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  • Articles  (10,679)
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  • 1985-1989
  • 1980-1984  (8,646)
  • 1960-1964  (2,033)
  • 1940-1944
  • 1981  (8,646)
  • 1961  (2,033)
  • Mathematics  (10,679)
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  • Books  (32)
  • Articles  (10,679)
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  • 1985-1989
  • 1980-1984  (8,646)
  • 1960-1964  (2,033)
  • 1940-1944
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  • 1
    Electronic Resource
    Electronic Resource
    Pie Eyed (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
    Details
    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00432.x
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  • 2
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    Electronic Resource
    Where Should We Start (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00431.x
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  • 3
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    Editorial (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00430.x
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  • 4
    Electronic Resource
    Electronic Resource
    An Analysis of the First 100 Years of Wimbledon Tennis Finals (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
    Details
    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Notes: The Wimbledon tennis tournament began in 1977 when the first men's singles final was played. In 1879 men's doubles were introduced and 1884 saw the advent of women's singles. Using elementary probability theory we can compare that the players and expected number of sets each match lasted under the hypothesis that the players were of equal ability. Consideration is also made of a “back-to-the-wall” effect which favours the player(s) trailing in the match.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00433.x
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  • 5
    Electronic Resource
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    Deriving Regression Lines without Calculus (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00434.x
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  • 6
    Electronic Resource
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    How to get a Higher Grade: Beware of Averages (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00435.x
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  • 7
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    Rapid Calculation of Standard Deviation of Small Samples (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00436.x
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  • 8
    Electronic Resource
    Electronic Resource
    An Alternative Approach to the Mean (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00437.x
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  • 9
    Electronic Resource
    Electronic Resource
    A Sample is a Sample is a Sample (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Notes: The exercise described has been used for students on the BEC Higher Diploma Course. They average 19 years of age and rarely arrive arrive with Mathematics beyond O-level. Teachers on BEC National and A-level Statistics courses should also find the exercise of interest.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00438.x
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  • 10
    Electronic Resource
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    A Variation on Coin Tossing Experiments (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00439.x
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  • 11
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    Book Reviews (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Notes: Books Reviewed in this Article:Elementary Statistics. By DONALD R. BURLESON.The Readable Maths and Statistics Book. By B. EDWARDS.The Penguin Book of Mathematical and Statistical Tables. By R. D. NELSONReason by Numbers. By PETER G. MOORE.Do you Panic about Maths. By LAURIE BUXTON
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00440.x
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  • 12
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    Letters (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00441.x
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  • 13
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    Equipment Review (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00442.x
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  • 14
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    Editorial (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00419.x
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  • 15
    Electronic Resource
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    Probability Experiments in the Secopndary School (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00420.x
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  • 16
    Electronic Resource
    Electronic Resource
    Take Your Pickle (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00421.x
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  • 17
    Electronic Resource
    Electronic Resource
    Why Kendall Tau? (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
    Details
    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00422.x
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  • 18
    Electronic Resource
    Electronic Resource
    Calculating Probabilities Using Recurrence Relations (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00423.x
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  • 19
    Electronic Resource
    Electronic Resource
    Reasonable Averages that Wrong Answers (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00425.x
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  • 20
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    Teaching Hypothesis Testing as a Six Step Process (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00424.x
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  • 21
    Electronic Resource
    Electronic Resource
    Have You Ever Found a Sampling Distribution? (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00426.x
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  • 22
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    Book Reviews (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Notes: Books Reviewed in this Article:Statistics in Your World. By the SCHOOLS COUNCIL PROJECT ON STATISTICAL EDUCATION.Statistics: A First Course. By JOHN E. FREUND.Exercises and Worked Examples in Statistics. By G. E. SKIPWORTHStatistical Concepts, with Applications to Business and Economics. By RICHARD W. MADSEN and MELVIN L. MOESCHBERGERCheck up Tests in Mathematics. MELVYN NOLAN
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00427.x
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  • 23
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    News and Notes (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00428.x
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  • 24
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    Erratum (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00429.x
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  • 25
    Electronic Resource
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    Editorial (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00404.x
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  • 26
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    Making Students Aware of Bias (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00405.x
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  • 27
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    Statistics Across the Curriculum (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00406.x
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  • 28
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    The Birth and Development of a Yearbook on the Teaching of Statistics and Probability (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00408.x
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  • 29
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    Birthdays and Breakdowns Revisited (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00409.x
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  • 30
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    Judgement by Sampling (2) (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00407.x
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  • 31
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    A Useful Display of a Normal Population (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00410.x
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  • 32
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    Sixteen Left Feet (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00411.x
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  • 33
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    A Tribute to Jack Selkirk (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00412.x
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  • 34
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    Book Reviews (1981)
    Oxford, UK : Blackwell Publishing Ltd
    Teaching statistics 3 (1981), S. 0 
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
    Notes: Books Reviewed in this Article:Statistics and Experimental Design Geoffrey ClarkeBasic Statistics, a Modern Approach. By M. HAMBURGStatistics for Geographers. By MARTIN MOWFORTH
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    URL: http://dx.doi.org/10.1111/j.1467-9639.1981.tb00413.x
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  • 35
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    Letters (1981)
    Oxford, UK : Blackwell Publishing Ltd
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
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  • 36
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    The Birthday Problem (1981)
    Oxford, UK : Blackwell Publishing Ltd
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
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  • 37
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    Letters (1981)
    Oxford, UK : Blackwell Publishing Ltd
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
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  • 38
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    News and Notes (1981)
    Oxford, UK : Blackwell Publishing Ltd
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
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  • 39
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    The Birthday Problem (1981)
    Oxford, UK : Blackwell Publishing Ltd
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    ISSN: 1467-9639
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Mathematics
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  • 40
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    An hypothesis of freese and the DNA-protein coding problem (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 305-318 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Freese’s Hypothesis states that a single specific alteration in the sequence of nucleotides of an information-bearing DNA molecule results in a specific mutational effect. Within the framework of the DNA-protein coding problem developed elsewhere, and assuming the quasi-ergodicity of the general coding process, it is shown that Freese’s Hypothesis allows us to derive expressions for the length of the smallest mutable DNA molecule and to obtain a bound for the maximal number of allelic molecules of fixed length. To illustrate these ideas, calculations are carried out on appropriate data from bacternophage and man, and the results are shown to differ by a factor of 10 (modulo the rather crude approximations used). It is further shown that, if ρ(N) and ϱ(N) are respectively the number of information-bearing words of lengthN in a given code and the number of words of lengthN, then the number lim ρ(N)/ϱ(N) depends sensitively on the parameter ∈ which specifiesN→∞ the given code. The implications of this result for the spontaneous aggregation of a sufficient number of information-bearing words to characterize an organism are discussed.
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    URL: http://dx.doi.org/10.1007/BF02476743
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  • 41
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    Post-and predoctoral fellowships in mathematical biology (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 319-319 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476744
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  • 42
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    The perfect learner (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 321-335 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract As a “base line” of memorization performance, the behavior of a “perfect learner” is considered. He is characterized by a perfect memory and by the ability to choose the best search procedure in problems where the correct response from a given repertoire is to be found to each of several stimuli under the condition of “right” and “wroing” promptings by the experimenter. Expected learning curves are derived for the case of disjoint response repertoires associated with the stimuli under cyclic and random presentation of the stimuli and for the case of a single response repertoire (a one-to-one matching problem) under cyclic presentation.
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    URL: http://dx.doi.org/10.1007/BF02476490
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  • 43
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    A method for the computation of aortic distensibility in the living human patient and its use for the determination of the aortic effects of aging, drugs, and exercise (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 337-353 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Detailed equations are given for the computation of aortic distensibility in the intact living human patient from measurements of systolic and diastolic arterial pressures, heart rate and cardiac output. From these equations, the aortic characteristics of a large series of normal men of different ages were computed. Comparing these results with measurements on excised aortas, a more pronounced trend toward increasing aortic stiffness with increasing age is evident in living than in dead aortas. Nor-epinephrine and exercise apparently cause the living aortas to constrict but to become more distensible. The same change occurs after 30 minutes of high spinal anesthesia. The ganglionic blocking agents hexamethonium, pentamethonium, and tetraethylammonium usually cause the living aorta to become stiffer, presumably due to dilatation. The aortas of patients with pulmonary disease usually react in different fashion to exercise or tetraethylammonium. The increased aortic distensibility that occurs with the hypertension induced by nor-epinephrine or exercise acts as a compensatory mechanism by decreasing systolic pressure.
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    Dimensional analysis in mathematical biology I. General discussion (1961)
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    Bulletin of mathematical biology 23 (1961), S. 355-376 
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    Notes: Abstract Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.
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    A minimal principle in biomechanics (1961)
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    Bulletin of mathematical biology 23 (1961), S. 377-391 
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    Notes: Abstract Expenditure of energy under several simultaneous forms (mechanical, chemical, etc.) is associated with all muscular activity. The energy is directly related to what is commonly called exertion or effort. This paper defines “muscular effort” quantitatively in terms of some of the elements of the dynamics of the human (and animal) body. It postulates that in all likelihood the individual will, consciously or otherwise, determine his motion (or his posture, if at rest) in such a manner as to reduce his total muscular effort to a minimum consistent with imposed conditions, or “constraints”. The principle, formulated in mathematical terms, is sufficient to ascribe to the moments at all body joints—a matter generally of free choice on the part of the individual—their most likely magnitudes. It therefore renders the equations of human (and animal) motion determinate within this context. The paper also describes briefly an iteration method for the solution of these equations, once they have been made determinate. A simple illustrative application of the principle is included.
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    On the role of chemical systems in the microphysical aspects of primary genetic mechanisms (1961)
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    Bulletin of mathematical biology 23 (1961), S. 393-403 
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    Notes: Abstract It is pointed out that two fundamentally different views of primary genetic processes occur in the literature which are frequently confused. The first is a true communication-theoretic view, which regards the genetic apparatus as containing a real information-source and a transducer which converts that information to useful form. The second view is generally expressed as a template scheme based on the Watson-Crick model; it is shown that in this model there is actually no such thing as genetic information in a communication-theoretic sense. Both views are then discussed on the basis of microphysical principles developed in previous work of the author (Bull. Math. Biophysics,22, 227–255, 1960) in an attempt to find which approach is in closer accord with the biological facts. It is shown that, if the communication-theoretic view is correct, then the information-bearing object must act as a “catalyst,” but it is pointed out that the type of catalysis involved must be of a fundamentally different nature than that occurring in familiar enzyme-catalyzed reactions. On the basis of general considerations of irreversible changes in microphysical measuring systems, it is shown that any type of template must suffer a gradual and irreversible denaturation, which seems to make it unlikely that a template could play a primary role in fundamental genetic processes.
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    On imitative behavior. II. Time course of change from one behavior to another (1961)
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    Bulletin of mathematical biology 23 (1961), S. 405-411 
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    Notes: Abstract The theory developed in previous papers and based on distribution curves of definite form is generalized to any form of unimodel distributions. The time course of the change from one behavior to another is discussed and a general theorem about the time course is established.
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    Abstract mathematical molecular biology (1961)
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    Bulletin of mathematical biology 23 (1961), S. 417-417 
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    Multibreath tracer species dynamics in the lung (1981)
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    Bulletin of mathematical biology 43 (1981), S. 1-19 
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    Notes: Abstract By studying the behavior of various tracer species in the lungs, one can assess many important characteristics which distinguish normal and abnormal function. Quantitative evaluation of function depends on the use of an appropriate model in conjunction with experimental data. A multi-compartment model is derived from mass balances to describe dynamic as well as (breath-averaged) steady-state transport processes between the environment and pulmonary capillary blood. The breathing cycle is divided into three time periods (inspiration, expiration, and pause) so that the model equations are discrete in time. No other model of tracer species transport in the lungs deals simultaneously with species dynamics, variable breathing pattern, distribution inhomogeneities, and non-equilibrium between alveolar gas and capillary blood. Models currently in the literature are shown to be special cases of the model presented here.
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    Global stability of population models (1981)
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    Bulletin of mathematical biology 43 (1981), S. 47-58 
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    Notes: Abstract Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.
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    The logistic equation with a diffusionally coupled delay (1981)
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    Bulletin of mathematical biology 43 (1981), S. 125-140 
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    Notes: Abstract The asymptotic behaviour of a logistic equation with diffusion on a bounded region and a diffusionally coupled delay is investigated. An equivelent parabolic system is derived for certain types of delays. Using a Layapunov functional, sufficient conditions for the global asymptotic stability of the constant steady state are obtained. When the global stability is lost, using Hopf's bifurcation theory, existence of travelling waves is shown for ring-like and periodic one dimensional habitats.
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    Uncertainty in physiology and physics (1981)
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    Bulletin of mathematical biology 43 (1981), S. 141-149 
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    Notes: Abstract It was hypothesized in an earlier work that sensory perception can occur only when the perceiving system is uncertain about the nature of the event being perceived. In the absence of any uncertainty, perception will not take place. The response of the sensory afferent neuron (impulse transmission rate) was calculated using Shannon's measure of uncertainty or entropy. It will now be shown that when the event being perceived is the position and momentum of a particle, Shannon's measure of uncertainty leads to the Heisenberg Uncertainty relationship.
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    Consensusn-trees (1981)
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    Bulletin of mathematical biology 43 (1981), S. 239-244 
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    Notes: Abstract It is not unusual for several classifications to be given for the same collection of objects. We present a method, called majority rule, which can be used to define a consensus of these classifications. We also discuss some mathematical properties of this consensus tree.
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    Morphogen pattern formation and development in growth (1981)
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    Bulletin of mathematical biology 43 (1981), S. 259-270 
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    Notes: Abstract The dependence of the spatial concentration profiles of morphogens on a characteristic dimension is obtained by continuation techniques for Gierer and Meinhardt's activator-inhibitor model of morphogenesis. The study of the behaviour of the system during growth, where the linear and exponential increase of the characteristic dimension is considered, revealed that more complex patterns of morphogen spatial concentrations appear regularly in a reproducible way.
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    Simulation experiments with a model for cellular growth (1981)
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    Bulletin of mathematical biology 43 (1981), S. 271-278 
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    Notes: Abstract Computer models have been used by various authors to simulate both the growth of normal cellular tissue and the development of cancerous cells within normal tissue. As these models were the result of considerable idealization, the purpose of the present paper is to propose a model in which the degree of simplification is relaxed: the features of simultaneous growth, and cell growth whose rate depends on the free absorbing periphery of the cell are introduced. Simulation experiments have been conducted using the model, and the results are presented.
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    Complementary extremum principles for a nonlinear model of heat conduction in the human head (1981)
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    Bulletin of mathematical biology 43 (1981), S. 341-346 
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    Notes: Abstract The theory of complementary variational principles is used to obtain maximum and minimum principles for a nonlinear model of heat conduction in the human head. Accurate variational solutions are obtained in illustrative calculations. The effect of nonlinearity is seen to be significant from a comparison with the linearized model.
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    On Zeeman's equations for the nerve impulse (1981)
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    Bulletin of mathematical biology 43 (1981), S. 279-325 
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    Notes: Abstract A model for the nerve impulse due to Zeeman (1972) and based on catastrophe theory is compared with alternative models and criticisms of Zeeman's model by Sussmann and Zahler (1977, 1978) are assessed. The criticisms of Zeeman's motivation for his model are found to carry some weight. Sussmann and Zahler (1977, 1978) list numerous features of Zeeman's model which, they state, are not in agreement with experiment. These statements as they stand are largely erroneous, and the model still remains to be tested by a critical series of experiments. However, a detailed analysis reveals defects in Zeeman's model, not among those claimed by Sussmann and Zahler, showing that the explicit equations of the model cannot be correct. The possibility of a modified approach along similar lines and its ultimate adoption remains open.
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    Asymptotic analysis of the Michaelis-Menten reaction equations (1981)
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    Bulletin of mathematical biology 43 (1981), S. 375-388 
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    Notes: Abstract The irreversible Michaelis-Menten reaction is studied by the use of the method of multiple scales. Three stages of the reaction are identified, one of which is studied in detail. The results are compared with those of two earlier analyses.
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    Pulse evolution on coupled nerve fibres (1981)
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    Bulletin of mathematical biology 43 (1981), S. 389-400 
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    Notes: Abstract A numerical study of the coupled nerve fibre problem is given which verifies and extends the perturbation theory of Luzader. Pulses on adjacent fibres can couple together with two possible stable pulse separations.
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    Field-induced forces at dielectric interfaces as a possible mechanism of RF hearing effects (1981)
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    Bulletin of mathematical biology 43 (1981), S. 401-413 
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    Notes: Abstract A possible mechanism for effects of microwave radiation on the auditory system is the generation of field-induced forces at interfaces that divide materials of dissimilar electrical properties. A general expression for these “Maxwell stresses” is derived and then used to calculate the approximate magnitude of field-induced force within the organ of Corti during microwave exposure. Comparison of the results with data on the force needed to excite cochlear hair cells indicates auditory responses could be evoked by this mechanism at power densities near the threshold of rf hearing sensations.
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    The construction of homogeneous nets (1981)
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    Bulletin of mathematical biology 43 (1981), S. 415-426 
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    Notes: Abstract A definition of homogeneity for neural networks is given which permits their construction as group quotients. The significance of this for neural dynamics is discussed.
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    Contractile analysis and kinetic energy of the myocardium (1981)
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    Bulletin of mathematical biology 43 (1981), S. 447-461 
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    Notes: Abstract The left ventricle is represented as a cylinder contracting both radially and longitudinally. A simple method is indicated to derive an expression for the rate of change of the kinetic energy of this three-dimensional model, which quantity can be used as an index for the study of the contractile behaviour of the myocardium. An application to the study of muscle mechanics is also indicated.
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    Limit cycles in periodically perturbed population systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 463-485 
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    Notes: Abstract A perturbation method is proposed to calculate approximately the limit cycle type nonequilibrium steady-state resulting from periodic perturbation of coefficients of stable population systems; the two species Lotka-Volterra competition system is explicity studied and the results are formulated for general multi-species population systems. Avoidance of competitive or other types of exclusion of species in a periodic environment is indicated.
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    On a unified approach to pattern generation (1981)
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    Bulletin of mathematical biology 43 (1981), S. 513-516 
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    Mathematical studies of the interaction of respiratory gases with whole blood II. CO2 absorption (1961)
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    Bulletin of mathematical biology 23 (1961), S. 1-14 
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    Notes: Abstract Some progress has been made on the problem of the interaction of respiratory gases with whole blood. A working mathematical model for the O2−CO2 interaction phenomena has been developed from mathematical studies of the data. The Edsall-Wyman (1958) model for CO2 absorption is improved upon in this paper by consolidating it with the O2 absorption model developed in paper I of this set (Bernard, S. R.,Bull. Math. Biophysics,22, 391–415, 1960). This improved model assumed the effect of O2 on CO2 absorption is mediated through the electrical charge possessed by the hemoglobin molecule,i.e., O2 molecules bound to hemoglobin displace protons from the hemoglobin thereby increasing the negative charge on the hemoglobin and at the same time increasing the acidity of the solution. The model is tested against the data.
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    On the mutual influence of individuals in a society (1961)
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    Bulletin of mathematical biology 23 (1961), S. 15-18 
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    Notes: Abstract On the basis of previously proposed mathematical models of social behavior, the present note investigates the possibility of the control of behavior remaining permanently in the hands of one class, if this class possesses sufficient means for influencing mass behavior. The conclusion is reached that, with the assumptions made, if the behavior imposed by the controlling class leads to sufficiently strong dissatisfaction, the control will pass to another class, no matter how strong the controlling power of the first.
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    Contribution to the mathematical biophysics of automobile driving (1961)
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    Bulletin of mathematical biology 23 (1961), S. 19-29 
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    Notes: Abstract Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.
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    Glucose metabolism—A “mixed-cycle” system (1961)
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    Bulletin of mathematical biology 23 (1961), S. 99-103 
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    Notes: Abstract Emphasis upon the importance of homeostatic feedback has drawn attention away from the complexity of biological processes. A study of glucose metabolism indicates the importance of open-cycle as well as closed-cycle mechanisms. Besides the glucose-dependent mechanism of insulin secretion, many open-cycle processes involving the liver, adrenal glands and kidneys, play important roles in the variation of blood glucose. In addition, glucose utilization by the tissues is essentially open-cycle in nature.
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    Southern Regional Graduate Summer Session in Statistics at Virginia Polytechnic Institute (1961)
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    Bulletin of mathematical biology 23 (1961), S. 105-106 
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    On the unreliability of judgments of certain social situations by minority groups (1961)
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    Bulletin of mathematical biology 23 (1961), S. 413-416 
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    Notes: Abstract In certain situations like the aftermath of a revolution when discontent rises amongst certain groups of the population, it is frequently observed that the discontented groups are firmly convinced that their point of view is shared by the majority of the population. Yet future events prove that this is far from being the case. This effect is partly attributable to “wishful thinking,” partly to a purely social mechanism. The wishful-thinking effect may be considered as a case of psycho-physical discrimination in which a bias is introduced proportional to the degree of satisfaction anticipated from a given situation. H. D. Landahl's well-known equations can be applied to this case. The social factor is based on the circumstance that an individual associates by preference with such other individuals as have similar opinions. This results in an actual error of estimation of the relative minority or majority because of different frequencies of contact with individuals of the two opposing groups. Both factors may be combined into one equation.
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    Announcements (1961)
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    Bulletin of mathematical biology 23 (1961), S. 421-422 
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    Limits on the computing power of biological systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 59-67 
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    Notes: Abstract The theory of computational complexity and certain explicitly-stated hypotheses imply limitations on the information processing power of biological systems. Parallelism, special purpose organization, and analog mechanisms may provide speedup critical for life processes, but have little power in the face of exponential growth. We show that “polynomially simulatable” biological systems cannot exhibit dynamic behavior which produces the solution of an intractable problem. The argument implies that parallelism does not allow biological systems to defeat the exponential explosion, but rather is important because it allows polynomial time algorithms to be used more efficiently.
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    A topological study of the base sequences of MS2 and ϕX174 genoma (1981)
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    Bulletin of mathematical biology 43 (1981), S. 81-88 
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    Notes: Abstract A correlation matrix analysis is applied to the base sequence of MS2 and ϕX174 in comparison with sets of simulated sequences with different degrees of constaint Significant differences between a codified sequence, and a statistical one in terms of the “correlation matrix” for sets of different length cannot be found. This result is analysed in terms of nucleotide sequences with different levels of informational content.
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    Volume of the conducting airways calculated from morphometric parameters (1981)
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    Bulletin of mathematical biology 43 (1981), S. 101-109 
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    Notes: Abstract A method of calculating the volume of a tree distal to a cut at the origin of a branch, using branching, diameter and length ratios, has been developed. The method was applied to bronchial tree casts from human, dog, sheep, hamster, and rat lungs. It was found that the exponenta in the equation weight=k×diameter a is approximately equal to 3.0 in sheep lung casts, as found by Hooper (1977), but it is greater than 3.0 in casts from the other four species.
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    Variability in a single compartment system: A note on S. R. Bernard's model (1981)
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    Bulletin of mathematical biology 43 (1981), S. 111-116 
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    Notes: Abstract In this note we examine a continuous time version of a compartmental model introduced in a discrete time setting by S. R. Bernard. The model allows for more than one particle to leave the system at any time. This introduces additional randomness into the system, over the pure death system and this is reflected in the variance function.
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    The diffusion transit time; A simple derivation (1981)
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    Bulletin of mathematical biology 43 (1981), S. 89-99 
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    Notes: Abstract The mean first passage time for free diffusion can be derived directly by solving a simple analogue steady state problem. In this problem the diffusion starting region is considered as a time independent source of diffusing particles and the diffusion target assumes the behaviour of a perfectly absorbing sink. It is shown here that the transit time between the source and the sink, which in this particular problem is equal to the ratio between the holdup of the system and the total flux, is identical to the Brownian movement concept of the mean first passage time for free diffusion. This established identity considerably facilitates the derivation and investigation of the timing of diffusion in complicated structures such as those commonly found in living organisms.
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    Nonlinear oscillations in biology (1981)
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    Bulletin of mathematical biology 43 (1981), S. 121-123 
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    A comment on Lin's paper on nonlinear diffusion in biological systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 117-120 
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    On the stochastic theory of compartments: The leaving process of the two-compartment systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 201-211 
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    Notes: Abstract In this paper three stochastic models are developed for a class of two-compartment systems to analyse the randomness of the leaving process of the particles in the system. Results in closed form for the distribution of the leaving process of the particles in the system are given both for general and exponential sojourn time distributions and also in association with forward recurrence time distributions with and without Poisson input.
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    Models on prebiotic polymer competition: A deterministic approach (1981)
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    Bulletin of mathematical biology 43 (1981), S. 213-232 
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    Notes: Abstract Two simple models are proposed and analysed, in which it is shown that the formation of a new polymer, resulting from an “error” in the template action mechanism of production of an old polymer, may compromise the stability of the initial system under specific conditions, in the context of prebiotic evolution. Autocatalysis is shown to be a “selective advantage”, enabling the “mutant” to dominate in concentration and even replace the initial polymer. The addition of a third molecule playing the role of a catalyst causes hysteresis effects.
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    Extinction probabilities in microbial predation: A birth and death approach (1981)
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    Bulletin of mathematical biology 43 (1981), S. 165-181 
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    Notes: Abstract The problem of extinction of the prey population in a microbial predator-prey interaction in a chemostat has been examined. Usual deterministic lumped parameter models were used for the dynamics of the chemostat for large numbers of the two populations; the generalized birth and death stochastic process was employed for the description of the random variations at small prey numbers. Extinction probabilities of the prey population were calculated for different holding times and chemostat volumes, and their dependence upon the growth parameters of the two populations was studied. It was found that extinction was possible when the Monod model was used for the specific growth rate of the predators as a function of the prey number density. On the other hand, the decrease of the feeding activity of the predators at low prey densities predicted by the multiple saturation model acts as a regulatory factor that prevents extinction of the prey. In view of the fact that extinction of the prey has never been observed in the laboratory, the latter model seems more appropriate to describe the dynamics of microbial predation.
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    Prediction of the effect of the choroidal circulation on the retinal temperature during illumination (1981)
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    Bulletin of mathematical biology 43 (1981), S. 233-238 
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    Notes: Abstract During exposures of the eye to light, the choroidal circulation may have a regulatory influence on the retinal temperature. This is investigated using a mathematical model and a finite-difference technique. It is predicted that the choroidal blood flow a small effect on retinal temperature, which may be important.
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    Spatially localized interactive neural populations—I. A mathematical model (1981)
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    Bulletin of mathematical biology 43 (1981), S. 427-446 
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    Notes: Abstract A probabilistic model of a spatially localized, mutually exitatory (inhibitory) population of neurons is formulated to help explain average evoked potential and post-stimulus time histogram measurements. The model is based on the stochastic activity of single neurons within interactive masses of neurons which exhibit co-operative behavior. Macrostate variables corresponding to the above measurements are related through the model to features of neural operation at the individual and ensemble level. Steady-state solution are obtained and their physiological implications are discussed.
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    A one-compartment model with stochastic parameter (1981)
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    Bulletin of mathematical biology 43 (1981), S. 503-512 
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    Notes: Abstract We consider a one-compartment system with stochastic transfer rate characterized either by Gaussian or by two-level jump process and study the time evolution of the (statistical) moments of the (random) amount of the substance present in the system. The effect of the coloured as well as of the white noise is investigated and it is found that the presence of stochasticity in the transfer rate parameter increases the relaxation time of the system. Finally, we obtain the conditions for the stability of the system in the moment sense.
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    A dose-response model for estimating lifetime tumor risks when cell killing occurs (1981)
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    Bulletin of mathematical biology 43 (1981), S. 487-501 
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    Notes: Abstract A model is described in which damage to a single intracellular locus can lead to a tumorigenic transformation. Assuming a large number of independent intracellular loci to be at risk and assuming that damage to a locus sufficient to cause a tumorigenic transformation occurs with probability greater than zero for all doses greater than zero, leads to the use of the Weibull distribution to characterize the probability of a nonspontaneous tumorigenic cellular transformation occurring after exposure to a given dose of carcinogen. The excess lifetime tumor incidence (i.e., the proportion of tumor bearers) above the spontaneous incidence is used as an estimate of the non-spontaneous incidence and is characterized by a tumor incidence function that represents the probability of occurrence of one or more non-spontaneous tumorigenic cellular transformations amongN(D) independent surviving cells per individual, after exposure to a doseD of carcinogen. The tumor incidence function is fitted to published data for the excess tumor incidence after exposure of animals or humans to ionizing radiation and after exposure of animals to chemical carcinogens.
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    A stochastic model for then-compartment irreversible system (1981)
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    Bulletin of mathematical biology 43 (1981), S. 549-561 
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    Notes: Abstract This paper deals with a stochasticn-compartment irreversible system with a non-homogeneous Poisson input and arbitrary residence time for each of the compartments. Results relating to the number of particles present in each of the compartments as well as the total number of particles present in the system at any time are derived. Further, explicit expressions for the auto covariance function for each compartment and the cross-covariance function between any two compartments with a given time lag are obtained. As a particular case, then-compartment irreversible system is analyzed with homogeneous Poisson input and exponential residence time distribution for each of the compartments. The possible applications of the model are discussed.
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    Pulsatile blood flow in a rigid pulmonary alveolar sheet with porous walls (1981)
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    Bulletin of mathematical biology 43 (1981), S. 563-577 
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    Notes: Abstract This paper deals with the pulsatile blood flow in the lung alveolar sheets by idealizing each of them as a channel covered by porous media. As the blood flow in the lung is of low Reynolds number, a creeping flow is assumed in the channel. The analytical and numerical results for the velocity and pressure distribution in the porous medium are presented. The effect of an imposed slip condition is also studied. Comparisons with the corresponding results for the steady-state case are made at the end.
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    Dynamic structure theory: A structural approach to social and biological systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 579-591 
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    Notes: Abstract The relationships that define the structure of a given ecosystem, social system, or even a physiological function can only exist if certain parameters are confined to a certain range of values. As the values change and exceed this given range the relationships are forced to change, and so produce a new pattern of relationships. The concept of a dynamic structure captures this potential for structural change in relation to a set of parameters. The precise definition of structure and allowable transformation constitutes the definition of a category. The total range of parameters associated with all the relevant structures provides a parameter space which is assumed to be a manifold. Maps with extra structure from the manifold to the category define dynamic structures. The domain of differential dynamic systems is the manifold, and a flow or movement across the manifold is associated with a series of structural transformations in the category. In some cases a structure outruns its parameter range, to be faced with an obstruction—an absence of possible transformations. Ways of studying such “obstructions” are considered along with the related problem of extending a dynamic structure beyond a previously given set of parameters. The cost or resistance of transformations is also studied. The concepts of dynamic structures are illustrated by the structural change of food webs and they are used in a necessarily qualitative fashion to study dominance structures of social orders and finally to speculate on the qualitative nature of evolutionary change of functional aspects of organisms.
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    Peak drug levels in linear pharmacokinetic systems—II. Conditions for a paradoxical injection rate effect with rectangular input (1981)
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    Bulletin of mathematical biology 43 (1981), S. 705-715 
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    Notes: Abstract The preceding paper (Thorn, 1981) has shown that in a linear pharmacokinetic system with a multimodal impulse response the peak drug level may sometimes be smaller with slower rates of injection. This paper presents two theorems on this paradoxical injection rate effect where the injection is a constant infusion of finite duration. The first theorem establishes a graphical method for determining whether a given impulse response will give a paradoxical injection rate effect; and the second establishes that the maximum paradoxical increase in peak drug level is by a factor of two. It is further shown that in order to approach this maximum paradoxical increase the impulse response must contain two isolated, sharp, narrow pulses of approximately equal area. Some examples of bimodal arterial dye-dilution curves from the literature are discussed as impulse responses; and there is also a discussion of the behavior of drug level maxima and minima at different injection rates.
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    Peak drug levels in linear pharmacokinetic systems—I. Effect of rate of injection (1981)
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    Bulletin of mathematical biology 43 (1981), S. 693-703 
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    Notes: Abstract This paper presents three theorems on the peak drug levels that result from injection into a linear pharmacokinetic system. As a preliminary, the “rate of injection” is defined in terms of time expansion or time contraction of the injection function (input). The first theorem then states that the peak drug level will not be greater when the rate of injection is slow than when it is fast, if the impulse response is unimodal. The second theorem sets limits for the time of the maximum drug level, in relation to the time of the maximum of the (unimodal) impulse response and the duration of the input. The third theorem defines conditions which assure a definitely lower peak drug level if the rate of injection is slower. A graphical method is suggested for determining the times and magnitudes of the peak drug levels that result from constant infusions of a fixed dose at different rates. An example is provided to show that if the impulse response is multimodal then the peak drug level may sometimes increase with a decrease in the injection rate.
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    The elements of an electrocardiographic lead tensor theory (1961)
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    Bulletin of mathematical biology 23 (1961), S. 31-42 
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    Notes: Abstract This report describes a method by which the concepts of the lead vector and the lead field may be extended to include equivalent cardiac singularities above the first order. The theoretical treatment results in generalization of the well-known lead vector principle into a lead tensor apparatus. The specific electrocardiographic contribution of annth order multipole is found to be the product of annth rank contravariant (“heart”) tensor by a covariant (“lead”) tensor of the same rank, and with identical indices. Methods are also described whereby lead tensor components may be cast into a form which relates them directly to the more usual, spherical harmonic notation of multipolar current generators. Some attention is devoted to the possibility of applying lead tensors to biomedical problems, particularly to the prospect of quantitatively evaluating the significant multipolar components of the human heart.
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    The formation of cell assemblies (1961)
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    Bulletin of mathematical biology 23 (1961), S. 43-53 
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    Notes: Abstract A simple model is presented for the formation of functional groups in a random neural net. They show the following characteristics: 1. They can maintain autonomous activity which might serve as temporary memory traces. 2. Early in the process of formation they become resistant to contraction. 3. Later they become resistant to expansion. 4. Nearby groups inhibit one another. 5. Two groups may contain some cells in common.
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    Contributions to quantum biology. I. Mobile electronic characteristics of riboflavin radicals (1961)
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    Bulletin of mathematical biology 23 (1961), S. 55-68 
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    Notes: Abstract Quantum biology is the quantum mechanical study of electrons in molecules of biological interest. This requires the solution of problems involving many electrons. Approximation methods are therefore necessary and are discussed. The present study, concerned with the mobile electrons in riboflavin (FMN) and its radicals (FMN−, FMNH and FMNH2 +), is based on the approximation method, developed by B. Pullman and A. Pullman. The solution of the eigenvalue problems so obtained gives the energy levels of the mobile electron systems involved. The corresponding eigenvectors yield the mobile electronic charges of the atoms of riboflavin radicals which have contributed mobile electrons. Important differences of the net charge distributions of these radicals are emphasized. The longest wave length of light absorption is calculated from the obtained energy levels and agrees, within the accuracy of the method, with corresponding experimental results. From the appropriate calculated results, electronic assignments are obtained for the experimental transitions involved.
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    Quantitative analysis of metabolic processes III. A model-system for enzyme synthesis by sequential induction and mathematical formulation of the process (1961)
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    Bulletin of mathematical biology 23 (1961), S. 69-89 
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    Notes: Abstract A postulated scheme for sequential enzyme synthesis coupled with linear chain reaction is proposed. Mathematical calculations are carried out for the transient-state formation of enzyme coupled with enzyme-substrate reactions. Examples are presented to demonstrate the relations between various operational parameters during enzyme synthesis as a function of time. Significance of the phenomenon of sequential synthesis is discussed in relation to abnormal growth and cellular injury.
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    A note on mathematical models for the interaction of neural elements (1961)
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    Bulletin of mathematical biology 23 (1961), S. 91-97 
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    Notes: Abstract By introducing a plausible model for the initiation of axonal impulses the output is obtained as a function of the input incoming impulses. If the temporal aspects of the excitatory process resulting from the afferent impulses are sufficiently rapid one obtains the discontinuous or microscopic model of McCulloch-Pitts. If these are sufficiently slow a continuous model, such as Rashevsky’s one or two factor theory, is a natural model. But the linear relation between the strength of excitation of one axon and excitatory factor of the next will not in general hold. However, under conditions which are not too restrictive the linear relation with threshold can be considered as satisfactory approximation over a fairly wide range of values.
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    Symposium on Mathematical Theories of Biological Phenomena (1961)
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    Bulletin of mathematical biology 23 (1961), S. 107-108 
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    On the roots of a certain type of matrix (1961)
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    Bulletin of mathematical biology 23 (1961), S. 217-221 
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    On relations between sets (1961)
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    Bulletin of mathematical biology 23 (1961), S. 233-235 
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    Notes: Abstract With a view to future applications in relational biology, the notion of relations between sets is introduced and several theorems are demonstrated.
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    Transport of gases through hemoglobin solution (1961)
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    Bulletin of mathematical biology 23 (1961), S. 223-232 
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    Notes: Abstract A mathematical theory is developed to explain the observed enhancement of O2 transport through solutions by hemoglobin. At high partial pressures of O2, ordinary diffusion through the solvent accounts for all transport of O2, but at low partial pressures the transport may be increased manyfold by the presence of hemoglobin. This phenomenon is explained and its possible role in living organisms is discussed. The theory also indicates a new method of determining dissociation curves from diffusion experiments.
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    The equations of population growth (1961)
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    Bulletin of mathematical biology 23 (1961), S. 261-262 
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    Notes: Abstract A solution in quadratures is found for a special class of differential equations suggested by studies in the growth of competing populations.
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