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A possible physical interpretation of the Λ operator in the prigogine theory of irreversibility
Volume 110A, number 2
PHYSICS LETTERS
8 July 1985
A POSSIBLE PHYSICAL INTERPRETATION OF THE A OPERATOR IN T H E P R I G O G I N E T H E O R Y O F I R R E V E R S I B I L I T Y S. M A R T I N E Z 1 and E. T I R A P E G U I 2,3 Facultb des Sciences, UniversitbLibre de Bruxelles, CP. 226, 1050 Brussels, Belgium Received 28 March 1985; accepted for publication 29 April 1985
We argue that if observations take a finite time At and if one can only distinguish points in phase space separated by distances greater than ¢then the A operator is determined by these conditions.
It is known since the work of Prigogine and collaborators [ 1 - 3 ] that one can associate to a class of dynamical systems, namely the K-systems [4] a Markov process going strongly to equilibrium. Let us consider a K-shift (I2,/~, ~ , o) where I2 = Sz, S = {rl, r 2 ... rs) is the alphabet,/a the invariant measure, q5 the product o-algebra and o: I2 --, I2, (ox)/=x/+ 1 is the shift transformation of the point x = (... X_l, x0, Xl , x 2 .... ) ~ [2, x i E S. In L 2 ~ ) the adjoint U* of the unitary operator U f = f o o , f E L 2 ~ ) , gives the evolution of probability densities p E q9 = {p: p(x) /> 0, f p dg = 1). We recall that we can def'me a distance in [2 by
d ( x , y ) = ~ 2-~nl dS(Xn,Yn) , nEZ where d s ( x n , y n ) = 0 if x n = Yn ~ S and one if x n 4=Yn (for notations and details see refs. [5,6]). Then the associated Marker process has an evolution operator W* associated to U* by a bounded, positive, invertible, nonunitary operator A : L2(~) -+ L 2 ~ ) by the formula W* = AU*A -1 [ 1 - 3 ] . Our aim here is to give a possible physical interpretation of this operator A. If we are given an initial 1 On leave of absence from Departamento de Matematicas, Facultad de Cieneias Fisicas y Matematicas, Universidad de Chile, Santiago, Chile. 2 On leave of absence from Departamento de Fisica, Universidad de Chile, Santiago, Chile. 3 Supported in part by Fondo Nacional de Ciencias and by PNUD project for Chile. 0.375-9601/85[$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
probability density Po(X') at time zero it will evolve at time one to p(x') = U*Po(X' ). Our first claim is that if Po(X' = x) 4=0 then one should also have Po(X") d=0 in points x" of the stable manifold XSt(x) of x, XSt(x) = {y : d(ony, onx) ~ 0, n -~ +,,o}. We arrive at this conclusion if we admit that an observation is done in a finite time At > 0 and with a precision e > 0 in space, (i.e. we can only distinguish points with distances greater than e) since then ff we say that the system is in x we have to admit that it can be in any point x " such that d(onx ", onx) < e, 0 < n ~< At. This means that in fact one does not observe Po(X') but rather an extended version ~'0(x') obtainable from P0 by the action of an operator A such that A/J{x}(X" ) 4= 0 for points x" E XSt(x) (~A = characteristic function of the set A C I2, ~{x} = 5-function centered at x), i.e. supp(A/j {x }) = XSt("x). This last equality is to be interpreted in the sense that points far away from x should be given very small (or zero) weight since they will take a long time to be within distance e o f x . We conclude then that the evolution we observe is P0 = AP0 -~ Pl = APl = It*P0, with W* given by W* = AU*A -1 if A is invertible~ The operators A and W must satisfy some obvious properties, namely they should both map the space ~ of densities into itself (A I> 0, 14/* t> 0, A* 1 = W1 = 1) and leave the constant functions invariant (A1 = It'* 1 = 1). We shall see that these requirements together with supp (Mj{x}) = XSt(x) are sufficient to determine the form of A. Let ot be the partition ot = {X/: i E S), X/= {x: x 0 81
PHYSICS LETTERS
Volume 110A, number 2
= i}, cB n the o-algebra o£,, = on_** aict, cB ** = cB, _ ** = {~, ~2}, R n = E:r~'n the conditional expectation with respect to the o-algebra cB ~, i.e. R n is the orthogonal projector on the space L~'~ CBn) of L20a) functions which are ~ n measurable, R** = id, R_** = projector on the constant functions, and E n = R n - Rn - 1 the orthogonal projector on L20alCBn) L20al°Bn_ 1). One has Z n e z E n +R_** = 1. It is easy to convince oneself that A is necessarily of the form
A--
..
one has
: An -
= 1 , x . = 0,
and {An) is a decreasing sequence in terms of which A can be written
A=
n~Z
X.en + R _ . .
(2)
In order that the observed evolution operator W* is positive we must still impose Vn = Vn - Vn+l >~ 0 with v n - knlAn+l which implies v_** = 1 and 0 <~ v,. < 1. At this point we have then recuperated the A operator of the Prigogine theory and we know then that the observed evolution W* is a Markov process converging strongly to equilibrium. We note for future use that if v** = c > 0 then for large n the sequence Xn is approximately of the form An = c n. We shall see now that the interpretation we propose of A gives supplementary information on the sequence {Xn). In order to do this we shall study ~{x)(X ) = A~(x)(X ) which is only different from zero for points x ' in a set of/a.measure zero, namely XSt(x). The operator A is doubly stochastic and hence generates a Markov transition probability~Q(x, B) = A/JB(X), B C c B , and it is easy to see that ~{x}(X') is just the generalized (in the sense that it is a distribution in x ' ) density of the transition probability Q'(x, B) of the reverse (adjoint) process [7], which is given b y Q'(x, 13) = A*~B(X ) = Q(x, B) since here A is self-adjoint. We specialize now to a Bernoulli shift with probability vector lr = (,rrl .... *rrs) and we study there Q(x, B). In a shift the stable manifold is the disjoint union XSt(x) = {x} W :KSt(x), where XSt(x) is the dis•
82
~(ff(X) = 0; ~ ~'2:.v] =xj,/>~k,y]_ 1 :#X]_l}.
~
t
t
(3)
Using the notation nk(X ) = IrxkWe find (the calculation is done with the same technique as in ref. [5]) O(x, {x}) = O, and
Qk = Q(x, )(~t(x)) = (/rk_l(X) -1 - 1)
u=0
nEZ
since this implies supp A/J(x ) = XSt(x) as well as the other properties of A (the condition Z-An = 1 is to have a total probability one in XSt(x), see after formula (4)). We have R n = Zi< n E/and defming An zj
joint union XSt(x) = U k e z x~t(x), where the (contracting) fibers :~¢t are
X ~
X.R., n~Z
:
8 July 1985
ffk_l(X)...ffk+u_l(X)~k_(k+u) ,
(4)
with Z k e z Qk = 1 (this being a consequence of Z ~ n = 1), which shows that indeed the probability measure Q(x, B) is concentrated in :KSt(x) as we wanted. Let us consider the case of the Baker transformation for simplicity (the discussion can be done in a general I Bernoulli shift). Then S = (0, 1), lr 0 = lr 1 = ~, and oo
Qk = Z) n=O
k e Z-
(5)
Using the distance function d ( x , y ) defined before we shall define an average distance d k of the fiber x~t(x) to x as dk = ~(5 z + 8~) with 8 ; = min d(x, x'), 8~ = sup d(x, x'), x ' e :K~t(x). One easily finds d k = 1, k 1> 1 and dk = 3 X 2 k - 2 < 1, k ~< 0. A first restric. tion on the sequence {An) appears immediately. Let e < 1, then due to omx~t(x) = ) ( ~ t m ( a m x ) we see that fibers :~[~t(x)'with k - At > 0 remain at time At at an average distance one of qAtx and we can give them zero weight, i.e. Qk = 0, k > At. Moreover if k - At ~< 0 the average distance o f : ~ t at time At will be 3 X 2 - ( a t - k + 2 ) , then i f k ' ~ At is the greatest value o f k ~< At such that 3 × 2 - ( a t - k ' + 2 ) ~ e we can take Qk = O, k > k'. From (5) we see that this can be realized taking7~ n = O, n > k', which means An = 1, n <~ - k ' . For n > k' we can take An = c n+k', c < 1, which according to our previous remarks modelizes the case v.. = c > O, i.e. exponential decreasing of the Xn for large n (one should note that it is in fact the large n behavior what is important in this construction). Let us consider for simplicity, the case k' = 0, i.e. An = 1,n ~<0, and, An = c n , n > O . ThenQn = O , n > O , and n
Q - n = ~(1 - c) .~_ ck]2 n - k , k=O
n >1 O,
(6)
Volume 110A, number 2
PHYSICS LETrERS
which is an increasing function o f n as it should be since the weight we give to points in XSt(x) should increase when their distance to x decreases. The invertible operator A is now given by A = (1 - c)Z n ~ 0 cnRn and the evolution operator W* o f the Markov process will be (Vn = Vn - Vn+l) [5] W*=vo.U*+~_IU*R
1,
(7)
with v.. = c, ~-1 = 1 - c > 0. The transition probability Qw(x, B) = W~B(X) o f the process is concentrated in the stable manifold XSt(ox) of ox and one has Qlc(x, Xff(ox)) = 0 if k / > 1, Qle(x, )(S_tk(OX)) = 2 - ( k + l ) 7_ 1 if k >i O, Qlv(x, { o x ) ) = c. We now can calculate using (6), the mean distance ¢d) of the points in the stable manifold XSt(x) to x as
(d) ~ dk Q-k = (l - c)/(2 - c ) . k=O
(8)
After the time At which lasts the observation this distance is reduced by a factor 2 - A t (this is due to o m Xff(x) = x~ctm(omx) and d g = 3 X 2 k - 2, k ~< 0), and we can Fm our parameter c b y asking 2 - A t (d) = e which gives for c the value c = (1 - 2r/)/(l - "0),
r/=
2ZXte,
(9)
where r~ measures the lack of precision of our observation. If r/~ 0 then c -~ I and we recuperate from (7) the deterministic evolution. We recall that we have proved [8,9] that vt = c t is the probability o f observing the deterministic trajectory up to time t, then writing c = e - l / r we define the time z up to which the classical trajectory can be observed. One has r = (ln[(1 - 77)/(1 - 2n)]) -1 -~ ~o, ~ 0. Moreover d r ] d r / < 0 thus showing that r increases when ~7decreases. We remark that when the " when r/ = 2 At e < ~ it has probability v** > ~, Le. been proved b y one o f us [ 10] that the K o l m o g o r o v Sinai entropy of the Markov process W* is equal to that o f the deterministic evolution U*, thus giving a
8 July 1985
precise meaning to the statement that there is no loss o f information in the change o f representation U* I4/* due to the invertibility o f A. We note finally that from (7) we can see that the observable evolution is the deterministic one with probability v** = c perturbed with probability 3 1 = 1 - c by a term U * R 1- The former generates a transition probability Ql(X, 13) = U~13(x),Ql(X, ( a x } ) = 1, and the latter a transition probability Q2(x, B) = R _ 1 U~B(X) which is concentrated on the stable manifold XSt(ox) (see after formula (7)). The authors thank Professors I. Prigogine and B. Misra for many interesting discussions. They are also indebted to Professor I. Prigogine for his hospitality at the University of Brussels and to the International Solvay Institute of Physics and Chemistry for financial support.
References [1] B. Misra, I. Prigogine and M. Courbage, Physiea 98A (1979) 1. [2] B. Misra and I. Prigogine, Time, probability and dynamics, in: Longtime prediction in dynamics, eds. L.E, Reichl and A.G. Szebehely (Wiley, New York, 1983). [3] S. Goldstein, B. Misra and M. Courbage, J. Stat. Phys. 25 (1981) 111. [4] V.I. Arnold and A. Avez, Probl6mes ergodiques de la m6canique classique (Gauthier-Villars, Paris). [5] S. Martinez and E. Tirapegui, J. Stat. Phys. 37 (1984) 173. [6] S. Marfinez and E. Ti~apegui, Physica 122A (!983) 593. [7] E. Nelson, Duke Math. J. (1958). [8] S. Martinez and E. Tirapegui, Irreversible evolution of dynamical systems, in: Lecture notes in physics, Vol. 179, Prec. Sitges (1982), ed. L. Garrido (Springer, Berlin, 1982). [9] S. M~tinez and E. Tirapegui, Phys. Lett, 95A (1983) 143. [ 10] S. Martinez, Equilibrium entropy for stationary Markov processes, Prepfint Universidad de Chile MA-84-B-303 (August 1984).
83
PHYSICS LETTERS
8 July 1985
A POSSIBLE PHYSICAL INTERPRETATION OF THE A OPERATOR IN T H E P R I G O G I N E T H E O R Y O F I R R E V E R S I B I L I T Y S. M A R T I N E Z 1 and E. T I R A P E G U I 2,3 Facultb des Sciences, UniversitbLibre de Bruxelles, CP. 226, 1050 Brussels, Belgium Received 28 March 1985; accepted for publication 29 April 1985
We argue that if observations take a finite time At and if one can only distinguish points in phase space separated by distances greater than ¢then the A operator is determined by these conditions.
It is known since the work of Prigogine and collaborators [ 1 - 3 ] that one can associate to a class of dynamical systems, namely the K-systems [4] a Markov process going strongly to equilibrium. Let us consider a K-shift (I2,/~, ~ , o) where I2 = Sz, S = {rl, r 2 ... rs) is the alphabet,/a the invariant measure, q5 the product o-algebra and o: I2 --, I2, (ox)/=x/+ 1 is the shift transformation of the point x = (... X_l, x0, Xl , x 2 .... ) ~ [2, x i E S. In L 2 ~ ) the adjoint U* of the unitary operator U f = f o o , f E L 2 ~ ) , gives the evolution of probability densities p E q9 = {p: p(x) /> 0, f p dg = 1). We recall that we can def'me a distance in [2 by
d ( x , y ) = ~ 2-~nl dS(Xn,Yn) , nEZ where d s ( x n , y n ) = 0 if x n = Yn ~ S and one if x n 4=Yn (for notations and details see refs. [5,6]). Then the associated Marker process has an evolution operator W* associated to U* by a bounded, positive, invertible, nonunitary operator A : L2(~) -+ L 2 ~ ) by the formula W* = AU*A -1 [ 1 - 3 ] . Our aim here is to give a possible physical interpretation of this operator A. If we are given an initial 1 On leave of absence from Departamento de Matematicas, Facultad de Cieneias Fisicas y Matematicas, Universidad de Chile, Santiago, Chile. 2 On leave of absence from Departamento de Fisica, Universidad de Chile, Santiago, Chile. 3 Supported in part by Fondo Nacional de Ciencias and by PNUD project for Chile. 0.375-9601/85[$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
probability density Po(X') at time zero it will evolve at time one to p(x') = U*Po(X' ). Our first claim is that if Po(X' = x) 4=0 then one should also have Po(X") d=0 in points x" of the stable manifold XSt(x) of x, XSt(x) = {y : d(ony, onx) ~ 0, n -~ +,,o}. We arrive at this conclusion if we admit that an observation is done in a finite time At > 0 and with a precision e > 0 in space, (i.e. we can only distinguish points with distances greater than e) since then ff we say that the system is in x we have to admit that it can be in any point x " such that d(onx ", onx) < e, 0 < n ~< At. This means that in fact one does not observe Po(X') but rather an extended version ~'0(x') obtainable from P0 by the action of an operator A such that A/J{x}(X" ) 4= 0 for points x" E XSt(x) (~A = characteristic function of the set A C I2, ~{x} = 5-function centered at x), i.e. supp(A/j {x }) = XSt("x). This last equality is to be interpreted in the sense that points far away from x should be given very small (or zero) weight since they will take a long time to be within distance e o f x . We conclude then that the evolution we observe is P0 = AP0 -~ Pl = APl = It*P0, with W* given by W* = AU*A -1 if A is invertible~ The operators A and W must satisfy some obvious properties, namely they should both map the space ~ of densities into itself (A I> 0, 14/* t> 0, A* 1 = W1 = 1) and leave the constant functions invariant (A1 = It'* 1 = 1). We shall see that these requirements together with supp (Mj{x}) = XSt(x) are sufficient to determine the form of A. Let ot be the partition ot = {X/: i E S), X/= {x: x 0 81
PHYSICS LETTERS
Volume 110A, number 2
= i}, cB n the o-algebra o£,, = on_** aict, cB ** = cB, _ ** = {~, ~2}, R n = E:r~'n the conditional expectation with respect to the o-algebra cB ~, i.e. R n is the orthogonal projector on the space L~'~ CBn) of L20a) functions which are ~ n measurable, R** = id, R_** = projector on the constant functions, and E n = R n - Rn - 1 the orthogonal projector on L20alCBn) L20al°Bn_ 1). One has Z n e z E n +R_** = 1. It is easy to convince oneself that A is necessarily of the form
A--
..
one has
: An -
= 1 , x . = 0,
and {An) is a decreasing sequence in terms of which A can be written
A=
n~Z
X.en + R _ . .
(2)
In order that the observed evolution operator W* is positive we must still impose Vn = Vn - Vn+l >~ 0 with v n - knlAn+l which implies v_** = 1 and 0 <~ v,. < 1. At this point we have then recuperated the A operator of the Prigogine theory and we know then that the observed evolution W* is a Markov process converging strongly to equilibrium. We note for future use that if v** = c > 0 then for large n the sequence Xn is approximately of the form An = c n. We shall see now that the interpretation we propose of A gives supplementary information on the sequence {Xn). In order to do this we shall study ~{x)(X ) = A~(x)(X ) which is only different from zero for points x ' in a set of/a.measure zero, namely XSt(x). The operator A is doubly stochastic and hence generates a Markov transition probability~Q(x, B) = A/JB(X), B C c B , and it is easy to see that ~{x}(X') is just the generalized (in the sense that it is a distribution in x ' ) density of the transition probability Q'(x, B) of the reverse (adjoint) process [7], which is given b y Q'(x, 13) = A*~B(X ) = Q(x, B) since here A is self-adjoint. We specialize now to a Bernoulli shift with probability vector lr = (,rrl .... *rrs) and we study there Q(x, B). In a shift the stable manifold is the disjoint union XSt(x) = {x} W :KSt(x), where XSt(x) is the dis•
82
~(ff(X) = 0; ~ ~'2:.v] =xj,/>~k,y]_ 1 :#X]_l}.
~
t
t
(3)
Using the notation nk(X ) = IrxkWe find (the calculation is done with the same technique as in ref. [5]) O(x, {x}) = O, and
Qk = Q(x, )(~t(x)) = (/rk_l(X) -1 - 1)
u=0
nEZ
since this implies supp A/J(x ) = XSt(x) as well as the other properties of A (the condition Z-An = 1 is to have a total probability one in XSt(x), see after formula (4)). We have R n = Zi< n E/and defming An zj
joint union XSt(x) = U k e z x~t(x), where the (contracting) fibers :~¢t are
X ~
X.R., n~Z
:
8 July 1985
ffk_l(X)...ffk+u_l(X)~k_(k+u) ,
(4)
with Z k e z Qk = 1 (this being a consequence of Z ~ n = 1), which shows that indeed the probability measure Q(x, B) is concentrated in :KSt(x) as we wanted. Let us consider the case of the Baker transformation for simplicity (the discussion can be done in a general I Bernoulli shift). Then S = (0, 1), lr 0 = lr 1 = ~, and oo
Qk = Z) n=O
k e Z-
(5)
Using the distance function d ( x , y ) defined before we shall define an average distance d k of the fiber x~t(x) to x as dk = ~(5 z + 8~) with 8 ; = min d(x, x'), 8~ = sup d(x, x'), x ' e :K~t(x). One easily finds d k = 1, k 1> 1 and dk = 3 X 2 k - 2 < 1, k ~< 0. A first restric. tion on the sequence {An) appears immediately. Let e < 1, then due to omx~t(x) = ) ( ~ t m ( a m x ) we see that fibers :~[~t(x)'with k - At > 0 remain at time At at an average distance one of qAtx and we can give them zero weight, i.e. Qk = 0, k > At. Moreover if k - At ~< 0 the average distance o f : ~ t at time At will be 3 X 2 - ( a t - k + 2 ) , then i f k ' ~ At is the greatest value o f k ~< At such that 3 × 2 - ( a t - k ' + 2 ) ~ e we can take Qk = O, k > k'. From (5) we see that this can be realized taking7~ n = O, n > k', which means An = 1, n <~ - k ' . For n > k' we can take An = c n+k', c < 1, which according to our previous remarks modelizes the case v.. = c > O, i.e. exponential decreasing of the Xn for large n (one should note that it is in fact the large n behavior what is important in this construction). Let us consider for simplicity, the case k' = 0, i.e. An = 1,n ~<0, and, An = c n , n > O . ThenQn = O , n > O , and n
Q - n = ~(1 - c) .~_ ck]2 n - k , k=O
n >1 O,
(6)
Volume 110A, number 2
PHYSICS LETrERS
which is an increasing function o f n as it should be since the weight we give to points in XSt(x) should increase when their distance to x decreases. The invertible operator A is now given by A = (1 - c)Z n ~ 0 cnRn and the evolution operator W* o f the Markov process will be (Vn = Vn - Vn+l) [5] W*=vo.U*+~_IU*R
1,
(7)
with v.. = c, ~-1 = 1 - c > 0. The transition probability Qw(x, B) = W~B(X) o f the process is concentrated in the stable manifold XSt(ox) of ox and one has Qlc(x, Xff(ox)) = 0 if k / > 1, Qle(x, )(S_tk(OX)) = 2 - ( k + l ) 7_ 1 if k >i O, Qlv(x, { o x ) ) = c. We now can calculate using (6), the mean distance ¢d) of the points in the stable manifold XSt(x) to x as
(d) ~ dk Q-k = (l - c)/(2 - c ) . k=O
(8)
After the time At which lasts the observation this distance is reduced by a factor 2 - A t (this is due to o m Xff(x) = x~ctm(omx) and d g = 3 X 2 k - 2, k ~< 0), and we can Fm our parameter c b y asking 2 - A t (d) = e which gives for c the value c = (1 - 2r/)/(l - "0),
r/=
2ZXte,
(9)
where r~ measures the lack of precision of our observation. If r/~ 0 then c -~ I and we recuperate from (7) the deterministic evolution. We recall that we have proved [8,9] that vt = c t is the probability o f observing the deterministic trajectory up to time t, then writing c = e - l / r we define the time z up to which the classical trajectory can be observed. One has r = (ln[(1 - 77)/(1 - 2n)]) -1 -~ ~o, ~ 0. Moreover d r ] d r / < 0 thus showing that r increases when ~7decreases. We remark that when the " when r/ = 2 At e < ~ it has probability v** > ~, Le. been proved b y one o f us [ 10] that the K o l m o g o r o v Sinai entropy of the Markov process W* is equal to that o f the deterministic evolution U*, thus giving a
8 July 1985
precise meaning to the statement that there is no loss o f information in the change o f representation U* I4/* due to the invertibility o f A. We note finally that from (7) we can see that the observable evolution is the deterministic one with probability v** = c perturbed with probability 3 1 = 1 - c by a term U * R 1- The former generates a transition probability Ql(X, 13) = U~13(x),Ql(X, ( a x } ) = 1, and the latter a transition probability Q2(x, B) = R _ 1 U~B(X) which is concentrated on the stable manifold XSt(ox) (see after formula (7)). The authors thank Professors I. Prigogine and B. Misra for many interesting discussions. They are also indebted to Professor I. Prigogine for his hospitality at the University of Brussels and to the International Solvay Institute of Physics and Chemistry for financial support.
References [1] B. Misra, I. Prigogine and M. Courbage, Physiea 98A (1979) 1. [2] B. Misra and I. Prigogine, Time, probability and dynamics, in: Longtime prediction in dynamics, eds. L.E, Reichl and A.G. Szebehely (Wiley, New York, 1983). [3] S. Goldstein, B. Misra and M. Courbage, J. Stat. Phys. 25 (1981) 111. [4] V.I. Arnold and A. Avez, Probl6mes ergodiques de la m6canique classique (Gauthier-Villars, Paris). [5] S. Martinez and E. Tirapegui, J. Stat. Phys. 37 (1984) 173. [6] S. Marfinez and E. Ti~apegui, Physica 122A (!983) 593. [7] E. Nelson, Duke Math. J. (1958). [8] S. Martinez and E. Tirapegui, Irreversible evolution of dynamical systems, in: Lecture notes in physics, Vol. 179, Prec. Sitges (1982), ed. L. Garrido (Springer, Berlin, 1982). [9] S. M~tinez and E. Tirapegui, Phys. Lett, 95A (1983) 143. [ 10] S. Martinez, Equilibrium entropy for stationary Markov processes, Prepfint Universidad de Chile MA-84-B-303 (August 1984).
83