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Fred Brauer: The Man and His Mathematics

  • Christopher M. Kribs-Zaleta
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 126)

Abstract

When I first met Fred Brauer, he was warning a class of first-semester calculus students about the dangers of applying techniques without thinking. As an example, he wrote on the chalkboard the expression\( \frac{{\sin x}}{n} \). He then proceeded to cancel the n’s from numerator and denominator:
$$ \frac{{si\not{n}x}}{{\not{n}}} $$
, leaving six.

Keywords

Asymptotic Stability Population Model Successive Approximation Epidemic Model Local Uniqueness 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Fred Brauer and Shlomo Sternberg (1958). Local uniqueness, existence in the large, and the convergence of successive approximations, Amer. J. Math. 80: 421–430.MathSciNetzbMATHCrossRefGoogle Scholar
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Papers by Fred Brauer

  1. 1.
    Fred Brauer (1958). Singular self-adjoint boundary value problems for the differential equation Lx = λMx, Trans. Amer. Math. Soc. 88: 331–345.MathSciNetzbMATHGoogle Scholar
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  4. 4.
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  6. 6.
    Fred Brauer (1960). Spectral theory for linear systems of differential equations, Pacific J. Math. 10: 17–34.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1961). Global behavior of solutions of ordinary differential equations, J. Math. Anal. Appl. 2: 145–158.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1962). Asymptotic equivalence and asymptotic behavior of linear systems, Mich. Math. J. 9: 33–43.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1963). Lyapunov functions and comparison theorems, in International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics (J. P. LaSalle & S. Lefschetz, eds.). New York: Academic Press. pp. 435–441.CrossRefGoogle Scholar
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    Fred Brauer (1963). Bounds for solutions of ordinary differential equations, Proc. Amer. Math. Soc. 14: 36–43.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1963). On the asymptotic behavior of Bessel functions, Amer. Math. Monthly 70: 954–957.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1965). Some refinements of Lyapunov’s second method, Can. J. Math. 17: 811–819.MathSciNetzbMATHCrossRefGoogle Scholar
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  16. 16.
    Fred Brauer (1966). The use of comparison theorems for ordinary differential equations, in Stability Problems of Solutions of Diff. Equations (Proc. NATO Advanced Study Inst., Padua, 1965). Gubbio: Edizione “Oderisi”. pp. 29–50.Google Scholar
  17. 17.
    Fred Brauer (1966). The asymptotic behavior of perturbed nonlinear systems, in Stability Problems of Solution of Diff. Equations (Proc. NATO Advanced Study Inst., Padua, 1965). Gubbio: Edizione “Oderisi”. pp. 51–56.Google Scholar
  18. 18.
    Fred Brauer (1966). The solution of non-homogeneous systems of differential equations by undetermined coefficients, Can. Math. Bull. 9: 81–87.Google Scholar
  19. 19.
    Fred Brauer (1967). Green’s functions for singular ordinary differential operators, Can. J. Math. 19: 571–582.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1967). Perturbations of nonlinear systems of differential equations, II., J. Math. Anal. Appl. 17: 418–434.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fred Brauer (1968). A class of nonlinear eigenvalue problems, in U.S.-Japan Seminar on Differential & Functional Equations. New York: W. A. Benjamin, Inc. pp. 429–433.Google Scholar
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    Fred Brauer (1968). Nonlinear perturbations of Sturm-Liouville boundary-value problems, J. Math. Anal. Appl. 22: 591–598.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fred Brauer and James S. W. Wong (1969). On asymptotic behavior of perturbed linear systems, J. Diff. Equations 6: 142–153.zbMATHCrossRefGoogle Scholar
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    Fred Brauer and James S. W. Wong (1969). On the asymptotic relationships between solutions of two systems of ordinary differential equations, J. Diff. Equations 6: 527–543.zbMATHCrossRefGoogle Scholar
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  26. 26.
    Fred Brauer (1972). Perturbations of nonlinear systems of differential equations, IV, J. Math. Anal. Appl. 37(1): 214–222.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Fred Brauer (1972). A nonlinear variation of constants formula for Volterra equations, Math. Systems Theory 6: 226–235.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer and David A. Sánchez (1976). Cosecha de poblaciones en competencia [Harvesting of competing populations], in Mathematical Notes and Symposia, Vol. 2: Ecuaciones Differenciales (Proc. Third Mexico-U.S. Symposium) (C. Imaz, ed.). Mexico City: Fondo de Cultura Económica. pp. 171–176.Google Scholar
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    Fred Brauer (1976, October). Perturbations of the nonlinear renewal equation, Advances in Math. 22(1): 32–51.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1977). Stability of some population models with delay, Math.Biosciences 33: 345–358.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1979). Decay rates for solutions of a class of differential-difference equations, SIAM J. Math. Anal. 10: 783–788.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer and A. C. Soudack (1979). Stability regions and transitions phenomena for harvested predator prey systems, J. Math. Biol. 7: 319–337.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1979). Boundedness of solutions of predator-prey systems, Theor. Pop. Biol. 15: 268–273.MathSciNetCrossRefGoogle Scholar
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    Fred Brauer and A. C. Soudack (1981). Constant-rate effort harvesting and stocking in a class of predator-prey systems, in Differential Equations and Applications in Ecology, Epidemics and Population Problems (Stavros N. Busenberg and Kenneth L. Cooke, eds.). New York: Academic Press. pp. 131–144.CrossRefGoogle Scholar
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    Fred Brauer (1983). Constant-rate harvesting of age-structured populations, SIAM J. Math Anal. 14 (1983), 947–961.MathSciNetCrossRefGoogle Scholar
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    Fred Brauer (1984). The effect of harvesting on population systems, in Trends in Theory & Practice of Nonlinear Differential Equations (Proc., Arlington, TX, 1982, V. Lakshmikantham, ed.). New York: Marcel Dekker. pp. 81–89.Google Scholar
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    Fred Brauer (1984). Constant-yield harvesting of population systems, in Mathematical Ecology, Proc. Trieste 1982 (S. A. Levin and T. G. Hallam, eds.), Lec. Notes in Biomathematics 54. Berlin: Springer-Verlag. pp. 238–246.CrossRefGoogle Scholar
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    Fred Brauer, Stephen Ellner and M. D. Krom (1988). Modeling and management for seawater fishponds, in Proc. 1986 Trieste Research Conference on Mathematical Ecology (T. G. Hallam, S. A. Levin, and L. J. Gross, eds.). Teaneck, NJ: World Scientific Press. pp. 215–235.Google Scholar
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    Fred Brauer (1988). Coexistence and survival of invading species, in Proc. 1986 Trieste Research Conference on Mathematical Ecology (T. G. Hallam, S. A. Levin, and L. J. Gross, eds.). Teaneck, NJ: World Scientific Press. pp. 599–610.Google Scholar
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    Fred Brauer, David Rollins and A. C. Soudack (1988). Harvesting in population models with delayed recruitment and age-dependent mortality, Nat. Res. Modeling 3(1): 45–62.MathSciNetzbMATHGoogle Scholar
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    Fred Brauer (1988). Some topics in population biology involving delay equations, 38 pp.; translated into Chinese by Ma Zhien. Xian, China: Department of Mathematics, Xian Jiaotong University. 58 pp.Google Scholar
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    Fred Brauer (1989). Epidemic models in populations of varying size, in Mathematical Approaches to Problems in Resource Management and Epidemiology (C. Castillo-Chávez, S. Levin, and C. Shoemaker, eds.), Lecture Notes in Biomathematics 81. Berlin: Springer-Verlag. pp. 109–123.CrossRefGoogle Scholar
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    Fred Brauer (1989). Multi-species interactions and coexistence, in Proc. Int. Conf. on Theory and Applications of Differential Equations (A.R. Aftabizadeh, ed.). Athens, OH: Ohio Univ. Press. pp. 91–96.Google Scholar
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    Fred Brauer (1990). Models for the spread of universally fatal diseases, J. Math. Biology 28: 451–462.MathSciNetzbMATHCrossRefGoogle Scholar
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    Fred Brauer (1990). Some infectious disease models with population dynamics and general contact rates, J. Diff. Integral Equations 5: 827–836.MathSciNetGoogle Scholar
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    Fred Brauer (1991). Stability of equilibria in some infectious disease models, in Differential Equations: Stability and Control (S. Elaydi, ed.). New York: Marcel Dekker. pp. 53–62.Google Scholar
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    S. P. Blythe, Fred Brauer and Carlos Castillo-Chávez (1995). Demographic recruitment in sexually transmitted disease models, in Computational Medicine, Public Health and Biotechnology: Building a Man in the Machine (M. Witten, ed.). Singapore: World Scientific Press. pp. 1438–1457.Google Scholar
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    Fred Brauer (1995). Models for diseases with exposed period, Rocky Mountain J. Math. 25: 57–66.MathSciNetzbMATHCrossRefGoogle Scholar
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    S. P. Blythe, Fred Brauer, Carlos Castillo-Chávez, and Jorge X. Velasco-Hernández (1995). Models for sexually transmitted diseases with recruitment, in Mathematicl Population Dynamics: Analyses of Heterogeneity (O. Arino, D. Axelrod, M. Kimmel, and M. Langlais, eds.). Vol 1. Winnipeg: Wuerz Publishing Co. pp. 197–207.Google Scholar
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    Fred Brauer (1995). Models for diseases with vertical transmission and nonlinear population dynamics, Math. Biosci. 128: 13–24.zbMATHCrossRefGoogle Scholar
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    Fred Brauer (1996). Variable infectivity in communicable disease models, in Proc. First World Congress of Nonlinear Analysts, Tampa, Florida, 1992 (V. Lakshmikantham, ed.). Berlin: de Gruyter. Vol 4, pp. 3201–3210.Google Scholar
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    Fred Brauer, Carlos Castillo-Chávez and Jorge X. Velasco-Hernández (1996). Recruitment effects in heterosexually transmitted diseases, Int. J. App. Science & Computation 3: 78–90.Google Scholar
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    Fred Brauer and Carlos Castillo-Chávez (1995). Basic models in epidemiology, in Ecological Time Series (T.M. Powell and J.H. Steele, eds.). Chapman & Hall. pp. 410–447.CrossRefGoogle Scholar
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    Fred Brauer, Jorge X. Velasco-Hernández and Carlos Castillo-Chávez (1996). Effects of treatment and prevalence-dependent recruitment on the dynamics of a fatal disease, IMA J. Math. Applied to Medicine and Biology 13: 175–192.zbMATHCrossRefGoogle Scholar
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    Fred Brauer (1996). A characteristic equation arising in models for diseases with vertical transmission and without immunity, in Differential Equations and Applications to Biology and to Industry (M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme, eds.). Singapore: World Scientific Press. pp. 41–48.Google Scholar
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    Fred Brauer (1996). A model for populations with variable maturation period, Dynamics of Continuous, Discrete, and Impulsive Systems 2: 41–50.MathSciNetzbMATHGoogle Scholar
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    Fred Brauer, Carlos Castillo-Chávez and Jorge X. Velasco-Hernández (1997). Recruitment with a core group and its effect on the spread of a sexually transmitted disease, in Advances in Mathematical Population Dynamics: Molecules, Cells, and Man (O. Arino, D. Axelrod, and M. Kimmel, eds.). Singapore: World Scientific Press. pp. 477–486.Google Scholar
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    Fred Brauer (1997). Continuous and discrete delayed, recruitment population models, Dynamics of Continuous, Discrete and Impulsive Systems 3: 245–252.MathSciNetzbMATHGoogle Scholar
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    Fred Brauer (1999). General recruitment models for sexually transmitted diseases, in Differential Equations with Applications to Biology (S. Ruan, G. S. K. Wolkowicz, and J. Wu, eds.), Fields Institute Communications No. 21. Providence, RI: Amer. Math. Soc. pp. 45–50.Google Scholar
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    Fred Brauer (1999). Continuous and discrete population models with age-dependent mortality, Dynamics of Continuous, Discrete, and Impulsive Systems 5: 107–113.MathSciNetzbMATHGoogle Scholar
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    Fred Brauer (in press). Time lags in disease models with recruitment, Mathematical and Computer Modelling, to appear.Google Scholar
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    Fred Brauer (in press). Infectious disease models with chronological age structure, this volume.Google Scholar
  92. 92.
    Fred Brauer (in press). Basic ideas of mathematical epidemiology, this volume.Google Scholar
  93. 93.
    Fred Brauer (in press). Extensions of the basic models, this volume.Google Scholar
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    Fred Brauer and Pauline van den Driessche (in press). Models for the transmission of disease with immigration of infectives, to appear.Google Scholar
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    Fred Brauer (in press). What goes up must come down, eventually, American Mathematical Monthly, to appear.Google Scholar
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    Fred Brauer (in press). A model for an SI disease in an age-structured population, to appear.Google Scholar

Books by Fred Brauer

  1. 1.
    Fred Brauer and John A. Nohel (1967). Ordinary Differential Equations: A first course. New York: W. A. Benjamin, Inc. 1st ed. xvi + 457 pp. 2nd ed., 1973, ix + 470 pp.zbMATHGoogle Scholar
  2. 2.
    Fred Brauer and John A. Nohel (1968). Elementary Differential Equations: Principles, problems, solutions. New York: W. A. Benjamin, Inc. xi + 222 pp.Google Scholar
  3. 3.
    Fred Brauer and John A. Nohel (1968). Problems and Solutions in Ordinary Differential Equations. New York: W. A. Benjamin, Inc. x + 267 pp.zbMATHGoogle Scholar
  4. 4.
    Fred Brauer and John A. Nohel (1969). Qualitative Theory of Ordinary Differential Equations. New York: W. A. Benjamin, Inc. xi + 314 pp. Reprinted, Dover, 1989.zbMATHGoogle Scholar
  5. 5.
    Fred Brauer, John A. Nohel and Hans Schneider (1970). Linear Mathematics. New York: W. A. Benjamin, Inc. xii + 347 pp.zbMATHGoogle Scholar
  6. 6.
    Fred Brauer (1976). Some Stability and Perturbation Problems for Differential and Integral Equations, Monografías de Matemática no. 25. Rio de Janeiro: Instituo de Matemática Pura e Aplicada, iii + 163 pp.Google Scholar
  7. 7.
    Fred Brauer and John A. Nohel (1985). An Introduction to Differential Equations with Applications. New York: Harper & Row. xii + 620 pages.Google Scholar
  8. 8.
    Fred Brauer and Carlos Castillo-Chávez (2001). Mathematical Models in Population Biology and Epidemiology. (Texts in Applied Mathematics 40), Springer-Verlag. New York, (c) 2001, 416 pages, ISBN 0–387–98902–1.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Christopher M. Kribs-Zaleta
    • 1
  1. 1.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

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