10.1007/BF02476570
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  • 12
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    Contribution to the mathematical biophysics of automobile driving (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 19-29 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.
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    URL: http://dx.doi.org/10.1007/BF02476571
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  • 13
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    Glucose metabolism—A “mixed-cycle” system (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 99-103 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Emphasis upon the importance of homeostatic feedback has drawn attention away from the complexity of biological processes. A study of glucose metabolism indicates the importance of open-cycle as well as closed-cycle mechanisms. Besides the glucose-dependent mechanism of insulin secretion, many open-cycle processes involving the liver, adrenal glands and kidneys, play important roles in the variation of blood glucose. In addition, glucose utilization by the tissues is essentially open-cycle in nature.
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    URL: http://dx.doi.org/10.1007/BF02476577
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  • 14
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    Southern Regional Graduate Summer Session in Statistics at Virginia Polytechnic Institute (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 105-106 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476578
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  • 15
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    On the unreliability of judgments of certain social situations by minority groups (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 413-416 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In certain situations like the aftermath of a revolution when discontent rises amongst certain groups of the population, it is frequently observed that the discontented groups are firmly convinced that their point of view is shared by the majority of the population. Yet future events prove that this is far from being the case. This effect is partly attributable to “wishful thinking,” partly to a purely social mechanism. The wishful-thinking effect may be considered as a case of psycho-physical discrimination in which a bias is introduced proportional to the degree of satisfaction anticipated from a given situation. H. D. Landahl's well-known equations can be applied to this case. The social factor is based on the circumstance that an individual associates by preference with such other individuals as have similar opinions. This results in an actual error of estimation of the relative minority or majority because of different frequencies of contact with individuals of the two opposing groups. Both factors may be combined into one equation.
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    URL: http://dx.doi.org/10.1007/BF02476496
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  • 16
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    Announcements (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 421-422 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476499
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  • 17
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    The elements of an electrocardiographic lead tensor theory (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 31-42 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This report describes a method by which the concepts of the lead vector and the lead field may be extended to include equivalent cardiac singularities above the first order. The theoretical treatment results in generalization of the well-known lead vector principle into a lead tensor apparatus. The specific electrocardiographic contribution of annth order multipole is found to be the product of annth rank contravariant (“heart”) tensor by a covariant (“lead”) tensor of the same rank, and with identical indices. Methods are also described whereby lead tensor components may be cast into a form which relates them directly to the more usual, spherical harmonic notation of multipolar current generators. Some attention is devoted to the possibility of applying lead tensors to biomedical problems, particularly to the prospect of quantitatively evaluating the significant multipolar components of the human heart.
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    URL: http://dx.doi.org/10.1007/BF02476572
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  • 18
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    The formation of cell assemblies (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 43-53 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A simple model is presented for the formation of functional groups in a random neural net. They show the following characteristics: 1. They can maintain autonomous activity which might serve as temporary memory traces. 2. Early in the process of formation they become resistant to contraction. 3. Later they become resistant to expansion. 4. Nearby groups inhibit one another. 5. Two groups may contain some cells in common.
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  • 19
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    Contributions to quantum biology. I. Mobile electronic characteristics of riboflavin radicals (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 55-68 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Quantum biology is the quantum mechanical study of electrons in molecules of biological interest. This requires the solution of problems involving many electrons. Approximation methods are therefore necessary and are discussed. The present study, concerned with the mobile electrons in riboflavin (FMN) and its radicals (FMN−, FMNH and FMNH2 +), is based on the approximation method, developed by B. Pullman and A. Pullman. The solution of the eigenvalue problems so obtained gives the energy levels of the mobile electron systems involved. The corresponding eigenvectors yield the mobile electronic charges of the atoms of riboflavin radicals which have contributed mobile electrons. Important differences of the net charge distributions of these radicals are emphasized. The longest wave length of light absorption is calculated from the obtained energy levels and agrees, within the accuracy of the method, with corresponding experimental results. From the appropriate calculated results, electronic assignments are obtained for the experimental transitions involved.
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  • 20
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    Quantitative analysis of metabolic processes III. A model-system for enzyme synthesis by sequential induction and mathematical formulation of the process (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 69-89 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A postulated scheme for sequential enzyme synthesis coupled with linear chain reaction is proposed. Mathematical calculations are carried out for the transient-state formation of enzyme coupled with enzyme-substrate reactions. Examples are presented to demonstrate the relations between various operational parameters during enzyme synthesis as a function of time. Significance of the phenomenon of sequential synthesis is discussed in relation to abnormal growth and cellular injury.
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  • 21
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    A note on mathematical models for the interaction of neural elements (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 91-97 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract By introducing a plausible model for the initiation of axonal impulses the output is obtained as a function of the input incoming impulses. If the temporal aspects of the excitatory process resulting from the afferent impulses are sufficiently rapid one obtains the discontinuous or microscopic model of McCulloch-Pitts. If these are sufficiently slow a continuous model, such as Rashevsky’s one or two factor theory, is a natural model. But the linear relation between the strength of excitation of one axon and excitatory factor of the next will not in general hold. However, under conditions which are not too restrictive the linear relation with threshold can be considered as satisfactory approximation over a fairly wide range of values.
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  • 22
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    Symposium on Mathematical Theories of Biological Phenomena (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 107-108 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476579
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  • 23
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    On the roots of a certain type of matrix (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 217-221 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476735
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  • 24
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    On relations between sets (1961)
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    Bulletin of mathematical biology 23 (1961), S. 233-235 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract With a view to future applications in relational biology, the notion of relations between sets is introduced and several theorems are demonstrated.
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  • 25
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    Transport of gases through hemoglobin solution (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 223-232 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical theory is developed to explain the observed enhancement of O2 transport through solutions by hemoglobin. At high partial pressures of O2, ordinary diffusion through the solvent accounts for all transport of O2, but at low partial pressures the transport may be increased manyfold by the presence of hemoglobin. This phenomenon is explained and its possible role in living organisms is discussed. The theory also indicates a new method of determining dissociation curves from diffusion experiments.
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    The equations of population growth (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 261-262 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A solution in quadratures is found for a special class of differential equations suggested by studies in the growth of competing populations.
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  • 27
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    Abstract mathematical molecular biology (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 237-260 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The tremendous complexity of even the simplest living unit makes a correct theoretical guess as to its mechanism very difficult. It is therefore suggested that, following the example of the physical sciences, a number of purely abstract cases in molecular biology be studied mathematically at first. Subsequent comparison of the different conclusions of such an abstract theory with available data would enable us to decide which of the conceivable situations are likely to occur in reality. As a first step toward such a study the problem of the minimal size of a living unit is studied. Usually the minimal size is considered to be determined by information-theoretical requirements. It is shown that the minimal size may have a very different origin. It may be determined by the possibility that too small a system, even though performing all necessary biological functions, may not be viable unless it is a member of a large group of other similar systems. This approach is developed both from a metric and from a relational point of view. Some relational characteristics of systems of reactions which possess the elementary metabolic properties of organisms are studied.
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  • 28
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    Eccentric multipole representations of current generators in a spherical volume conductor (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 263-276 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In Yeh, Martinek and de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958), a method is presented for determining successively better central multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. This paper generalizes the method such that the multipoles may be located at any chosen point in the conductor. The spherical harmonic expansion is advantageously used and the “interior sphere theorem” of Ludford, Martinek and Yeh (Proc. Cambridge Philos. Soc.,51, 389–93, 1955) makes possible disturbance potential expressions in closed forms. A method for approximate determination of the eccentricity is also presented. In the theory of electrocardiography, the eccentric multipoles can more accurately represent the heart as a current generator with fewer surface potential measurements than the central multipoles.
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  • 29
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    A derivation of latency equation (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 297-303 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Given three primary events, occurrences or nonoccurrences of two response classes and nonoccurrence of both response classes, with their respective probabilities to be known, and assuming the first two events are disjoint, this paper deduces the error equation of the incorrect response and the latency equation of the correct response under the condition of correlated reward situation.
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  • 30
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    Quantitative analysis of metabolic processes. II. A model-system for the synthesis of the dissociable enzyme and mathematical formulation of the process (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 277-295 
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    ISSN: 1522-9602
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    Notes: Abstract Schemes are presented for induced synthesis of the dissociable enzyme in which repeated use of the template is made. The role of the inducer is to release the repression. A mathematical analysis is carried out and expressions are obtained to describe the kinetics of enzyme formation. A practical case (penicillinase synthesis) is compared with theoretically derived equations by using an analogue computer to simulate an enzyme forming system. A good correlation between theoretical and experimental data is obtained.
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    Post- and Predoctoral Fellowships in mathematical biology (1961)
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    Bulletin of mathematical biology 23 (1961), S. 419-419 
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    The influence of noninstantaneous mixing on the measurement of fluxes of solute and water across a biological membrane (1970)
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    Bulletin of mathematical biology 32 (1970), S. 1-24 
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    Notes: Abstract The usual method of tracer analysis for calculating the flow across a biological membrance is based on the assumption that the compartments on either side are well-stirred. Thus, the validity of the rate of flow determination is questionable for cases where the distribution of tracer is not homogeneous. In this study, a mathematical model is developed for the purpose of estimating the effect of slow mixing on the calculation of the flow rate. The model is applied to the measurement of the rate of flow of aqueous humor through the living eye by use of a fluorescent dye as a tracer. A transit time of several minutes for the passage of fluorescein through the posterior chamber and an extended period of nonuniform distribution of fluorescein in the anterior chamber was observed. The effect of slow mixing on the calculated flow rate is compared to rates derived from equations based on the assumption of rapid mixing. Aqueous flow rates determined by the two methods were found to agree to within ≈20%.
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    The dispersion of indicator in the cardio-pulmonary system (1970)
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    Bulletin of mathematical biology 32 (1970), S. 25-43 
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    Notes: Abstract This paper presents a three-parameter model of the mechanism of dispersion of an indicator in the cardio-pulmonary system, based on the postulates that this dispersion can be described by the one-dimensional diffusion equation and that dispersion continues past the sampling site. The model is tested using indicator dilution curves obtained from dogs, and the coefficient of diffusion is thus measured. It is found that this coefficient increases in magnitude non-linearly with increasing blood speed.
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    The effect of distension of the left ventricle of the heart on the length of the individual myocardial fibers (1970)
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    Bulletin of mathematical biology 32 (1970), S. 45-58 
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    Notes: Abstract Based on the ellipsoid model of the left ventricle and the helicoidal course of the left ventricular myocardial fibers, a theory has been developed for calculating the length of the individual myocardial fibers. Numerical solutions of the final equation show that when the left ventricle is distended, the increase in length of the myocardial fibers is not uniform throughout the thickness of the myocardial wall. It was shown that with increasing dimensions of the left ventricle, the distension of the myocardial fibers becomes smaller as one advances from the endocardium to the middle layer of fibers, whereas it increases as one advances from the middle layer to the epicardial layer. The mechanism by which this effect is brought about as well as its physiological implications are discussed.
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    The gas washout determination under a symmetry assumption (1970)
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    Bulletin of mathematical biology 32 (1970), S. 59-63 
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    Notes: Abstract In an open circuit gas washout determination the output of test substance is shown to be of the form ∑ i=0 ∞ A i λ i k on thekth expiration whereA i 〉0,i=1,2,... and 1 〉 λ1 ≥ λ2...≥ 0 provided the transition from inspiration to expiration has certain symmetry properties with the transition from expiration to inspiration. In general, no direct physical interpretation such as volume for theA t ’s or fraction of retained gas on expiration for the λ i is justified.
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    Mathematical biology of automobile driving IV (1970)
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    Bulletin of mathematical biology 32 (1970), S. 71-78 
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    Notes: Abstract Previous studies by this author of the mathematical biology of automobile driving have emphasized only the biological aspects, except for such mechanical factors as the size of the car. Otherwise, the ideal case of an inertialess car was considered. In this paper the first step is made toward introducing the effects of the mass of the car and the side-slip of the tires when the direction of driving is even slightly altered and combining these with the previously studied biological aspects. Some tentative comparisons with available data are made.
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    Relaxation methods for the determination of drug receptor affinities (1970)
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    Bulletin of mathematical biology 32 (1970), S. 65-69 
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    Notes: Abstract According to the occupation theory of drug receptor interaction, the response is a functionf of the number of receptors occupied by drug molecules. Considerable controversy exists regarding assumptions about this function. Without knowledge of the nature of the function, it is not possible to determine directly the rate constants, and hence the affinity constant, in the reaction between the receptor and an agonist drug. Instead, indirect determinations involving the use of antagonists have been employed, limiting the determination of affinity to those agents for which specific antagonists exist. The present paper discusses a method for the direct determination of affinity of an agonist drug. It is a “relaxation method,” i.e., the equilibrium is perturbed and the kinetics of the restoration process are studied. Assuming only thatf is non-decreasing and approximately linear over a limited domain of concentrations, it is shown that the change in response obeys first order kinetics, permitting a determination of the rate constants from the time course of the restoration process.
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    A remark on the course of development of organismic sets (1970)
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    Bulletin of mathematical biology 32 (1970), S. 79-81 
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    Notes: Abstract The basic postulate la, which governs the development of organismic sets, introduced previously (Bulletin of Mathematical Biophysics,31, 159–198, 1969), is generalized so as to contain also the rates of changes of the number and variety of differentQ-relations which determine an organismic set. It is thus brought closer to the Lagrangian principle in physics. It is pointed out that the postulate also provides a criterion of stability of an organismic set.
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    In-vivo estimations of the nonlinear pressure-volume relationship of the aorta and instantaneous left ventricular volume (1970)
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    Bulletin of mathematical biology 32 (1970), S. 155-172 
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    Notes: Abstract A method is presented in this paper for the in-vivo estimation of the nonlinear pressure-volume relationship of the human aorta. The method is based on nonlinear elastic reservoir theory and utilizes clinical data that can be obtained with a high degree of accuracy, namely stroke volume, end diastolic ventricular volume and aortic pressure trace data. The computational procedure is described and then carried out for six cardiac patients. A method for the estimation of instantaneous left ventricular volume during the ejection period based on the considered nonlinear elastic reservoir theory is also presented. The method is applied for the six cardiac patients cited and the results compared with those obtained for the same subjects by a method of estimation based on linear elastic reservoir theory described in a previous paper by the author (1969).
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    The application of the minimal energy hypothesis to a casson fluid (1970)
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    Bulletin of mathematical biology 32 (1970), S. 249-262 
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    Notes: Abstract Deakin (1967b) suggested that flow of blood might obey a law of minimal energy dissipation. The present paper presents a simpler derivation of Deakin’s equations pointing out several previously unrecognized features. It is shown that these equations are unlikely to be applicable. In particular, the solution obtained by Deakin and Jones (1968) does not yield a true minimum for energy dissipation. The solution for which energy dissipation is actually minimized is shown to possess features which render it unlikely to apply to a real flow.
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    Some remarks on the Kedem-Katchalsky equations for non-electrolytes (1970)
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    Bulletin of mathematical biology 32 (1970), S. 237-247 
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    Notes: Abstract The Kedem-Katchalsky equation for the flow of a non-electrolyte through a homogeneous membrane is shown to be a first order expansion of an exact integral of the Spiegler-Bearman-Kirkwood frictional equations under the assumption that the partial frictional coefficients, ζ ij , are concentration independent. The equations are solved in terms of volume flow; there are no water-to-volume flow correction terms for the permeability, ω, or the reflection coefficient, σ. The precision of the expansion depends upon the magnitude of the water flow. The frictional coefficientsf sm andf sw are given as functions of the experimentally determined parameters ω and σ; the frictions, are shown to be independent ofL p .
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    On the average cellular volume in synchronized cell populations (1970)
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    Bulletin of mathematical biology 32 (1970), S. 459-473 
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    Notes: Abstract As was done by Sinclair and Ross (1969(, we consider a cellular population that consists initially (at time zero) ofN 0 newborn cells, all with the same volumev o. It is assumed that the occurrence of cell division is determined only by a cell’s age, and not by its volume. The frequency function of interdivision times, τ, is denoted byf(τ). If cell death is negligible, the expected number of cells,N(t), will increase according to the laws of a simple age-dependent branching process. The expression forN(t) is obtained as a sum over all generations; thevth term of this sum, in turn, is a multiple convolution integral, reflecting the life history ofvth generation cells (i.e., the lengths of thev successive interdivision periods plus the age of the cell at timet). Assuming that cell volume is a given function of cell age, e.g., linear or exponential, and that cellular volume is exactly halved at each division, it is possible to calculate the volume of a cell with a given life history, and thus the average cellular volume of the whole population as a function of time. If at time zero the volumes differ from cell to cell, the final equation must be modified by averaging over initial volumes. In the case of linear volume increase with age, a very simple asymptotic expression is found for the average cellular volume ast→∞. The case of exponential volume increase with age also leads to a simple asymptotic formula, but the resulting volume distribution is unstable. The mean cellular volume at birth and the second moment of the volume distribution can be calculated in a similar manner.
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    Dynamic analysis, with feedback characterization, of hybrid human musculoskeletal frameworks by the linegraph-flowgraph procedure (1970)
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    Bulletin of mathematical biology 32 (1970), S. 499-520 
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    Notes: Abstract The dynamic response of human musculo-skeletal framework is treated by (i) idealization of the musculo-skeletal framework as hybrid structural networks possessing feedback characteristics and then (ii) employing linegraph-flowgraph procedures for the feedback characterization of the hybrid structural networks. Topological procedures are used in which a “tree” of a network furnishes the skeleton upon which the “linkage” (muscle representing) members provide interaction. Feedback characterization (representing the sensitivity of the skeletal members to the tensile forces) is defined, between the internal “linkage” and “tree” members, by means of the flowgraph. Mikusinski operational calculus is used to facilitate representation of inertia effects by dynamic feedback characterization, with inclusion of initial conditions.
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    Organismic supercategores: II. On multistable systems (1970)
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    Bulletin of mathematical biology 32 (1970), S. 539-561 
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    Notes: Abstract The representation of biological systems in terms of organismic supercategories, introduced in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–70) is further discussed. To state more clearly this representation some new definitions are introduced. Also, some necessary changes in axiomatics are made. The conclusion is reached that any organismic supercategory has at least one superpushout, and this expresses the fact that biological systems are multistable. This way a connection between some results of Rashevsky’s theory of organismic sets and our results becomes obvious.
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    On the mass distribution model for microbial cell populations (1970)
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    Bulletin of mathematical biology 32 (1970), S. 521-537 
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    Notes: Abstract The full implications of a statistical model for growth of a microbial cell population using cell mass as the index of physiological state have been examined by solving the partial differential integral equations resulting from the model. Calculations reveal that a lag phase is predicted during the initial stages of batch growth although no specific cellular mechanism for the phenomenon of lag had been incorporated into the model. The model predicts several situations of batch and continuous growth in which the population density and biomass concentration show opposing trends due to significant variation in the cell mass distribution with time.
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    Traitement des informations chez les etres vivants: Systemes a services par essais successifs (1970)
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    Bulletin of mathematical biology 32 (1970), S. 83-148 
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    Notes: Abstract Information treatment, especially in a natural medium, may be done by a random system: given an input, possible solutions are randomly selected and successively tested until the right one is found and then emitted. This paper is a mathematical study of this type of system and its use as a model of some natural (biological) systems. The theoretical development is illustrated by several examples of possible applications to biological cases. We study, under different assumptions, the distribution of the number of trials needed to treat the input, the distribution of the time for this treatment, and the distribution of the average amount of information treated in a trial or in a unit of time. We consider the case of error, and study some optimal conditions of operation and the occurrence of memory influencing the treatment of an input according to previously processed inputs.
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    Energy of nerve impulse (1970)
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    Bulletin of mathematical biology 32 (1970), S. 151-153 
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    Comments concerning P. C. Voukydis’ “Physiological significance of the geometrical shape of the left ventricle: Course and curvature of the individual myocardial fibers” (1970)
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    Bulletin of mathematical biology 32 (1970), S. 149-149 
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    Mathematical biology of automobile driving: V (1970)
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    Bulletin of mathematical biology 32 (1970), S. 173-178 
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    Notes: Abstract The role of some inertial properties of the car, studied previously only for the case when the stimulus for the corrective turn is the perception of the angle between the direction of the car and the direction of a straight lane (Bull. Math. Biophysics,32, 71–78, 1970), is generalized to include such stimuli as the nearness to the edge of the lane and the anticipatory effect for a corrective turn, as well as the combination of all three stimuli. Conditions for stability of driving are deduced and discussed. They now depend on both biological parameters and such parameters as the position of the center of gravity of the car, its mass, and the side slip of the tires.
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    Definition d’une distance entre deux mobiles se deplacant de facon aleatoire sur une droite (1970)
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    Bulletin of mathematical biology 32 (1970), S. 179-195 
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    Description / Table of Contents: Résumé Pour le biologiste, la notion dedistance entre deux ou plusieurs éléments d’un ensemble est très utile car elle sert à mesurer laproximité, laparenté, laressemblance qui existe entre ces éléments ou parties selon le caractère auquel on s’intéresse (position géographique, situation chronologique, aptitudes, phénotypes, composition chimique etc...).
    Notes: Abstract Adistance between two mobiles performing a random walk in one dimension is defined. At a given time this distance is directly related to theprobability of encounter for the mobiles. This definition is used when the motion of the mobiles is a Wiener-Levy process, first in the case of an unrestricted random walk, then if a reflecting or absorbing barrier is introduced.
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    Effects of anisotropy and nonhomogeneity on left ventricular stresses in the intact heart (1970)
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    Bulletin of mathematical biology 32 (1970), S. 197-213 
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    Notes: Abstract The qualitative effects of anisotropy and nonhomogeneity are considered in the evaluation of left ventricular stresses in the intact heart. Maximum stresses and their location are significantly dependent on the nonhomogeneity factors and to a lesser degree on anisotropy of the ventricular wall material. If the circumferential elastic modulus is assumed to vary in a parabolic manner through the wall thickness, maximum stresses occur within the endocardial layers, a result in qualitative agreement with experimental studies.
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    A correction for the effect of the dead space in pulmonary gas washout tests (1970)
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    Bulletin of mathematical biology 32 (1970), S. 215-218 
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    Notes: Abstract Under the assumption that there is complete mixing in the dead space a series of equations are given which give the values for a washout test of the functional component of the pulmonary bellows (without a dead space) in terms of the washout values of the bellows with a dead space.
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    Respiration of tissue as a medium of heterogeneous permeability (1970)
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    Bulletin of mathematical biology 32 (1970), S. 219-235 
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    Notes: Abstract Expressions are derived for the overall oxygen consumption and the O2 penetration depth into a tissue section in terms of the basic parameters, of the system under steady-state conditions. The approach differs from many previous analyses in so far as the oxygen molecules are regarded as reaching their sites of chemical assimilation by diffusion through extracellular fluid followed by bulk diffusion into irregular cells of significantly lower permeability. This “two-phase” model would seem to be compatible with the major experimental features of steady-state respiration, and gives a ratio of cellular to extracellular diffusion coefficients of the same low order as that found for inert gases under transient conditions. The greater oxygen penetration predicted by this model is discussed in relation to the survival of ischemic tissue and is shown to be consistent with data for myocardial infarction.
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    Analytical solutions for the specific activities of a traced substance within the component parts of a simulated kidney-ureter system (1970)
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    Bulletin of mathematical biology 32 (1970), S. 263-278 
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    Notes: Abstract Two models for a kidney-ureter system are considered: one model of one vessel in which a traced substance, undergoing exchange between the vessel and an external compartment, is emptying into the ureter; the second model of two approximately parallel, identical vessels in which a traced substance, undergoing exchange between each vessel and an external compartment, is emptying into the ureter. A single impulsive input of label into a vessel is assumed. For mathematical simplicity, the major conditions imposed on each system are: (1) rapid mixing transverse to a vessel axis and no mixing longitudinal to a vessel axis within the plasma; (2) small variation of the specific activity within the plasma in the longitudinal direction to a vessel axis; (3) constant flow rate of urine into the ureter and (4) constant exchange coefficients, tubule flow velocity and traced substance concentrations within individual compartments.
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    The motion of the left ventricle—III (1970)
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    Bulletin of mathematical biology 32 (1970), S. 279-291 
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    Notes: Abstract We use the concept of a layered wall, where each separate layer is to be homogeneous, isotropic, and incompressible, to derive stress-strain relations for the middle layer muscle ring at the transverse midsection of the left ventricle; a convenient method of formulation is that based on the elastic potential function. The hoop or circumferential stress in all three layers is found using dimensional and mechanical parameters derived earlier. The various parameters are expressed as Fourier series so that their behavior over a complete ventricular cycle is known analytically. The cases of simple elongation and what we termcurvilinear simple elongation are considered for the middle layer muscle ring strain, and the resulting stress-strain relations are derived. The results are compared with an incompressible rubber-like material known as a Mooney material.
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    Transient thermal behavior in biological systems (1970)
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    Bulletin of mathematical biology 32 (1970), S. 303-314 
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    Notes: Abstract The Peaceman-Rachford finite difference method is applied to cylindrically symmetric, transient heat conduction problems in biological media. Inhomogeneous media and internal sources which vary in both space and time are permitted. Boundary conditions are satisfied without sacrificing high local resolution by means of an exponentially stretched grid. Computation time on a Philco 2000/210 computer is approximately 5 msec per grid point per time step.
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    International Symposium on Mathematical Biology (1970)
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    Bulletin of mathematical biology 32 (1970), S. 601-601 
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    Physiological significance of the geometrical shape of the left ventricle: Course and curvature of the individual myocardial fibers (1970)
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    Bulletin of mathematical biology 32 (1970), S. 599-600 
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    A concentric layer model for estimating the energy expenditure of the left ventricle (1970)
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    Bulletin of mathematical biology 32 (1970), S. 581-598 
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    Notes: Abstract The energy cost of the left ventricle is quantitatively analyzed on the basis of the following assumptions: (1) The left ventricle is assumed to be an isotropic, homogeneous elastic, thick, spherical shell. (2) The ventricular wall is made up of a finite number of thin concentric shells. (3) The energetics of the left ventricle is in accordance with the second law of thermodynamics. An expression for the work done during ventricular contraction is derived according to the definition of physical work. The energy liberation during isovolumic contraction is formulated parallel to the concepts of heat production in skeletal muscle during isometric contraction. This expression gives the total work done per stroke in terms of mean systolic pressure, end diastolic volume, stroke volume and wall thickness during diastolic phase.
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    Ernesto Trucco (1970)
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    Bulletin of mathematical biology 32 (1970), S. i 
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    ISSN: 1522-9602
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    Topics: Biology , Mathematics
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    Die Distributivität in der Intervallarithmetik (1970)
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    Computing 5 (1970), S. 6-16 
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    Topics: Computer Science
    Description / Table of Contents: Summary This paper deals with a characterization of special cases when the distributive law is satisfied in interval arithmetic, a question first formulated byMoore [1]. To find a solution, the basic problem is subdivided into eight cases; in three of them distributivity ever holds, while in other three cases the law is failing. The conditions concerning the remaining cases are combined to a general characterization.
    Notes: Zusammenfassung Diese Arbeit behandelt die vonMoore [1] aufgeworfene Frage nach einer Charakterisierung der Sonderfälle, für die das distributive Gesetz in der Intervallarithmetik Gültigkeit hat. Das Ausgangsproblem wird in acht Fälle unterteilt, die einzeln behandelt werden; in dreien gilt das Gesetz immer, in weiteren dreien versagt es. Die Bedingungen, die sich für die beiden restlichen Fälle angeben lassen, werden zu einer allgemeinen Charakterisierung zusammengefaßt.
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    Über einen Automaten mit Pufferspeicherung (1970)
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    Computing 5 (1970), S. 57-70 
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    Topics: Computer Science
    Description / Table of Contents: Summary A class of automata with tape has been defined and considered as recognition devices. Our automaton can use its working tape only as a buffer, i. e. in a “first-in-first-out” manner. Some modifications of the general model are considered, especially the deterministic, real-time version. The position in theChomsky hierarchy of the class of languages accepted by such automata and some operations with these languages have been investigated. Some results on decidability questions are cited.
    Notes: Zusammenfassung Es wird ein Automat mit Speicherband eingeführt und in seiner Eigenschaft als erkennender Automat untersucht. Er kann sein Arbeitsband nur als Puffer benutzen, d. h. was zuerst eingelesen wird, wird auch zuerst wieder ausgelesen. Es werden verschiedene Modifikationen des Grundmodells, insbesondere aber die deterministische, in Realzeit arbeitende Version betrachtet, sowohl hinsichtlich der Lage der durch den Automatentyp definierten Sprachklasse in derChomsky-Hierarchie, als auch das Verhalten dieser Sprachen gegenüber gewissen Operationen. Einige Resultate über Entscheidbarkeitsfragen werden zitiert.
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    Buchbesprechungen (1970)
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    Computing 5 (1970), S. 89-96 
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    Zur Einschließung der Eigenwerte vollstetiger positiver Operatoren in separablen Hilbert-Räumen (1970)
    Springer
    Computing 5 (1970), S. 17-26 
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    Topics: Computer Science
    Description / Table of Contents: Summary In the first part of this paper a method is given, which permits enclosing the first eigenvalue of positive completely continuous Operators in a separableHilbert space. The interval of enclosing, constructed by the orthogonal-invariants does not need arbitrary initial elements but needs knowledge of variety of the first eigenvalue. For numerical computation of the variety a method is proposed which supposes knowledge of estimation of the quotient of the first and second eigenvalue. The problem of enclosing higher order eigenvalues is treated in the second part of this paper.
    Notes: Zusammenfassung Im ersten Teil dieser Arbeit soll ein Verfahren angegeben werden, das es gestattet, den ersten Eigenwert eines positiven, vollstetigen Operators in einem separablenHilbert-Raum einzuschließen. Das mit Hilfe der Orthogonalinvarianten konstruierte Einschließungsintervall bedarf keinerlei willkürlicher Anfangselemente, benötigt aber die Kenntnis der Vielfachheit des ersten Eigenwertes. Zur numerischen Berechnung der Vielfachheit wird ein Verfahren vorgeschlagen, das die Kenntnis einer Abschätzung des Quotienten aus dem ersten und zweiten Eigenwert voraussetzt. Die Einschließung der höheren Eigenwerte wird im zweiten Teil der vorliegenden Arbeit behandelt.
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    Algorithmus 12 Ein ableitungsfreies Verfahren zur Lösung endlich-dimensionaler nichtlinearer Gleichungssysteme (1970)
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    Computing 5 (1970), S. 82-88 
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    Zur Einschließung der Eigenwerte vollstetiger positiver Operatoren in separablen Hilbert-Räumen (1970)
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    Computing 5 (1970), S. 119-127 
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    Description / Table of Contents: Summary The construction of intervals of enclosing for higher order eigenvalues of positiveHilbert-Schmidt operators in a separableHilbert space is reduced to the problem of the interval of enclosing for the first order eigenvalue of a positiveHilbert-Schmidt operator.
    Notes: Zusammenfassung Die Konstruktion von Einschließungsintervallen für die höheren Eigenwerte positiverHilbert-Schmidtscher Operatoren in separablenHilberträumen wird auf die Aufgabe zurückgeführt, ein Einschließungsintervall für den ersten Eigenwert eines positivenHilbert-Schmidtschen Operators anzugeben.
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    Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle (1970)
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    Computing 5 (1970), S. 184-184 
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    Zur Mehrdeutigkeit bei der Approximation in der L p -Norm mit Hilfe rationaler Funktionen (1970)
    Springer
    Computing 5 (1970), S. 349-355 
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    Description / Table of Contents: Summary In 1967Cheney andGoldstein presented a paper on mean square approximation by generalized rational functions. The problem of uniqueness of the best approximation remained unsolved. In this paper it is shown, that there may be several different rational functions which are bestL p -approximation (1≤p〈∞) of a given function.
    Notes: Zusammenfassung In einer Arbeit vonCheney undGoldstein aus dem Jahre 1967 wurden die Probleme untersucht, die sich im Zusammenhang mit der rationalen Approximation einer gegebenen stetigen Funktion in derL 2-Norm stellen. In der vorliegenden Arbeit soll unter anderem die vonCheney undGoldstein offengelassene Frage der Eindeutigkeit der besten rationalen Approximation untersucht werden.
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    Die Frequenzkennlinien (1970)
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    Computing 5 (1970), S. 394-394 
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    Domains of uncertainty for perturbed operator equations (1970)
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    Computing 5 (1970), S. 312-323 
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    V. Internationaler Kongreß über Anwendungen der Mathematik in den Ingenieurwissenschaften in Weimar (V. IKM) (1970)
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    Computing 5 (1970), S. 332-332 
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    Ableitungsfreie Verfahren ohne Auflösung linearer Gleichungen (1970)
    Springer
    Computing 5 (1970), S. 71-81 
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    Topics: Computer Science
    Description / Table of Contents: Summary Iteration methods are given for the solution of nonlinear equations in normed spaces requiring functionvalues and first-order divided differences. The latter may be replaced by derivations. The linear equations of each iteration step are solved only approximatively but this does not diminish the order of convergence.
    Notes: Zusammenfassung Zur Lösung von nichtlinearen Gleichungen in normierten Räumen werden Iterationsverfahren angegeben, die Funktionswerte und Steigungen erster Ordnung benutzen. Es ist möglich, die Steigungen durch Ableitungen zu ersetzen. Die linearen Gleichungen, die in jedem Iterationsschritt auftreten, werden nur näherungsweise gelöst. Dabei ist es wichtig, daß hierdurch keine Verringerung der Konvergenzgeschwindigkeit eintritt.
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    Mechanisierung von Dreieckskonstruktionen (1970)
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    Computing 5 (1970), S. 128-135 
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    Topics: Computer Science
    Description / Table of Contents: Summary The intelligent behavior of a highschool student, who constructs triangles from three given pieces shows that he generates auxiliary points of a tree-like structure by means of geometric loci. The formalization of this leads to a system of points, geometric loci and rules. In each case the mechine is able to decide whether and how the triangle can be contructed and this leads to the decision-procedure we called the „Dreiecksalgorithmus”.
    Notes: Zusammenfassung Analysiert man das Verhalten eines intelligenten Schülers bei der Konstruktion eines Dreiecks aus drei gegebenen Stücken, so erkennt man, daß er mittels geometrischer Örter Hilfspunkte eines baumartigen Gebildes erzeugt. Eine Formalisierung dessen führt auf Listen von Punkten, geometrischen Örtern und auf ein Regelsystem. Die Maschine kann damit in jedem Fall entscheiden, ob und wie das Dreieck aus den gegebenen Stücken konstruierbar ist, und das ist das Entscheidungsverfahren, das wir „Dreiecksalgorithmus” genannt haben.
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    Topological invariants of matrices (1970)
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    Computing 6 (1970), S. 97-103 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine Methode zur Definition und Charakterisierung der Zusammenhangs- und Repetitionsstruktur von Matrizen vorgeschlagen.—Im Anschluß an die Definition des “Vokabulars” der einzelnen Zeilen (Spalten) werden die numerischen Invarianten (Betti-Zahlen) eines simplizialen Komplexes bestimmt, der als der sogenannte “Nerv” des Systems der Zeilen-(Spalten-)Vokabulare erhalten wird. Die Berechnung erfolgt über die Inzidenzmatrizen des Nervs. Einige Hinweise auf Anwendungsmöglichkeiten und ein elementares Beispiel beschließen die Note.
    Notes: Summary A method is proposed for characterising the connectivity- and repetitivity-structure of matrices. After defining the “vocabularies” of the single rows (columns) we calculate the numerical invariants (Betti-numbers) of a simplicial complex, the “nerve” of the system of vocabularies, via incidencematrices.
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    Über eine Klasse komplexer Intervallgleichungssysteme mit iterationsfähiger Gestalt (1970)
    Springer
    Computing 6 (1970), S. 104-106 
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    Topics: Computer Science
    Description / Table of Contents: Summary In two previous papers [3] and [4] were deduced some statements for certain interval equations over the real number field concerning the existence and uniqueness of the solution and the convergence of iteration methods for the evaluation of the solution. This paper shows, how these statements and their proofs can be extended to analogous interval equations over the complex number field.
    Notes: Zusammenfassung In früheren Arbeiten [3] und [4] wurden für gewisse Intervallgleichungssysteme über dem Körper der reellen Zahlen Kriterien für die Existenz und Eindeutigkeit einer Lösung sowie die Konvergenz von Iterationsverfahren zur Bestimmung der Lösung bewiesen. Die vorliegende Arbeit zeigt, wie diese Kriterien und ihre Beweise zu verallgemeinern sind, damit sie auch für entsprechende Intervallgleichungssysteme über den komplexen Zahlen gelten.
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    Ein kontinuierliches Modell für Kraftfahrzeugströmungen (1970)
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    Computing 6 (1970), S. 107-120 
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    Topics: Computer Science
    Description / Table of Contents: Summary Mathematical description of traffic flow has been tried repeatedly. Two types of theories are to be distinguished: Discrete traffic flow theories start from a system of differential equations for functionsx n (t) describing the routes of single cars. Continuous flow models require solution of a partial differential equation for the velocityv (x, t) and determination of trajectoriesx (t; c) of the velocity field thus obtained. In this paper these integrations for general initial and boundary conditions of a continuous flow model are performed. Treatment of a special case is added, yielding a flexible graphic method for solution of traffic flow problems. As an example of application, a frequently occurring traffic situation is analysed with this theory.
    Notes: Zusammenfassung Die Beschreibung des Kraftfahrzeugverkehrs in mathematischer Form ist mehrfach versucht worden. Es lassen sich zwei Typen von Ansätzen unterscheiden: Die diskreten Fahrzeugfolgetheorien gehen von einem Differentialgleichungssystem für die Bahnkurvenx n (t) der einzelnen Fahrzeuge aus. Die kontinuierlichen Strömungsmodelle verlangen die Lösung einer partiellen Differentialgleichung für die Strömungsgeschwindigkeitv (x, t) sowie die Ermittlung der Trajektorienx (t; c) im so entstandenen Geschwindigkeitsfeld, die als Bahnkurven der Fahrzeuge interpretiert werden. In der vorliegenden Arbeit werden für ein kontinuierliches Modell die genannten Integrationen bei allgeneinen Anfangs- und Randbedingungen geschlossen durchgeführt. Daran schließt sich die Behandlung eines Spezialfalles an, aus dem sich ein flexibles graphisches Verfahren zur Lösung von Kraftfahrzeugströmungsproblemen gewinnen läßt. Als Beispiel einer Anwendung wird ein im Verkehr häufig auftretender Effekt mit Hilfe dieser Theorie graphisch untersucht.
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    Investigation of a certain stray process applied to the optimum synthesis problems (1970)
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    Computing 6 (1970), S. 139-160 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird ein neues Stichprobenverfahren vorgeschlagen, das zur Untersuchung von zulässigen Gebieten in der Optimalsynthese dient. Die Methode ist gekennzeichnet durch einen größeren Wirkungsgrad der Stichproben und bevorzugt die Grenze und ihrer Nachbarschaft gegenüber dem Inneren des zulässigen Gebietes. Der Algorithmus des Irren-Prozesses und die Untersuchungsresultate werden beschrieben. Die Anwendung wird durch Beispiele illustriert. Die Resultate bei der Anwendung verschiedener Methoden werden verglichen. Die verwendeten Rechenautomaten waren GIER und ZAM-2 Computer.
    Notes: Summary A new method is proposed for the random checking of the region of admissible solutions of synthesis problems by means of digital computers. This method is characterized by increasing efficiency of random checking and by preference of the boundary and its neighbourhood to the interior of the region. The algorithm of the stray process is described. The paper contains also the test results of the influence of various parameters on the process. The application of the method is illustrated by examples. The results obtained by different methods are confronted. The computers used were the GIER and ZAM-2 computers.
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    Über die aus monoton zerlegbaren Operatoren gebildeten Iterationsverfahren (1970)
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    Computing 6 (1970), S. 161-172 
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    Topics: Computer Science
    Description / Table of Contents: Summary By means of monoton decomposable operators,Collatz, Albrecht andSchröder have considered a variety of iterative methods, by which they get inclusion sets for the solution of system of simultaneous linear or nonlinear equations for example. It is shown, that these methods are special cases of basic iterative methods for equations with interval coefficients. In the concluding part of this paper a general convergence theorem in aRieszian-space is given.
    Notes: Zusammenfassung Vor allem vonCollatz, Schröder undAlbrecht wurden mit Hilfe des Begriffes des monotonen Operators verschiedene Iterationsverfahren betrachtet, welche Einschließungsmengen zum Beispiel für die Lösung eines linearen oder nichtlinearen Gleichungssystems liefern. Im folgenden wird der Zusammenhang zwischen diesen Verfahren und der Intervallrechnung hergestellt. Es zeigt sich, daß diese Verfahren als Spezialfälle in den auf Intervallbasis gebildeten Iterationsverfahren enthalten sind. Im abschließenden Teil der Arbeit wird ein allgemeiner Konvergenzsatz in einemRieszschen Raum angegeben.
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    A finite difference method for analyzing the compression of poro-viscoelastic media (1970)
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    Computing 6 (1970), S. 342-348 
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    Description / Table of Contents: Zusammenfassung Das Zunehmen der Kompression von wassergesättigten, porösen Medien läßt sich aus der Kontinuitätsgleichung, demDarcyschen Gesetz und einer passenden Druck-Ausdehnungsbeziehung herleiten. Theorien einer sekundären Verdichtung nehmen an, daß die tatsächliche Druck-Ausdehnungsbeziehung zeitabhängig ist. Eine Form dieser Charakterisierungsart ist ein System von linearen, viscoelastischen Modellen. Das ausgewählte Model besteht aus einem elastischen Element in Serie mit einer beliebigen Anzahl vonKelvin-Einheiten. Formuliert wird dieses System als Differential-Integral-Gleichung. Es wird ein endliches Differenzenschema entwickelt, in dem die beherrschende gewöhnliche Differential-Integralgleichung in ein Gleichungssystem übergeführt wird, das aus WärmeleitungsundEuler-Gleichungen besteht. Ein Stabilitätstheorem wird bewiesen.
    Notes: Summary The progress of compression of a water saturated porous medium is derivable from the equation of continuity,Darcy's law, and an appropriate effective stressdilation relationship. Theories of secondary consolidation assume that the effective stress-dilatation relationship is time-dependent. One form of this type of characterization is a system of linear viscoelastic models. The system chosen consists of an elastic element in series with an arbitrary number ofKelvin units. The formulation of this system is a differential-integral equation. The integral portions of the equation are a series of convolution integrals. A finite difference scheme is developed in which the single governing differential-integral equation is broken up into a system of equations of the heat conduction andEuler types. A stability theorem is proved.
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    Abschätzungen für die Anzahl der konvexen Körper, die einen konvexen Körper berühren (1961)
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    Monatshefte für Mathematik 65 (1961), S. 74-81 
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    Majorization of vectors and inequalities for convex functions (1961)
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    Monatshefte für Mathematik 65 (1961), S. 159-169 
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    Sui tacnodi armonici (1961)
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    Monatshefte für Mathematik 65 (1961), S. 201-208 
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    Buchbesprechungen (1961)
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    Monatshefte für Mathematik 65 (1961), S. 183-200 
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    Sur deux congruences associées à une congruence W (1961)
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    Monatshefte für Mathematik 65 (1961), S. 220-226 
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    Der Hjelmslevsche Mittelliniensatz und verwandte Sätze (1961)
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    Monatshefte für Mathematik 65 (1961), S. 249-251 
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    Ein Beitrag zur konstruktiven Differentialgeometrie (1961)
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    Monatshefte für Mathematik 65 (1961), S. 265-269 
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    Eine zusammenhängende Ebene ohme nichtgerade Kurven (1961)
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    Monatshefte für Mathematik 65 (1961), S. 233-235 
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    Über windschiefe Flächen mit konischer Zentraltorse (1961)
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    Monatshefte für Mathematik 65 (1961), S. 337-350 
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    Flächen mit ebenen Fallinien (1961)
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    Monatshefte für Mathematik 65 (1961), S. 291-300 
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    Buchbesprechungen (1961)
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    Monatshefte für Mathematik 65 (1961), S. 391-400 
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    On certain factors of automorphy for the modular group of degree n (1961)
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    Monatshefte für Mathematik 65 (1961), S. 82-87 
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    Buchbesprechungen (1961)
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    Monatshefte für Mathematik 65 (1961), S. 88-100 
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    Eine Bemerkung zum Wilsonschen Satze (1961)
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    Monatshefte für Mathematik 65 (1961), S. 143-149 
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    Die Äquivalenz konstanter und verschwindender Christoffelklammern zweiter Art (1961)
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    Monatshefte für Mathematik 65 (1961), S. 259-264 
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    On strictly convex curves and linear monotonicity (1961)
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    Monatshefte für Mathematik 65 (1961), S. 213-219 
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    Cremonatransformation von Folgen modulo 1 (1961)
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    Monatshefte für Mathematik 65 (1961), S. 227-232 
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    Die kinematischen Abbildungen im dreidimensionalen Raum (1961)
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    Monatshefte für Mathematik 65 (1961), S. 252-258 
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    Eine Bemerkung zur Methode von D. A. Baschkirow für die numerisch-graphische Lösung von Differentialgleichungen (1961)
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    Monatshefte für Mathematik 65 (1961), S. 315-322 
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    Die Stellung der projektiven Geometrie im relativistischen Weltbild (1961)
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    Monatshefte für Mathematik 65 (1961), S. 323-336 
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    Das Maximum des Selbstbehaltes in der Lebensversicherung (1961)
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    Monatshefte für Mathematik 65 (1961), S. 351-357 
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