OF
COUNTABLE MODELS NI-CATEGORICAL THEORIES BY
MICHAEL MORLEYO)
Countable models of ~rcategorical theories are classi...
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OF
COUNTABLE MODELS NI-CATEGORICAL THEORIES BY
MICHAEL MORLEYO)
Countable models of ~rcategorical theories are classified. It is shown that such a theory has only a countable number of nonisomorphic countable models.
A theory (formulated in the predicate calculus) is categorical in power r (rcategorical) if it has a model of power x and any two such are isomorphic. In [3] I proved the conjecture of Los' that a theory categorical in one uncountable power is necessarily categorical in every uncountable power. The example of the theory of algebraically dosed fields of characteristic 0 shows that such a theory need not be No-categorical. However, one might expect that the isomorphism types of countable models of such theories could be classified in some particularly neat fashion. In the example mentioned the isomorphism type can be characterized by the number of algebraically independent elements. Thus, it is possible to find an increasing sequence of length t9 + 1 of models:
0~o ~ ¢~1 -~ "'" --- %o such that every countable model is isomorphic to some member of the sequence and, for each n, %+1 is the "next larger" model than %. The results of this paper, together with a more recent result of Marsh [2], show that such a sequence of models can be found for every theory which is N1- but not No-categoricaL In every known instance of such a theory no two members of this sequence are isomorphic. It is an open question whether this is true in general. Indeed, it is not known whether a theory which is NI- but not N0-categorial must have an infinite number of isomorphism types of countable models. We have adopted the following compromise with respect to prerequisites. The definitions and statements of theorems assumes only a basic knowledge of model Received August 3, 1966. (1) The author was partially supported by NSF grants GP-1621 and GP-4257 dtLring the period those results were obtained.
65
66
MICHAEL MORLEY
[April
theory but an understanding of the proofs requires an understanding of the methods of [3]. Some of the results of this paper were announced in [4]. 1. Preliminaries. A relation system, 92, is a set A, the universe of 92, together with an indexed set of finitary relations, finitary functions and distinguished elements. (We shall adopt the now common convention of denoting relation systems by Gothic letters, 92, and their universe by the corresponding Roman letter A.) Two relation systems are similar if they have the same index set and corresponding relations and functions have the same degree. For similar systems one defines 92 a subsystem of ~3 (92 _~~3) and 92 isomorphic to ~3 (92 ~ ~3) in the obvious fashion. Corresponding to each similarity type of relation systems there is a first order with identity language having relation symbols, function symbols, and individual constants corresponding to the relations, functions and distinguished elements of the relation systems. We shall always assume that this language is countable. The reader is presumed familiar with the notions of formula and sentence in such a language and what it means for a sequence of elements of 92 to satisfy a formula in 92 or for a sentence to be valid in 92. A relation system 92 is a model of a set of sentences if each of the sentences is valid in 92. A complete theory (in a given language) is a maximal consistent set of sentences in that language. For each relation system 92 there is a complete theory, Th(9.l), consisting of all sentences valid in 92. We write 92-~3 to indicate Th(92) = Th(~3). If 92 is a relation system and X _ A we denoted by (92,x)x~x the new relation system formed by taking the elements of X as distinguished elements. The language corresponding to (92,x)x~ x differs from that corresponding to 92 by the addition of new individual constants corresponding to the elements, of X. If the letter a denotes an element of X we shall denote the new individual constant by bold face a. Suppose 9~ and ~B are similar relation systems, X _ A, and f is a function defined on X into B. Then f is an elementary map if (92,x)~ x = (~,f(x)):~ x. In particular, if 92~B and the identity map on 92 into ~ is an elementary map then 92 is an elementary subsystem of ~ (92 --95 and ~ has power ~1- ~B is equal to the union of an increasing chain of countable models {~3~;~ >col}; so it is isomorphic to the union of a subsequence of length to i of the 95-tower. That is, ~3 is isomorphic to 95,0lFinally, suppose ~3 and ~3' are two models of T of power 1'¢1. From the results of the preceding paragraph it follows that there must be countable models 9/ and 95' such that ~B and ~3' are isomorphic to the last member of the 95-tower and 95'-tower respectively. There exists a countable model ~ such that both 95 and 9~' may be mapped by elementary maps into ~ (cf. lemma 1.2 of [5]). So isomorphic images of ~ must appear in both the 95-tower and ~'-tower. Then ~3 and ~3' are both elementary extensions of isomorphic images of so they are both isomorphic to the last member of the g-tower. This shows that T is Nl-categofical and Theorem 1 is proved. Trmov,~M 2. Suppose T is Nlcategorical and 9~ a countable model of T.
1967]
COUNTABLE MODELS
69
Then every prime elementary extension of 9~ is also a minimal elementary extension of 9~ and a prime elementary extension is therefore unique up to an isomorphism leaving 9~fixed. Proof. Suppose 9/had a minimal elementary extension E and ~3 was a prime elementary extension of 9;[. By definition there would be an elementary map of ~3 into ~ which was the identity on ~. But this map must be onto ~ for otherwise would not be a minimal extension. Thus, in order to prove the theorem it will suffice to show that 9/has a minimal elementary extension. Consider the prime elementary extension ~3 of 9/constructed in the proof of Theorem 1. Suppose it were not a minimal elementary extension. Then there would be some model "interpolated" between ~ and ~3. But since ~3 is a prime extension of 9.[ there would be an elementary map f of ~3 into this interpolated model such that f is the identity on 9)[.A fortiori, the image o f f is not all of ~3. To complete the proof we shall show that the assumption that such an f exists leads to a contradiction. The argument is somewhat lengthy. So, as in the proof of Theorem 1, assume that ~3 >- ~ are countable models of Nl-categorical theory T, p is the only limit point in some neighborhood of S(A), b ~ realizes p, and ~3 is a model prime over X = A [,.J{b}, f is an elementary map of ~3 into a proper subsystem of itself and f is the identity on ~. Let ~3' =(~3,x)x~x and T ' = Th(~3'). The theory T' is Nl-categorical since X is countable and every model of T of power N1 is saturated. The theory T' is not No-categorical since by a result of Ryll-Nardzewski [6] no No-categorical theory can have an infinite set of distinguished elements. By definition ~3' is a prime model of T'.A theorem of Vaught 17] says that a prime model of an N~-categorical but not No-categorical theory is also a minimal model so ~3' is a minimal model of T'. Denote by ~ the image of~B under f. I assert that b ¢~. For if b e(~, then X ___C and E' = (~, x)x ~x would be a proper elementary subsystem of ~B'. In particular, letting f ( b ) = c we have c # b. Since f is the identity on A, c realizes the same point p ~ S(A) as b does. Indeed ~ is the prime and minimal model of T " = Th((~,x)x~). where Y = A L J { c }. Since x ~ 3 and ~3 is prime over X, c must realize an isolated point is S(X). That is the formulas satisfied by c in ~3' are exactly the formulas of a principle dual ideal determined by a formula ff(Vo). Of course, ~/ is a formula in the language of the theory T'; but this language differs from the language of the theory T only in the addition of new individual constants. Thus there is no loss of generality in assuming there is some finite sequence a2, ..., a n in A such that ~b(v0)= ~(vo,b , a2, "",an) where ~(Vo, ..., vn) is a formula in the language of the theory T. For typographical convenience we shall henceforth show only the first two arguments of ~b explicitly, viz. ~(Vo, Vl). = ~k(Vo, Vl,. a2, "', an). I assert that if d E ~ and satisfies in ~ the formula ~(e, %) then d does not realize the same point p e S(A) which is realized by b. For suppose it did. Then the
70
MICHAEL MORLEY
[April
mapping of A U (b, c} onto A U {d, c} which is the identity on A LJ {c} would be an elementary map. Since ~ is prime over A U {b} it is also prime over A U {b, c}, so there would be an elementary map g : ~ ~ ff which is the identity on A U {c}. But g maps ~ onto a proper subsystem of itself and so the same argument given above to show that f ( b ) ~ b shows that g(c)~ c, a contradiction. On the other hand, c does realize the point p in S(A) and hence c has all the same first order properties with respect to 9~ that b does. In particular, any element satisfying ~(Vo, e) must realize p. Combining this with the result of the last paragraph we have the following anti-symmetry property, X(Vo), satisfied by c;
X(Vo) = (vl)(¢(Vo, vl) --, -7¢(vl, Vo)). Let U be some neighborhood of p in S(A) which contains no other limit point. Then there is some formula p(vo) such that U consists of all prime dual ideals containing p. (Of course, p may involve individual constants corresponding to the elements of 9~). The formula:
X(Vo)& p(vo) & g/(e, Vo) is satisfied by b so, since ~- n implies ~(d/, dn) and z(dn). This says that ~ determines a linear ordering of the dn's which contradicts theorem 3.9 of [3]. Theorem 2 is now proved. 3. The number of countable models. Suppose T is Nt-categorical. Vaught has proved that T must have a prime model, 9.L As in the proof of Theorem 1 we may construct an 9A-tower {9~ ;a < o~1} such that ~[o = ~, 9A~+1is the prime elementary extension of 9A~and for non-zero limit ordinals ~, 9~6= [,.J~ 0. Vaught [7] has also shown that no complete theory T can have exactly two isomorphism types of countable models. In doing so he proves that: if T is not No-categorical then no model
1967]
COUNTABLE MODELS
71
prime over a finite set is saturated. From this it follows that in our tower of models, 9/, is not saturated for any n e o9. On the other hand, Vaught has also shown that there must be a countable saturated model of T. So there is a countable ~ > o9 such that 9i~ is saturated. Since the saturated model is isomorphic to a proper elementary subsystem of itself there is a countable fl such that 9~= --- 9.I~+p. By induction therefore 9~=+~ ~ 9i=+#+~ for all countable 7. In particular, ! assert that 9I= ~ 9I=+p, for all countable 3. The proof is by induction on 3. The only difficulty in the induction occurs at limit ordinals; there we use the fact [5] that the union of a countable elementary chain of countable saturated models is a countable saturated model. Thus from 0t on the tower repeats isomorphism types with period ft. We have proved: THEOm~M 3. I f T is Nl-categorical but not No-categorical then the number of isomorphism types of countable models is countable, i.e.,finite or denumerably infinite. There are no examples known where the number of isomorphism types is finite, but whether any such exist is an open question. We may sharpen the results of the last theorem by: THEOREM 4. I f T is an N 1- but not No-categorical theory and {9~;~t < o91} the tower described above, then 9~ is saturated for every limit ordinal fi > O. Proof. We shall prove something slightly stronger, namely, if t~ is a limit ordinal and ~ < t5 then every point of S(A~) is realized in 9ia. To do this is it sufficient to prove the following: If {~,; n ~ co) is an increasing elementary chain of countable models of Tthen every point of S(Bo) is realized in ~,.J, ~ , ~ , . From [3] we know that S(Bo) is countable and hence must contain a point Po which is the only limit point in some neighborhood. This point must be realized in ~ . As discussed in [3] the identity map i: ~ o ~ induces a continuous, onto projection i*:S(B1)~S(Bo). Since S(B1) is countable and compact i*-l(po) must contain some point p~, which is the only limit point in some neighborhood of S(B~). Then there must be some element of ~B2 which realizes p~ and therefore also realizes Po. Proceeding inductively we may find for each n ~ to a p. ~ S(B,) and an element b , e ~.+~ such that b. realizes p, and p,+~ projects onto p.. By Lemma 4.4(a)(i) of 1-3] there is an no e o9 such that all the p,'s with n > no have the same transcendental degree and rank. It is shown in the proof of theorem 4.6 of [3] that this implies that the corresponding b,'s are indiscernible over B o . Thus, U . ~ , ~ , is a model of Tcontaining an infinite set of indiscernibles over Be. It is shown in Theorem 5.4 of [3] that underthese circumstances if some q~ S(Bo)were not realized in ~ , ~ , ~ , then T would have an uncountable model which is not saturated. But this is impossible by Theorem 5.5. While this paper was in preparation the following theorem was proved by Marsh [2].
72
MICHAEL
MORLEY
TrmOREbi. I f T is Nt-categortcal then every elementary extension of a saturated model is saturated. Combining this with Theorem 4 we see that in our tower of countable models they are all saturated and hence isomorphic to each other from the coth step on. The remaining open question is whether the first co models of the tower must all be distinct isomorphism types. In every known example of NI- but not Nocategorical theories they are distinct. Suppose T is an N1- but not N0-categodcal theory in a countable language L, L' a language which extends L by the addition of a countable number of individual constants and T ' a complete extension of T in L'. Then T ' is also N l- but not No-categorical. Marsh [2] has proved the following theorem, THEOREM. I f T is N~- but not No-categorical then there is a complete extension T" of T in a language which extends the language of T by only a finite number of new individual constants such that in the tower of countable models of T' the first co models are of distinct isomorphism types. REFERENCES 1. J. Los', On the categoricity in power of elementary deductive theories, Colloq. Math. 3 (1954),5-62. 2. W. Marsh, On O~rcategorical but not co-categorical theories, Doctoral dissertation, Dartmouth College, June, 1966. 3. M. Morley, Categoricityin power, Trans. Amer. Math. Soc. 114 (1965), 514-538. 4. M. Morley, A conditionequivalentto Nl-categoricity,Notices Amer. Math. Soc. 11 (1964), 687. 5. M. Morley and R. L. Vaught, Homogeneous universal models, Math. Scand. 11 (1962), 37-57. 6. C. Ryff-Nardzewski, On theories categorical in power No, hr(m) < hT(NO).
THEOREM B. Assume the GCH (generalized continuum hypothesis). If N1 0, and assuming the GCH, hr(m) > 0 for all infinite m. Some other results on homogeneous models are given in [4]. For results on the total number of models of T of a given power see [5], [6], and [9]. In this paper we shall use 9.I,~3,~ to denote models with universe sets A, B, C. T(9.0 stands for the complete theory of 9~, that is, the set of all sentences of our ReceivedNovember 23, 1966. (*) This work was supported in part by NSF contracts GP 4257 and GP5913. 73
74
H. JEROME KEISLER AND MICHAEL D. MORLEY
[April
given logic L which are true of 9"[. If a is an a-termed sequence of elements of A, then (9'[2,a) is a model for the language L with ~ additional individual constants; in this case, T(92, a) is called the type of the sequence a in 9"[. We shall be especially interested in types of finite sequences of elements, and we let S(9~ be the set of all types of finite sequences o elements of A, S(92) = {T(92, a): a ~ I,.Jn [ U , , A , I . Then using the fact that the ~p are homogeneous, we may choose the isomorphisms fp so that
The union of all the fp is the desired isomorphism from ~ to an elementary submodel of ~.
Lv.u.v_x 3. I f 9.[,~ are homogeneous, and ~ are isomorphic.
l al = IBI,
and S(9.D = S(fB), then
Proof. Similar to Lemma 2. Going back and forth between 9.[ and ~ , an isomorphism can be constructed with exhausts both A and B. LEM~L~ 4. For any complete theory T and any infinite cardinal m, hr(m) _- No. The theory of a model (A, Co, c~, c2, -.-) where the c~ are distinctconstants is a T
with
hT(No) =
No, hr(Nl) = i.
W e do not know whether there is a complete theory T which does any of the following: hr(No) = No, I < hz(N1) < No. hr(No) = Ni, hT(Ni)< No. hT(NO) > No and hr(N1) > hz(N2),
hr(N2) > hT(N3). REFERENCES
1. W. Hanf, Some fundamental problems concerning languages with infinitely long expressions, Doctoral dissertation, Berkeley, 1962. 2. B. J'6nsson, Homogeneous universal relational systems. Math. Scand. 8 (1960), 137-142. 3. C. Karp, Languages with expressions of infinite length. Amsterdam, 1964. xix q- 183 pp. 4. H. J. Keisler, Some model-theoretic results for to-logic. Israel Journal of Math, to appear. 5. M. Morley, Categoricity in power. Trans. Amer. Math. Soc. 114 (1965), 514-538. 6. M. Morley, Countable models of Rl-categorical theories. Israel Journal of Math. This issue, pp. 65-72. 7. M. Morley and R. L. Vaught, Homogeneous universal models. Math. Scand. 11 (1962), 37-57. 8. A. Tarski and R. L. Vaught, Arithmetical extensions of relational systems. Compositio Math. 13 (1957), 81-102. 9. R. L. Vaught, Denumerablemodels of complete theoriespp. 303-321, in Infmitistic Methods, Warsaw 1961. UNIVERSITYOF WISCONSIN) MADISON)WISCONSIN
A CHARACTERIZATION OF THE SIMPLE GROUPS PSL (2, p), p > 3 BY MARCEL HERZOG
ABSTRACT Let G be a finite group, containing a self-centralizing subgroup of prime order p. If G is non-solvable, contains more than one class of conjugate elements of order p, and satisfies an additional condition, then G is isomorphic to PSL (2,p), p > 3.
Introduction. The purpose of this paper is to prove the following THEOREM. Let G be a finite group containing a cyclic subgroup M of prime order p and satisfying the following conditions: (i) C6(m)~_M for all m E M ~ (ii) INn(M): 3/1] ~ p - 1 (iii) I f z E M # and xy = z, where xP = yV= 1, then x e M, except possibly in the case that both x and y are conjugate to z - 1 in G. Then one of the following statements is true. (I) G is a Frobenius group with M as the kernel. (II) There exists a nilpotent normal subgroup K of G such that: G = No(M)K, K ~ No(M ) = 1.
(III) G is isomorphic to PSL(2,p), p > 3. As an immediate consequence of the theorem we get the following characterization of the simple groups PSL(2, p), p > 3, which are known to satisfy the assumptions of the theorem. COROLLARY. Let G be a finite non-solvable group containing a cyclic subgroup M of prime order p which satisfies conditions (i)-(iii). Then G is isomorphic to PSL(2, p) and p > 3. Conditions (i) and (ii) certainly exclude the case p = 2, and ff p = 3 they allow only the trivial situation Na(M) = M, thus forcing G to be of type (II). However, Received December 21, 1966. 79
80
MARCEL HERZOG
[April
the case p = 3 was investigated by W. Feit and J. G. Thompson in [3], under the single assumption (i). They classified the groups in question and proved that if G is a simple group, then it is isomorphic to either PSL(2,5) or PSL(2,7). If no exceptions are allowed in condition (iii), then it follows from [5], Theorem 5 that G is either of type (I) or of type (II). Groups G containing a subgroup M of order p which satisfies condition (i) were studied by R. Brauer in [1]. Among other results he proved that if G = G' and [G: No(M)l < I~P + 3)/2 + 1 then G is isomorphic either to PSL(2,p), p > 3 or to PSL(2, p - 1), where p - 1 = 2n, n > 1. In our proof this result serves as the concluding argument. The methods of this paper are similar to those applied in [4] and [5], which rely heavily on the work of W. Feit [2]. But in the present case the results of Brauer [1] are available, simplifying the necessary notation as well as many of the arguments. We therefore repeat the necessary definitions from [4] and [5] (not always identically) and prove everything except for the results form [1] and [2], which are summarized. Consequently, this work can be read independently of
E41 and [51. If Tis a subset of a group G, Co(T), No(T), [ T[ and T # will denote respectively: the centralizer, normalizer, number of elements and the non-unit elements of T. The subscript G will be dropped in cases where it is clear from the context which group is involved. The commutator subgroup of G will be denoted by G', and 1 will be the notation for the trivial subgroup. Proof of the Theorem. It will be assumed that G satisfies the assumptions of the theorem, but is not of type (I) or (II). It suffices to show that G satisfies (III). M is certainly a trivial -intersection- set in G and it follows easily from (i) that M is a Sylow p-subgroup of G. Since G is not of type (II), No(M) ~ M. Thus the results of W. Felt [2, §2] and R. Brauer [1, pp. 59-60] are applicable, and the relevant ones will be summarized below, together with the corresponding notation. As N = No(M) ~ M, N is a Frobenius group with M as its kernel and it is well known that there exists a subgroup Q of N such that:
N=QM,
Q~M=I.
Let the order of Q be q; then q divides p - 1, and t = (p - 1) ]q is the number of conjugate classes 0~ of elements of order p in N. Since M is a Sylow subgroup of G, t is also the number of conjugate classes C~ of elements of order p in G and 0t = C~ n M after rearrangement, if necessary. Let ml, ..., mt be a set of representatives of ~ , i = 1, ..., t; they also represent the Cz, i = 1, ..., t. It follows from the Sylow Theorem that order g of G can be expressed by the formula g = qp(np + 1). As G is not of type (I) n > 0 and consequently (1)
g > qp2.
1967]
A CHARACTERIZATION OF THE SIMPLE GROUPS PSL(2,p) p > 3
81
The irreducible characters of N fall under two categories. The first one consists of t characters ~, i = 1, ..., t of degree q, vanishing outside M. The second category consists of q linear characters which contain M in their kernel. It follows that ~,
~ ( m i ) ~ ( m f 1) = ~ijP -- q
Z
~s(mi)=-I
where 1 < i, j < t and the summation ranges over s = 1, ..., t. The index of summation s will have the above meaning throughout this paper. The exceptional characters of G associated with ~i will be denoted by X,, i = 1,..-,t. We have: Xi(1) = x = (wp + ~)/t where w is a positive integer and 6 = + 1; hence x > q. Also: X t ( m ) = e~i(rn) + c for all m e M #,
i = 1,..., t
where c is a rational integer and e = + 1. The non-exceptional irreducible characters o f G non-vanishing on M # will be denoted by R~, i = 1, ..., q. Each of these characters is constant on M # , the values being either 1 or - 1 . Let R~(1)= r i and let R l ( m ) = ci for all m e M #. Then c i = + 1 and ri - ci (mod p). R1 will denote the principal character of G. Since all the remaining irreducible characters of G vanish on M #, none of them is linear; hence [G: G'] < q + t. We will need also the following inequalities. It follows immediately from the fact that if ci = - 1 then r~ > p - 1 that q
(2)
s = X
c3~ /r, >= 1 -- (q -- 1)/(p -- 1).
t=l
Suppose that c~ = - 1 ,
i = 2,...,q. Then: t
0 =
Z
q
X~(ml)x +
t=l
Z
c,r
1=1 q
= x ( t c - 8) + 1 -
Z
1"t
1---2
and therefore tc - 8 ~_ O. Thus if tc - ~ < 0 then at least two c~ are equal to 1. Consequently (3)
S > I - (q - 2)/(p - I)
if
tc - 8 < 0
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MARCEL HERZOG
[April
Let siih, 1 < i,j, k < t denote the coefficient of ~g in ~ j and let cok, 1 < i,j,k < t denote the coefficient of Ck in C~Cj. Then it is well known that for all 1 =< i , j , k = < t
(4)
Sijk = (qp /p2) (Buk + q) = (q /p) (Bij k + q)
(5)
ctlk = (g /p2) (A~jk + S)
where
B~s~ = O/q) Z$
~ ( m 3 ~ ( m l) ~ ( m Z 1 )
A,jk= (l/x) X$ X~(mi)X,(ml)Xs(mk 1 ). Let finally t~(i,j, k) -- ~ik + 6j~ + t~tj. 1 < i,j, k < t tc 3 - 3c2~ - 3cq
K=
E -- {(i,j, k) l 1 _- 3
83
Finally (8) follows from:
p=lC~(ml)] = Z
$
X~(mi)Xs(m ~ 1) + q
Zdt (e~+(ml) + c)(e~(mT') + c) + q = p-q--2ec+t&+q.
LEMMA 2.
and G ' = G .
q=(p--1)/2
Proof. Suppose that q < ( p - 1)/2; then t > 3 and by (8) c = 0. Hence by (6) and (7) all B,~k, (i,j, k)eE, satisfy the same linear equation:
qBuk + q2 = g(e.qBijk + XS)/px.
(9)
Since q = geq/px would imply that g = px p - 1 - (q - 1) = p - q and p > 3q, it follows that: g < p2q2/( _ q + p _ q) < p~q
in contradiction to (1). Thus q = ( p - 1)/2. As rG: G'] ___q + t < p, the order of G' is of the form q'p(np + 1). That follows from the fact that the number of Sylow p-subgroups of G' equals to that of G. If G' satisfies (I), then obviously G satisfies (I), in contradiction to our assumptions. If G' is of type (II), then the normalizer N1 of M in G' has a nilpotent normal complement K in G'. Let F be the Fitting subgroup of G'; then dearly K ~ F and M n F = I . Since G ' = N 1 K , F = ( N I t ~ F ) K . Let x ~ N ~ n F , m e M # ; then x - lm- ~xm e M n F = 1, hence x E CG(m) t~ F = M • F = 1. Thus F = K and K is characteristic in G', hence normal in G. It follows that G is of type (II), again a contradiction. Therefore G' satisfies the same assumptions as G does, and consequently by the first part of this proof q' = ( p - 1 ) / 2 = q, G ' = G.
LEMMA 3. I f q = ( p - - 1 ) / 2 is odd, then: (10)
p2(p _ 3)x g=-2(q+l)+4xS
'
2¢ - 8 = - 1.
84
MARCEL HERZOG
If q = (p-
1)/2 is even, then: p 2 ( p _ 1)x g=-2q+4xS '
(11) Proof.
[April
2 c - e = 1.
By (6), (4), (5), and (7) for all (i,j, k ) e E
(12)
qB~jk + q2 = g(eqBuk + K + 6 ( i , j , k ) c p + x S ) / p x .
As t = 2, it is easy to check that if q is o d d then 6(i,j, k) = 2 for all (i,j, k) ~ E a n d if q is even then 6(i,j, k) = 1 for all (i,j, k) ~ E. Since q ~ g s q / p x , in each case all t h e B~j~ are equal to each other for all (i,j, k ) ~ E; so are the corresponding St~k. Thus if q is odd q = 1 + s122 dr" S 2 2 2 , ( q B 1 2 2 - 1 -
q2)/p = S122 = ( q - - 1)/2 = (p - 3)/4
and (12) yields p2(p -- 3)X g = e[p(p -- 3) -- (p -- 1) 2] + 4 K + 8cp + 4 x S " I f q is even, then: q = s212 + s112, (qB1~2 + q2)/p = s112 = q / 2 = (/7 - 1)/4. Consequently, (12) yields: p 2 ( p _ 1)x g = e[p(p - 1) - (p - 1) 2] + 4 K + 4cp + 4xS" We will n o w show that (11) holds; the p r o o f of (10) is similar and it is left to the reader. It suffices to show that: e(p-1)+4K+4cp=-2q=l-pand2c-8=l. N o w K = 2c a - 3c28 - 3cq and c 2 = c8; hence: 4 K = - 4e28 - 12cq = - 4c28 - 6cp + 6e = 2c - 6cp and n(p - 1) + 4 K + 4ep = (8 - 2c)p + (2c - 8). Thus it suffices to show that 2c - ~ = 1. But (2c - 8) 2 = 4c 2 - 4c8 + 1 = 1 and consequently it remains to prove that 2c - 8 # - 1. Suppose that 2c - 8 = - 1 then: g = p 2 ( p - 1)x p 2 ( p - 1) p - 1 ~'4"~S =< 4[1 - ( q - 1)/(p
1)] < (p - 1)p2/2 = qp2
in contradiction to (1). T h e p r o o f of the l e m m a is complete.
1967]
A CHARACTERIZATION OF THE SIMPLE GROUPS PSL (2,p) p > 3
85
We will continue now with the proof of the theorem.Lemmas 2 and 3 yield, in view of (2), (3) and the fact that x >=q, that: p 2 ( p _ 1) = ( p - 1)2p 2/4. g - [" -- 2(q + 1)/q],+ 411 - (q - 2)/(p - 1)]
As g = ( p - 1)p(np + 1)/2, it follows that n < ( p - 1)/2. Consequently by Brauer [1, Corollary, p. 70] either G is isomorphic to P S L ( 2 , p ) , p > 3 or it is isomorphic to P S L ( 2 , p - 1), where p - 1 = 2m> 2. Since q = ( p - 1)/2, the second case may occur only if q = 2, p = 5. But PSL(2, 4) is isomorphic to PSL(2,5); hencep > 3 and G is isomorphic to P S L ( 2 , p ) for allp. The proof of the theorem is complete. REFERENCES 1. R. Brauer, On permutation groups of prime degree and related classes o f groups, Ann. of Math. 44 (1943), 5.5-79. 2. W. Fcit, On a class of doubly transitive permutation groups. Ill. J. Math. 4 (1960), 170--186. 3. W. Feit and J. G. Thompson, Finite groups which contain a self-centralizing subgroup of order 3. Nagoya J. Math. 21 (1962), 185-197. 4. M. Herzog, On finite groups which contain a Fri~benius subgroup. To appear in Journal of Algebra. 5. M. Herzog, A characterization o f some projective special linear groups. To appear. UNIVERSITYOFILLINOIS, U ~ A , ILLINOIS
ANALYTIC ITERATION AND DIFFERENTIAL EQUATIONS* BY
M E I R A LAVIE ABSTRACT
In this paper we study some mapping properties of analytic iterations W(a, z). Our purpose is to establish a sufficient condition for W(a, z) to be conformal and univalent in z for z e D, where D is a given domain and for sufficiently small I a I. To this end we consider the differential equation a W(a, z)/aa = L[W(a, z)] with the condition W(O, z) = z. A sufficient condition for the solution W(a, z) of this system to be conformal and univalent in D for I a I < a0 (for some a0 > 0), and to satisfy the iteration equation, is established.
1. Introduction and plan. We are concerned with functions in a and z, which satisfy the iteration equation
(1)
W(a, z) analytic
W[a, W(b, z)] = W(a + b, z),
with
(2)
W(O,
z) = z,
for z eD, where D is a given domain, and for sufficiently small [a [,[bland
[a+b[. Putting
@W(a,z) [
(3)
aa
= L(z),
a=o
it is known [31 [2], that the function W(a, z) satisfies simultaneously the following three differential equations: (4)
~W(a, aa z) = L[W(a, z)],
(5)
OW(a,z) OW(a,z) L(z), Oa = az
and hence also: Received December 27, 1966. * This paper is based on a part of the author's thesis towards the D.Sc. degree, under the guidance of Professor E. Jabotinsky at the Teclmion Israel Institute of Technology, Haifa.
86
1967]
ANALYTIC ITERATION AND DIFFERENTIAL EQUATIONS
87
OW(a,z) L[W(a,z)] Oz
(6)
=
L(z)
Evidently it follows from the definitions, that if the mappings W(a, z) (for sufficiently small [ a [ ) map D conformally on Da, then L(z) is necessary regular in D. Our main purpose is to establish a sufficient condition for the mappings W(a, z) to be conformal in D for [a [ < ao, for some ao > 0. Next we ask about the sufficient condition for the mappings W(a, z) to be univalent in D. It turns out that the answer to both questions is the same, namely: If L(z) is regular and single-valued in the closure b of the domain D, with a double pole at most at z = oo (in the case oo ~/)), then W(a, z), for sufficiently small [a [, map D conformally and univalently onto Da. Now, let W(a, z) (for z ~ D) be a single-valued analytic function of a for a E A, where A is a bounded domain in the a plane including the origin. Moreover, let W(a, z) satisfy equations (1) and (2) for z ~ D and a, b, a + b ~ A. Consider now the mapping of D given by W(a*, z), where a* ~ A. If there exists a continuous curve C c A connecting a* with the origin and such that for every a ~ C, L(z) is regular and single-valued in/3o (with a double pole at most at z = oo), then W(a*, z) maps D conformally and univalently onto
Da*. In the following we assume that it is the function L(z), rather that W(a, z), that is given, and we shall use the differential systems (4) and (2) or (5) and (2) to generate the function W(a, z), which is obtained as the solution of either system. We shall prove that this solution, W(a, z), satisfies equation (1) and is conformal and univalent in D for I a] < a o for some ao > 0. We do not treat the differential equation (6), as this has been done (at least in the special case when L(0) = L'(0) = 0) in [2], but we obtain Theorem 1 of [2] as an immediate corollary of our Theorem 1. We first suppose D to be bounded and later the results are extended, with some modifications, to the case of an unbounded domain D. 2. The ease of a bounded domain D. We consider the differential equation (4) with the initial condition (2). Equation (4) can be regarded as an ordinary differential equation and we may apply the classical existence and uniqueness theorem
[q: Let L(W) be a regular and single-valued function in the neighborhood [ W - z[ < p(z) of the initial value z, then there exists one and only one analytic function W(a,z) which is regular for [a I < ao(z) and such that: (i)
W(O,z)= z.
(ii)
Iw(a, )-zl
for [a[ 0. But both functions are regular for
90
MEIRA LAVIE
[April
l el < ao, ao > 0 and they coincide in the disk I c - b * ] < ~o. Hence, by the monodromy theorem, they coincide everywhere, and in particular for c = 0. Thus, W(O, z l ) = W(0,z2) which by (2) implies zl = z2. RE~4ARK. The estimate ao = R/M(D, R) obtained by Picard's method is sharp in the sense that it cannot be improved by a multiplicative constant. Indeed, let L(W) = W -1In n = 1 , 2 , 3 , . . , and let D be a bounded domain contained in the half plane Re{z} > 1, with z = 1 as a boundary point. By solving equation (4) with condition (2) we obtain:
W(a,z) = z {1 +
(n + 1)a2-(n+ l)/n[ nl(n+l) n
-j
, where 1n/(*+lJ = 1.
W(a,z) is regular for lal < n/(n + 1)lzl 1, we use induction as follows. Let v be any point of a tree G of order p with deg v = 1, and let u be the point adjacent with v. Ifdeg u 2, we observe that the graph Go obtained from G by the removal of v is a tree of order p - 1. The points a and b of Go whose existence has been hypothesised are also effective for G. Before stating Theorem 1, we mention that a star graph of order p is a tree all of whose p - 1 lines are incident with one point. THEOREM 1. For r > 2, s > 1, we have f ' ( r , s ) > (r - 1)(s - 1) + 1. Proof. For each pair of positive integers r and s, r > 2, s > 1, we shall construct a tree of order ( r - 1) ( s - 1) + 1 which is not in ~/(r, s). Let G~, G2, "", and Gs- 1 be s - 1 copies of the star graph of order r. For each i, i = 1, 2,..., s - 2, "identify" a point of degree 1 of G~ with a point of degree 1 of G~+t, to obtain a tree T o f order (r - 1)(s - 1) + 1. (See Fig. 1 for an illustration in which r = 6 and s = 5.)
Fig. 1 Since Tdoes not contain any point of degree r, in order to show that Tis not in ~/(r, s), it suffices to observe that disjoint lines of Tnecessarily come from distinct G~'S.
In the proof of the next theorem we shall need the class of all trees of order f ( r , s) - 1 which are not in ~ ( r , s). For this reason we define ~ ( r , s), r > 2, s > 1. For a fixed r > 2 let ~(r,2) be the class consisting of the single star graph of order r. Having constructed ~(r, s - 1), we define ~ ( r , s) as follows. Let Tbe any member of ~ ( r , s - 1). We "identify" any point of degree 1 of Twith a point of degree 1 of the star graph of order r to obtain a tree G. The class ~ ( r , s) is the set of all such trees as G. We note that every member of ~ ( r , s) is of order ( r - 1)(s - 1) + 1. As an illustration we mention that the trees given in Fig. 1 and Fig. 2 are, up to isomorphism, the only members of ~(6,5). TI-I~OI~M 2.
For r > 2 ,
s>l,
we have f ' ( r , s ) = ( r - 1 ) ( s - 1 ) +
2.
1967]
THE LINE ANALOG OF RAMSEY NUMBERS
95
Fig. 2 Proof. By Theorem 1, f ' ( r , 2 ) > r. Since every tree of order r + 1 is either the star graph of order r + 1 or else it contains two disjoint lines, we have f ' ( r , 2 ) = r + 1 for all r > 2. We note that for every r > 2 the star graph of order r is the only tree of this order which is not in ~¢(r, 2). We now proceed by induction on s. For a fixed r, r > 2, assume that the members of g ( r , s - 1) are the only trees of order greater than or equal to (r - 1 ) ( s - 2) + 1 which are not in ~¢(r, s - 1). Now, by the definition of f '(r, s) and by Lemma 1, there exists a tree of orderf'(r,s) - 1 not belonging to s/(r,s). Suppose G is any such tree. Let a and b be the two points of G determined by Lemma 2. By the assumption on G, deg b < r - 1; hence the removal from G of all the lines incident with b will result in a tree Go of order Po, Po > f ' ( r , s ) - r, together with some isolated points. It follows from Theorem 1 that Po > ( r - 1 ) ( s - 2 ) + 1. The induction hypothesis now implies that Go is in ~ ( r , s - 1). Hence Po = f ' ( r , s ) - r = ( r - 1 ) ( s - 2 ) + 1, from which it follows that i) f ' ( r , s ) = ( r - 1 ) ( s - 1 ) + 2 , and ii) the degree of b in G is r - 1, implying that G is in ~(r,s). Before stating the main result we observe that: 1) f ' ( r , 1 ) = 2 for every r, 2) f ' ( 1 , s ) = 2 for every s, and 3) f'(2, s) = 3 for all s > 1. THEOREM 3. For r = 2 and s > 1 we always have f ' ( 2 , s ) = 3. For all other positive integers r and s the formula f ' ( r , s ) = ( r - 1 ) ( s - 1 ) + 2 holds. It is perhaps worth mentioning that, in contrast with the case of f ( r , s), the symmetricity in r and s of the function f ' ( r , s), just established for almost all values of r and s, is not at all self-evident. In conclusion we would like to thank Professors E. A, Nordhaus and B. M. Stewart of Michigan State University for pointing out an error in the original manuscript.
96
MEHDI BEHZAD AND HEYDAR RADJAVI
BIBLIOGRAPHY 1. P. Erd~s, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292-294. 2. P. Erd6s, Remarks on a theorem of Ramsey, Bull. Res. Coanc. Israel 7F (1957), 21-24. 3. P. ErdOs, Graph theory and probability I, Canad. J. Math. 11 (1959), 34-38. 4. P. Erdos, Graph theory and probablity II, Canad. J. Math. 13 (1961), 346-352. 5. P. Erd6s and G. Szekeres, A combinatorial problem in geometry, Composito Math. 2 (1935), 463-470. 6. R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math. 7 (1955), 1-7. 7. F. Harary, A seminar on graph theory, Holt, Rinehart, and Winston, New York, (1967). 8. O. Ore, Theory of graphs, Amer. Math. Soe. Colloq. 38, Providence (1962). PAHLAVlUmv~m'Y, SHIRAZ, I R ~
A GENERALIZED MOMENT PROBLEM BY D. LEVIATAN
ABSTRACT Let {2,} (n _> 0) satisfy (1.1) we are considering the following problems: What are the-"necessaryand sufficientconditions on a sequen.ce{p~}(n ~_0) in order that it should possess the representation (1.2) where aft) is oI oounaea variation or the representation (1.3) wheref(t) E LM[O, 1 ] o r f(t) is essentially bounded. 1. Introduction and definitions. lowing properties:
Let the sequence {;ti} (i > O) possess the fol-
oo (1.1)
O 0) define:
Dk#s = ] IAs,as, l , ' " ,
a,,k
#s+k, as+k,1, ""', as+ k,k (when k = 0 D °ps = #,). We denote after Schoenberg [9] (O,m + 1,.--,n)
(1.4)
~'nm "-~ (m + 1,.-., n) (m,..., n)
D"-ml~m
0
< m < n = 1,2,-.-
and
(o)
(1, m + 1, ..., n)
tnm=
(0, m + 1 , . . . , n )
0 < m < n = 1,2,...
and tnn = 1. We shall use the function {~b,(x)} (n >_-0) defined by Schoenberg [91 where it was proved that the functions ~b,(x) are continuous convex functions and that 0 = t.o < t , l < "" < t,,, = 1. If A is an infinite Vandermonde, i.e.
A=lla.ll,
"-1
. =0,1,2,.
m=1,2,..
where {2,} satisfies Condition (I.I) then it was shown in Schoenberg [9.1 that (2.1)
for n >_- 0
d?,,(x) = x c;~-x°)/(x'-a°)
and that t..
= (-i)"--(,~,+,
- ~o) .....
(t. - ,Io)[~.,
..., ~.],
where (2.2)
L~.,...,~,.] = ~=.~ ( & - ~ . 3
]-/t
..... (~, - & - , ) ( &
- x,+,) ..... ( & - x , , )
(see also Jakimovski [5] (11.3)). 2, The main results. First we shall generalize Hausdorff's solutions [3] b y solving
the first problem for 20 > 0.
19671
A GENERALIZED MOMENT PROBLEM
99
THEOREM 2.1. Let {2~) (i _>_0) satisfy Condition (1.1). The sequence {/~,,} (n >=O) possesses the representation (1.2), if, and only if: sup ~ )~m+l . . . . . )~n I t.., n~O m=O
(2.3)
..., ..] I -- n
0) by the equations (3.1)
~[o = 0, /~o = ~(1) - ~(0), J~n = 2n-i, /~, =/~,,-1
(n > 1)
by (1.2) and (3.1) we have (3.2)
/~, =
fo tiC'dot(t)
n = 0,1, 2,-.-
Hence by Hausdorff's Theorem VI [3] (3.3)
sup ~ ~,+I ..... ][,]~m,'",/~n]l
= L < o0.
n_~O m=0
By an easy calculation we get from (2.3) that for 1 -< m -< n = 1,2,... (3.4)
[/~,,,'",/~J = [/L, - 1, " " , # , - i ] .
Therefore by (3.3) we get sup n ~ 0 m---0
.....
. 0:
~.+l ..... x.I Ea.,
m=O
,a,31 z K < ~o
where K does not depend on n. Hence by Hausdorff's Theorem VI [3]: 1
(3.6)
/~, =
f0
tX"dot(t)
n = 0,1, 2,...
where ~(t) is of bounded variation in [0,1]. Now by (3.1) and (3.6)
!% =
ta"de(t)
n = 0,1, 2,...
Q.E.D.
Proof of Theorem 2.2. (i) By corollary 8.1 of Schoenberg the proof is as that of Berman [I], but now the results of Schoenberg [9] are used. (ii) In order to prove necessity, let us assume that {~,} (n > 0) possesses the representation (2.4) where f(t) e M[0,1]. We have 12,,]
=0) has a subsequence {kj} (j _~ 0) such that limg-.oo ~'tkj.m~_x~ 2,~,m(t)dot(t) = ~(x) for each point t = x where ot(t) is continuous. Hence lim~.. • ~,t,,,.~_xfXo2m(t)doc(t)= ~(x) for each point t = x where ~(t) is continuous and we obtain lira
2.,,m(t)dt = y - x .
f--+ o0 m = r + l
Therefore ] ~(y) - ct(x)] < H for any two points x , y , 0 < x < y < 1, hence y-x n(x) = c + f~ f ( O d t where f(t) ~ M[0,1] and by (3.8): 1
#~ =
~0
$n(t)f(t)dt
n = 0,1, 2,....
Q.E.D
Proof of Theorem 2.3. The proof of the necessity is similar to that of Theorem 2.2 using, instead of (3.7) formula (I 1) p. 46 of Lorentz [7] {] (-1)'-'~1~+1 ..... 2n[t~,...,t ~"] = 1 for 0 < t < l . m----O
We prove now the sufficiency. As in the proof of Theorem 2.2 we get
1967]
A GENERALIZED MOMENT PROBLEM sup n~_O
.....
103
K
0, 8 > 0 et N on peut trouver p satisfaisant la condition (Q') et dent le support est de diam~tre < r/. (Q) Quels que soient 8 > 0, N e t J on peut trouver p e CA(E))* et k tels que II P I[a" = 1 et clue quel que soit l e Z, H, poss~de au plus un seul 616ment. Remarquons que chacune de ces trois conditions sont inchang6es si on remplace p(t) par ge ~=p(t), I~ I = 1. En effectuant cette transformation on pourra toujours se ramener au cas oi~ p satisfait de plus/t ~0(0) > ½.
1967] ANALYSE HARMONIQUE DANS QUELQUES ALGEBRES HOMOGENES
109
Les trois conditions (Q'), (Q") et (Q) sont rang6es dans l'ordre inverse de leurs implications. Ceci r6sultera de
L~MME 1. La condition (Q) implique la condition (Q~). Preuve. Donnons nous r/> 0, e > 0 et l'entier N. Notons par 6 la fonction dont le support est contenue dans [ - r//2, 17/2] et 6gale sur son support /t
21 1 -1) Soit J un entier tel que Igl>Z
13(q) l
1 pour certaines valeurs de t. D'autre part,
pto(nk + I) = ~, ~(nk + I + q)~( - q)e -i~° q
=
X
l¢l~_Z
+
Y..,
[~[>J
D'apr~s le choix de J la deuxi~me somme est inf6rieure/l e; en ce qui conceme la premiere on a l+J
Y~ = ]ql>J
E
p(nk + s)~(l - s)e -i°-s)'°
s=l-J
Si n ~ Ht tons les termes de fi figurant d a m cette somme sent en module inf&ieur /l ~, par suite, comme ~[~(m)[ = 2r/-1,
[pto(nk + l ) l < t si n¢H,. Comme d'apr6s (Q), Ht ne poss&le au plus qu'un 616ment, alors (Q ~) est v6riti~.
Y. KATZNELSON ET P. MALLIAVIN
110
[April
H. Une condition pour qne A E = A. On sait que les fonctions qui op~rent sur A(E) sont celles qui opfirent sur Ar. Par suite, un corollaire imm6diat d'un r6sultat du type A t = A sera que seules les fonctions analytiques op~rent sur A(E). On a
TI-I~I~ME 1. Supposons que E satisfasse dt la condition (Q'), alors A E = A. Prenve. Pour 6valuer la norme dans (AE)* nous utiliserons le lemme:
LEm~m 2. Soit p~(A(E))* telle que /~(0)= 1, soit f ~ A ( T ) , alors
Ilffx) d~ I1,~ --< sup tlS<x + t)p<x)ll,.. t
Prenve.
Consid6rons la fonction ~ valeurs dans A* ~( t) = f ( x ) p ( x - O .
Notons par (g, S) le produit scalaire d'une fonction g e A e t d'une pseudomesure S. Alors pour tout g e A l e produit scalaire (g, ~b(t)) est une fonction continue de t. De plus, on a
I? (g, alp(t))~" I=?
"
/(x)g(x) 2rc "
D 'o1~1, eomnle
[If(x)dxllA~=supf~ogf(x)g(x)Y-~ g~A,
IlgllA~ < 1
on en d6duit que
Notons par Et le translat6 de E d'amplitude t, alors ~b(t)e (A(Et))*. Par suite
D'oh le lemme. D6monstration du th~or~me 1. Nous montrerons que pour tout polyn6me trigonom6trique on a
~1)
II P II.- -- II P IIA.
Soit N le degr6 de P. Notons par h k l'application P ( x ) ~ P(kx), on a
Ilell~=supllh,ell~, •
1967] ANALYSE HARMONIQUE DANS QUELQUES ALGEBRES HOMOGENE$
111
D'autre part, il existe R, polyn6me de degr6 N tel que I!R [L4*= 1 et que
IIe I1~ = IIhkP 11~= < h : , h~R>. Soit p e t k la pseudomesure et l'entier satisfaisant la condition (Q') avec N e t
= ~l/N, avec de plus 1
:(0) > T " On a alors, d'apr6s le lemme 2, 11hkR IIc
N 1Z Ic.(u) l" NOUS noterons par ea et Nq un hombre positif et un entier tels que quel que soit u, u > 0, u < 2 s'2.-x-3, on ait e~ < exp( - 2u)
zx+(u) < exp(- ul/2). Supposons maintenant que k l , - " , kq_ j aient 6t6 choisis. Nous allons choisir alors r/g > 0 tel que pour tout intervalle I de longueur inf6deure/t tlq on air 11e ' ' I -
111 ,,, < 2
0 < u < 2 a.2,-,-a
Ceci 6tant on appliquera la condition (Q") avec q = t/q, e = eq et N = Nq. On notera par #a la pseudo-mesure correspondant/t ces donn6es dont l'existenee est assur6e par ( Q 3 et kq sera l'entier associ6/l #~ pour clue l'on ait Card(Gl) ~ 1 pour tout I e Z. La construction de f est ainsi effectu6e.
1967] ANALYSE HARMONIQUE DANS QUELQUES ALGEBRES HOMOGENES
113
Ceci 6tant, nous noterons dans toute la suite par q = q(u) la fonction/l valeur enti6re d6finie par la relation -t
< 16u,/a
-t
aq(u) + i =
< aq(u) + 2.
Comme ~E a n < 2 a q + t n>q
on en d6duit que l'on a A(u) < [[d't-t[a; exp(2uaq+ i). Utilisons d'autre part le lemme 2, on a A(u) < sup
lle"%~,UA.eXp(2ua~+ i)
o0 /~ note la translat6e de la mesure #q. Enfin
IIe":q.;h. =< IIe":° ' A.., e
h"
oO I t e s t un intervalle de longueur r/~ contenant le support de #L En vertu du choix de r/q cette norme est inf6rieure /t 2, d'ofJ
II
A(u) < 2 sup e"~°#'x'#7
I[A,exp(2uaq+ 1).
t
Notons par dt(u) le/-i~me coefficient de Fourier du second membre, on a
dt = ~, t~(l+kqn)c_,, n
X+X ~--- n 6 G !
+ nl~G1
I.I~N
X
Inl>N.
La premi6re somme se compose au plus d'un seul terme, donc est major6e par 2~(ua~). La seconde somme est major6e par
La troisi6me somme est major6e par 2zN (u). On obtient ainsi A(u) < 20~(aqU)exp(2uaq+l) < 20 e x p ( - aq1/2u 1/z + 2aq+lu). Remarquons que darts l'intervalle de variation de u consid6r6 on a
2 aq+lU
al/2tjl/2
u l/a,
Y. KATZNELSON ET P. MALLIAVIN
114
[April
on obtient
[
A(u) < 20exp( - [a~u'~,/z] < 20 exp [ 36] ]
ul/6
--iT-}"
IV. Etude des enflomorphismes de An. Considerons un endomorphisme de l'alg~bre At. Si I'on note par O(t) l'image de e~t il est imm~diat que [ ~(t) l = 1 pour tout t e T et par cons6quent l'on peut 6crire q)(t) = ei*(° avec ~b(t) r~elle et 2g-p6riodique (rood 2r0. L'endomorphisme en question est alors donn~ par f(t)-~f(dp(t)) pour t o u t e f e AE. Comme e i"*(') est l'image de e ~"t, dont la norme dans AE est toujours ~gale /t 1, l'on voit qu'une condition n~eessaire pour que f~f(qb) d6finisse un endomorphisme est [[,,~,*llIta~ < Const uniform6ment en n. Le tMor~me 3 dit que, si E contient des progressions arithm6tiques arbitrairement longues, les endomorphismes qui donnent lieu ~ des changements de variables deux fois diff6rentiables sont n6cessairement triviaux.
THEOR~ME 3. Supposons que E contient des progressions arithm6tiques arbitrairement longues. Si qb(x) est rdelle, deux fois continuement d~rivable et si II~'uel'iiA~ est bornde pour lul-oo alors (best lin~aire. D6monstration. I1 est clair que E contient des progressions arithm6tiques arbitrairement longues port6es par des intervalles (arcs) arbitrairement courts. Supposons ~b" ¢ 0; il existe done un intervalle Idans lequel 4~" -~ ~/> 0. Rempla~ant ~(x) par q~(x + e) nous pouvons supposer que I contient une progression arithm&ique E s de E de longueur N. Le th6or~me r6sulte du lemme suivant: LEMME 3. Soit EI~ une progression arithmdtique de longueur N + 1, port~e par un intervalle I; ~p(x) r~elle deux fois d~rivable telle que
l ¢ ( x ) l --< c,
_>_n > 0
x e I,
alors
IIe"*l]~c~, > Cl,/-N
Sup I1
oh cl ne d~pend que de ct1-1. D~monstration. Sans changer Ie rapport o f f t ni les normes, nous pouvons 6taler Es sur ( - lr, r0 c'est /t dire supposer
[2r& l~ E~ = ~--N'h = ~
Prenons u =
N/IO0 c
et 6valuons
Ile",h,e.).
D'apr~s le lemme de Van der
Corput (ef.[4]) l'on a
II
The flow is supersonic, sonic or subsonic according as
(.5~~)+ + (C~
71_i) ~ 1.
Other classes of flows may be found by assigning appropriate functions to u3 in (5.9). Different classes of flows may be obtained by assigning appropriate values to /~. 6. Acknowledgement. This work was done while the author held a Visiting Research Appointment at the University of Queensland. He takes this opportunity to thank the University for support and Professor A. F. Pillow for providing a stimulating research enviroment. REFERENCES 1. N. Coburn, Intrinsic relations satisfied by the vorticity and velocity vectors in fluid flow theory, Michigan Math. J. I, (1952), 113-130. 2. R. P. Kanwal, Variation of flow quantities along stream lines and their principal normals and binormals in three dimensional gas flows, J. Maths. Mech. 6 (1957), 621--628. 3. R. H. Wasserman, Formulations and Solutions of the Equations of Fluid Flow, Unpublished Thesis, University of Michigan (1957). 4. D. Truesdell, Intrinsic equations of spatial gas flow, Z. Angew. Math. Mech. 40 (1960), 9-14. 5. J. Serrin, Mathematical Principles of Classical fluid Mechanics, Handbuch der Physik, Band VIII/l, Springer Verlag (1959), 186. 6. J. A. Schouten, and D. J. Struik, Einfuhriing in die Neuren Methoden der Differential geometrie, P. Noordhoff, Groningen, Batavia, (1938), 33. 7. E. R. Surayanarayan, On the Geometry of the steady, Complex-Lamellar Gas Flows, J. Math. Mech. 13 (1964), 163-170. 8. C. E. Weatherburn, Differential Geometry, I, Cambridge University Press (1955), 68. 9. C. Truesdell, The classical field theories, Handbuch der Physik, Band III/1, Springer Verlag (1960) 404. 10. Reference 8, 258.
U
~
or ~
(AVSrRA~)
U~av~Rsrrv or RHODE ISXZ~D (IT.S. A.)
CONVERGENCE IN PROBABILITY OF RANDOM POWER SERIES AND A RELATED PROBLEM IN LINEAR TOPOLOGICAL SPACES(1) BY
LUDWIG ARNOLD ABSTRACT
A linear topological space is said to have the circle property if every power series with coefficients in it has a circle of convergence. Every complete locally convex or locally bounded space has the circle property, but not a certain class of F-spaces including the space of all random variables on a non-atomic probability space, endowed with the topology of convergence in probability. 1. Introduction. Let (f~,F,P) be a probability space and {an(to)}~=o an arbitrary sequence of complex-valued random variables defined on it. The formal power series
F(z, 09)= ~ an(og)z" n=0
where z is an element of the complex plane C, is called a random power series. Such a series is said to converge (in any mode considered in probability theory) at the point z if the sequence of its partial sums converges at z. Recently [1, 21 we gave an example of a r a n d o m power series converging in probability only at the points z = 0 and z = 1, and nowhere else(2). Therefore, in general, for the convergence in probability of a random power series there exists no so-called circle of convergence (i.e. a circle around z = 0 such that we have convergence inside but divergence outside). On the other hand, such a circle always exists for almost sure convergence and convergence in the pth mean (p > 0). The first aim of this note is to characterize the class of probability spaces (f~, F, P), for which every random power series which can be defined on it has a circle of convergence in probability. Furthermore, the set M(f~, F, P) of all equivalence classes of complex-valued r a n d o m variables defined on (f~, F, P), endowed with the topology of convergence Received August 23, 1966. (1) Research sponsored by the National Science Foundation under Grant No. GP 6035. (2) Professor H. Rubin pointed out that the constructions given in [1], p. 86 and [2l, P. 6 can be generalized to give a random power series converging in probability at z = 0 and in a prescribed denumerable set of complex numbers having no finite limit point, but nowhere else. 127
128
LUDWIG ARNOLD
[April
in probability, forms a linear topological space, in particular an F-space (see e.g. [4], p. 329). This leads our attention to the power series o0
f(z) = ~, anz~
(1)
n=O
whose coefficients a~ are elements of an arbitrary linear topological space X over the complex field C, and z ~ C. Such a space X is said to have the circle property, if every power series (1) possesses a circle of convergence. For instance, every Banach space has the circle property ([9], or [4] pp. 224-232), and in this case the circle of convergence of (1) has radius (lim sup ~ - 1 . Our second aim is to give some sufficient conditions for the circle property and to describe a class of spaces which fail to have this property. 2. Probability spaces with the circle property. Let (f~, F, P) be an arbitrary probability space. A set A e F is called an atom if P(A) > 0, and if B e F, B c A, then either P(B)= P(A) or P(B)= 0. If {A,} is the (at most countable) family of disjoint atoms of (f~,F,P) and if P(f~ - uAn) = 0, the probability space is called atomic. T r m o ~ 1. Every random power series with coefficients defined on a fixed probability space (~,F,P) has a circle of convergence in probability if and only if (f~,F,P) is atomic. Proof. (a)Suppose (f~,F,P) is atomic. Then convergence in probability is equivalent to almost sure convergence. But for the latter there always exists a circle of convergence. (b) Suppose P(B)> 0 where B = f ~ - u A , . In this case we can construct a random power series without a circle of convergence in probability. To avoid redundance, let us assume that B = f~. By a theorem of S. Saks (see [4], p. 308), for every 8 > 0 there exist finitely many disjoint sets Bt, ...,Bm¢F with uB~ = t~ and 0 < P(B~) < 8. We set 8 = e~ where ~ > 0, ek ~ 0(k ~ oo), and arrange the elements of the resulting partitions off~ for k = 1,2, ... in a sequence {Cn}. Now let So(tO) = 0 VtO and
sn(tO) = n~Ic,,(tO)
(n > 1),
where IA denotes the indicator function of a set A. The random power series ?Es,z" cannot converge in probability at any z # 0. For, if z # 0 is fixed and C,o, "", C~l(no < "" < ni) are the elements of a complete partition of f~ with
(nol
"°
x, we have
{ol I
rio
1967]
RANDOM POWER SERIES
129
Now let us consider the series Y~a.z" where ao(o~) = 0¥o~ and a. = s. - s._ 1 (n ~ 1). We have .--1
akz k = S.Z" + (1 -- Z) ]E SkZk 0
0
and s,z" ~ 0 in probability Y z ( n ~ oo), since ek "-*0. Hence, lEa.z" converges in probability at z = 0 and z = 1 but at no other point, Otherwise
) ( l - z ) -1
n--1
~.. akz k - s.z n = 0
~ SkZk 0
would converge, in contradiction to what was proved above, q.e.d. 3. Power series with coefficients in a linear topological space. As mentioned above, in the F-space M = M(f~, F, P) with norm
Ilxll =E Ix r(A). (b) O at zl implies anZ~ - + 0. A convergent sequence in a linear topological space is bounded, so that for every U e W there exists an e > 0 such that o t a . z ] e U V n whenever I~'1 = c
J ll--
and the space X is complete, q.e.d. Theorem 5 characterizes a class of F-spaces containing arbitrarily short straight lines (i.e. for every neighborhood V of 0 e X there corresponds some x # 0 for which ctx e V V ~te C). These spaces are necessarily of infinite dimension. On the other hand, we have Trmor.EM 6. Every F-space having arbitrarily short straight lines contains an infinite-dimensional subspace which has the circle property with r(O) = r(A). Proof. By Theorem 9 of [3], an arbitrary F-spaces has arbitrarily short straight lines if and only if it contains a subspace isomorphic to (s). But (s) has the circle property, with r(O) = r(A), q.e.d. REFERENCES 1. L. Arnold, t)ber die Konvergenz einer zuf'dlligen Potenzreihe, J. Reine Angcw. Math. 222 (1966), 79-112. 2. L. Arnold, Random Power Series, Statistical Laboratory Publications No. 1 (1966), Michigan State University, East Lansing, Michigan.
134
LUDWIG ARNOLD
[April
3. C. Bessaga, A. Pelczy~ski and S. Rolewicz, Some Properties of the Space (s), Colloq. Math. 7 (1959), 45-51. 4. N. Dunford and J. T. Schwartz, Linear Operators, Part 1, New York, 1958. 5. A. Dvoretzky, On series in linear topological spaces, Israel J. Math. 1 (1963), 37-347. 6. J. L. Kelley and I. Nanfioka, Linear Topological Spaces, Princeton, N. J., 1963. 7. R. S. Phillips, Integration in a convex linear topological space, Trans. Amer. Math. Soc. 47 (1940), 114-145. 8. S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Pol. Sci. CI. III, 5 (1957), 471-473. 9. N. Wiener, Note on a paper of M. Banach, Fund. Math. 4 (1923), 136-143. MXCmOANSTATIKUNIVERSITY,EASTLANSING AND TeCtImSCHE HOCHSCHUL~,STUT'fGART