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  • Elsevier  (546,680)
  • Springer  (348,921)
  • Taylor & Francis  (19,057)
  • 2020-2020
  • 2005-2009  (756,568)
  • 1955-1959  (79,674)
  • 1950-1954  (45,498)
  • 1945-1949  (14,343)
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  • 1
    Keywords: Air ; Chlor ; GSM-Postponed Project ; Manual ; Oxide ; Water ; bacteria ; microorganism ; pollution ; production ; soil ; toxicity
    Description / Table of Contents: Environmental Chemistry: Fundamentals, by Jorge Ibanez et al., is an exceptionally useful and well organized book. After reviewing basic chemical concepts, Environmental Chemistry: Fundamentals quickly progresses to more advanced and contemporary applications including ozone depletion, physiochemical and biological treatment of pollutants, and green chemistry. The chemistry of processes of the atmosphere, lithosphere and hydrosphere are covered in detail and the effects of pollutants on each of these chemical processes are extensively considered, as are their effects on the biosphere. The book also has an experimental companion, Environmental Chemistry: Microscale Laboratory Experiments, which includes an array of environmental chemistry experiments that can be readily performed at the microscale level. Ideas for additional open-ended projects are provided for all experiments, and they impart a thorough introduction to environmental experimentation. I strongly recommend Environmental Chemistry: Fundamentals and its experimental accompaniment, Environmental Chemistry: Microscale Laboratory Experiments. Dr. Zvi Szafran Vice President for Academic Affairs and Professor of Chemistry Southern Polytechnic State University Our Earth is a remarkable reaction vessel. It is of paramount importance that students grow in their understanding and awareness of the astounding effects that chemistry and biochemistry have on our environment…and why they are so significant to our present and future hopes as a civilization. Environmental Chemistry: Microscale Laboratory Experiments, intended to complement lessons in the companion textbook Environmental Chemistry: Fundamentals, covers the chemical and biochemical processes that take place in air, water, soil, and living systems. The corresponding experiments range from the characterization of aqueous media to pollutant-treatment schemes. For increased safety, as well as for reduced costs, wastes, and environmental damage, the experiments are presented at the microscale level. Pre- and post-laboratory exercises and open-ended projects accompany each experiment, to develop problem-solving skills and initiative among students.
    Pages: Online-Ressource (XII, 238 pages)
    ISBN: 9780387494937
    Language: English
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  • 2
    Keywords: Atom ; Fulleren ; Fullerene ; Nanocar ; Nanomaterial ; Nanotube ; Transport ; carbon nanotubes ; electricity
    Description / Table of Contents: The 2007 ARW “Using Carbon Nanomaterials in Clean-Energy Hydrogen Systems” (UCNCEHS’2007) was held in September 22–28, 2007 in the remarkable town Sudak (Crimea, Ukraine) known for its heroic and unusual fate. In the tradition of the earlier conferences, UCNCEHS’2007 meeting served as an multidisciplinary forum for the presentation and discussion of the most recent research on transition to hydrogen-based energy systems, technologies for hydrogen production, storage, utilization, carbon nanomaterials processing and chemical behavior, energy and environmental problems. The aim of UCNCEHS’2007 was to provide the wide overview of the latest scientific results on basic research and technological applications of hydrogen interactions with carbon materials. The active representatives from research/academic organizations and governmental agencies could meet, discuss and present the most recent advances in hydrogen concepts, processes and systems, to evaluate current progress and to exchange academic information, to identify research needs and future development in this important area. This ARW should help further the progress of hydrogen-based science and promote the role of hydrogen and carbon nanomaterials in the energy field.
    ISBN: 9781402088988
    Language: English
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  • 3
    Keywords: Assessment ; Malaria ; Public Health ; Scale ; Weather ; climate change ; public health policy ; temperature
    Description / Table of Contents: Awareness that many key aspects of public health are strongly influenced by climate is growing dramatically, driven by new research and experience and fears of climate change and the research needed to underpin policy developments in area is growing rapidly . This awareness has yet to translate into a practical use of climate knowledge by health policy-makers. Evidence based policy and practice is the mantra of the health sector. If climate scientists are to contribute effectively to health policy at local and global scales then careful empirical studies must be undertaken – focused on the needs of the public health policy and decision-makers. Results presented at the Wengen conference make clear that the science and art of integrating climate knowledge into the control of climate sensitive diseases on a year to year time frame as well as careful assessments of the potential impacts of climate change on health outcomes over longer time frames is advancing rapidly on many fronts. This includes advances in the empirical understanding of mechanisms, methodologies for modeling future impacts, new partnership developments between the health and climate community along with access to relevant data resources, and education and training. In a rapidly evolving field this book provides a snapshot of these emerging themes.
    Pages: Online-Ressource (X, 232 pages)
    ISBN: 9781402068775
    Language: English
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  • 4
    Keywords: Biomass conversion ; Biotechnology ; Chemical Engineering ; Chemistry industry ; Industrial Chemistry ; Kent ; Riegel ; biochemical engineering
    Description / Table of Contents: Substantially revising and updating the classic reference in the field, this handbook offers a valuable overview and myriad details on current chemical processes, products, and practices. No other source offers as much data on the chemistry, engineering, economics, and infrastructure of the industry. The Handbook serves a spectrum of individuals, from those who are directly involved in the chemical industry to others in related industries and activities. It provides not only the underlying science and technology for important industry sectors, but also broad coverage of critical supporting topics. Industrial processes and products can be much enhanced through observing the tenets and applying the methodologies found in chapters on Green Engineering and Chemistry (specifically, biomass conversion), Practical Catalysis, and Environmental Measurements; as well as expanded treatment of Safety, chemistry plant security, and Emergency Preparedness. Understanding these factors allows them to be part of the total process and helps achieve optimum results in, for example, process development, review, and modification. Important topics in the energy field, namely nuclear, coal, natural gas, and petroleum, are covered in individual chapters. Other new chapters include energy conversion, energy storage, emerging nanoscience and technology. Updated sections include more material on biomass conversion, as well as three chapters covering biotechnology topics, namely, Industrial Biotechnology, Industrial Enzymes, and Industrial Production of Therapeutic Proteins.
    Pages: Online-Ressource (XIV, 1562 pages)
    ISBN: 9780387278438
    Language: English
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  • 5
    Pages: Online-Ressource (XII, 543 Seiten)
    Edition: 3. Aufl.
    ISBN: 9783540345251
    Language: German
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  • 6
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    London ; New York : Taylor & Francis
    Keywords: Fluid mechanics.
    Pages: x, 696 p.
    Edition: 8th ed
    ISBN: 0-203-01232-1
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  • 7
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    London ; New York : Taylor & Francis
    Keywords: Canals, Design and construction. ; Canals, Maintenance and repair.
    Pages: xx, 389 p.
    ISBN: 0-203-01242-9
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  • 8
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    New York : Springer
    Keywords: Computer engineering. ; Computer security. ; Electronic data processing, Distributed processing.
    Pages: xx, 239 p.
    ISBN: 0-387-23917-0
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  • 9
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    Abingdon [England] ; New York : Taylor & Francis
    Keywords: Courtyard houses.
    Pages: xix, 248 p., [8] p. of plates
    ISBN: 0-203-64672-X
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  • 10
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    Amsterdam ; Boston : Elsevier
    Sustainable world series  
    Keywords: Coal-fired furnaces. ; Coal-fired power plants. ; Coal.
    ISBN: 1-417-55267-0
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  • 11
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    Publication Date: 2020-02-12
    Keywords: 550 - Earth sciences
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  • 12
    Publication Date: 2020-02-12
    Keywords: 550 - Earth sciences
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  • 13
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    In:  International Association of Geodesy Symposia
    Publication Date: 2020-02-12
    Description: The lAG International Symposium on Gravity, Geoid, and Space Missions 2004 (GGSM2004) was lield in the beautiful city of Porto, Portugal, from 30 August to 3 September 2004. This symposium encompassed the themes of Commission 2 (Gravity Field) of the newly structured lAG, as well as interdisciplinary topics related to geoid and gravity modeling, with special attention given to the current and planned gravi- dedicated satellite missions. The symposium also followed in the tradition of mid-term meetings that were held between the quadrennial joint meetings of the International Geoid and Gravity Commissions. The previous mid-term meetings were the International Symposia on Gravity, Geoid, and Marine Geodesy (Tokyo, 1996), and Gravity, Geoid, and Geodynamics (Banff, 2000). GGSM2004 aimed to bring together scientists from different areas in the geosciences, working with gravity and geoid related problems, both from the theoretical and practical points of view. Topics of interest included the integration of heterogeneous data and contributions from satellite and airborne techniques to the study of the spatial and temporal variations of the gravity field. In addition to the special focus on the CHAMP, GRACE, and GOCE satellite missions, attention was also directed toward projects addressing topographic and ice field mapping using SAR, LIDAR, and laser altimetry, as well as missions and studies related to planetary geodesy.
    Language: English
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  • 14
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  • 15
    Publication Date: 2020-02-12
    Keywords: 550 - Earth sciences
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  • 16
    Publication Date: 2020-02-12
    Keywords: 550 - Earth sciences
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  • 17
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  • 18
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  • 19
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    Publication Date: 2020-02-12
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  • 20
    Electronic Resource
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    Amsterdam : Elsevier
    Physics Letters B 294 (1992), S. 466-478 
    ISSN: 0370-2693
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
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  • 21
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    Amsterdam : Elsevier
    Physics Letters B 317 (1993), S. 474-484 
    ISSN: 0370-2693
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
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  • 22
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    Bulletin of mathematical biology 15 (1953), S. 311-338 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The partial differential equation of the random walk problem with persistence of direction and external bias is derived. By persistence of direction or internal bias we mean that the probability a particle will travel in a given direction need not be the same for all directions, but depends solely upon the particle's previous direction of motion. The external bias arises from an anisotropy of the medium or an external force on the particle. The problem is treated by considering that the net displacement of a particle arises from two factors, namely, that neither the probability of the particle traveling in any direction after turning nor the distance the particle travels in a given direction need be the same for all directions. A modified Fokker-Planck equation is first obtained using the assumptions that the particles have a distribution of travel times and speeds and that the average time of travel between turns need not be zero. The fional equation incopporating the assumption of a persistence of direction and an external bias is then derived. Applications to the study of diffusion and to long-chain polymers are then made.
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  • 23
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    Bulletin of mathematical biology 15 (1953), S. 383-383 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
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  • 24
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    Bulletin of mathematical biology 15 (1953), S. 385-385 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
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  • 25
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    Bulletin of mathematical biology 15 (1953), S. 367-381 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.
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  • 26
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    Bulletin of mathematical biology 15 (1953), S. 387-394 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A situation is considered in which a fluid containing a substance flows through a vessel at a constant rate, the substance being permeable to the vessel wall. In the region outside the vessel there is supposed to be rapid mixing in the direction perpendicular to the axis of the vessel but no mixing longitudinally. The solution for the spatial distribution at any time is given for the case of an arbitrary initial distribution along the vessel length in the absence of an input. The solution is also given for the case of a single impulsive input, the concentration being initially zero everywhere.
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  • 27
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    Bulletin of mathematical biology 15 (1953), S. 431-476 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Organisms orient themselves to a stimulus by two general methods. One method is by directed orientation (taxis); the other is by undirected locomotory reaction (kinesis). An equation, and the methods for finding the necessary parameters of this equation, is derived for the distribution of organisms within a container, with the following limitations: (1) the organisms have no accommodation, (2) they are always active, and (3) the stimulus changes slowly with position. Necessary modifications of the equation are then derived, so that the last two limitations may be eliminated. The equation cannot be solved excatly because of its complexity; hence an approximation method must be used. This method is discussed, an approximate solution is found, and a time constant for equilibrium to be established is derived. Applications tovarious experiments in the literature are then made with fairly satisfactory results. A new interpretation of the theory of klino-kinesis with accommodation is found upon application of the equations developed to experimental work. Further limitations and uses of these equations are then discussed.
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  • 28
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    Bulletin of mathematical biology 15 (1953), S. 501-507 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Certain parameters are defined which roughly characterize the internal structure of networks. A given network structure uniquely determines the values of the parameters, but the reverse is not true. The parameters therefore define certain classes of networks. One of the parameters, thedispersion D(S) gives an indication of the “compactness” of the internal structure. Addition theorems and inequalities are derived relating the dispersions of sub-systems to the dispersion of the complete structure.
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  • 29
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    Bulletin of mathematical biology 15 (1953), S. 489-500 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A mathematical theory is developed which permits the determination of certain parameters of an inhomogenous tissue, such as a nerve trunk without its epineurium. The parameters are the permeability coefficients for entrance into an exit of a substance from the nerve fibers, and the diffusion coefficient of the interstitial material. The experimental data required are the dimensions of the cross-section, the average diameter of the fibers, and the ratio of the cross-sectional are of the fibers to the total cross-section, as well as the time course of the decrease of the fraction of the substance left in the nerve trunk, when the trunk is immersed in a bathing solution containing none of it.
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  • 30
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    Bulletin of mathematical biology 15 (1953), S. 509-522 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A model based on enzyme localization is developed which gives rise to an apparent active transport of a metabolite into or out of cells. The model is applied to three simple situations, using Fick's equation and the Rashevsky approximation. It is shown that the apparent efficiency can be made as large as desired if, for constant reaction, the outer cell region is made sufficiently small, or, for autocatalytic reaction, if the metabolite concentration in the outer region is sufficiently small. The physical limitations imposed by this mechanism are developed for all three situations.
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  • 31
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    Bulletin of mathematical biology 15 (1953), S. 523-533 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.
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  • 32
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    Bulletin of mathematical biology 15 (1953), S. 535-546 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The assumption of transitivity treated in part I is modified in various ways to describe an information-diffusion process, in which a certain amount of randomness of contact does occur. In one model a parameter is introduced which is indicative of a tendency to go beyond one's immediate vicinity to spread the information as the vicinity becomes saturated with knowers. In another model the randomness appears in the assumption that new knowers are uniformly distributed among the knowers. Two of the equations thus derived, each with two free parameters are in good agreement with experimental results.
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  • 33
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    Bulletin of mathematical biology 20 (1958), S. 71-93 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A somewhat different approach to the principle of biotopological mapping, discussed in previous publications, is given. The organism is considered as a set of properties, each of which is in its turn a set of numerous subproperties which are logically included in the corresponding properties. Topology is introduced by an appropriate definition of neighborhoods, and four postulates are stated which concern the mapping of the spaces corresponding to higher organisms on those of lower ones. A number of conclusions are drawn from the postulates. Some of them correspond to well-known facts. For example, in man and some higher organisms appropriate emotional stimuli should produce gastrointestinal or cardiovascular disturbances; or some microorganisms should produce substances harmful to other microorganisms (antibiotics). Some other conclusions are still awaiting verification. One of them is, for example, that there must exist unicellular organisms which produce antibodies to appropriate antigens.
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  • 34
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    Bulletin of mathematical biology 20 (1958), S. 25-32 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Zusammenfassung Für die Praxis der Pflanzenernährung ist es wichtig, zu wissen, in welcher Weise die Ertragsbildung von der Konzentration eines mineralischen Nährstoffes in der Umgebung der Pflanze abhängt. Da nur diejenigen Nährstoffmengen das physiologische Geschehen in der Pflanze unmittelbar zu beeinflussen vermögen, die sich in der Pflanze befinden, wird angenommen, dass das Wachstum zum Zeitpunktt, d.h. die Geschwindigkeit der Trockensubstanzzunahme zu diesem Zeitpunkt, eine Funktion der zur Zeitt in der Pflanze enthaltenen Nährstoffmenge ist. Diese Nährstoffmenge wird natürlich im Intervall vor dem Zeitpunktt aufgenommen. Deshalb und auch noch aus anderen Gründen hängt das Wachstum zur Zeitt davon ab, wie die in der Umgebung der Pflanze herrschende Konzentration des betrachteten Nährstoffes in demjenigen Zeitintervall verläuft, das sich von der Aussaat bis zum Zeitpunktt erstreckt. Die angegebene Annahme fürhrt zusammen mit einigen weiteren naheliegenden Annahmen zu einem Ansatz, der Ergebnisse liefert, die in verschiedener Hinsicht gut mit der Erfahrung übereinstimmen. Jedoch gibt es auch noch Widersprüche zwischen Theorie und Erfahrung. Durch weitere Ausgestaltung der Theorie lassen sich diese Widersprüche beseitigen. Es wird angeregt, Versuche durchzuführen, deren Resultate Hinweise für die weitere Ausgestaltung der Theorie liefern.
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    Bulletin of mathematical biology 20 (1958), S. 33-70 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The dynamics of cell multiplication and differentiation in tissues in asteady state and the kinetics of isotope incorporation into the DNA have been theoretically analyzed. Equations have been derived, with the aid of which thegeneration time, thelife span, and the distribution or rate of death of the cells can be obtained if the tissue is in asteady state, i.e., if the number of cells is maintained constant by constant, equal rates of cell division and cell death and if the mean DNA content per cell is also constant. An equation has also been derived which gives thegeneration time in the case of logarithmic multiplication of cells. Two special cases have been analyzed: InCase 1, the isotope is considered as being introduced into the metabolic system at zero time only; inCase 2, the specific activity of the DNA precursor is considered as being maintained constant. The use of the method has been illustrated by an example in which thegeneration time and themean, themedian, and themode life span, as well as the curve of the rate of death of leukocytes in a patient with chronic leukemic granulocytic leukemia, have been obtained from the rate of P32 incorporation into the DNA. The merits and the limitations of the method are discussed.
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    Bulletin of mathematical biology 20 (1958), S. 95-95 
    ISSN: 1522-9602
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    Bulletin of mathematical biology 21 (1959), S. 1-11 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract By means of the Laplace transform, the behavior of a simplified model of the cardiovascular system is mathematically formulated. This formulation allows mathematical expression of the periodicity of the cardiac output and the systemic response. With the cardiac output represented as half of a sine function cycle, the systolic aortic pressure becomes the sum of a sine term and exponential terms, while the sum of the exponential terms alone represents the diastolic pressure. The characteristics of the mathematical expressions for systole and diastole are analyzed, and some relationships of potentially practical value are derived. Variation in the parameters of the system yields mathematical results consistent with the expected physical ones.
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    Bulletin of mathematical biology 21 (1959), S. 19-32 
    ISSN: 1522-9602
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    Notes: Abstract A generalization of Landahl's approximation method (H. D. Landahl,Bull. Math. Biophysics,15, 49–61, 1953) for non-linear diffusion problems is suggested. The method is applied to sorption, desorption, and free diffusion problems involving concentration-dependent diffusion coefficients. With some limitations, the results compare favorably with those obtained by numerical methods.
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    Bulletin of mathematical biology 21 (1959), S. 33-60 
    ISSN: 1522-9602
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    Notes: Abstract Recently a theorem for representing current generators in a volume conductor by the superposition of a central dipole, quadrupole, octopole, etc., has been established by G. C. K. Yeh, J. Martinek, and H. de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958). This theorem makes possible the representation of any discrete or line, surface- or volume-distributed current source by a unique model which can be determined for each given case by surface potential measurements and closed form analysis. In this paper the multipole representations of an eccentric dipole and an eccentric double-layer are obtained in terms of the various parameters of the assumed singularities, and the contributions to surface potentials due to each of the multipoles are compared. Certain numerical results corresponding to those of E. Frank (Amer. Heart J.,46, 364–78, 1953) are carried out and compared. Furthermore, the multipole representation of a partially damaged double-layer is also determined and compared with that of an undamaged one. It is concluded that within the range of parameters corresponding to human subjects the higher-order multipoles can contribute significantly to the surface potentials compared with the dipole.
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    Bulletin of mathematical biology 21 (1959), S. 97-100 
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    Notes: Abstract In line with a recent suggestion by the author (Bull. Math. Biophysics,20, 267–73, September, 1958) that not only does the organism as a whole map on the primordial, but that each organ can also be thus mapped, it is shown that the previously introduced abstract spaces, which represent an organism, contain subspaces which map continuously on the space of the primordial. Several theorems about those subspaces are proven.
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    Bulletin of mathematical biology 21 (1959), S. 71-95 
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    Notes: Abstract The DNA-protein coding problem is given a general algebraic formulation, the consequences of which are then explored by standard mathematical methods. To keep the treatment self-contained, the mathematical techniques to be used are explained in detail. It is demonstrated that there exista priori a countably infinite number of different abstract DNA-protein codes, thereby showing that inductive attempts to construct such a code will most likely be fruitless. A notion of ergodicity is then introduced, which imposes a number of restrictions on the admissible codes, and, in fact, these considerations enable us toderive a small portion of a code which, if our hypothesis of ergodicity is correct, must occur in nature. Finally, we discuss briefly the problem as to whether there can exist more than one DNA-protein code in nature.
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    Notes: Abstract The present-day practices of electrocardiography and vectorardiography are based upon the theory that the surface potential differences can be assumed to be due to a single dipole inside the body. It is shown in this paper that a dipole cannot account for all the surface potentials due to realistic current generators, and hence the determination of the current generator from surface potential measurements based upon such a theory will lead to inconsistent representations of the heart for one and the same subject. To demonstrate this point two eccentric dipoles of different strengths and locations representing two muscle fibers are taken to be the current generator in a homogeneous spherical conductor. The exact surface potentials are then expressed by means of the “interior sphere theorem” of the authors. With these expressions the magnitude, direction, and location of the resultant dipole are determined by the method of D. Gabor and C. V. Nelson (J. App. Physics,25, 413–16, 1954). The surface potentials due to this resultant dipole are again exactly expressed by means of the “interior sphere theorem” and compared with those due to the eccentric dipoles assumed. It can be seen that the differences can be considerable. It is suggested that the multipole model of the authors (Bull. Math. Biophysics,20, 203–16, 1958) be used as a more accurate and the only unique representation of the heart.
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    Bulletin of mathematical biology 21 (1959), S. 101-106 
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    Notes: Abstract In a preceding paper (Bull. Math. Biophysics 20, 71–93, 1958) the principle of biotopological mapping was formulated in terms of a continuous mapping of an abstract space, made from the set of biological properties which characterize the organism, by an appropriate definition of neighborhoods. In this paper it is shown that we may consider directly the mappings of the different sets of properties which characterize different organisms without taking recourse to abstract spaces. All the verificable conclusions made in the preceding paper remain valid. A serious difficulty mentioned previously is, however, avoided and the possibility of more general predictions is established.
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    Bulletin of mathematical biology 21 (1959), S. 107-107 
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    Bulletin of mathematical biology 21 (1959), S. 109-128 
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    Notes: Abstract The general Theory of Categories is applied to the study of the (M, R)-systems previously defined. A set of axioms is provided which characterize “abstract (M, R)-systems”, defined in terms of the Theory of Categories. It is shown that the replication of the repair components of these systems may be accounted for in a natural way within this framework, thereby obviating the need for anad hoc postulation of a replication mechanism. A time-lag structure is introduced into these abstract (M, R)-systems. In order to apply this structure to a discussion of the “morphology” of these systems, it is necessary to make certain assumptions which relate the morphology to the time lags. By so doing, a system of abstract biology is in effect constructed. In particular, a formulation of a general Principle of Optimal Design is proposed for these systems. It is shown under what conditions the repair mechanism of the system will be localized into a spherical region, suggestive of the nuclear arrangements in cells. The possibility of placing an abstract (M, R)-system into optimal form in more than one way is then investigated, and a necessary and sufficient condition for this occurrence is obtained. Some further implications of the above assumptions are then discussed.
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    Bulletin of mathematical biology 21 (1959), S. 141-151 
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    Notes: Abstract The transient stage of the random dispersal of logistic populations is investigated, using a Sturm-Liouville series leading to an infinite system of non-linear integral equations. These equations are then solved via a successive approximation scheme. R. A. Fisher's (steady-state) velocity of advance paradox is discussed. An illustrative example is worked to the second order of approximation.
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    Bulletin of mathematical biology 21 (1959), S. 153-159 
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    Notes: Abstract An approximation method using a sine function is used to solve the second degree growth equation for the case in which an organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.
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    Bulletin of mathematical biology 21 (1959), S. 129-140 
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    Notes: Abstract Diffusion through a flat pore into a large open region is proportional to the linear dimension of the pore and not to its area. This was first explained by Brown and Escombe (1900) for a circular pore and is here generalized, by means of a dimensional argument, to include any type of regular opening. The problem is further generalized to include diffusion through pores of finite thickness, finite distance apart, and into finite regions. Since this problem cannot be solved exactly, an approximation method is introduced. Reasons for the credibility of the approximation are presented. It is then shown, by means of the approximation method, that the diffusive flow through a pore is equal to the total concentration difference divided by the resistance of the system. The resistance, in turn, is the sum of the resistances of all portions of the system, each of which is calculated. The result is compared with results which have been calculated exactly for limiting cases and found to agree very well. The results are then applied to a standard method of computing pore size in membranes, and it is shown that the correction factor is negligible.
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    Bulletin of mathematical biology 21 (1959), S. 161-183 
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    Notes: Abstract V. S. Ivlev [Experimental Ecology of Nutrition of Fishes, 1955, Moscow (in Russia)] has shown that the food uptake by fishes during a fixed interval of time is an exponential function of the concentration of food. Ivlev's equation is derived here, and it is shown that it can hold only for non-stationary conditions, such as prevailed in Ivlev's experiments. For a stationary state, the rate of food uptake should tend asymptotically to a limiting value as the concentration increases, but the variation is not exponential. Different other aspects of the problem are investigated, and definite new experimental procedures suggested. The implications of Ivlev's findings on the effect of non-uniformity of food distribution upon the rate of food consumption are studied from a mathematical point of view. The conclusion is reached that whereas a fish does not, in the process of eating, move directly to an individual food particle which it perceives, it does move more or less directly to large aggregates of particles, if the latter are distributed nonuniformly.
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    Bulletin of mathematical biology 21 (1959), S. 185-193 
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    Notes: Abstract Some relational aspects of the property of self-reproduction of biological systems are studied. If in addition to the requirement of the property of self-reproduction we add also the requirement of adaptability of the organism to changing environment, this imposes certain conditions on the topology of the graphs which represent such systems. A further study of the relational properties of such systems seems to offer the possibility of deriving the principle of biological mapping from the requirement of self-reproduction and adaptability. An examination of the problem of the original formation of such self-reproducing systems in connection with the established fact of impossibility of spontaneous generation leads to the conclusion that an organism must inhibit such processes which, in the absence of organisms, would lead to spontaneous generation.
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    Bulletin of mathematical biology 21 (1959), S. 195-216 
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    Notes: Abstract In the human, the antagonistic, extensor-flexor system of the leg is an example of a common type of neurophysiological feedback system. After a brief introduction to the neuroanatomy and physiology of this feedback system, the paper formulates transfer functions from temporal response data available in the literature. A feedback stability analysis, based on the extension of Nyquist's stability criteria to multiple-loop systems and utilizing flow-graph techniques, demonstrates the stable behavior of the system. Expressions are given relating the sensitivity of the system to variations in muscle response and Golgi tendon organ (tension receptor) response. By considering the events taking place at synapses and end-plates during “isometric tension-small knee angle excursion” conditions as stationary stochastic processes, an external “noise” input to the system is given, whose spectrum is derived from the statistics of a shot-process representation of these events. The paper concludes with some correlations between the analytical results and clinical syndromes.
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    Bulletin of mathematical biology 21 (1959), S. 217-255 
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    Notes: Abstract In this continuation of a previous report it is shown how the Volterra population dynamics, which underlies the statistical theory, can be based on a variational principle; how the dynamics can be generalized as regards both the behavior of total populations and migration phenomena; and how many directly observable data, such as amplitudes and frequencies of oscillation of a population, fit into the statistical theory and can test it. Such a test is carried out in some detail using the fox-catch data of Elton, with a clear indication that the theory is capable of comprehending the major statistical properties of population-time curves. A final section sketches an extension of the theory to cover secular variations of external conditions such as temperature of the environment.
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    Bulletin of mathematical biology 11 (1949), S. 59-82 
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    Notes: Abstract A theory of linkage of autopolyploids is developed under consideration ofm loci andr alleles. The simplifying assumption of chromosome segregation, which may be considered as an approximation to the more adequate theory of chromatid segregation, is made throughout. Random mating and distinct, non-overlapping generations are assumed. Under these assumptions the problem is determined by three basic probability distributions—the distributions of genotypes and of gametes, and the segregation distribution. The segregation distribution is assumed to be the same for males and for females. The aim of the paper is to establish recurrence formulas (which allow to find the distributions of gametes and of genotypes from generation to generation, if the distribution of genotypes for an initial generation is known) and to investigate the limit behavior of these distributions as the number of generations increases indefinitely. In the present paper (hereafter referred to as I) the problem is explained, and the three characteristic distributions are introduced for the general case of a 2s-ploid,m loci, andr alleles. Recurrence relations are established for tetraploids,s=2 andm=2 loci, while the recurrence relations for the general case as well as the limit theorems will be given in the second part of this paper (hereafter referred to as II).
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    Bulletin of mathematical biology 11 (1949), S. 83-95 
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    Notes: Abstract Steady state kinetic relations previously developed forclosed, homogeneous systems are extended toopen systems consisting of geometrically confined regions of arbitrary shape. The generalized system of consecutive reactions is considered to occur within the cell, and the cell plus environment are treated as an open system. The diffusion condition is imposed upon the kinetic solution for various special cases and the method ofgeneral solution when all products, reactants and intermediates, may enter and leave the cell is indicated.
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    Bulletin of mathematical biology 11 (1949), S. 97-103 
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    Notes: Abstract When two polished metal spheres in contact with each other are immersed in a corrosive solution, their surfaces gradually corrode, leaving, however, a non-corroded zone around the point of contact. The size of this zone is a function of the size of the spheres, concentration of the corrosive, and of the time. The phenomenon has been first observed by F. C. Besic on human teeth immersed in hydrochloric acid, and studied on metal balls. In the present paper it is shown that such a phenomenon may be due to gradients of concentration of the corrosive in the neighborhood of the point of contact, caused by the chemical reaction which consumes the corrosive. Approximate expressions for the size of the uncorroded zones as a function of the size of the balls and as a function of time are derived and found to be in fair agreement with F. C. Besic's data.
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    Bulletin of mathematical biology 11 (1949), S. 105-113 
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    Notes: Abstract When an individual grows up in a society, he learns certain behavior patterns which are “accepted” by that society. He may in general have a tendency toward behavior patterns other than those which are “accepted” by the society. This tendency toward such unaccepted behavior may be due to a process of cerebration which results in doubt as to the “correctness” of the accepted behavior. Thus, on the one hand, the individual learns to follow the accepted rules almost automatically; on the other hand, he may tend to consciously break those rules. Using a neural circuit, suggested by H. D. Landahl in his theory of learning, a neurobiophysical interpretation of the above situation is outlined. Mathematical expressions are derived which describe the social behavior of an individual as a function of his age, social status, and some neurobiophysical parameters.
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    Bulletin of mathematical biology 11 (1949), S. 139-144 
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    Notes: Abstract Analysis is made to show that the mitotic index is simply proportional to the ratio of the duration of mitosis (T) to the intermitotic time only under special conditions. In the case of exponential growth of cell population the simple proportionality hold if the product ofT and the growth constant is small. For power law (t n ) growth of cell population the simple proportionality holds only when a steady state of growth has existed for at least ten intermitotic periods. The simple proportionality does not apply in conditions of transient growth.
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    Bulletin of mathematical biology 11 (1949), S. 115-138 
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    Notes: Abstract Food and metabolic waste products, insofar as they act upon the hereditary substrate of cells, are the most important factors governing tissue growth. Equations describing the growth of tissues are derived in consideration of this fact. A quantity is found in these equations which, if slightly changed, results in very great changes in the growth rate of the tissue, where such very great changes are interpretable as neoplastic growth. The relationship between our equations and similar equations which others have proposed is discussed.
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    Bulletin of mathematical biology 11 (1949), S. 145-147 
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    Bulletin of mathematical biology 15 (1953), S. 1-13 
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    Notes: Abstract The impressed field, “Window Field” (WF), due to a half-wave action potential on a muscle fiber, has been calculated on the basis of potential theory. It has been shown that in spite of the small intensity of the field, its integrated action can transfer the energy needed to induce, contraction from the membrane to the interior of the fiber. The energy of polarization has been found to be sufficient to exceed the energy of, thermal agitation on that length of fiber, which can be identified as the length of a sarcomere. The changes of ion concentration, caused by the WF, if calculated on the assumption of the semipermeability of theZ membranes, was found to be equal to the changes necessary to induce contraction of actomyosinin vitro.
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    Bulletin of mathematical biology 15 (1953), S. 15-21 
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    Notes: Abstract Some general properties of the solution of the diffusion equation are deduced for the steady-state, spherically symmetric system. On the basis of these developments some results of N. Rashevsky (Bull. Math. Biophysics,11, 15, 1949) are discussed and the results of a previous investigation (Hearon,Bull. Math. Biophysics,12, 135, 1950b) are extended to more general conditions. In particular these extensions apply to the flow of a soluteagainst its concentration gradient, the nonzero gradient of an inert metabolite, and theaccumulation or exclusion of an inert metabolite in a metabolic system.
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    Bulletin of mathematical biology 15 (1953), S. 23-31 
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    Notes: Abstract The approximation method of N. Rashevsky is discussed and reviewed. It is shown that in addition to theexplicit assumptions and approximations there is involved the assumption that the rate of metabolism is the same at every point in the cell and that theaverage rate of metabolism is different from zero. An expression is given for the error in the approximate method when the rate of metabolism is any function of the concentration. It is also shown that a solution in theform of that obtained by the approximate method is not possible if the generalized laws of diffusion are assumed to apply.
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    Bulletin of mathematical biology 15 (1953), S. 33-42 
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    Notes: Abstract Rate equations for the enzymatic oxidation of succinic acid are derived on the assumption that when a single molecule of substrate combines with an enzyme molecule, it can do so with either one or two sites on the enzyme, and that oxidation occurs only in the second case. In addition it is assumed that the product of the reaction, fumaric acid, combines reversibly with the enzyme. With certain enzyme preparations the data fitted such an equation satisfactorily. In others the rate was that of a first-order reaction, but addition of cytochrome changed it to the former type. It was concluded that the transfer of hydrogen to oxygen was a first-order reaction and dominated the whole rate when enzyme preparations were used which had been washed relatively free of cytochrome. When the limiting factor was succino-dehydrogenase the rates followed the new equation. Criteria for recognizing noncompetitive inhibition are given, and inhibition by di-tertiary butyl peroxide was shown to be of this type.
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    Bulletin of mathematical biology 15 (1953), S. 43-47 
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    Notes: Abstract This paper deals with the following question: Which distributions of radiosensitivity in a population can lead to an exponential survival curve? The problem is solved exactly, with statistical fluctuations in dose fully accounted for. It is shown that only an exponential distribution of sensitivities can give rise to an exponential survival curve.
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    Bulletin of mathematical biology 15 (1953), S. 49-61 
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    Notes: Abstract An approximation method is introduced which enables a number of diffusion-type problems to be solved in an approximate but simple manner. Many cases require only the solution of a simple first-order differential equation. The method is applied to a number of cases in which the exact solutions are available. A comparison shows that the method is quite satisfactory in these cases. The method is applied to diffusion problems with rate of consumption proportional to concentration or to the square of the concentration. In the latter case, the result obtained is essentially the same as that found by H. G. Landau (1950) after elaborate calculations.
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    Bulletin of mathematical biology 15 (1953), S. 83-91 
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    Notes: Abstract It is shown that a slight modification of a model of excitatory phenomena in irritable tissues, which has been treated before, exhibits spontaneous oscillations. The frequency of these oscillations and the time-course of the potential across the model membrane have been determined, together with the dependence of some of their characteristics on some important parameters, particularly (Ca++).
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    Bulletin of mathematical biology 15 (1953), S. 73-81 
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    Notes: Abstract It is demonstrated that, if the variations of viscosity throughout a cell are considered, swelling stresses may produce elongation and division. To do this it is necessary to generalize Betti's theorem to cover systems containing viscosity gradients and such a generalization is presented. On the basis of two special assumptions it is shown that most of the results of the diffusion drag theory of cell division may be duplicated by the present theory.
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    Bulletin of mathematical biology 15 (1953), S. 63-71 
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    Notes: Abstract The theory of imitative behavior, developed previously, is applied to the case of two social groups which are separated spatially. If the information of each group as to the behavior of the other is complete, the case reduces to that of a single group. When any information is lacking at all, the two groups are independent. If we have two mutually exclusive behaviorsA andB, all four combinationsAA, AB, BA, andBB are possible. If the mutual information gradually increases from zero, then for a certain value of it, the group which is more informed about the behavior of the other will change to that behavior if it did not already exhibit it. If for constant information the size of the group increases, then above a certain threshold value, the larger group imposes its behavior on the smaller.
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    Bulletin of mathematical biology 15 (1953), S. 103-104 
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    Bulletin of mathematical biology 15 (1953), S. 107-107 
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    Bulletin of mathematical biology 15 (1953), S. 105-106 
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    Bulletin of mathematical biology 15 (1953), S. 93-101 
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    Notes: Abstract Local regulation of blood flow as determined by capillary diameter and the number of open capillaries in a region is considered. The local changes in capillary diameter and in the number of open capillaries are assumed to be due to concentration changes of a diffusible, nonspecified metabolite. This metabolite is produced in the tissue and carried away by the blood stream. Using these assumptions and applying pertinent data on capillaries, deductions are made concerning: (a) the law of blood flow as a function of temperature and capillary radius for the hyperemia of high temperature, (b) high flow as it depends on metabolism during strenuous exercise of muscle, and (c) a first approximation to the time duration of occlusion hyperemia.
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    Bulletin of mathematical biology 15 (1953), S. 109-109 
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    Bulletin of mathematical biology 15 (1953), S. 143-148 
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    Notes: Abstract The necessary and sufficient condition is given forn integers to be the score structure of a society with a dominance relation. A proof is also given for a theorem showing that there are members who dominate every other member either directly or indirectly through a single intermediate member.
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    Bulletin of mathematical biology 15 (1953), S. 111-119 
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    Notes: Abstract On the basis of a previous general formulation (Bull. Math. Biophysics,15, 21–29, 1953a) a discussion is given of the error in the approximation method of N. Rashevsky. This error, inherent in the method when the metabolic rate is different at each point in the cell, is discussed in detail and numerical values are presented for two particular cases: the rate proportional to the concentration and the rate a prescribed function of the spatial coordinates. It is shown that the formulation for the first case also applies to several other cases, that the error is negligible provided the rate is sufficiently small, and that the error is fairly sensitive to the cell size. If the rate depends upon the coordinatesalone a small rate is not sufficient to insure a negligible error. The relations between the exact method, the standard approximate method, an earlier approximate method (Physics,7 260, 1936), and a more recent refinement (Bull. Math. Biophysics,10, 201, 1948) of the standard method are discussed.
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    Bulletin of mathematical biology 15 (1953), S. 121-141 
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    Notes: Abstract It is shown on the basis of (1) conservation of mass, (2) positive concentrations, and (3) the principle of detail balancing that periodic reactions cannot occur in a closed system described bylinear differential equations. The matrix,A, of the rate equations must be such that |A|=0,a ij〉0 fori≠j,a ii〈0, andVAV −1=B, whereV is diagonal andB is symmetric. These properties ofA imply that the latent roots are real and non-positive and that neither catalysis nor inhibition can be described bylinear equations. It is further shown that periodic reactions cannot occur in anopen system for which the matrix associated with the chemical reactions has the above properties and in which thesimple law of diffusion is obeyed. The relation of these results to Onsager's reciprocal relations and to previous work on periodic and cyclic chemical reactions is discussed. The utility of certain of these results for the treatment of isotope kinetics is indicated.
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    Bulletin of mathematical biology 15 (1953), S. 149-152 
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    Notes: Abstract It is demonstrated that an explanation of the small radius effect or the so-called sigma phenomenon may be obtained by noting that one of the effects of the presence of suspended particles in a flowing fluid is to increase the velocity of flow near the wall over that existing in the absence of particles. This effect may be considered equivalent to relaxing the boundary conditions at the wall. An expression for the viscosity is compared with data and fit is found to be good.
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    Bulletin of mathematical biology 15 (1953), S. 153-159 
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    Notes: Abstract The solution for the spatial distribution of ions in a Donnan equilibrium has been given by J. H. Bartlett and R. A. Kromhout (1952). The present note gives an explicit solution for the case in which the length of the region containing the membrane is large; in biological situations this requires only that the length considered should be greater than a few hundred Ångstrom units. The Donnan equilibrium may be considered to be a special case of a situation in which forces other than electrical act upon the ions; in particular, it represents the case in which only one ion is acted upon and the energy difference on the two sides of the membrane is infinite. An expression is given for the difference in energy of theith in terms of the electrical potential and of the ion concentrations. As an illustration, the results are applied to nerve membrane potentials.
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    Bulletin of mathematical biology 15 (1953), S. 161-165 
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    Notes: Abstract A mechanism is described which accounts for the active transport of Na+ ions through a membrane. It is assumed that at one side of the membrane the ion combines with a carrier ion, the resulting carrier compound then diffuses through the membrane and decomposes at the other side of the membrane. The free diffusion of the ions is also taken into account. The time rate of accumulation of the ion in question at the latter side of the membrane is calculated in terms of the concentrations of the ion at both sides of the membrane.
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    Bulletin of mathematical biology 15 (1953), S. 167-171 
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    Notes: Abstract The recent extension of the approximation method is applied to enable us to arrive at the time course of the concentrations at both sides of a membrane. From the differential equations which govern these, the steady-state solution is obtained in terms of the parameters, which are determined by the thickness of the diffusion layers, the chemical composition and reactions, and the diffusion constant of the membrane.
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    Bulletin of mathematical biology 15 (1953), S. 173-183 
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    Notes: Abstract An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.
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    Bulletin of mathematical biology 15 (1953), S. 235-235 
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    Bulletin of mathematical biology 15 (1953), S. 185-195 
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    Notes: Abstract The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.
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    Notes: Abstract The question is raised concerning the possible causes of abnormally small standard deviations found in counting samples in which particles are distributed at random (e.g., blood cells, fat globules in milk, etc.). The effect of discarding abnormal samples is discounted inasmuch as small standard deviations occur even when all samples are counted. An approximation method is used to calculate the effect of finite particle size, of known repulsive forces between particles and of convection currents. This calculation shows that neither finite size nor the known repulsive forces are sufficient to account for the observed abnormality of standard deviation, but that convection currents can possibly account for it. The possible presence of long-range repulsive forces cannot, however, be excluded.
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    Bulletin of mathematical biology 15 (1953), S. 245-250 
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    Notes: Abstract The effect of finite particle size on the standard deviation in sample counts is computed for the one-dimensional case. To a first order of approximation the correction is found to be identifical with that found by H. de Vries (1953) using a general approximation method.
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    Bulletin of mathematical biology 15 (1953), S. 251-260 
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    Notes: Abstract A critical examination of the “classical” theories of photoreception in view of more recent experimental findings yields the result that these theories do not possess the property to describe all the more significant phenomena of photoreception correctly, and to some extent suffer the lack of more general applicability. The basis for a new and presumably more general theory of photoreception based on dynamical aspects is laid out. Emphasis is put on the time course of afferent and efferent excitation in the photoreception model, consisting of a receptor element, an afferent and an efferent neuron of the one-factor Rashevsky-type, and an effector organ.
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    Bulletin of mathematical biology 15 (1953), S. 197-234 
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    Notes: Abstract A mathematical model for the development of human society, beginning with the earliest stages of urban cultures, is outlined. In the early stages of history, behavior was characterized largely by adherence to a number of beliefs and prejudices of diffeirent kinds, which were accepted on faith and not subject to critical rational analysis. Due to psychobiological variability a very small number of individuals spontaneously appear at all times who challenge the accepted beliefs and prejudices and do not follow the accepted patterns of social behavior. The effect of these individuals upon the rest of the society, especially upon the younger generation, depends on the facilities with which information spreads in society. In earliest societies, when modern methods of mass communication were unknown, the channels of communication were practically identical with the channels of economic transport. The latter in its turn depended on the nature of the roads, and especially on the presence of waterway, which facilitated transportation. The sizes of the earliest cities and the distances between them were largely determined by relative ease of transporation. Expressions are derived for the average size of the earliest cities and for the average distance between them. The calculated average populations of the earliest cities are of the order of 103; the distance of the order of 102 km. Both are in agreement with some archaeological findings. An expression for the time spaon required for the development from the earliest stages of urban cultures to the present time is derived and shown to depend on the specific shoreline of the country, that is, the length of the shorline divided by the area of the country. It is pointed out that western Europe's specific shoreline, including land bordering both seas and rivers, is ten times as large as the shoreline area of other parts of the world. It is shown that this greater specific shoreline may account quantitatively for the faster social and technological development of western Europe in the last few centuries. The calculated total span of time of development from earliest urban cultures to our days is found to be of the order of magnitude of ten thousand years. It is shown that the model accounts for the existence at the present time of primitive cultures. A number of suggestions is made in regard to other possible applications of mathematics to history.
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    Bulletin of mathematical biology 15 (1953), S. 269-276 
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    Notes: Abstract The assumptions latent in the derivation of the integral equation of Branson are rendered explicit and discussed. It is shown that the equation is valid only for systems in which the substance disappears according to a linear rate law.
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    Bulletin of mathematical biology 15 (1953), S. 261-268 
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    Notes: Abstract It is shown that the validity of Branson's integral description of metabolizing systems is subject to severe limitations. The validity is insured only in cases where the reaction is of first order, or quasi of first order. In all other cases Branson's equation has to be modified to insure general applicability. The consequences of a different definition of the metabolizing functionF have also been investigated. With the new definitionF describes the pure effect of metabolization. It is found that in this case the integral equation is only capable of describing first-order reactions. With a slight modification of the integral equation it is possible to describe metabolites “with age”, which do not have reactions of definite order, but which satisfy the superposition principle.
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    Bulletin of mathematical biology 15 (1953), S. 277-292 
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    Notes: Abstract A problem in probability is stated with included the problem of the distribution of bacterial mutants as a special case. This problem is solved exactly but since the resulting expressions are too complicated for practical use, various approximate expressions for the distribution are considered, especially for the bacterial mutation case.
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    Bulletin of mathematical biology 15 (1953), S. 293-300 
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    Notes: Abstract Simple reaction and discrimination reaction, under the influence of imitation, are considered for the situation in which the stimulus or the stimuli vary slowly with time. The result is analogous to hysteresis under certain conditions. The calculations are facilitated by the solution of $$x = \int_{ - \infty }^{a + \beta x} {g\left( \xi \right)d} \xi ,$$ g(ξ) being the normal error function. Values ofx(α, β) are given in a table.
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    Bulletin of mathematical biology 15 (1953), S. 301-309 
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    Notes: Abstract On the basis of simple physical considerations the blood flow in a branching circulatory system is studied. The case of two groups of parallel vessels is treated. The vessels of the same group are supposed to be identical. The resistance of each group is determined by the resistance of each vessel in the group and by the number of vessels in the group. From the dependence of the resistance of each vessel on its radius an expression is obtained for the blood flow through each group of vessels in terms of the numbers and sizes of the vessels in each group. The number of open vessels in an organ and the radius of each of those vessels are assumed to depend on the metabolic rate of that organ. The relations so obtained, together with the expression above, are applied to derive the blood flow through an organ as a function of the metabolic rate of that organ. It is indicated that the relations obtained might describe the shifting of blood from one organ to another if the activity of one of them changes. A way is pointed out to treat neural regulation of this phenomenon.
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    Bulletin of mathematical biology 15 (1953), S. 361-365 
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    Notes: Abstract A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1−σ), 0≤σ〈1, of the organism dividing into two at the end of its lifetime, and probability σ that division will not take place.
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    Bulletin of mathematical biology 15 (1953), S. 339-359 
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    Notes: Abstract In a previous paper, in which a possible mathematical approach to history was outlined, it was shown that urbanization plays an important part in the propagation of new ideas. The rate of such propagation influences the rate of historical developments. The present paper deals in more detail with possible mechanisms of formation of earliest cities. Equations are derived which give the limiting size of such cities and their rate of growth. Of particular importance for the spread of new ideas is the spread of information. The latter largely depends on the fraction of individuals who travel between city and country. Expressions for this quantity are derived. An approach is outlined to the mathematical study of the earliest social classes, which may have been formed as a result of military, religious, or economic stratifications.
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    Bulletin of mathematical biology 15 (1953), S. 395-409 
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    Notes: Abstract The velocity of propagation of a disturbance wave in a liquid flowing in a distensible tube is computed. The mathematical model is more general than those used in previous analyses: the tube wall properties are realistic; the convective part of the axial inertia forces is taken into account; radial inertia forces of both the fluid and tube wall are present; viscous stresses are present. Four parameters influencing the velocity of propagation are obtained and discussed. Curves are plotted illustrating the effects of the parameters. Contrary to the results of previous analyses, viscous effects are shown to be appreciable in blood flow. It is also shown that radial inertia effects can be important in laboratory set-ups.
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    Bulletin of mathematical biology 15 (1953), S. 411-429 
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    Notes: Abstract A general theory of the drying of frozen tissue is developed and applied to the measurement of the drying rate of frozen guinea pig liver. It is shown that for a given temperature of the subliming ice crystals the mininum drying time of a piece of guinea pig liver is greater than the minimum sublimation time of a piece of ice of the same size and shape by a factor of the order of one thousand. This fact has many implications in the design of freeze-dry apparatus which will be developed in a following paper.
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    Bulletin of mathematical biology 15 (1953), S. 477-488 
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    Notes: Abstract The equations governing the time course of the exchange of substances between the blood in the capillaries and the extracellular space are solved for the case of substances which do not penetrate the cells. The equations given relate the time course of the exchange process to the various tissue and circulation parameters such as the specific capillary wall area, the pore area, the inter-capillary distance, the size of the extra-vascular, extra-cellular space, the diffusion coefficient in this space, and the velocity of blood in the capillaries. Some experimental work on capillary exchange is discussed in relation to the theory and estimates are made of the relative importance of the various tissue and circulation parameters in the exchange of substances in different tissues.
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    Notes: Abstract Previous work (Macey, 1952) in the application of the one-factor theory to the heart is extended. The rate of production of the excitatory state is assumed to be linear. Two possible mechanisms are indicated whereby such a situation might arise. Assumptions are made regarding the mode of action of the chemical mediators on the heart, and an equation is derived relating the heart rate to the frequency of nerve impulses traveling along the cardiac nerves. This result compares favorably with the experimental findings of A. Rosenblueth and F. A. Simeone (1934). Other experimental results are interpreted in terms of the theory.
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    Bulletin of mathematical biology 15 (1953), S. 561-563 
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    Bulletin of mathematical biology 20 (1958), S. 1-23 
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    Notes: Abstract To account for some of the more important aspects of drug interaction we shall consider a model which can also account for certain general properties of the action of a single drug. A simple model in which there may be enzymatic detoxification of a drug is studied theoretically. The relation between time for appearance of an effect due to the drug and the size of the dose is found to contain the same parameters as the relation between the effectiveness of paired doses and the interval of time between doses. A similar situation holds when the drug is given at a constant rate. When two drugs are administered together, their effect will depend on the manner of interaction, how much of each drug is given, which is given first, and on the interval of time between each administration. A number of plausible types of interaction is considered theoretically in terms of the model, analytical expressions being given for a number of cases. The interaction may be synergistic or antagonistic. In the former case the potentiation may be more than or less than additive depending on the order of delivery and on the time between injections. Methods for the estimation of the parameters from data are discussed.
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