118) "Entropy, pressure, ground states and calibrated sub-actions for linear dynamics",

joint work with V. Vargas

preprint (2021)

117) "Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry"

joint work with R. Ruggiero

preprint (2021)

116) "Explicit Bivariate Rate Functions for Large Deviations in AR(1) and MA(1) Processes with Gaussian Innovations"

joint work with M. Karling and S. R. C. Lopes

preprint (2021)

115) "Decision Theory and Large Deviations for Dynamical Hypotheses Test: Neyman-Pearson, Min-Max and Bayesian Tests"

joint work with H. H. Ferreira and S. R. C. Lopes

preprint (2021)

114) "On information gain, Kullback-Leibler divergence, entropy production and the involution kernel"

joint work Jairo Mengue

preprint (2020)

113) "Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs Measures: theory and examples"

joint work with L. Cioletti, L. Hataishi and M. Stadlbauert

preprint (2018)

112) "The sectional curvature of the infinite dimensional manifold of Holder equilibrium probabilities"

joint work with R. Ruggiero,

preprint (2018)

111) "Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism"

joint work A. Messaoudi, M. Stadlbauert and V. Vargas

to appear in Nonlinearity

110) "Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism"

joint work S. R. C. Lopes and P. Varandas

to appear in Journ. of Statis. Physics

109) "Lyapunov exponents for Quantum Channels: an entropy formula and generic properties"

joint work Jader E. Brasil, and Josue Knorst

to appear in Journal of Dynamical Systems and Geometric Theories

108) "Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs Channels and generic properties"

joint work with Jader Brasil and Josue Knorst,

to appear in Communications in Contemporary Mathematics

107) "Pentadiagonal Matrices and an Application to the Centered MA(1) Stationary Gaussian Process",

joint work with M. Karling and S. R. C. Lopes

to appear in International Journal of Applied Mathematics and Statistics

106) "An iterative process for approximating subactions"

joint work with Hermes H. Ferreira and Elismar R. Oliveira,

Modeling, Dynamics, Optimization and Bioeconomics IV" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag, pp 187--212 (2021)

105) "On Bertelson-Gromov Dynamical Morse Entropy ",

joint work with Marcos Sebastiani,

"Modeling, Dynamics, Optimization and Bioeconomics IV" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag, 297--321, (2021)

104) "Haar systems, KMS states on von Neumann algebras and C^*-algebras on dynamically defined groupoids and Noncommutative Integration"

joint work with G. Castro and G. Mantovani

"Modeling, Dynamics, Optimization and Bioeconomics IV" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag 79--137 (2021)

103) "Thermodynamic Formalism for Haar systems in Noncommutative Integration: transverse functions and entropy of transverse measures"

joint work with Jairo Mengue,

Erg. Theo. and Dyn. Systems, (2021), 41, 1835–1863

102) "Quantum Spin probabilities at positive temperature are Holder Gibbs probabilities"

joint work with J. Brasil, J. Mengue and Carlos G. Moreira

Comm. in Contemporary Mathematics Vol. 23, No. 1 (2021) 1950050 (32 pages)

101) "Explicit examples in Ergodic Optimization"

joint work with Hermes H. Ferreira and Elismar R. Oliveira,

Sao Paulo Journal of Math. Sciences - Vol 14 - pp 443-489 (2020)

100) "The Ruelle operator for symmetric beta-shifts"

joint work with Victor Vargas,

Publicacions Mathematiques, 661 - 680 - Volume 64, Issue 2 (2020)

99) "Gibbs States and Gibbsian Specifications on the space R^N"

joint work with Victor Vargas,

Dyn. Systems., Volume 35, Issue 2, 216-241 (2020)

98) "Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures",

joint work with L. Cioletti and M. Stadlbauert

Disc and .Cont. Dyn. Syst. A Vol 40, N. 8, 4625-4652 (2020)

97) "Correlation Inequalities and Monotonicity Properties of the Ruelle Operator",

joint work with L. Cioletti

Stoch. and Dyn, 19 (2019), no. 6, 1950048, 31 pp

96) "Continuous groupoids on the symbolic space, quasi-invariant probabilities for Haar systems and the Haar-Ruelle operator"

joint work with Elismar Oliveira

Bull of the Braz. Math. Soc., Vol 50. Issue 3, pp 663–683 (2019)

95) "The KMS Condition for the homoclinic equivalence relation and Gibbs probabilities",

joint work with G. Mantovani

Sao Paulo Journ. of Math. Sciences, Volume 13, Issue 1, pp 248–282 (2019)

94) "A general renormalization procedure on the one-dimensional lattice and decay of correlations",

Stoch. and Dyn., Vol. 19, No. 01, 1950003 (2019)

93) "The calculus of thermodynamical formalism",

joint work with B. Kloeckner, P. Giulietti and D. Marcon,

Journ. of the European Math Society, Vol 20, Issue 10, pages 2357-2412 (2018)

92) "Large Deviations for Quantum Spin probabilities at temperature zero",

joint work with Jairo K. Mengue, Joana Mohr and Carlos G. Moreira,

Stochastics and Dynamics, Vol. 18, No. 06, 1850044 (2018)

91) "Selection of calibrated subaction when temperature goes to zero in the discounted problem",

joint work with Renato Iturriaga and Jairo Mengue

Discrete and Cont Dyn. Syst., Series A, Vol 38, n 10, 4997-5010 (2018)

90) "A formula for the Entropy of the Convolution of Gibbs probabilities on the circle",

Nonlinearity, 31 , 3441-3459 (2018).

89) "Uma breve Introdução à Matemática da Mecânica Quântica"

- XXXI Coloquio Brasileiro de Matematica - 2017

88) "Introdução à Matemática da Mecânica Quântica"

On line text

87) "On the thin boundary of the fat attractor"

joint work with E. Oliveira,

"Modeling, Dynamics, Optimization and Bioeconomics III" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag, 205-246 (2018)

86) "Transport and large deviations for Schrodinger operators and Mather measures",

joint work with Ph. Thieullen, "Modeling, Dynamics, Optimization and Bioeconomics III" Editors: Alberto Pinto and David Zilberman, Proceedings in Mathematics and Statistics, Springer Verlag, 247-255 (2018)

85) "An introduction to Coupling"

"Modeling, Dynamics, Optimization and Bioeconomics II" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, pp 307-335, (2017) Springer Verlag

84) "A detailed proof of the von Neumann's Quantum Ergodic Theorem",

joint work with Marcos Sebastiani,

Quantum Studies: Mathematics and Foundations - Vol 4 - Issue 3, 263-285 (2017).

83) "Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice"

joint work with L. Cioletti, Discrete and Cont. Dyn. Syst. - Series A, Vol 37, Number 12, 6139 -- 6152 (2017)

82) "Spectral Properties of the Ruelle Operator for Product Type Potentials on Shift Spaces",

joint work with L. Cioletti, M. Denker and M. Stadlbauer, Journal of the London Mathematical Society - Volume 95, Issue 2, 684–704 (2017)

81) "Semiclassical limits, Lagrangian states and coboundary equations",

joint work with J. Mohr,

Vol 17. N 2, 1750014 (19 pages) Stoch. and Dynamics (2017)

80) "Duality between Eigenfunctions and Eigendistributions of Ruelle and Koopman operators via an integral kernel"

joint work with P. Giulietti and V. Pit

Stoch. and Dynamics, 16 , 1660011 - 22 -pages - vol 3 (2016).

79) "Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps",

joint work with Benoit Kloeckner and Manuel Stadlbauert,

Nonlinearity, 28, Number 11, 4117–4137 (2015)

78) "Generic properties for random repeated quantum iterations",

joint work with Marcos Sebastiani,

Quantum Studies: Mathematics and Foundations, Vol 2, pp 389-402 (2015)

77) "Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport"

joint work with J. K. Mengue, J. Mohr and R. R. Souza - Bull. of the Brazilian Math. Soc. Vol 46 - N 3 - 353-389 (2015)

76) " Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature"

joint work with J. K. Mengue, J. Mohr and R. R. Souza

Erg. Theory and Dyn Systems, 35 (6), 1925–1961 (2015)

75) "The Dual Potential, the involution kernel and Transport in Ergodic Optimization"

joint work with E. Oliveira and Ph. Thieullen,

Dynamics, Games and Science -International Conference and Advanced School Planet Earth DGS II, Portugal (2013), Edit. J-P Bourguignon, R. Jelstch, A. Pinto and M. Viana, Springer Verlag, pp 357-398 (2015)

74) "Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach"

joint work with L. Cioletti,

Journ. of Statistical Physics, 159 - Issue 6, 1424–1455 (2015)

73) "Large Deviations for stationary probabilities of a family of continuous time Markov chains via Aubry-Mather theory"

joint work with A. Neumann,

Journ. of Statistical Physics. Vol. 159 - Issue 4 pp 797-822 (2015)

72) "On generic G-prevalent properties and a quantitative K-S theorem for C^{r} diffeomorphisms of the circle"

joint work with Elismar R. Oliveira

Dynamical Systems: an Int. Journ. Vol 30. Issue 1, 1-12 (2015)

71) "Ergodic Transport Theory, periodic maximizing probabilities and the twist condition"

joint work with G. Contreras and E. Oliveira,

"Modeling, Optimization, Dynamics and Bioeconomy I", Springer Proceedings in Mathematics and Statistics, Volume 73, Edit. David Zilberman and Alberto Pinto, 183-219 (2014)

70) "Selection of measure and a Large Deviation Principle for the general one-dimensional XY model"

joint work with J. Mengue,

Dyn. Systems: an Int. Jour. Vol 29, Issue 1 (2014) pp 24-39

69) "On the selection of subaction and measure for a subclass of potentials defined by P. Walters"

joint work with A. T. Baraviera and J. Mengue,

Erg. Theo. and Dyn. Systems, Volume 33, issue 05, pp. 1338--1362 (2013)

68) "Ergodic Optimization, Zero Temperature Limits and the Max-Plus Algebra"

joint work with A. Baraviera and R. Leplaideur, mini-course in XXIX Coloquio Brasileiro de Matemática (2013)

67) "A thermodynamic formalism for continuous time Markov chains with values on the Bernoulli Space: entropy, pressure and large deviations"

joint work with A. Neumann and Ph. Thieullen, Journ. of Statist. Phys. Volume 152, Issue 5, Page 894-933 (2013)

66) "The effective potential and transshipment in thermodynamic formalism at temperature zero"

joint work with E. Garibaldi,

Stoch. and Dynamics, Vol 13 - N 1, 1250009 (13 pages) (2013)

65) "Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics"

joint work with E. R. Oliveira and D. Smania,

Bull. of the Braz. Math Soc. Vol 43 (3) 467-512 (2012)

64) "Duality Theorems in Ergodic Transport"

joint work with J. Mengue, Journal of Statistical Physics. Vol 149, issue 5, pp 921–942 (2012)

63) "Wigner measures and the semi-classical limit to the Aubry-Mather measure"

joint work with Diogo Gomes and Joana Mohr,

Applied Mathematics Research Express, Volume 2012, Issue 2, 152-183 (2012)

62) "The potential point of view for renormalization",

joint work with A Baraviera and R. Leplaideur

Stochastic and Dynamics, Vol 12. N 4, (2012) 1250005(1-34)

61) "Selection of ground states in the zero temperature limit for a one-parameter family of potentials"

joint work with A. Baraviera and R. Leplaideur,

SIAM Journal on Applied Dynamical Systems, Vol. 11, n 1, 243-260 (2012)

60) "On the general one-dimensional XY Model: positive and zero temperature, selection and non-selection"

joint work with A. T. Baraviera, L. M. Cioletti, J. Mohr, R. R. Souza,

Reviews in Math. Physics. Vol. 23, N. 10, pp 1063–-1113 (2011).

59) "A dynamical point of view of Quantum Information: Wigner measures",

joint work with A. Baraviera, C. F. Lardizabal, and M. Terra Cunha.

Dynamics, Games and Science II, DYNA 2008, Edit. M. Peixoto, A. Pinto and D Rand, pp 161-186 Springer Verlag (2011)

58) "A dynamical point of view of Quantum Information: entropy and pressure",

joint work with A. Baraviera, C. F. Lardizabal, and M. Terra Cunha.

Dynamics, Games and Science I, DYNA 2008, Edit. M. Peixoto, A. Pinto and D Rand, pp 81-122 Springer Verlag (2011)

57) "Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics"

joint work with R. Ruggiero,

Disc. and Cont. Dyn. Syst. A. Vol 29, N 3, pp 1155-1174 -(2011).

56) "The Mather measure and a Large Deviation Principle for the Entropy Penalized Method",

joint work with D. Gomes and J. Mohr

"Communications in Contemporary Mathematics", Vol 13, issue 2, 235-268 (2011)

55) "Quantum Stochastic Processes, Quantum Iterated Function Systems and Entropy"

joint work with A. Baraviera, C. F. Lardizabal and M. Terra Cunha, Sao Paulo Journal of Math. Sciences, Vol. 5, N 1, pp 51-84 (2011).

54) "Zeta measures and Thermodynamic Formalism for temperature zero"

joint work with J. Mengue

Bull. of the Bras. Math. Soc 41 (3) pp 449-480 (2010)

53) "A Thermodynamic Formalism for density matrices in Quantum Information",

joint work with A. Baraviera, C. F. Lardizabal and M. Terra Cunha.

Applied Mathematics Research Express, Vol. 2010, No. 1, pp. 63–118 (2010)

52) "Poncelet pairs and the Twist Map associated to the Poncelet Billiard",

joint work with M. Sebastiani

"Real Analysis Exchange", Vol 35 , Issue 2, pp 355-374 (2010)

51) "A Ruelle Operator for continuous time Markov Chains"

joint work with A. Baraviera and R. Exel

Sao Paulo Journal of Mathematical Sciences, vol 4 n. 1, pp 1-16 (2010)

50) "Minimax Probabilities for Aubry-Mather problems",

joint work with Diogo Gomes and N. Jung

, Commun. Contemporary Mathematics, Vol 12, Issue 5, pp 789-813 (2010)

49) "On calibrated and separating sub-actions",

joint work with E. Garibaldi and P. Thieullen

Bull. of the Bras. Math. Soc. Vol 40, 577-602, (4)(2009)

48) "Entropy and variational principles for holonomic probabilities of IFS",

joint work with E. Oliveira

"Disc. and Cont. Dyn. Systems" (Series A) Vol 23, N 3, 937-955 (2009)

47) "KMS States, Entropy and a Variational Principle for Pressure",

joint work with G. Castro

"Real Analysis Exchange" Vol 34 , Issue 2, 333-346 (2009)

46) " Negative Entropy, Zero temperature and stationary Markov chains on the interval",

joint work with J. Mohr, R. Souza and Ph. Thieullen

Bull. Soc. Bras. Math. Vol 40 n 1, (2009), 1-52

45) "C*- Algebras and Thermodynamic Formalism",

joint work with R. Exel

Sao Paulo Journal of Mathematical Sciences 2, 1 (2008), 285–307

44) "Exponential decay of correlation for the Stochastic Process associated to the Entropy Penalized Method",

joint work with Diogo Gomes

Sao Paulo Journal of Mathematical Sciences 2, 1 (2008), 21-28

43) "Eigenfunctions of the Laplacian and associated Ruelle operator ",

joint work with P. Thieullen

, "Nonlinearity", Volume 21, Number 10, pp-2239-2254 (2008)

42) "On the Aubry-Mather theory for symbolic dynamics",

joint work with E. Garibaldi

Ergodic Theory and Dynamical Systems, v. 28, issue 3, pp 791-815 (2008)

41) "Functions for relative maximization",

joint work with E. Garibaldi

Dynamical Systems, v. 22, 511-528, 2007

40) "Cohomology and Subcohomology for expansive geodesic flows",

joint work with V. Rosas e R. Ruggiero

"Discrete and Continous Dynamical Systems" Volume: 17, N. 2, pp 403-422 (2007).

39) "A Large Deviation Principle for the equilibrium states of Holder Potentials: the zero temperature case",

joint work with A. Baraviera e P. Thieullen

"Stochastics and Dynamics" Vol 6, n.1,- pp 77-96 (2006)

38) ``Mather measures and

the Bowen-Series transformation,",

joint work with P. Thieullen

"Annal Inst Henry Poincare - Anal non-linear". pp 663-682, vol 5, (2006)

37) ``Subactions for Anosov

Flows",

trabalho conjunto com P. Thieullen

Erg Theo and Dyn Syst vol 25 Issue 2, pp605-628 (2005).

36) ``C^* Algebras, approximately proper equivalence

relations and Thermodynamic Formalism",

trabalho conjunto com R. Exel

Vol 24, pp 1051-1082 Erg Theo and Dyn Syst (2004).

35) ``Subactions for Anosov

Difeomorphisms",

trabalho conjunto com P. Thieullen

Asterisque Vol 287 pp 135-146 ( 2003).

34) ``Convergence in distribution

of the periodogram of chaotic systems,",

trabalho conjunto com S. Lopes

"Stochastic and Dynamics", vol 2, Issue 4 pp 609-624 (2002).

33) ``Exact bounds for the polynomial

decay of correlation, 1/f noise and the CLT for a non-Holder potential,",

trabalho conjunto com A. Fisher

"Nonlinearity", vol 14, pp 1071-1104 (2001).

32) ``Lyapunov minimizing measures

for expanding maps of the circle",

trabalho conjunto com G. Contreras and P. Thieullen

"Ergodic Theory and Dynamical Systems" , vol 21, Issue 5 pp 1379-1409 (2001).

31) `` The capacity costfunction of a hard constrained channel"

trabalho conjunto com M. Craizer

publicado em Int Journal of Appl Math Vol 2, N 10 pp 1165-1180 (2000) .

30) ``Large deviations for expanding transformations with additive white noise"

trabalho conjunto com S. Carmona, publicado em

J. Statist. Phys. 98 (2000), no. 5-6, 1311–-1333

29)"On the minimal action function of an autonomous Lagrangian associated with a magnetic phenomena"

trabalho conjunto com Mario Carneiro, "Annal Inst Henry Poincare - Anal Non-lin" Vol 16, N 6 , pp 667-690 (1999)

28) ``A level-1 LDP for the autocovariances

of uniquely ergodic transformations with aditive noise,",

joint work with S. Carmona, C. Landim and S. Lopes

"Jour. of Statist. Physics", Vol. 91, N. 1/2, pp. 395-421 (1998).

27) ``Parametric estimation and spectral analysis of piecewise linear maps of the interval. "

trabalho conjunto com Silvia Lopes,

Adv. in Appl. Probab. 30 (1998), no. 3, 757–776.

26)" Generic Properties of open billiards"

trabalho conjunto com M. Mauldin, "Journal of Mathematical Analysis and Applications", Vol 205 pp 251-258 (1997)

25) "A billiard in the hyperbolic plane with decay of correlation of type n^-2"

trabalho conjunto M. Bauer, "Discrete and Continuous Dynamical Systems", v. 3, n.1, p. 107-116, 1997.

24) "On the spectral density of a class of chaotic time series",

trabalho conjunto com S.Lopes e R. Souza , "Journal of Times Series", Vol 18 N 5, spt pp 465-474 (1997)

23) "Spectral Analysis of one-dimensional chaotic transformations" "

trabalho conjunto com S. Lopes e R. Souza, "Random and Computational Dynamics Journal" Vol 3, N 1 pp 107-117 (1997)

22) ``Open biliards: invariant and conditionally invariant probabilities on

Cantor sets",

trabalho conjunto com R. Markarian,

"SIAM Journal of Applied Mathematics", Vol 56, Issue 2, april, pp 651-680, 1996.

21) ``On the dimension of the predictive process of a memoryless channel",

trabalho conjunto com Silvia Lopes , "Stochastic Analysis and Applications",

Vol 14, N 4 (1996)

20)) `` Spectral analysis of chaotic transformations.",

trabalho conjunto com S. Lopes and R. Souza,

Rebrape 10 (1996), no. 2, 151–179.

19) ``The Theta Group and the Continued

Fraction Expansion with Even Partial Quotients",

trabalho conjunto com C. Kraaikamp

Geometricae Dedicatae , vol 59, pp 293-333 1996.

18) ``Maximum Likelihood and Minimum Entropy

Estimation of Grammars",

trabalho conjunto com P. Collet

e A. Galves. Random and Computational Dynamics, 3, pp-241-256, 1995" .

17)``Absolutely Continuous Invariant Measures for a Class of Affine Interval Exchange Maps",

trabalho conjunto com Z. Coelho e L. F. da Rocha, " Proceedings of the American Mathematical Society", 123, N. 11, pp. 3533-3542, 1995.

16) ``Invariant Measure for the Gauss Map Associated with Interval Exchange Maps",

trabalho conjunto com L. F. da Rocha, Indiana University Mathematics Journal, Vol. 13, N. 4, pp. 1399-1438, (1994).

16) ``Entropy, Pressure and Large Deviation'',

in Cellular Automata, Dynamical Systems and Neural Networks, E. Goles e S. Martinez (eds.), Kluwer, pp. 79--146, (1994)

15) "Weight-balanced measures and free energy for one-dimensional dynamics",

joint work with W. Withers, "Forum Math.", Vol. 5, pp. 161-182 (1993) (1993).

14) "The Zeta Function, Non-Differentiability of Pressure and The Critical Exponent of Transition",

"Advances in Mathematics", Vol. 101, pp. 133-167, (1993).

13) `On the Dynamics of Real Polynomials on the Plane",

Computers and Graphics, Vol. 16, N. 1, pp. 15-23, (1992).

12) ``A First-Order Level-2 Phase Transition in Thermodynamic Formalism",

Journal of Statistical Physics, Vol. 60, N. 3/4, pp. 395-411, (1990).

11) ``Dynamics of Real Polynomials on the Plane and Triple Point Phase Transition",

Mathematical and Computer Modelling, Vol. 13, N. 9, pp. 17-32, (1990).

10)"Entropy and Large Deviation",

"NonLinearity", Vol. 3, N 2, pp. 527-546, (1990).

9) "Dimension Spectra and a Mathematical Model for Phase Transitions",

Adv. in Appl. Math. 11, pp 475-502 (1990)

8) ``A note on delta equilibrium for rational maps",

Mathematische Zeitschrift, 202, 261-269 (1989)

7) `The Dimension Spectrum of the Maximal Measure",

SIAM Journal of Mathematical Analysis, Vol. 20, N. 5, pp. 1243-1254, (1989).

6)"An Analogy of The Charge Distribution on Julia Sets with the Brownian Motion",

"Journal of Mathematical Physics", Vol. 30, N 9, pp. 2120-2124, (1989).

5)" The Complex Potential generated by the maximal measure for a family of rational maps",

"Journal of Statistical Physics", Vol. 52, N 3/4, pp. 571-575, (1988).

4) `Equilibrium Measures for Rational Maps",

Ergodic Theory and Dynamical Systems, Vol. 6, Part 3, pp. 393-399, (1986)

3) "Orthogonality and the Hausdorff Dimension of the Maximal Measure",

" Proceedings of the American Mathematical Society", Vol. 98, N 1, 51-55, (1986).

2) `An Invariant Measure for Rational Maps",

Boletim da Sociedade Brasileira de Matem\'atica, Vol. 14, N. 1, pp. 45-62, Alexandre Freire, Artur O. Lopes and e Ricardo Mane (1983)

1)`Structural Stability and Hyperbolic Attractors",

Transactions of the American Mathematical Society, Vol. 252, pp. 205-219, (1979).