Publications, Talks, and Lecture Notes in Mathematics

Richard S. Ellis

Professor, Department of Mathematics and Statistics
Adjunct Professor, Department of Judaic and Near Eastern Studies

• 2012
  
  
  1. Conditional Gaussian Fluctuations and Refined Asymptotics of the Spin in the Phase-Coexistence Region by Richard S. Ellis and Jingran Li. 36-page Latex manuscript. Submitted, Journal of Statistical Physics.
  
  
  
  2. Refined Asymptotics of the Finite-Size Magnetization via a New Conditional Limit Theorem for the Spin by Richard S. Ellis and Jingran Li. 78-page Latex manuscript.
     • This unpublished paper contains details of proofs and calculations omitted from the paper listed in the preceding item. It is posted at http://arxiv.org/abs/1205.0970.
  
  
• 2011
  
  
  3. Monte Carlo Methods for Rough Free Energy Landscapes: Population Annealing and Parallel Tempering by Jonathan Machta and Richard S. Ellis. Journal of Statistical Physics, Volume 144, pages 541–553 (2011).
  
  
  
• 2010
  
  
  4. The Theory of Large Deviations and Applications to Statistical Mechanics by Richard S. Ellis. Long-Range Interacting Systems: Les Houches 2008 Session XC, pages 227–277. Edited by T. Dauxois, S. Ruffo, and L. F. Cugliandolo. Oxford University Press (New York), 2010.
     • This article is based on lectures that I gave during Session XC, 4–29 August 2008 at École d'Été de Physique Theorique in Les Houches, France. An updated version of the lecture notes on which this paper is based is referenced in the fourth item under 2008.
  
  
  5. Asymptotic Behavior of the Finite-Size Magnetization as a Function of the Speed of Approach to Criticality by Richard S. Ellis, Jonathan Machta, and Peter Tak-Hun Otto. Annals of Applied Probability, Volume 20, Number 6, pages 2118–2161 (2010).
  
  
  
• 2008
  
  
  6. Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points via Ginzburg-Landau Polynomials by Richard S. Ellis, Jonathan Machta, and Peter Tak-Hun Otto. Journal of Statistical Physics, Volume 133, Number 1, pages 101–129 (2008).
  
  
  
  7. Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points by Richard S. Ellis, Jonathan Machta, and Peter Tak-Hun Otto. 75-page Latex manuscript.
     • This unpublished paper contains details of proofs and calculations omitted from the paper listed in the preceding item. It is posted at http://arxiv.org/abs/0803.0178.
  
  
  8. Global Optimization, the Gaussian Ensemble, and Universal Ensemble Equivalence by Marius Costeniuc, Richard S. Ellis, Hugo Touchette, and Bruce Turkington). Geometry, Probability and Integrable Systems. For Henry McKean's Seventy-Fifth Birthday. Edited by Mark Pinsky, Hugo Rossi, and Pierre van Moerbeke. MSRI Publications, Volume 55, 131–165. Cambridge University Press (New York), 2008.
  
  
  
  9. The Theory of Large Deviations and Applications to Statistical Mechanics by Richard S. Ellis. An updated version of the lecture notes for three lectures given August 5-8, 2008 at École d'Été de Physique Theorique in Les Houches, France during the August 2008 summer school devoted to long-range, interacting systems (123 pages).
     • A talk based on chapters 3–5 of these lecture notes is available online. It is titled “From Large Deviations to Statistical Mechanics: What Is the Most Likely Way for an Unlikely Event To Happen?”
     
     
     
     • At the end of the talk I give a much simpler derivation of the canonical ensemble than that given in chapter 5 of the lecture notes. This simpler derivation is based on standard ideas in information theory, which were pointed out to me by Neri Merhav, Department of Electrical Engineering at the Technion – Israel Institute of Technology in Haifa, Israel.
  
  
• 2007
  
  
  10. Multiple Critical Behavior of Probabilistic Limit Theorems in the Neighborhood of a Tricritical Point by Marius Costeniuc, Richard S. Ellis, and Peter Tak-Hun Otto. Journal of Statistical Physics, Volume 127, Number 3, 495–552 (2007).
      • A talk based on this paper is available online.
  
  
• 2006
  
  
  11. Entropy, Large Deviations, and Statistical Mechanics [Grundlehren der mathematischen Wissenschaften, Volume 271, 364 pages (1985)], reprinted by Springer-Verlag in their Classics in Mathematics series (2006).
  
  
  
  12. Generalized Canonical Ensembles and Ensemble Equivalence by Marius Costeniuc, Richard S. Ellis, Hugo Touchette, and Bruce Turkington. Physical Review E, Volume 73, 026105 (8 pages), (2006).
  
  
  
  13. Metastability within the Generalized Canonical Ensemble by Hugo Touchette, Marius Costeniuc, Richard S. Ellis, and Bruce Turkington. Physica A, Volume 365, 132–137 (2006). Proceedings of the 3rd International Conference on News, Expectations and Trends in Statistical Physics (NEXT-ΣΦ 2005) Held in Kolymbari, Crete, Greece, 13–18 August 2005.
  
  
  
  14. Nonconcave Entropies from Generalized Canonical Ensembles by Marius Costeniuc, Richard S. Ellis, and Hugo Touchette. Physical Review E, Volume 74, Rapid Communications, 010105(R) (4 pages), (2006).
  
  
  
  15. The Theory of Large Deviations and Applications to Statistical Mechanics by Richard S. Ellis. Lecture notes for the International Seminar on Extreme Events in Complex Dynamics, October 23–27, 2006 (115 pages). Max-Planck-Institut für Physik komplexer Systeme, Dresden, Germany.
  
  
  
• 2005
  
  
  16. Analysis of Phase Transitions in the Mean-Field Blume-Emery-Griffiths Model by Richard S. Ellis, Peter Otto, and Hugo Touchette. Annals of Applied Probability, Volume 15, Number 3, 2203–2254 (2005).
  
  
  
  17. Complete Analysis of Phase Transitions and Ensemble Equivalence for the Curie-Weiss-Potts Model by Marius Costeniuc, Richard S. Ellis, and Hugo Touchette. Journal of Mathematical Physics, Volume 46, 063301 (25 pages), (2005).
  
  
  
  18. The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble by Marius Costeniuc, Richard S. Ellis, Hugo Touchette, and Bruce Turkington. Journal of Statistical Physics Volume 119, Numbers 5/6, 1283–1329 (2005). A talk based on this paper is available online.
  
  
  
  19. Nonequivalent Ensembles and Metastability by Hugo Touchette and Richard S. Ellis. Complexity, Metastability and Nonextensivity, 81–87 (2005). Proceedings of the 31st Workshop of the International School of Solid State Physics Held in Erice, Sicily, Italy, 20–26 July 2004. Edited by C. Beck, G. Benedek, A. Rapisarda, and C. Tsallis. World Scientific Publishers (Hackensack, NJ).
  
  
  
• 2004
  
  
  20. An Introduction to the Thermodynamic and Macrostate Levels of Nonequivalent Ensembles by Hugo Touchette, Richard S. Ellis, and Bruce Turkington. Physica A, Volume 340, pages 138–146 (2004). Proceedings of the Second Sardinian International Conference on News and Expectations in Thermostatistics (NEXT 2003) Held in Villasimius (Cagliari), Sardegna, Italy.
  
  
  
  21. A Statistical Approach to the Asymptotic Behavior of a Class of Generalized Nonlinear Schrödinger Equations by Richard S. Ellis, Richard Jordan, Peter Otto, and Bruce Turkington. Communications in Mathematical Physics, Volume 244, pages 187–208 (2004).
  
  
  
  22. Thermodynamic versus Statistical Nonequivalence of Ensembles for the Mean-Field Blume-Emery-Griffiths Model by Richard S. Ellis, Hugo Touchette, and Bruce Turkington. Physica A, Volume 335, pages 518–538 (2004).
  
  
  
• 2003
  
  
  23. Large Deviations for a Random Walk Model with State-Dependent Noise by Michelle Boué, Daniel Hernández-Hernández, and Richard S. Ellis. SIAM Journal on Control and Optimization, Volume 42, Number 3, 810–838 (2003).


• 2002
  
  
  24. Analysis of Statistical Equilibrium Models of Geostrophic Turbulence by Richard S. Ellis, Kyle Haven, and Bruce Turkington. Journal of Applied Mathematics and Stochastic Analysis, Volume 15, Number 4, 341–361 (2002).
  
  
  
  25. Nonequivalent Statistical Equilibrium Ensembles and Refined Stability Theorems for Most Probable Flows by Richard S. Ellis, Kyle Haven, and Bruce Turkington. Nonlinearity, Volume 15, 239–255 (2002).
      • On February 27, 2002, my co-authors and I were informed that this “article has been selected for inclusion on the Journal Information Page for Nonlinearity as a Featured Article. . . . Featured Articles are chosen by the journal for their high quality and interest to readers.”


• 2000
  
  
  26. Derivation of Maximum Entropy Principles in Two-Dimensional Turbulence via Large Deviations by Christopher Boucher, Richard S. Ellis, and Bruce Turkington. Journal of Statistical Physics, Volume 98, Numbers 5/6, 1235–1278 (2000).
  
  
  
  27. Entropy as a Measure of Randomness: A Fundamental Concept in the Mathematical Sciences by Richard S. Ellis. Newsletter, Department of Mathematics & Statistics, University of Massachusetts Amherst, Volume 15, Academic Year 1999–2000, pages 5–6.
  
  
  
  28. Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles by Richard S. Ellis, Kyle Haven, and Bruce Turkington. Journal of Statistical Physics, Volume 101, Numbers 5/6, 999–1064 (2000).
      • A talk based on this paper is available online.
      
      
  29. Large Deviations for Small Noise Diffusions with Discontinuous Statistics by Michelle Boué, Paul Dupuis, and Richard S. Ellis. Probability Theory and Related Fields, Volume 116, Number 1, 125–149 (2000).


• 1999
  
  
  30. Spatializing Random Measures: Doubly Indexed Processes and the Large Deviation Principle by Christopher Boucher, Richard S. Ellis, and Bruce Turkington. Annals of Probability, Volume 27, Number 1, 297–324 (1999). Erratum, Annals of Probability, Volume 30, Number 4, 2113 (2002).
  
  
  
  31. The Theory of Large Deviations: From Boltzmann’s 1877 Calculation to Equilibrium Macrostates in 2D Turbulence by Richard S. Ellis. Physica D, Volume 133, Numbers 1–4, 106–136 (1999).
      • A talk based on this paper is available online.
      
      
      
      • This paper was reprinted in Predictability: Quantifying Uncertainty in Models of Complex Phenomena (Special issue originating from the 18th Annual International Conference of the Center for Nonlinear Studies, Los Alamos, NM, May 11–15 1998), 106–136 (1999). Edited by Shiyi Chen, Len Margolin, and David Sharp. Elsevier (Amsterdam).
  
  
• 1997
  
  
  32. A Weak Convergence Approach to the Theory of Large Deviations by Paul Dupuis and Richard S. Ellis. Research monograph. Wiley Series in Probability and Statistics, 479 pages (1997). John Wiley & Sons (New York).
  
  
  
• 1995
  
  
  33. Large Deviation Analysis of Queueing Systems by Paul Dupuis and Richard S. Ellis. Stochastic Networks, 347-365. Edited by F. P. Kelly and R. J. Williams. IMA Volumes in Mathematics and Its Applications, Volume 71 (1995). Springer-Verlag (New York).
  
  
  
  34. The Large Deviation Principle for a General Class of Queueing Systems, I by Paul Dupuis and Richard S. Ellis. Transactions of American Mathematical Society, Volume 347, Number 8, pages 2689–2751 (1995).
      • Irina Ignatiouk-Robert pointed out a gap in our proof of the large deviation upper bound that appears on page 2730 of this paper. She gives a correct proof of the large deviation upper bound in her paper, “Large Deviations for Processes with Discontinuous Statistics,” published in Annals of Probability 33:1479–1508 (2005). She also proves that for any absolutely continuous path—not just for piecewise linear paths as in our paper—the rate function has the form given in equation (4.8) of our paper.
  
  
  35. An Overview of the Theory of Large Deviations and Applications to Statistical Mechanics by Richard S. Ellis. Scandinavian Actuarial Journal, Number 1, pages 97–142 (1995).
      • My paper was solicited for this special memorial volume commemorating the centennial of the birth of the Swedish mathematician Harald Cramér, who was the editor of the journal for many years.


• 1993
  
  
  36. The Large Deviation Principle for Measures with Random Weights by Richard S. Ellis, John Gough, and Joseph V. Pulé. Reviews in Mathematical Physics Volume 5, Number 4, pages 659–692 (1993).


• 1992
  
  
  37. Large Deviations for Markov Processes with Discontinuous Statistics II: Random Walks by Paul Dupuis and Richard S. Ellis. Probability Theory and Related Fields, Volume 91, pages 153–194 (1992).
  
  
  
  38. Limit Theorems for Maximum Likelihood Estimators in the Curie-Weiss-Potts Model by Richard S. Ellis and Kongming Wang. Stochastic Processes and Their Applications, Volume 40, Number 2, pages 251–288 (1992).
  
  
  
  39. Review of James Bucklew’s book Large Deviation Techniques in Decision, Simulation, and Estimation by Richard S. Ellis. Bulletin (New Series) of the American Mathemati­cal Society, Volume 26, Number 1, pages 160–171 (1992).


• 1991
  
  
  40. Large Deviations for Markov Processes with Discontinuous Statistics, I: General Upper Bounds by Paul Dupuis, Richard S. Ellis, and Alan Weiss. Annals of Probability, Volume 19, Number 3, 1280–1297 (1991).


• 1990
  
  
  41. A Unified Approach to Large Deviations for Markov Chains and Applications to Statistical Mechanics by Richard S. Ellis. Stochastic Processes, Physics and Geometry—Ascona/Locarno, Switzerland, 4–9 July 1988, 392–433 (1990). Edited by S. Albeverio, G. Casati, U. Cattaneo, D. Merlini, and R. Moresi. World Scientific Publishers (Teaneck, NJ).
  
  
  
  42. Limit Theorems for the Empirical Vector of the Curie-Weiss-Potts Model by Richard S. Ellis and Kongming Wang. Stochastic Processes and Their Applications, Volume 35, Number 1, 59–79 (1990).


• 1989
  
  
  43. Uniform Large Deviation Property of the Empirical Process of a Markov Chain by Richard S. Ellis and Aaron Wyner. Annals of Probability, Volume 17, Number 3, 1147–1151 (1989).


• 1988
  
  
  44. Inequalities for Multivariate Compound Poisson Distributions by Richard S. Ellis. Annals of Probability, Volume 16, Number 2, 658–661 (1988).
  
  
  
  45. Large Deviations for the Empirical Measure of a Markov Chain with an Application to the Multivariate Empirical Measure by Richard S. Ellis. Annals of Probability, Volume 16, Number 4, 1496–1508 (1988).
  
  
  
  46. Multiple Phase Transitions in the Generalized Curie-Weiss Model by Theodor Eisele and Richard S. Ellis. Journal of Statistical Physics, Volume 52, Numbers 1/2, 161–202 (1988).


• 1985
  
  
  47. Entropy, Large Deviations, and Statistical Mechanics. Research monograph. Grundlehren der mathematischen Wissenschaften, Volume 271, 364 pages (1985). Springer-Verlag (New York).
  
  
  
  48. Large Deviations and Statistical Mechanics by Richard S. Ellis. Particle Systems, Random Media, and Large Deviations, 101-123 (1985). Proceedings of the 1984 AMS Summer Research Conference Held at Bowdoin College, Brunswick, Maine, June 24–30, 1984. Edited by Richard Durrett. Contemporary Mathematics, Volume 41. American Mathematical Society (Providence).


• 1984
  
  
  49. Large Deviations for a General Class of Random Vectors by Richard S. Ellis. Annals of Probability, Volume 12, Number 1, 1–12 (1984).


• 1983
  
  
  50. Continuous Symmetry Breaking in a Mean Field Model by Richard S. Ellis and Theodor Eisele. Journal of Physics A: Mathematical and General, Volume 16, Number l, 195–199 (1983).
  
  
  
  51. Symmetry Breaking and Random Waves for Magnetic Systems on a Circle by Richard S. Ellis and Theodor Eisele. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, Volume 63, Number 3, 297–348 (1983).


• 1982
  
  
  52. Asymptotics of Certain Random Fields on a Circle by Richard S. Ellis and Jay S. Rosen. Proceedings of Colloquium on Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory (Esztergom, Hungary, June 24-30, 1979), 279-321 (1982). Edited by J. Fritz, J. L. Lebowitz, and D. Szasz. Seria Colloquia Mathematica Societatis Janos Bolyai, Volume 27. North Holland (New York).
  
  
  
  53. Laplace’s Method for Gaussian Integrals with an Application to Statistical Mechanics by Richard S. Ellis and Jay Rosen. Annals of Probability, Volume 10, Number 1, 47–66 (1982). Correction: Annals of Probability, Volume 11, Number 2, 456 (1983).
  
  
  
  54. Asymptotic Analysis of Gaussian Integrals, I: Isolated Minimum Points by Richard S. Ellis and Jay Rosen. Transactions of the American Mathematical Society, Volume 273, Number 2, 447–481 (1982).


• 1981
  
  
  55. Asymptotic Analysis of Gaussian Integrals, II: Manifold of Minimum Points by Richard S. Ellis and Jay Rosen. Communications in Mathematical Physics, Volume 82, Number 2, 153–181 (1981).
  
  
  
  56. Discussion of Karl F. Freed’s paper “Polymers as Self-Avoiding Walks” by Richard S. Ellis. Annals of Probability, Volume 9, Number 4, 551–554 (1981).
      • This article was prepared at the invitation of Associate Editor Thomas G. Kurtz.
  
  
  
  57. The GHS Inequality for a Large External Field by Richard S. Ellis, Charles Newman, and Michael O'Connell. Journal of Statistical Physics, Volume 26, Number 1, 37–50 (1981).


• 1980
  
  
  58. Asymptotic Expansions of Gaussian Integrals by Richard S. Ellis and Jay Rosen. Bulletin (New Series) of the American Mathematical Society, Volume 3, Number 1, 705–709 (1980).
  
  
  
  59. Limit Theorems for Sums of Dependent Random Variables Occurring in Statistical Mechanics, II: Conditioning, Multiple Phases, and Metastability by Richard S. Ellis, Charles M. Newman, and Jay S. Rosen. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, Volume 51, 153–169 (1980).
  
  


• 1978
  
  
  60. Extensions of the Maximum Principle: Exponential Preservation by the Heat Equation by Richard S. Ellis and Charles Newman. Journal of Differential Equations, Volume 30, Number 3, 365–379 (1978).
  
  
  
  61. Fluctuationes in Curie-Weiss Exemplis by Richard S. Ellis and Charles Newman. Mathematical Problems in Theoretical Physics, 313–324 (1978). Proceedings of the International Conference Held in Rome, June 6-15, 1977. Edited by G. Dell'Antonio, S. Doplicher, and G. Jona-Lasinio. Springer Lecture Notes in Physics, Number 80. Springer-Verlag (Berlin).
  
  
  
  62. Limit Theorems for Sums of Dependent Random Variables Occurring in Statistical Mechanics by Richard S. Ellis and Charles M. Newman. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, Volume 44, 117–139 (1978).
  
  
  
  63. Necessary and Sufficient Conditions for the GHS Inequality with Applications to Analysis and Probability by Richard S. Ellis and Charles Newman. Transactions of the American Mathematical Society, Volume 237, 83–99 (March, 1978).
  
  
  
  64. The Statistics of Curie-Weiss Models by Richard S. Ellis and Charles Newman. Journal of Statistical Physics, Volume 19, Number 2, 149–161 (1978).


• 1977
  
  
  65. Diffusion Approximation for Transport Processes with Boundary Conditions by Richard S. Ellis and Walter Rosenkrantz. Indiana University Mathematics Journal, Volume 26, Number 6, 1075–1096 (1977).
  
  
  
  66. Monotone Decrease of Characteristic Functions by Richard S. Ellis. Journal of Statistical Physics, Volume 16, Number 1, 117–118 (1977). Correction: Journal of Statistical Physics, Volume 18, Number 1, 107 (1978).


• 1976
  
  
  67. The GHS and Other Correlation Inequalities for a Class of Even Ferromagnets by Richard S. Ellis, James Monroe, and Charles Newman. Communications in Mathematical Physics, Volume 46, Number 2, 167–182 (1976).
  
  
  
  68. Quantum Mechanical Soft Springs and Reverse Correlation Inequalities by Richard S. Ellis and Charles Newman. Journal of Mathematical Physics, Volume 17, Number 9, 1682–1683 (1976).
  
  
  
  69. Volume of an N-Simplex by Multiple Integration by Richard S. Ellis. Elemente der Mathematik, Volume 31, Number 3, 57–59 (1976).


• 1975
  
  
  70. Aspects of the Krook Model of the Boltzmann Equation. Indiana University Mathematics Journal, Volume 24, Number 10, 915–923 (1975).
  
  
  
  71. Asymptotic Equivalence of the Linear Navier-Stokes and Heat Equations in One Dimension by Richard S. Ellis and Mark Pinsky. Journal of Differential Equations, Volume 17, Number 2, 406–420 (1975).
  
  
  
  72. Asymptotics and Limit Theorems for the Linearized Boltzmann Equation by Richard S. Ellis. Probabilistic Methods in Differential Equations, 143–151 (1975). Proceedings of the Conference Held at University of Victoria, August 19–20, 1974. Edited by Mark A. Pinsky. Springer Lecture Notes in Mathematics, Number 451. Springer-Verlag (Berlin).
  
  
  
  73. Concavity of Magnetization for a Class of Even Ferromagnets by Richard S. Ellis. Bulletin of the American Mathematical Society, Volume 81, Number 5, 925–929 (1975).
  
  
  
  74. The First and Second Fluid Approximations to the Linearized Boltzmann Equation by Richard S. Ellis and Mark Pinsky. Journal de Mathématiques Pures et Appliquées, Volume 54, Number 2, 125–156 (1975).
  
  
  
  75. The Projection of the Navier-Stokes Equations upon the Euler Equations by Richard S. Ellis and Mark Pinsky. Journal de Mathématiques Pures et Appliquées, Volume 54, Number 2, 157–182 (1975).
  
  
  
  76. A Simple Proof of the GHS and Further Inequalities by Richard S. Ellis and James Monroe. Communications in Mathematical Physics, Volume 41, Number 1, 33–38 (1975).


• 1974
  
  
  77. An Application of Stochastic Optimal Control Theory to the Optimal Rescheduling of Airplanes by Richard S. Ellis and Ray Rishel. IEEE Transactions on Automatic Control, Volume AC-19, Number 2, 139–142 (1974).
  
  
  
  78. Asymptotic Nonuniqueness of the Navier-Stokes Equations in Kinetic Theory by Richard S. Ellis and Mark Pinsky. Bulletin of the American Mathematical Society, Volume 80, Number 6, 1160–1164 (1974).
  
  
  
  79. The Asymptotic Behavior of the First Real Eigenvalue of Second Order Elliptic Operators with a Small Parameter in the Highest Derivatives, II by Allen Devinatz, Richard S. Ellis, and Avner Friedman. Indiana University Mathematics Journal, Volume 23, Number 11, 991–1011 (1974).
  
  
  
  80. Chapman-Enskog-Hilbert Expansion for the Ornstein-Uhlenbeck Process and the Approximation of Brownian Motion by Richard S. Ellis. Transactions of the American Mathematical Society, Volume 199, 65–74 (1974).
  
  
  
  81. Limit Theorems for Random Evolutions with Explicit Error Estimates by Richard S. Ellis. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, Volume 28, Number 3, 249–256 (1974).


• 1973
  
  
  82. Chapman-Enskog-Hilbert Expansion for a Markovian model of the Boltzmann Equation by Richard S. Ellis. Communications on Pure and Applied Mathematics Volume 26, Number 3, 327–359 (1973).
  
  
  
  83. Limit Theorems for Model Boltzmann Equations with Several Conserved Quantities by Richard S. Ellis and Mark Pinsky. Indiana University Mathematics Journal, Volume 23, Number 4, 287–307 (1973).