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  • Articles  (4,432)
  • Springer  (4,432)
  • Cambridge University Press
  • Frontiers Media
  • National Academy of Sciences
  • 2020-2022
  • 1980-1984
  • 1975-1979  (4,432)
  • 1970-1974
  • 1977  (4,432)
  • Mathematics  (4,432)
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  • Articles  (4,432)
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  • 2020-2022
  • 1980-1984
  • 1975-1979  (4,432)
  • 1970-1974
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  • 1
    Electronic Resource
    Electronic Resource
    Observation and biological systems (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 663-678 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A number of apparently different lines of inquiry into fundamental biological processes point to the central role played by the notion of observation in the theory of biological systems. Not only do we use the results of our own observations to obtain the system descriptions which are the starting-points for an understanding of biological processes, but it is a basic postulate of physics that the interactions between biological systems themselves can be regarded as observations. On this basis, it is clear that we cannot properly understand biological interactions unless the observables we employ for system description are the same as those involved in the interactions we are describing. To do this requires a general theory of observables and system description, establishing the relationship between different modes of description. A sketch of such a theory is developed in the present paper, using only two postulates: (a) that all interactions are determined by the values of observables of a system evaluated on specific states, and (b) that real-valued observables suffice. As an application, a specific test is proposed whereby it can be determined whether the observables employed to describe interacting systems are sufficient to specify the observables involved in the interaction itself.
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    URL: http://dx.doi.org/10.1007/BF02461776
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  • 2
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    Cross-sectional shape of flexible tubes (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 679-691 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Differential equations are derived whose solution gives the cross-sectional shape of a flexible tube as a function of the transmural pressure. These equations are solved digitally to produce a series of closed curves, each curve representing the shape of a cross section for a particular set of conditions. These are then applied to the case of systemic arteries, pulmonary arteries, and large veins. The results predict that systemic arteries must always be circular, even when the internal and external pressures are equal. In veins, a small positive internal pressure causes them to become circular, regardless of their initial state, with negligible stretching. Further increases in internal pressure cause the area of the cross section to increase due only to stretching, the shape remaining essentially circular. With pulmonary arteries, known to be noncircular, changes in the cross-sectional area result from a combination of stretching and changes of shape.
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    URL: http://dx.doi.org/10.1007/BF02461777
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  • 3
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    Some techniques in building and validating stochastic models of human reproduction (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 693-704 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Stochastic models of human reproduction are beginning to play significant roles in the evaluation of family planning programs. A class of stochastic processes called absorbing, agedependent, semi-Markov processes frequently arises in the construction of such models. The paper begins with a discussion of some technicalities regarding absorbing, age-dependent, semi-Markov processes. Then, an algorithm due to Littman, which makes possible the computerization of this class of stochastic processes, is presented. Briefly, Littman’s algorithm provides an efficient method for numerically solving systems of renewal type integral equations, provided the system does not contain a large number of equations. After setting down a concrete model for a large clinical trial of intrauterine devices conducted in Taiwan, the paper concludes with a discussion of a method for validating the model based on the data collected in the clinical trial.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461778
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  • 4
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    On peristaltic flow and its efficiency (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 743-743 
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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/BF02461781
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  • 5
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    Detection and analysis of recognition and selection in the immune response (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 705-719 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Some theoretical requirements for the valid use of hemolytic plaque inhibition as a method for studying cellular selection and recognition in an immune response are reviewed. Aside from providing a rational basis for the use of this technique, the theory resolves a growing body of apparently conflicting results on the affinity regulation of IgM secreting cells and can also be used to deduce structural (in particular, morphological) features of the IgM molecule. The diverse predictions of the theory are examined in light of experimental results and find support in a wide data base. Additional specific experiments are proposed which can be used in conjunction with the theory to help clarify the relation between cellular proliferation and maturation.
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    URL: http://dx.doi.org/10.1007/BF02461779
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  • 6
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    A theory of cell swelling in high concentrations of KCl and other chloride salts (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 721-741 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A theory of the cell volume is presented with emphasis on the swelling effect of high concentrations of KCl and other chloride salts. In this theory a particular cell volume represents a state of balance between the tendency of the cell water to build deeper layers of polarized water and the restraining forces exerted by the salt linkages and H-bonds. Taking into account also the different structure-breaking effects of different salts, theoretical curves can be constructed which describe the complex multiple peak-plateau of swelling curve observed in frog muscle in response to increasing concentrations of different chloride salts.
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    URL: http://dx.doi.org/10.1007/BF02461780
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  • 7
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    Stability analysis of predator-prey models via the liapunov method (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 339-347 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract As is well-known from the classical applications in the electrical and mechanical sciences, energy is a suitable Liapunov function: thus, by analogy, all energy functions proposed in ecology are potential Liapunov functions. In this paper, a generalized Lotka-Volterra model is considered and the stability properties of its non-trivial equilibrium are studied by means of an energy function first proposed by Volterra in the context of conservative ecosystems. The advantage of this Liapunov function with respect to the one that can be induced through linearization is also illustrated.
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    URL: http://dx.doi.org/10.1007/BF02462913
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  • 8
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    The stability of entrainment conditions forRLC coupled van der pol oscillators used as a model for intestinal electrical rhythms (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 359-372 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A commonly accepted mathematical model for the slow wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected relaxation oscillators. The method of harmonic balance is used here to obtain analytical results for the entrained frequencies and amplitudes of two oscillators coupled with a parallelRLC network. By perturbations and linearisation about these values the conditions for stable limit-cycles are found and regions in theRLC parameter space which give one or two stable limit-cycle conditions are derived. These analytical results are compared with simulated results and found to creelate well for a waveshape factor of ε=0.1 and fairly well for ε=1.0. The single limit-cycle region corresponds to the requirement for a single mode having a frequency higher than the uncoupled value in small-intestinal data, while the double limit-cycle region corresponds to the two rhythms found in human large-intestinal activity.
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    URL: http://dx.doi.org/10.1007/BF02462915
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  • 9
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    Simulation of nonlinear reaction-diffusion equations (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 391-396 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Discrete particle simulation techniques developed for problems in plasma physics have been adapted to investigate one-dimensional dissipative structures. The results of the model are found to be consistent with bifurcation analysis of nonlinear reaction-diffusion equations. Two distinct type of behavior are observed corresponding to a localized steady-state solution and a time-dependent solution.
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    URL: http://dx.doi.org/10.1007/BF02462918
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  • 10
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    Kinetics of isobaric counterdiffusion (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 587-595 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Isobaric inert gas counterdiffusion has been demonstrated to produce gas lesions in man (Lambertsen and Idicula, 1975) and lethal gas embolism in animals (Lambertsen, Cunnington and Cowley, 1975). Equations have been derived for the stable-state supersaturation pressures developing at interfaces during inert gas counterdiffusion (Graveset al., 1973). The present analysis is a mathematical treatment of the kinetics of the isobaric counterdiffusion of a pair of gases through a membrane consisting of two layers composed of substances with different diffusion coefficients and solubilities for each of the gases involved. The time to reach the stable supersaturation state due to isobaric counterdiffusion, even when circulatory transport and pulmonary washout times are included, is found to be at least an order of magnitude smaller than the time required for visible bubble formation and tissue distortion.
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    URL: http://dx.doi.org/10.1007/BF02461204
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  • 11
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    Prediction of light-induced transient thermal response of the retina (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 607-612 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Suppose that a collimated beam of light traveling along the axis of the eye deposits energy, and that the rate at which energy is deposited by the beam is known at every point of the eye. The thermal response in the chrioretinal region of the eye is predicted by convolving this rate of deposition per unit volume with the fundamental solution of the associated heat operator.
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    URL: http://dx.doi.org/10.1007/BF02461206
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  • 12
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    Exact solutions for a special prey-predator or competing species system (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 619-622 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Exact solutions for the two-species Volterra prey-predator equations and for the Volterra equations for competition are obtained in the special case that the rate of increase of both species in the absence of interaction is the same.
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    URL: http://dx.doi.org/10.1007/BF02461208
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  • 13
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    A graphical analysis of biological evolution in a prey-predator ecology (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 623-627 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 14
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    Structural stability and morphogenesis (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 629-632 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 15
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    International Conference on Energy Use Management (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 633-633 
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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/BF02461211
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  • 16
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    Henry Laurence Lucas, Jr. 1916–1977 (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 635-636 
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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/BF02461773
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  • 17
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    Perspectives of mathematical biology presidential address 1976 annual program of the society for mathematical biology (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 637-641 
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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/BF02461774
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  • 18
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    A physical model of nerve axon—I. Ionic distribution, potential profile, and resting potential (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 1-22 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A theory which is based on a set of assumptions different from those of the sodium theory is developed. Here the mobile ions are considered to be distributed at Donnan equilibrium and the axoplasm is regarded as an analog of a cation-exchanger. Following the spirit of the Debye-Hückel theory, some important features of the ionic distribution and electrical potential of the nerve fiber have been calculated. The results appear to be in better agreement with the experimental observations than the Goldman-Hodgkin-Katz equation.
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    URL: http://dx.doi.org/10.1007/BF02460678
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  • 19
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    The leading edge approximation to the nerve axon problem (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 43-58 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A negative resistance piece-wise linear model, and one which is the sum of two sine terms are used to solve the nerve axon problem for leading edge waveshape, pulse velocity, maximum rate of rise, and rise time for the “Hodgkin-Huxley axon.” The results are compared analytically and numerically to experiment and the calculated results of Hodgkin and Huxley, a fuse model, and a cubic solution.
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    URL: http://dx.doi.org/10.1007/BF02460680
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  • 20
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    Local models of cell aggregation kinetics (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 23-42 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Mathematical models of the phenomena of the sorting out of embryonic cells are derived which predict multiple clustering of like cell types into many small islands. The experiments of Trinkaus and Lentz, and other essentially two-dimensional cell sorting results, provide examples. Two continuous models are introduced, both based on the Smoluchowski theory of Brownian motion. A mass-interaction model predicts local clumping of the internally segregating cell type within a continuum of the other type. A diffusion model simulates the suspension-aggregate experiments of Roth and Weston. A discrete model of sorting out, the exchange model, is re-derived in a clearer manner than before, and certain recent criticisms are responded to. Results in recent two-dimensional cell sorting experiments are discussed and a stability criterion for patterns of multiple clusters is introduced. The site frequency model is re-examined and criticized in terms of its application to the Roth-Weston experiments.
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    URL: http://dx.doi.org/10.1007/BF02460679
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  • 21
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    Spatially distributed stochasticity and the constancy of ecosystems (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 157-166 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Ifconstancy is a measure of an ecosystem's (in) variability through time andstability is a measure of the system's ability to damp and recover from environmental perturbations, then constancy depends not only on stability but also on the frequency and amplitude of perturbations—theenvironmental “noise level”. The stability of an ecosystem reflects its texture, extent, and viscosity (fine-scale structure); the noise level experienced by the system (“effective” noise level) reflects the level at any point (“ambient” noise level), the spectrum of stochastic scale (regional distribution of stochasticity), and the system's spatial extent (size, or number of patches included). The coefficient of variation of a limiting stochastic variate is a measure of the effective noise level. Ifp is the total number of patches in the system (its extent) andn is the number of contiguous patches with noise signals correlated through time (its stochastic scale), then the coefficient of variation is directly proportional to $$\sqrt {(n/p)} $$ whenevern〈p. Thus ecosystems of small stochastic scalen or large sizep damp out environmental noise by “spreading the risk” in space, thereby reducing their variability in time.
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    URL: http://dx.doi.org/10.1007/BF02462855
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  • 22
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    Pumping action on blood by a magnetic field (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 385-390 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We report here an analysis of pumping of blood by means of a non-invasive circulatory-assist device using the principle of Magnetohydrodynamics. This study shows that such a pump would require the application of a slowly moving axial magnetic field of strength of about 108 amp turns/m (approximately 106 oersteds). The results indicate that the temperature rise on account of induced currents is within permissible range.
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    URL: http://dx.doi.org/10.1007/BF02462917
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  • 23
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    Effects of diffusion on the stability of the equilibrium in multi-species ecological systems (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 373-383 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.
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  • 24
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    Transfers between biological and environmental systems (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 397-406 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A newM γ categorical representation of biological systems is developed which is an extension of previously reported work (Leguizamón, 1975a, b; 1976). In this representation, each material physical nature going to a component or produced by it, is given by a set whose cardinality corresponds to the quantity of the material physical nature. Biological and environmental diagrams, respectively represented byM γ andE categories, are used in order to obtain a theory of transfers between both systems. From the mathematical representation of this theory, the transfers are classified as: material-energetic, exclusively energetic and theoretical material-theoretical energetic. These concepts are incorporated into a diagram, and later, a set of axioms is given, in order to have the previous developments as a new categoryT.
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    URL: http://dx.doi.org/10.1007/BF02462919
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  • 25
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    A mathematical model of the dynamic heat transfer from the respiratory tract of a chicken (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 415-433 
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    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model of the dynamic (periodic) heat exchange from the respiratory tract of a chicken is postulated and solved analytically. The model expresses the periodic respiratory heat loss as a function of respiration rate, respiratory air velocity, ambient temperature and humidity ratio, and body (trachea) temperature. It is unique in that previous models have been formulated for steady state heat transfer. The processes of sensible and latent heat exchange are considered as uncoupled processes.
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    URL: http://dx.doi.org/10.1007/BF02462921
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  • 26
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    The bio-environmental system and the realizability of pre-biological systems (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 407-413 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A set of axioms is given which embed theM γ E andT categories developed previously (Leguizamón 1975a, b; 1976, 1977) into a new categoryK. The categoryK represents a bioenvironmental system with the properties of the three previous developments. This bioenvironmental system is believed to be useful in order to study the realization of prebiological systems. The categorical structure of most simple biological system of “zero life” is given, on the basis of two specific biological conditions. This system is proposed as starting point for study of the organization of living systems.
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  • 27
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    Exact fundamental solutions of linear parabolic equations with spatially varying coefficients (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 435-451 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.
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    URL: http://dx.doi.org/10.1007/BF02462922
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  • 28
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    On the information received by sensory receptors (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 453-461 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is hypothesized that a sensory neuron, a neuron issuing from a sensory receptor, encodes the rate at which entropy or uncertainty is removed at the receptor level. This hypothesis is tested for the case of the entropy associated with the magnitude of a signal (stimulus) applied at the sensory receptor. A simple mathematical model of the process is presented and a number of well-known stimulus-response relationships are seen to emerge. For example, the adaptation of a receptor may be seen to occur as a consequence of reduced uncertainty regarding stimulus intensity. A general equation relating stimulus and response is developed, and this equation will simplify, depending upon the ratio of signal power to noise power, to either a logarithmic or a power law.
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  • 29
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    Precipitation of submicron charged particles in human lung airways (1977)
    Springer
    Bulletin of mathematical biology 39 (1977), S. 471-478 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Precipitation of charged particles in a tube by their own space charge is investigated theoretically, when the number density of the particles is large enough so that the potential is a smooth function given by Poisson’s equation, and when the number density is small so that only the image force is important. These two approaches have been applied to the data given by Weibel for the human lung, to determine the deposition probabibilities at different generations for submicron particles when the particle density is 1×105 particles/cm3. The results indicate that the electrostatic dispersion can only lead to a small effect on the lung deposition, the predominant effect is due to the image force exerted on the particles.
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    The struggle for life—V. One species living in a limited environment (1977)
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    Bulletin of mathematical biology 39 (1977), S. 479-485 
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    Notes: Abstract The general properties of the equations describing a single species living in a limited environment in the presence of its own pollutant are discussed.
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    An urn model study of variability within a compartment (1977)
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    Bulletin of mathematical biology 39 (1977), S. 463-470 
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    Notes: Abstract Formulas are derived for the mean and variance of the number of radioactive atoms present in a compartment (or urn). Initally,n 1 radioactive atoms andb stable atoms are placed in the urn; and subsequently,r stable atoms are added and an equal number,r, of a random mixture of stable and radioactive atoms is removed per unit time. The expected number of radioactive atoms,E(t), present at timet is, as expected,n 1 e−λt where λ=(r/Δt)/(b+r+n 1). The relative variance, σ2(t)/n 1 2 , vanishes to zero forr=1, atoms per unit time and for a large number ofn 1 radioactive atoms; but for a large number of bothr andn 1 atoms the relative variance is ∼e −λt , equal to the fractional retention, fort〉1/λ. Thus in studies where radionuclides are injected into animals and a single compartment represents the data, if a large variance is observed it might be due to the fact that large numbers of atoms are transferred out in unit time. When a small variance is observed, this is probably due to the fact that few atoms are transferred in smaller units of time (such that λ is the same in both cases).
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    Immune surveillance and neoplasia—II A two-stage mathematical model (1977)
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    Bulletin of mathematical biology 39 (1977), S. 487-497 
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    Notes: Abstract In a previous paper (DeLisi and Rescigno, 1977) a model for the interaction of tumor cells and killer lymphocytes was presented. Although that model was highly simplified, the qualitative behavior was in accord with intuitive expectations and a wide range of data. It could not however account forde novo tumor development. In this paper a slightly more realistic model is presented by introducing a delay in the formation of killer lymphocytes. This is done by requiring two stages in the production of a killer. We show that introduction of this second stage allows tumor development from even a single cell, thus removing an important limitation of two variable systems.
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    Realization of dependency structures (1977)
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    Bulletin of mathematical biology 39 (1977), S. 499-503 
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    Notes: Abstract In this paper we show how to construct block diagram realizations of a given dependency structure and investigate some elementary properties.
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    Errata (1977)
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    Bulletin of mathematical biology 39 (1977), S. 507-507 
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    Bifurcation analysis of nonlinear reaction-diffusion equations—I (1977)
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    Bulletin of mathematical biology 39 (1977), S. 505-506 
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    Optimization of diameters and bifurcation angles in lung and vascular tree structures (1977)
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    Bulletin of mathematical biology 39 (1977), S. 509-520 
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    Notes: Abstract An optimization model is described for lung and vascular tree structures. The model extends Murray's model, which is derived from minimal power dissipation due to the frictional resistance of laminar flow and the volume of the duct system. Instead of just laminar flow, it takes into account all types of steady flow, e.g. turbulent and laminar flow, and predicts which structural changes will occur among different parts of trees having different types of flow. The sensitivity of the optimal values is indicated and the model, predictions are compared with literature data.
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    The stochastic theory of compartments—A mammillary system (1977)
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    Bulletin of mathematical biology 39 (1977), S. 533-542 
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    Notes: Abstract A stochastic mammalillary model with arbitrary holding times for each of the compartments is analyzed here both with and without input. Several results found earlier by other authors for a special case of this model are generalized and some new results are also added. In particular the distribution of the system content is determined at a finite time and also in the steady state. Computable bounds for the probability of exceeding a given threshold in the peripheral compartments are given. The time until the system becomes empty is also discussed.
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    The Green's function for the convection-diffusion equation in an analytic lung model (1977)
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    Bulletin of mathematical biology 39 (1977), S. 543-563 
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    Notes: Abstract The method of Green's functions is applied to the convection-diffusion equation describing gas transport in the human lung. Explicit analytic solutions are found for the zeroth approximation to the value of steady state alveolar oxygen concentration, given the frequency of breathing, tidal volume, oxygen consumption, alveolar volume, volume of the bronchial tree, and two modelling parameters describing the geometry. The results of the calculation are compared favorably with some experimental data.
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    Physio-chemical mathematical theory for transitions in cellular metabolism: An aspect of carcinogenesis (1977)
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    Bulletin of mathematical biology 39 (1977), S. 521-532 
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    Notes: Abstract It is pointed out that the non-monotonic character of the chemical reaction rate expressions, together with the relative magnitude of the diffusivity constants, is likely to engender a multiplicity of locally stable steady-state solutions to the system of reaction-diffusion equations for the concentration distributions of molecular species through the volume of a living cell. A transition in cellular metabolism, i.e., the dynamical evolution from an initial locally stable steady-state solution for the concentration distributions to another distinct locally stable steady-state solution, can be induced by an etiologic agent which modifies the rate expressions significantly during an interval of time. Global inequality analysis is employed to derive a condition on the modified rate expressions that is sufficient to guarantee the occurrence of such a transition in cellular metabolism. The possibility of a transition induced by a chemical carcinogen is investigated by applying the latter sufficient condition, and it is found that the statistical frequency of carcinogenesis should depend essentially on the magnitude of the grouping (T 2 − α D α) for a total doseD of carcinogen administered to an animal at a uniform rate (D/T) over a time interval of durationT, where α is a certain positive number less than 1. This theoretical result is shown to be supported by the available experimental evidence.
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    Bilinear programming: An exact algorithm (1977)
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    Mathematical programming 12 (1977), S. 173-194 
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    Notes: Abstract The Bilinear Programming Problem is a structured quadratic programming problem whose objective function is, in general, neither convex nor concave. Making use of the formal linearity of a dual formulation of the problem, we give a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution.
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    On the existence and uniqueness of solutions in nonlinear complementarity theory (1977)
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    Mathematical programming 12 (1977), S. 110-130 
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    Notes: Abstract A complementarity problem is said to be globally uniquely solvable (GUS) if it has a unique solution, and this property will not change, even if any constant term is added to the mapping generating the problem. A characterization of the GUS property which generalizes a basic theorem in linear complementarity theory is given. Known sufficient conditions given by Cottle, Karamardian, and Moré for the nonlinear case are also shown to be generalized. In particular, several open questions concerning Cottle's condition are settled and a new proof is given for the sufficiency of this condition. A simple characterization for the two-dimensional case and a necessary condition for then-dimensional case are also given.
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    Quadratic termination properties of Davidon's new variable metric algorithm (1977)
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    Mathematical programming 12 (1977), S. 141-147 
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    Notes: Abstract Davidon [1] provides a variable metric algorithm for minimization calculations that achieves quadratic termination without any line searches by a method that uses positive definite second derivative approximations. We give a new and elementary proof of these termination properties. It shows the great freedom allowed by the algorithm in the choice of search directions and it explains the effect of a restart procedure.
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    The Mathematical Programming Society (1977)
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    Mathematical programming 12 (1977), S. 155-155 
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    The zero pivot phenomenon in transportation and assignment problems and its computational implications (1977)
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    Mathematical programming 12 (1977), S. 226-240 
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    Notes: Abstract Transportation and assignment models have been widely used in many applications. Their use was motivated, among other reasons, by the existence of efficient solution methods and their occurrence as sub-problems in the solution of combinatorial problems. A previous study [10] observed that, in large-scale Transportation and Assignment Problems, 95 percent of the pivots were zero or degenerate pivots. This study investigates the ratio of zero pivots to the total number of pivots and verifies the above observation under conditions of small rim variability. Rules are introduced that pay special attention to the zero pivot phenomenon, and significantly reduce CPU time in both phase-1 (generating the initial basic feasible solution) and in phase-2 (selecting the variable leaving the base and the variable entering the base). When these rules were applied, they reduced the CPU time substantially: a 500×500 assignment problem was solved in 1.3 seconds.
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    Matrix factorizations in optimization of nonlinear functions subject to linear constraints — an addendum (1977)
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    Mathematical programming 12 (1977), S. 279-280 
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    The use of augmented lagrangian functions for sensitivity analysis in nonlinear programming (1977)
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    Mathematical programming 12 (1977), S. 281-284 
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    Keywords: Nonlinear programming ; Augmented Lagrangian functions ; Sensitivity analysis
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    Constructive proofs of theorems relating to:F(x) = y, with applications (1977)
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    Mathematical programming 12 (1977), S. 328-343 
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    Notes: Abstract Some theorems are given which relate to approximating and establishing the existence of solutions to systemsF(x) = y ofn equations inn unknowns, for variousy, in a region of euclideann-space E n . They generalize known theorems. Viewing complementarity problems and fixed-point problems as examples, known results or generalizations of known results are obtained. A familiar use is made of homotopies H: E n × [0, 1]→E n of the formH(x, t) = (1 −t)F 0 (x) + t[F(x) − y] where theF 0 in this paper is taken to be linear. Simplicial subdivisionsT k of E n × [0, 1] furnish piecewise linear approximatesG k toH. The basic computation is via the generation of piecewise linear curvesP k which satisfyG k (x, t) = 0. Visualizing a sequence {T k } of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curvesP on whichH(x, t) = 0. This paper builds upon and continues recent work of C.B. Garcia.
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    A practicable steepest-edge simplex algorithm (1977)
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    Mathematical programming 12 (1977), S. 361-371 
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    Keywords: Simplex Method ; Linear Programming ; Steepest-edge ; LU Factorization
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    Notes: Abstract It is shown that suitable recurrences may be used in order to implement in practice the steepest-edge simplex linear programming algorithm. In this algorithm each iteration is along an edge of the polytope of feasible solutions on which the objective function decreases most rapidly with respect to distance in the space of all the variables. Results of computer comparisons on medium-scale problems indicate that the resulting algorithm requires less iterations but about the same overall time as the algorithm of Harris [8], which may be regarded as approximating the steepest-edge algorithm. Both show a worthwhile advantage over the standard algorithm.
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    The Mathematical Programming Society (1977)
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    Mathematical programming 12 (1977), S. 423-423 
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    On monotonicity in parametric linear complementarity problems (1977)
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    Mathematical programming 12 (1977), S. 60-66 
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    Notes: Abstract This paper generalizes the answers that were given by R.W. Cottle to questions that were originally raised by G. Maier. Essentially, we give necessary and sufficient conditions for some notions of monotonicity of solutions for the parametric linear complementarity problem. Our proofs are direct ones and not algorithmic, as Cottle's proofs are, and also cover a broader class of matrices.
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    An algorithm for determining all extreme points of a convex polytope (1977)
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    Mathematical programming 12 (1977), S. 81-96 
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    Notes: Abstract An algorithm for determining all the extreme points of a convex polytope associated with a set of linear constraints, via the computation of basic feasible solutions to the constraints, is presented. The algorithm is based on the product-form revised simplex method and as such can be readily linked onto standard linear programming codes. Applications of such an algorithm are reviewed and limited computational experience given.
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    A monotone complementarity problem with feasible solutions but no complementary solutions (1977)
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    Mathematical programming 12 (1977), S. 131-132 
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    On some properties of L1 estimators (1977)
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    Mathematical programming 12 (1977), S. 139-140 
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    The Committee on Algorithms of the mathematical programming society: (1977)
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    Mathematical programming 12 (1977), S. 151-153 
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    The complementary unboundedness of dual feasible solution sets in convex programming (1977)
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    Mathematical programming 12 (1977), S. 392-405 
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    Keywords: Complementary Unboundedness ; Dual Feasible Solution Sets ; Convex Programming ; Geometric Programming ; Fenchel Duality ; Rockafellar Duality ; Ordinary Duality ; Quadratic Programming ; Optimal Location ; Traffic Equilibria
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    Notes: Abstract F.E. Clark has shown that if at least one of the feasible solution sets for a pair of dual linear programming problems is nonempty then at least one of them is both nonempty and unbounded. Subsequently, M. Avriel and A.C. Williams have obtained the same result in the more general context of (prototype posynomial) geometric programming. In this paper we show that the same result is actually false in the even more general context of convex programming — unless a certain regularity condition is satisfied. We also show that the regularity condition is so weak that it is automatically satisfied in linear programming (prototype posynomial) geometric programming, quadratic programming (with either linear or quadratic constraints),l p -regression analysis, optimal location, roadway network analysis, and chemical equilibrium analysis. Moreover, we develop an equivalent regularity condition for each of the usual formulations of duality.
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    Finite horizon approximations of infinite horizon linear programs (1977)
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    Mathematical programming 12 (1977), S. 1-17 
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    Notes: Abstract This paper describes the class of infinite horizon linear programs that have finite optimal values. A sequence of finite horizon (T period) problems is shown to approximate the infinite horizon problems in the following sense: the optimal values of theT period problems converge monotonically to the optimal value of the infinite problem and the limit of any convergent subsequence of initialT period optimal decisions is an optimal decision for the infinite horizon problem.
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    Isotone solutions of parametric linear complementarity problems (1977)
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    Mathematical programming 12 (1977), S. 48-59 
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    Notes: Abstract The parametric linear complementarity problem is given by the conditions:q + αp + Mz ⩾ 0,α ⩾ 0,z ⩾ 0,z T (q + αp + Mz) = 0. Under the assumption thatM is a P-matrix, Cottle proved that the solution mapz(α) of the above problem is montonically nondecreasing in the parameterα for every nonnegativeq and everyp if and only ifM is a Minkowski matrix. This paper examines whether a similar result holds in various other settings including a nonlinear case.
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    Optimality condition: Constraint regularization (1977)
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    Mathematical programming 13 (1977), S. 38-48 
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    Keywords: Constraint regularization ; Determinant cone ; Kuhn—Tucker constraint-qualification ; Lagrange regularity ; Optimality condition ; x 0-redundant constraints
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    Notes: Abstract Necessity and sufficiency conditions are given for the existence of a finite set of redundant constraints which together with a given set of constraints in an optimization problem satisfy the Kuhn—Tucker, and the Guignard Constraint qualifications when the original set of constraints fails to satisfy either or both of them.
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    Ergodicity and symmetric mathematical programs (1977)
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    Mathematical programming 13 (1977), S. 81-87 
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    Keywords: Symmetric mathematical programs ; Ergodicity ; S-concavity and majorization ; Stochastic matrices ; Cyclic symmetry
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    Notes: Abstract The paper provides new conditions ensuring the optimality of a symmetric feasible point of certain mathematical programs. It is shown that these conditions generalize and unify most of the known results dealing with optimality of symmetric policies (e.g. [2, 4, 6, 11]). The generalization is based on certain ergodic properties of nonnegative matrices. An application to a socio-economic model dealing with optimization of a welfare function is presented.
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    Restart procedures for the conjugate gradient method (1977)
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    Mathematical programming 12 (1977), S. 241-254 
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    Notes: Abstract The conjugate gradient method is particularly useful for minimizing functions of very many variables because it does not require the storage of any matrices. However the rate of convergence of the algorithm is only linear unless the iterative procedure is “restarted” occasionally. At present it is usual to restart everyn or (n + 1) iterations, wheren is the number of variables, but it is known that the frequency of restarts should depend on the objective function. Therefore the main purpose of this paper is to provide an algorithm with a restart procedure that takes account of the objective function automatically. Another purpose is to study a multiplying factor that occurs in the definition of the search direction of each iteration. Various expressions for this factor have been proposed and often it does not matter which one is used. However now some reasons are given in favour of one of these expressions. Several numerical examples are reported in support of the conclusions of this paper.
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    Line perfect graphs (1977)
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    Mathematical programming 12 (1977), S. 255-259 
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    Notes: Abstract The concept of line perfection of a graph is defined so that a simple graph is line perfect if and only if its line graph is perfect in the usual sense. Line perfect graphs are characterized as those which contain no odd cycles of size larger than 3. Two well-know theorems of König for bipartite graphs are shown to hold also for line perfect graphs; this extension provides a reinterpretation of the content of these theorems.
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    Testing for complete efficiency in a vector maximization problem (1977)
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    Mathematical programming 12 (1977), S. 285-288 
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    Keywords: Vector maximization ; Efficient points
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    Topics: Computer Science , Mathematics
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    An algebraic approach to assignment problems (1977)
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    Mathematical programming 12 (1977), S. 318-327 
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    Keywords: Combinatorial Optimization ; Assignment Problem ; Ordered Semigroups ; Bottleneck Objective ; Lexicographical Objective ; Labeling Method ; Admissible Transformation
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    Notes: Abstract For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.
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    Examples of non-convergence of conjugate descent algorithms with exact line-searches (1977)
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    Mathematical programming 12 (1977), S. 356-360 
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    Keywords: Conjugate ; Convergence ; Exact Line-search ; Optimization
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    Notes: Abstract Examples are given in which quasi-Newton and other conjugate descent algorithms fail to converge to a minimum of the object function. In the first there is convergence to a point where the gradient is infinite; in the second, a region characterized by a fine terraced structure causes the iterates to spiral indefinitely. A modification of the second construction gives convergence to a point where the gradient is non-zero, but the gradient is not continuous at this point. The implication of these examples is discussed.
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    Mathematical programming study 5 and 6 (1977)
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    Mathematical programming 12 (1977), S. 415-422 
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    Topics: Computer Science , Mathematics
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    Using shortest paths in some transshipment problems with concave costs (1977)
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    Mathematical programming 12 (1977), S. 18-25 
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    Topics: Computer Science , Mathematics
    Notes: Abstract This paper deals with the problem of solving an uncapacitated transshipment problem with either one source and several sinks or one sink and several sources. The cost function of the problem is concave in the amount shipped on each arc and thus local optima are possible. A characterization of adjacent extreme flows in terms of corresponding arborescences is given for this type of networks. This characterization together with shortest path methods is then used to attack the problems of finding local optima and of ranking extreme points. A real-world problem and computational evidence for the usefulness of the method are produced.
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    On the size of balls covered by analytic transformations (1977)
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    Monatshefte für Mathematik 83 (1977), S. 9-23 
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    Notes: Abstract Two quantitative forms of the inverse function theorem giving estimates on the size of balls covered biholomorphically are proved for holomorphic mappings of a ball in a Banach space into the space. Also, a Bloch theorem forK-quasiconformal mappings on the open unit ball of a Banach space is given and some mapping properties ofK-quasiconformal mappings are deduced.
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    Value-true walks in finite, connected, valuated graphs (1977)
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    Monatshefte für Mathematik 83 (1977), S. 1-8 
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    Notes: Abstract Starting from Problem 4 on p. 36 ofK. Wagner's book “Graphentheorie” the following question is investigated: Suppose an (undirected) graphG having its edges valuated with nonnegative integers; does there exist a closed walk inG such that each 0-valuated edge is passed the same number of times in both directions, while ak-valuated edge fork〉0, is passed exactlyk times in one direction and not at all in the other direction (without having the direction prescribed)? A general criterion and a sufficient condition for the existence of such “value-true walks” are proved. For the analogous question for directed graphs an algorithm for the construction of such walks is established.
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    Über die Kantenlängensumme von Dreieckspolyedern (1977)
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    Monatshefte für Mathematik 83 (1977), S. 25-36 
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    Notes: Abstract In 1953L. Fejes Tóth published the following conjecture: Among all triangle-faced convex polyhedra with a given inradius the regular tetrahedron and octahedron have the least possible total edge-length. In the following this conjecture is proved.
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    A comparison method for forced oscillations (1977)
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    Monatshefte für Mathematik 83 (1977), S. 37-41 
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    Notes: Abstract Consider the functional differential equations (1)x (n) (t)+a(t)f(x(g(t)))=Q(t) and (2)x (n) (t)+s(t)f(x(t))=0 wheres(t)≧γa(t) for all sufficiently larget, sayt≧T〉0 and for some γ, 0〈γ〈1. Recently,E. D. True proposed a new comparison theorem for equations (1) and (2) with which he proved some results on the oscillation of (1). The purpose of the present paper is to extend them to the more general case. The examples indicate that the results are applicable to interesting problems.
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    The automorphism group of a class of semi-groups (1977)
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    Monatshefte für Mathematik 83 (1977), S. 53-58 
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    Notes: Abstract Let (S,·) be a semi-group having the following properties: (1)S=∪S α where α is in some index setI andS α are subgroups isomorphic to each other, (2)S α∩S β=Ø, a void set for α≠β and (3) the identity ofS α is a left identity ofS for each α inI. Then the automorphism group Aut (S) ofS is studied from the point of category theory. It is proved that Aut (S) is determined by Aut (S α) and right multiplications by the identities of groupsS α.
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    Eine Bemerkung zu den Nielsen-Transformationen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 43-52 
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    Notes: Abstract LetG=H 1* A H 2 a free product with amalgamation with $$H_1 = \left\langle {s_1 ,...,s_m |s_1^{\alpha _1 } = ... = s_m^{\alpha _m } = 1} \right\rangle ,\alpha _i \geqslant 2,m \geqslant 2,$$ a free product ofm cyclic groups. If we ask for the generation ofG, the following questions are significant: 1) For which α i andm there is a set {x 1, ...,x m } of. generators ofH 1 withx 1=(s 1...s m )α, α≥2, and what can we say about α andx 2, ...,x m ? 2) for which α i andm there is a set {x 1, ...,x m } of generators ofH 1 withx 1=(s 1...s m )α x 2=h(s 1...s m )β h −1, α〉0, β〉0,h∈H 1, and what can we say about α, β,h andx 3, ...,x m ? In this note we give a complete solution of these questions.
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    Über die Häufigkeit großer Differenzen konsekutiver Primzahlen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 59-63 
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    Notes: Abstract The following theorem is going to be proved. Letp m be them-th prime and putd m :=p m+1−p m . LetN(σ,T), 1/2≤σ≤1,T≥3. denote the number of zeros ϱ=β+iγ of the Riemann zeta function which fulfill β≥σ and |γ|≤T. Letc≥2 andh≥0 be constants such thatN(σ,T)≪T c(1−σ) (logT) h holds true uniformly in 1/2≤σ≤1. Let ε〉0 be given. Then there is some constantK〉0 such that
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    Über Tuttes Cages (1977)
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    Monatshefte für Mathematik 83 (1977), S. 65-75 
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    Notes: Abstract An undirected graph Γ of valencyd and girth γ is called a (d, γ)-cage if its automorphism group acts transitively on the set of alls-paths in Γ ands≥(γ+1)/2. We discuss an elementary construction of two known families of cages which allows us to prove easily some facts about their automorphism groups. We give, for example, a new proof of the fact that the automorphism group ofSp 4(2 n ) contains elements which are not induced by ΓSp 4(2 n ).
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    Join operations in vector spaces (1977)
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    Monatshefte für Mathematik 83 (1977), S. 93-97 
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    Notes: Abstract LetX be a vector space over a totally ordered fieldF and Λ a subset ofF. Necessary and sufficient conditions are obtained on Λ for a join defined bya·b={λa+(1−λ)b|λ∈Λ} to make (X,·) (i) a convexity space (ii) a join system and (iii) a join space. For (X,·) to be a convexity space Λ is necessarily the set of elements between 0 and 1 for some ordered subfield ofF. In this setting join systems and join spaces are essentially the same but they remain quite distinct from convexity spaces.
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    Der quadratfreie Kern der Eulerschen φ-Funktion (1977)
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    Monatshefte für Mathematik 83 (1977), S. 89-91 
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    Notes: Abstract It is shown by a simple induction that any square-free natural number is the square-free kernel of Euler's φ-function of infinitely many natural numbers. Furthermore every square-free natural number 〉1 is the square-free kernel of Euler's φ-function of infinitely many square-numbers.
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    Buchbesprechungen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 77-88 
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    Density functions for prime and relatively prime numbers (1977)
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    Monatshefte für Mathematik 83 (1977), S. 99-112 
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    Notes: Abstract Letr *(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ϱ*(x) denote the maximum number ofprime integers in any interval (y,y+x] wherey≥x. Throughout this paper we assume the “primek-tuples hypothesis.” (This hypothesis could be avoided by using an alternative sievetheoretic definition of ϱ*(x); cf. the beginning of Section 1.) We investigate the differencer *(x)—ϱ*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr *(x)—ϱ*(x)〈x c for somec〉0 (whenx→∞). This improves the previous lower bound of logx. As an upper bound we getr *(x)—ϱ*(x)=o[x/(logx)2]. It is known that ϱ*(x)—π(x)〉const.[x/(logx)2];.; thus the difference betweenr *(x) and ϱ*(x) is negligible compared to ϱ*(x)—π(x). The results mentioned so far involve the “upper bound” or “maximizing” sieve. In Section 2, similar comparisons are made between two types of “minimum” sieves. One of these is the “erasing” sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdös and Selfridge [1], involves a kind of “minimax” for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.
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    Product integration in reflexive Banach spaces (1977)
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    Monatshefte für Mathematik 83 (1977), S. 113-119 
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    Notes: Abstract LetX denote a reflexive Banach space and {A(t)|t∈[0,T]} a time dependent family of accretive operators defined onX. Conditions are placed on {A(t)|t∈[0,T]} which are sufficient to guarantee the existence of solutions to the Cauchy initial value problem:u′(t,x)+A(t)u(t,x)=0; u(0,x)=x. These solutions are obtained via the method of product integration; however limits for the infinite product are taken with respect to the weak topology.
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    On the connectivities of finite and infinite graphs (1977)
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    Monatshefte für Mathematik 83 (1977), S. 121-131 
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    Notes: Abstract Results byW. Mader and the authors on the connectivity of finite graphs are generalized to include infinite graphs. In the infinite case a distinction must be made concerning the distribution of finite and infinite components with respect to the separating sets. Results are obtained relating these components to the atoms.
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    Striktionseigenschaften von Regelscharen imE n (1977)
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    Monatshefte für Mathematik 83 (1977), S. 133-146 
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    Notes: Abstract Let ℜ={e(u)|u∈I} be a one-parameter family of straight lines forming a ruledC r-2-surface Φ⊂E n (n≥2,r≥1) without singular generatorse(u) (u∈I). As a synopsis, a generalization and an improvement of various results already known about the strictional properties of ruled surfaces Φ⊂E n (especially in the casen=3) the author demonstrates a uniform geometrical way of defining and uniquely obtaining thestriction point S(u) and theparameter of distribution d(u) of a generatore(u)∈ℜ under the minimal assumptions thate(u)∈ℜ⊂E n (n≥2) be noncylindrical andr≥1. Other methods of obtainingS(u) andd(u) are discussed in comparison, and special strictional properties ofskew ruled surfaces Φ⊂E n are proved.
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    Un Exemple de Transition de Phase (1977)
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    Monatshefte für Mathematik 83 (1977), S. 147-153 
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    Notes: Abstract We give a new proof of the non uniqueness of the equilibrium state inF. Hofbauer's example. We extend it to obtain examples with any finite number of ergodic equilibrium states.
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    Das Existenzproblem multivalenter Lösungen bei Differentialgleichungen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 159-162 
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    Notes: Abstract We consider linear differential equations $$w^{(n)} + \sum\limits_{i = 0}^{n - 1} {\sigma _i w^{(i)} = 0 in |z|}〈 1.$$ If the coefficients σi inH ∞ the solutions of theis equations have only a finite number of zeros and therefore these solutions are multivalent in |z|〈1.
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    Eine Bemerkung über die Wurzelpolynome Galoisscher Gleichungen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 155-157 
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    Notes: Abstract By means of results due toKlieman andFlor it is shown that a Galois-polynomial of degree greater than two over a formally real field is determined uniquely by its root polynomials except for the degenerate case where the roots are linearly dependent.
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    Eine Bemerkung über die Werte der Funktion σ (n) (1977)
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    Monatshefte für Mathematik 83 (1977), S. 163-166 
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    Notes: Abstract By means of the Hoheisel—Montgomery prime number theorem it is shown that for every α≥1 the inequality $$|(\sigma (n)/n) - \alpha | \leqslant {1 \mathord{\left/ {\vphantom {1 {n^{({2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-\nulldelimiterspace} 5}) - \varepsilon } }}} \right. \kern-\nulldelimiterspace} {n^{({2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-\nulldelimiterspace} 5}) - \varepsilon } }}(\varepsilon 〉 0,\sigma (n) = \sum\limits_{d/n} d )$$ has infinitely many solutionsn∈N. It is highly probable that the exponent 2/5 can be replaced by 1.
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    Buchbesprechungen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 167-176 
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    Eine Note zum System PC (1977)
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    Monatshefte für Mathematik 83 (1977), S. 191-200 
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    Notes: Abstract Corresponding to a graduation of the comprehension predicateC there arise two systems PC1 and PC2. PC1 represents constructive mathematics whereas PC2 as a restricted enlargement of PC1 enables one to deal with full set theory. Unnatural definitions likeKuratowski's ordered pair and identification of 0 and $$\not 0$$ are avoided. In PC1 the postulate introduced for ordered pairs holds not only for sets but for classes generally. The comprehension condition in PC1 implicitly contains a set axiom for at most denumerable calsses and for ordered pairs. By introduction of the concept “ordered self-pair” the real numbers are interpreted as self-pairs of (non denumerable) equivalence classes of concentrated ratioal sequences and they can be shown to be sets in PC1.
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    Über Mittelwerte aus Potenzproduktsummen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 177-189 
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    Notes: Abstract Using symmetric forms $$M_{k_1 ,...,k_p } : = (1/A(n))\,\sum\limits_{v_1 ,...,v_p } {a_{v_1 }^{k_1 } ...a_{v_p }^{k_p } } $$ A(n)=number of terms of the sum,a ν〉0,k i≠0,i=1,...,n) the meansm k 1,...,kp:=(Mk 1,...,kp1/(k1+...+kp)(k1+...+kp≠0) are formed and investigated as to monotonicity under the hypothesis that the exponentsk 1,...,k p are certain linear functions of only one parameterk(k i =λ i k+β 1,λ i 〉0,β 1+...β p =0). (The means $$m_{\lambda _1 k_1 ,...,\lambda _1 k_p } $$ , e. g., are increasing ifk is increasing.) The proofs are elementary and use the known method of positive logarithmically convex (or concave) sequences and certain generalizations of Muirhead's theorem.
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    The cauchy problem of a degenerate parabolic equation (1977)
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    Monatshefte für Mathematik 83 (1977), S. 201-205 
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    Notes: Abstract This note is concerned with the existence of a weak solution for a degenerate Cauchy problem of parabolic type in then-dimensional spaceR n. The degenerate property is in the sense that the matrix (a ij(t,x)) involved in the differential operator is not necessarily positive definite. The essential idea is the construction of a suitable function spaceH and to prove the existence of a weak solution inH.
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    Buchbesprechungen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 253-264 
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    Über einen Satz von Shl. Sternberg in der Theorie der analytischen Iterationen (1977)
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    Monatshefte für Mathematik 83 (1977), S. 207-221 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let ℂ〚X〛=ℂ〚X 1,...,X n 〛 be the ring of formal power series inn indeterminates over ℂ. LetF:X→AX+B(X)=(F (1)(X),...,F (n)(X))∈(ℂ〚X〛) n denote an automorphism of ℂ〚X〛 and let ϱ1,...,ϱ n be the eigenvalues of the linear partA ofF. We will say thatF has an analytic iteration (a. i.) if there exists a family (F t (itX)) t∈ℂ of automorphisms such thatF t(X) has coefficients analytic int and such thatF 0=X,F 1=F,F t+t′=FtℴFt′ for allt,t′∈ℂ. Let now a setΛ=(lnϱ1,...,lnϱ n ) of determinations of the logarithms be given. We ask if there exists an a. i. ofF such that the eigenvalues of the linear partA(t) ofF t(X) are $$e^{\ln \varrho _1 t} ,...,e^{\ln \varrho _n t} $$ . We will give necessary and sufficient conditions forF to have such an a. i., namely thatF is conjugate to a semicanonical formN=T −1ℴFℴT such that inN (k)(X) there appear at most monomialsX 1 α1 ...X n n $$\ln \varrho _k = \sum\limits_{i = 1}^n {\alpha _i \ln } \varrho _i $$ . This generalizes a result of Shl.Sternberg.
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    URL: http://dx.doi.org/10.1007/BF01541637
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    Finite procedures for sofic systems (1977)
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    Monatshefte für Mathematik 83 (1977), S. 265-278 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A sofic system is a symbolic flow defined by a finite semigroup. We exhibit finite procedures, involving only the defining semigroup, for answering cetain questions about a sofic system and for constructing certain subshifts of finite type associated with a sofic system.
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    URL: http://dx.doi.org/10.1007/BF01387905
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    Über Distanzfunktionen mit Werten in angeordneten Halbgruppen (1977)
    Springer
    Monatshefte für Mathematik 83 (1977), S. 223-251 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a general investigation of the relations between structural properties of a totally ordered abelian semigroupS and the properties of various “topological” structures, such as topologies, bitopologies and (semi-)uniformities on a spaceX induced byS-valued distance functionsd∶X×X→S satisfyingd(x,y)=0 iffx=y and the triangular inequalityd(x,z)≤d(x,y)+d(y,z), for allx,y,z∈X. Since a linearly ordered abelian semigroupS need not be a topological semigroup with respect to its order topology we have to consider two cases: the case where addition inS is continuous at 0∈S, and the case where it is not. For both cases, we state several metrization theorems, examples and applications. In this connection, we are also concerned with some special basis-properties of topological spaces. Closely connected is the program stated byAlexandroff-Bourbaki (amongst others) to investigate to what extent countability inherent in matrization theory can be replaced by order-theoretic properties.—Distinguishing between symmetric and not necessarily symmetric distancesd S we obtain a theory containing the theory ofω µ -metrics andω µ- quasimetrics.—As far as it concerns not necessarily symmetric distancesd onX, it seems adequate to study the bitopological structure (τ e ,τ r ) induced onX byd and the “inverse” distanced −1 respectively. This is done in § 4 where, in this respect, we also generalize a well-known theorem ofSion andZelmer.
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    Paare und Tripel halbkonzentrischer Kreise der euklidischen Ebene (1977)
    Springer
    Monatshefte für Mathematik 83 (1977), S. 309-313 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In [1], semi-concentric circles have been defined and considered, also their special nets and pencils.—The isochordal curve and the isogonal curve of two circles, the isochordal lines and the isogonal points of three circles show many differences according to whether the two or three circles are semiconcentric or not. Results for semi-concentric circles are developed here, while results for other circles have been given in [2] and [5].
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    Lattice point theory of irractional ellipsoids with an arbitrary center (1977)
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    Monatshefte für Mathematik 83 (1977), S. 279-307 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We shall consider positive definite quadratic formsQ inr≥2 variables of the “almost diagonal shape” $$Q{\mathbf{ }}(u) = \sum\limits_{1 \leqslant j{\mathbf{ }} \leqslant \sigma } {\alpha j{\mathbf{ }}(u^{(j)} )} $$ where ϖ≥2, and for 1≤j≤ϖ,Q j is a positive definite quadratic form with integral coefficients inr j variables, α j is a positive real number,r j≥1 andr 1+...+r α=r Letb 1,...,b r be a system of real numbers with 0≤b j〈1. For x〉0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)≤x, letV(x) be the volume of this ellipsoid and letP(x)=A(x)-V(x). Our purpose is to find the exact order ofP(x); i. e., the numberf for which for each ɛ〉0P(x)=O(xf+ε) andP(x)=Ω(x f−ε).
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    Répartition de Suites et Équations Fonctionnelles Associées (1977)
    Springer
    Monatshefte für Mathematik 83 (1977), S. 315-329 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetX be a compact metric space, λ a probability measure onX, T a measure preserving transformation ofX into itself. Furthermore, letZ be a compact abelian group, andS the skew-product defined onX×Z by the formula:S(x,z)=(T x,z+ϕ(x)) where ϕ is some map fromX toZ. It is known that under the hypothesis of unique ergodicity for the triple (X, λ,T) and some suitable conditions of continuity onT and ϕ, the triple (X×Z,m,S) wherem is the product measure of λ by the Haar measure onZ is uniquely ergodic if and only if an associated functional equation has no non trivial solution.—We introduce here a generalization of unique ergodicity (which appears quite naturally when studying well distributed sequences) and prove about it the same kind of results.
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    Gewichtete Volumsmittelwerte von Simplices, welche zufällig in einem konvexen Körper des ℝ gewählt werden (1977)
    Springer
    Monatshefte für Mathematik 83 (1977), S. 331-337 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For any set ofn+1 pointsx 1, ...,x n+1F we denote byv(C(x 1,...,x n+1∈ℝ)) then-dimensional oriented volume of the convex hullC(x 1,...,x n+1) of these points. With a fixed symmetric functionf: ℝ→〉ℝ〉 strictly monotone increasing on the nonnegative real line, we consider the real functional RODEL on the set of all convex bodiesK of ℝn with absolute volume |v(K)|=1 and assert, that it takes its minimal value on the ellipsoids with absolute volume 1.
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    Eingegangene Bücher (1977)
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    Monatshefte für Mathematik 83 (1977), S. 348-348 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01387911
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    Buchbesprechungen (1977)
    Springer
    Monatshefte für Mathematik 83 (1977), S. 339-347 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01387910
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    Eine Bemerkung zu dem größten quadratfreien Teiler einer natürlichen Zahl (1977)
    Springer
    Monatshefte für Mathematik 84 (1977), S. 13-19 
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    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let γ(n) be the largest squarefree divisor of a natural numbern. In this note we give an asymptotic formula for $$\sum\limits_{n \leqslant x} {\gamma (n)/n^k } $$ wherek is an arbitrary integer.
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    URL: http://dx.doi.org/10.1007/BF01637021
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