Israel Journal of Mathematics, Volume 5, Issue 2 (1967)

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.

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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 -<~B). A system 92 is minimal if it has no proper elementary subsystems. It is prime if 92 ----~B implies that there is an elementary map of 92 into ~ . If X __qA we say 92 is prime over X if (92,x)~ x is prime.(2) It can be shown (cf. 17]) that if 9~ is prime then it is prime over every finite X~_A. ~ is a minimal elementary extension of 92 if ~ is a proper elementary extension of 92and for all if, 92 -< ~ ~(~B implies 92 = E or ~ = ~. ~5 is a prime elementary extension of 9A if ~ is a proper elementary extension of 92 and for all other proper elementary exten(2) Strictly speaking, this terminology is ambiguous since the notion depends not only on the set X but the formulas satisfied by X. In our usage, however, the meaning will always be clear from context. A similar remark applied to the notation S(X) introduced below.

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sions i f ) - 9 1 there is an elementary map of~3 into ~ which is the identity on

~l[,

Suppose T is a complete theory in some language and F the set of formulas having no free variable other than some fixed one, say Vo. Let S be the set of subsets of F maximal consistent with T. S may be given a compact, totally-disconnected topology(s) by taking as a basis the sets:

{p~S; d?ep}

(dpeF).

If ~3 is a model of T and b e ~3 then the set of formulas satisfied by b in ~3 is some p c S. We say b realizes p or b is of type p. In the particular case where T = Th((91,X)x~X) we shall write S as S(X) and if be~3, ~3 )-91 we shall say b realizes p e S ( X ) if b realizes p in (~3,x)x~ x - A system 91 is saturated if for every X _ .4 with the cardinality of X less than that of A, every p e S(JO is realized in 91. It can be shown (see [4] or [5]) that if 9.I = ~3, 91 and ~3 are of the same power and both are saturated then 9I ~ ~3. 2. T h e E l e m e n t a r y Tower.

LEMMA(4). Suppose T is a complete theory, 91 a countable model of T and p the only limit point in some neighborhood orS(A). I f there is a proper elementary extension ~ of 91 in which p is not realized then there is an uncountable elementary extension of 91 in which only a countable number of elements realize p. Proof. Suppose 91, ~ and p are as in the hypothesis of the lemma. Since 91 is a model of T every isolated point of S(A) must be realized in 91 (cf. lemma 4.1 of [3]). From this it follows that every isolated point of S(A) can be realised by only one point. Hence no isolated point of S(A) is realized in ~ - 91. Let ~b be a formula defining a neighborhood of p in which p is the only limit point. Since p is not realized in ~ and no isolated point is realized in ~ - 91 it follows that no point in the neighborhood defined by ~b is realized in ~ - 91. Thus, ,~ is satisfied by no element of ~ - 91. By Vaught's two-cardinal theorem [5] there is an uncountable model of Th(91, a)a ~w in which only a countable number o f elements satisfy ~b and hence a fortiori only a countable number realize p. THEOREM 1. A complete theory T is N1-categorical if and only if every countable model has a prime elementary extension. Proof. Suppose T is Nl-categorical and 91 is a countable model of T. Using the results of [3] we know that T is totally transcendental and therefore S(A) is countable and must have a point p which is the only limit point in some (3) This is no more than a thinly disguised version of the Stone representation theorem for Boolean algebras. For a more detailed discussion see [3]. (4) A stronger form of this lemma was announced as theorem 1 of [4l. However there was an error (pointed out to me by Charlotte Stark CheU) in my proof of that result.

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neighborhood of S(A). T is tq~-categorical so by theorem 5.5 of [3] every uncountable model of T is saturated and hence every uncountable elementary extension of 9.[ has an uncountable number of points realizing p, From the preceding lemma it follows that every proper elementary extension of 95 has an element realizing p. Let b be such an element in some elementary extension of 9[. Then for every proper elementary extension ~ 9 5 there is an elementary map ofA U { b ) into C which is the identity on A. But by theorem 4.3 of [3] A I,.J {b} has a model of T prime over it. This model is the desired prime elementary extension of 9~. Conversely, suppose every countable model of T has a prime proper elementary extension. Using the axiom of choice we choose for each isomorphism type of countable models a type of prime elementary extension. (We have not assumed that the prime elementary extension is unique, but see Theorem 2 below.) For each countable model 95 then we define an increasing sequence of models {95~; ct < to1} by 950 = 95, 95~+1 is the prime elementary extension of 95~ and for limit ordinals c5, 956 = ~J__95~. Let us call this the 95-tower. Notice that every member of this sequence is countable except the last one, 9~,o~, which has power ~ . Suppose ~3 is a countable elementary extension of 95. Letting fo be the identity map of 95 into ~ we may proceed inductivity to extend this to elementary maps f~ ; 95~~ ~ since each 95~+1 is a prime elementary extension of 95~. The induction will cease exactly at that ~ where f~ maps 95~ onto ~3. This must occur for some 0e-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.

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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

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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 ~-<~a; it must be satisfied by some element, say do, in ~. We have already shown that this implies that do does not realize the point p E S(A). Therefore do must realize an isolated point of S(A) and as mentioned in the proof of Lemma 1, therefore do e ~. Then the formula ~(Vo, do) determines a neighborhood of p in S(A). Repeating the above argument we can find a dl eg~ which satisfies:

X(Vo)&p(Vo) & ¢(Vo, do) & ~(c, Vo). Proceeding inductively we may find a sequence {dn ; ~ ~o} of elements of A such that m > 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~<~9~. Since 9A is a prime model the arguments used to prove Theorem 1 shows that every countable model of T is isomorphic to some member of this tower. It remains to consider which members of the tower are isomorphic. One possibility is that T is No-categorical so that all members of the tower are isomorphic. Let us exclude this case and suppose that T is Nl-categorical but not No-categorical. Vaught [7] has shown that under this hypothesis the prime model is also minimal. Hence 9~o is not isomorphic to 9A~for any ~ > 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

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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].

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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, Buff. Acad. Polon. Sci. C1. III 7 (1959), 545-548. 7. R. L. Vaught, Denumerable models of complete theories, Proc. Sympos. Foundations of Mathematics, lnfinitistic Methods, Warsaw, Pergamon Press, Krakow, 1961, pp. 303-321. UNIVERSrlN OF WISCONSIN, MADISON, WISCONSIN

ON THE NUMBER OF HOMOGENEOUS MODELS OF A GIVEN POWER* BY

H. JEROME KEISLER AND MICHAEL D. MORLEY

ABSTRACT

It is shown that, For each complete theory T, the nomber hT(rlt)of homogeneous models of T of power rrt is a non-increasing function of uneountabel cardinals 11t Moreover, if hT(NO)~ No, then the function hr is also non-increasing No to N1.

A model ~ is said to be homogeneous if, roughly speaking, any isomorphism between two small elementary submodels of 9~ can be extended to an automorphism of ~. The notion is due to Morley and Vaught [7], and is based on J6nsson [2] (a precise definition is stated below). Consider a complete theory T in a countable first order logic with identity. For each infinite cardinal cardinal N~, let hr(N,) be the number of (non-isomorphic) homogeneous models of T of power N~. We shall prove the following two theorems. THEOREMA.

If hr(No) < No, then for all m > No, hr(m) < hT(NO).

THEOREM B. Assume the GCH (generalized continuum hypothesis). If N1 <=m < n, then hr(m) _~ hr(n). COROLLARYC

(GCH). The function hr is non-increasing except for the single

possibility hr(No) = Nt, hr(N1) = N2. It is shown in [7] that if Thas any infinite models then hr(No) > 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

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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<,~A~}. The symbol 9"[ < ~ means that ~ is an elementary extension of 9"[ (or 9"[ is an elementary submodel of ~). We assume the reader knows the definition and basic properties of 9"[-< ~ , which are given by Tarski and Vaught in [8]. Note that 9"[ -< ~3

implies S(92) c_ S(~),

S(92) G S(~3) implies T(92) = T(~). DEfinITION [7].(2)

Let 9"[ be a model of infinite power

Ia I--m.

CAsE 1. m is regular. 9"[ is homogeneous if for all ~3,(~-< 9"[ of power less than m, any isomorphism f on ~3 onto ~ can be extended to an automorphism of 9"[. CASE 2. m is singular. 9"[ is homogeneous ff 9"[ is the union of an elementary chain

(*)

9"[0 "< 921 "<"" "< 92p ""

where each 92~ is a homogeneous model of a regular power less than m such that s(92,) =

The following lemma from [7] puts the definition of homogeneous model in a more convenient form.

LWMI~ 1. Let 11t be an infinite regular cardinal. The following is a necessary and sufficient condition for a model 92 of power m to be homogeneous: For all ordinals ~ < m , all s-termed sequences a, b e A ~, and all c e A , if

T(92, a) = T(92, b) then there exists d e A such that T(92,a,c) = T(92,b,d). We shall now prove a series of lemmas from which Theorems A--C will follow at once. LEI~IA 2. I f 9"[ is homogeneous, S(~) ~ S(9.D, and I B[ < I A [, then ~3 is isomorphic to an elementary submodel of 9"[. (2) This definition differs from the one in [7] when 11t is singular.

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Proof. First let the cardinal m of A be regular. By transfinite induction on it can be shown that for each cardinal n < m, for all b e B n there exists a e Au such that a) -- T ( $ , b).

The desired result follows when we take b e B m whose range is B. Now let m be singular and represent 9.[ as the union of a chain (*). Let ~ be the union of an elementary chain "< "'"

-< ...

where each ~Bp has power at most the power of 9J[~. Then for each fl there is an isomorphism fp of ~a onto an elementary submodel of ~p. We may assume that the sequence (.) has been thinned out so that for each fl, [A, I > [ 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) _-<2 2% and hr(m) <_-2m. Proof. There are only 2 % complete types of finite sequences, hence only 2 2% possible sets S(~). Moreover, there are only 2ra models of power m, up to isomorphism.

Lv.M~ 5. Suppose 9.[ is a homogeneous model, X ~_ S(9£), and l X I < Ro. Then there is a countable homogeneous model ~ - < 9.[ such that X c S(~). Proof. Let C be a countable subset of A such that for each x e X there is an n-tuple of elements of C of type x in 9.[. It suffices to find a countable homogeneous model ~ -< ~ such that C _~ B. First choose a countable ~3o -< 9.[ such that C _~ Bo. Let C~ be a countable set such that Bo ~ C~ ~ A and for all a, b ~ B~ and c e Bo, if

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T(91, a) = T(91, b) then there is a d v C1 such that

T(91,a,c) = T(91,b,d). Choose a countable ~1 "< 91-[such that C1 ~ BI. Continuing the process we choose countable models ~ o "<~l ~ 2

"<""

and let ~ be the union of the chain. Then ~ is homogeneous and X ~_ S(~). Theorem A now follows easily. We note that if ~ is countable then S(~) is countable. Hence if the theory T has a homogeneous model 9I such that S(91) is uncountable, then by Lemma 5 we have hT(RO)> Ro. Suppose, on the other hand, that S(91) is countable for every homogeneous model 91 of T. Then for each homogeneous model 9/ of T there is a countable homogeneous model ~ with S(~) = S(92) (by Lemma 5). Hence by Lemma 3, hr(m)=< hT(No) for every m~_~o. LEMMA 6 (GCH). Let A be a homogeneous model of power rt and let Ni <=p <=1t. Then there is a homogeneous model ~ .< 91 of power p such that S(~) S ( ~ . =

Proof. From the definition we may clearly assume that n is regular. It suffices to prove that for each C _ A and each regular cardinal m with Nl _-<] C[ ~ m =
T(91, a, c) = T(91, b, d). We choose ~ -< 91 of power m containing C1, and continue the process m times to form an elementary chain ~ o < ~ 1 "< "'" "< ~ , "< "'" of length m with each B, of power m. The union of the chain will be homogeneous of power m. If p is regular, take p = m and take C to contain sequences of each type in S(91); note that =
Is(ml_

1967] ON THE NUMBER OF HOMOGENEOUS MODELS OF A GIVEN POWER

77

regular m, N1 _--<m < p, with S(9~.T)= S(9~. Then the union is homogeneous of power p. This completes the proof. Theorem B is now immediate. By Lemma 6, for each homogeneous model 9~ of T of power 1t there is a homogeneous model ~3 of power m with S(~3) = S(~). By Lemma 3 we have ha(m) > ha(11). Corollary C follows from A, B, and Lemma 4. Our lemmas give more general versions of Theorems A, B, and Corollary C. Let L,o,,o be the logic which is like first order logic but has countably infinite conjunctions, and let L~,,o,l be the logic which has countably infinite conjunctions and quantifiers over countable sequences of variables (cf. [3]). If Tis a countable set of sentences of L,~:o, then the results A, B and C all hold for T. Theorem B still holds if Tis a set of sentences of L,o,~,, andre,11 are not cofinal withco. Moreover, both of the above remarks still hold true if the sentences of T have countably many second order existential quantifiers at the beginning. The point is that in Lemmas 5 and 6, if 9~ is a model of T then ~3 can be taken to be a model of T. Instead of giving direct proofs of Lemmas 5 and 6, we could have deduced them as special cases of L6wenheim-Skolem theorems for the appropriate infinitary logics in 13]. We conclude with some problems and examples. Assume the GCH from now on. By Theorem B, each complete first order theory T has a least cardinal ml(T) such that for all m, n > ml(T), ha(m)= hr(n). Since there are only 2%T's, there is a least ml such that ml ~ ml(T) for all T. What is the cardinal m f l All we know so far is that ml > N2 (from the examples below). There is also a least cardinal m2 such that for every homogeneous model 9~ of power > m2 and all rt > m2, there is a homogeneous model ~3 of power rt with S(~3) = S(9.t) (by Lemma 6). It follows from Lemmas 3 and 6 that rrt2 > rn~. The value of m2 is not known, but seems easier to get at than m~. The argument that the numbers m~,m2 exist works in a wide variety of situations and is due to Hanf [1]; such numbers are sometimes called Hanf numbers. For each finite or infinite cardinal rt < N~, there is a theory T such that ha, is the constant function with value 11. Any theory T categorical in power No has ha(m) = 1 for all m. For 3 ~ n < co, the theories Tn of Ehrenfeucht discussed in [9], p. 318, have hr,(m ) = n for all m. It is easy to find theories Twhere hr has the constant value No, or N~. The complete theory of the model (-4, I, < ) where I is the set of all integers, < the natural order on I, and A - I is countably infinite is an example of a theory T with ha(m) = 2 for all m. The complete theory of the countable tree in which each element has finitely many predecessors and exactly two immediate successors is a theory T with ha(No) = 3, hr(Nt) = 2, ha(N2) = 1, Whenever 3 > a > b => c we know an example of a theory T with ha(No) = a,

ha(Rt) = b,

ha(N2) -- e.

78

H. JEROME KEISLER AND MICHAEL D. MORLEY

We have not yet found any example of a theory T with hz(No) = 4, hT(NI) = 1,

or even with No > hr(No) > 3 hT(N1). The complete theory of a model with unary relations Ro, RI, ". such that each non-trivial Boolean combination of the R, is infinite is a theory T with hT(No) = N1, hT(m) = N2 for 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

82

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 _-
1. For all (i,j, k) ~ E

(6)

s~jk = c~jk

(7)

Aok = (1/x)(eqBij k + c3(i,j, k)p + K)

(8)

tc 2 = 2ec

Proof. Since M is a trivial-intersection-set in G, condition (iii) implies that s~jk = c,j~ whenever (i,j, k ) e E. To prove (7) notice that: xA~jk = ~

(e~(ml) + c)(e~,(mj) + c)(e~s(mk 1) + c)

$

= 8 ~, ~,(m3~(mi)~(m~ 1) s

+ c~

[~,(m3~,(mj) + ~,(m,)~(mk 1) + ~(mj)~,(m~ 1)]

8

+ ec2 ~$

C~,(m,) + ~(mj) + ~s(mkl)] + cat

= eqBll~ + c(3il*P + 6~P + 61kP -- 3q) -- 3C2e + cat = eqB~lk + c~(i,j, k)p + K.

1967]

A CHARACTERIZATION OF THE SIMPLE GROUPS PSI. (2,p) p > 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 <=Px/g, g <- p2 in contradiction to (1), all BUk , (i,j,k)eE, are equal to each other. Let Ca~C1,C-tl; then: t

q=E

s~la = t(qBll 3 + q2) /p.

i=l

Therefore qBll a + q2= pq/t, which when inserted into (9) yields:

p2qx p2q2 g = (pq - tq2)~ + txS = (q2e/x) + (p - 1)S As x => q, (p - 1)S > 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[
(fii) aW(a, Oa z) = L(W) for [ a [ < ao(z). Using Picard's successive approximation method, and the fact that the right

88

MEIRA LAVIE

[April

hand side of (4) does not contain the independent variable a explicitly (a fact which will play an important role in the sequel) one obtains ao(z) = p(z)/M(z,p), where M(z, p) is the maximum of I L(W) I for I W - z I = 0 from/3 to the nearest singularity of L(z). Denote by D(R) the domain consisting of points W, for which [ W - z [ < R, z e / ) and 0 < R < d, where R is chosen sufficiently small to ensure that L(z) is still regular and single-valued in the closure of D(R). (The existence of such R > 0, follows from the fact that L(z) is single-valued in the compact domain/3). Denote by M(D, R) the bound of IL(W) I for We D(R). We have then:

IL(w)l <_M(D,R)

(7)

for [ W - z ] < R and z e / ) .

We now prove: TKEOREM 1. Let L(z) be a regular single-valued function in the closure of the bounded domain D. Then there exists one and only one analytic function W(a,z), satisfying equation (4) with the initial condition ( 2 ) f o r z ~D. For z e D and lal < ao = R/M(D,R), this unique solution W(a,z) is regular in a, and regular and univalent in z. It satisfies equation (1)for lal, lbl and I a + b I < ao and z ~ D. Proof. It follows from the existence and uniqueness theorem that for every initial value z ¢D(R) there exists a unique solution of the differential equation (4) with the initial condition (2), which is regular for ] a ] < ao(z)= p(z)/M(z,p). If we restrict the initial values to/3, then W(a, z) is regular in a for Ia I < R/M(D,R) = a o and for every z~/3 and satisfies:

[ W ( a , z ) - z l < R , [a[ < a o ,

(8)

zE/3.

Note that we may apply the same argument to points z eD(R) and using the fact that the minimal distance from D(R) to the nearest singularity of L(z) is d - R, we obtain that W(a, z) is then regular in a for [ a [ < %, for some ~o > 0. It also follows from the classical proof of the existence and uniqueness theorem, that W(a, z) is analytic in the initial value z. We prove it here again in order to obtain something more, namely that W(a, z) is analytic in z for z ~ D and I a [ < ao. We expand W(a,z) as a power series in a with coefficients which are functions of z and put: ®

W(a,z)= •

(9)

e.(z)

-~. a',

n=O

where

Using (4) and (2) we obtain:

P,,(z) = a--O

1967]

ANALYTICITERATION A N D DIFFERENTIAL EQUATIONS

(to)

89

Po(z) = z, Pl(z) = L(z),

and generally: = Oa.

1°=o =

aa-- V-t/

Ta

o=o

It follows from (11) that Pn(z) is a polynomial in L(z) and its derivatives up to order (n - 1). P,(z) is, thus, a regular single-valued function for z e/3. The series in (9) is a series of regular functions, which converges uniformly for z e / 3 and lal < ao. Hence W ( a , z ) is regular in a and z for lal < ao and z e D . To prove that W(a, z) satisfies (1), we use the uniqueness property of the solution of the system (4) and (2). Suppose that }a I < %, Ibt < ao and ]a + b I < a° and put a + b = c. Consider the function W(c, z) once as a function of z and c, and once as a function of z and a with b kept constant. In the second case we have:

tgW(c, z) _ dW(c, z) = L[W(c, z)]. Oa OC Hence W(c, z), qua function of a and z satisfies the differential system:

(4')

aW(c, z) = L[Wc, z)],

0a

with: (2')

W(c, z)la =o = W(b, z),

where W(b, z) is uniquely defined as I b l < ao and z e D, and by (8) we have W(b, z)eD(R). But the unique solution of the system (4') and (2') can be written as W[a, W(b,z)], which is regular for la ] < %, and so W(c, z) = W[a, W(b, z)], which is (1). But as equation (1) holds for z e D, l a ] < Oto, I b l < ao, I c ] = ] a + b ] < ao, it holds whenever both sides can be defined, i.e. at least for z e D, [a l, [b l, ]a + b[ < ao. In fact we have used equation (1) to obtain an analytic continuation of W(a, z), where z ~ D b and [b[ < ao, for such values of a for which [a + b] < a o. As the function W(a, z), so continued satisfies equation (1), it also satisfies equation (4). The univalence of W(a, z) for z ~ D and I a ] < ao can be proved either by using equation (1), or as another consequence of the uniqueness of the solution. Suppose there exist in D two distinct points zt and z2, and a value [ b* ] < ao, such that: (12)

W(b*, zl) = W(b*, z2) = W*.

Put c = a + b* and consider the two functions W(c, zl) and W(c, z2). Both functions, as functions of a, satisfy equation (4) with the initial condition: W]a=o = W*. By the uniqueness property it follows that W(c,z~)= W(c, z2) holds for ] c - b * ] < %, for some % > 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'/~, so that W(a,z) is regular for z E D and ]a ] < n/(n + 1). In this case the value ao = n/(n + 1) is the best possible. On the other hand, for every 0 < R < 1 we have R[M(D,R) = R(1 - R) 1In. We easily get: Max M ( 6 , R ) = n ¥ 1 W - 4 - f }

= n + 1

where # = (1/(n + 1))1/~< 1 for every n, but limn-.~ /z = 1. Hence if we put ao = CR/M(D,R), we cannot take C > 1. We deduce now from Theorem 1 two corollaries regarding the solutions of the differential equations (5) and (6). COROLLARY 1. Consider the partial differential equation (5)with the boundary condition (2) and let L(z) be a regular single-valued function in the closure of the bounded domain D. Then, there exists a unique analytic solution W(a, z) of the system (5) and (2) and it is given by the power series (9); i.e. it is identical with the solution of the system (4) and (2). Hence all the results obtained in Theorem l for the solution of(4) and (2) are valid for the solution of(5) and (2). Proof. Assume there exists a solution of the system (5) and (2), given by a power series in a:

W(a,z)= ~, ~ a

(13)

~.

n=0

Inserting (13) into (5) and using (2) we obtain: (14)

Qo(z) = z, Q,(z) = Q i, - l ( z ) L ( 2) ,

n = 1,2, ...

By comparing (14) with (10) and (11) we see that Q,(z)=P,(z), (n = 0,1,2,...). COROLI.ARY 2.

(15)

Theorem 1 of I-2]. Let C(z) = l : 2 + l : 3 +

....

1967]

ANALYTICITERATION AND DIFFERENTIAL EQUATIONS

91

In Theorem 1 [21 the authors prove that if(15) converges for [ z [< r, r >0, then the series defines a function L(z) and permits the construction of a uniquely defined function W(a,z), satisfying equation (1). This function W(a,z) is then analytic in a and z for all finite complex a and for ]z[ < R(a), R(a) > 0. The construction of the function W(a, z) is carried out by proving the existence of a solution of the differential equation (6), which, in this case, has an inconvenient singularity at z = 0. We give another proof to this theorem: If L(z) is regular for I z I < r, r > 0, it follows from (15) that for [z[ __
R

1

a° = M(D, R--'-'--~)= > 4R2K = 4RK "

If R ~ 0, then R ]M(D, R) --+ co. This implies that for every finite a, there exists R(a) > 0, such that [a I < 1/4R(a)K < ao holds. Thus, we have proved that for every finite a, there exists a disk [ z [ < R(a), R(a) > 0, such that W(a, z) is regular (and even univalent) for Izl < R(a). 3. Extension to an unbounded domain D. Theorem 2. Let D be an unbounded domain in the z plane, such that [) contains the point at infinity, but not the whole z plane, and let L(z) be analytic and single-valued for z eD, with a double pole at most at z = oo. Then there exists a unique solution of equation (4) with the initial condition ( 2 ) f o r z eD. This solution W(a, z) is analytic in a for [a I < ao, ao > 0 and z ~D, conformal and univalent in z for z e D, I a I < 60, and has at most one simple pole at some point z o e D , (Zo may vary with a), and it satisfies equation ( I ) f o r [a 1, I bl and (a + b) < 60. Proof. Without loss of generality we may assume 0 4/). Setting 1 (16)

co =

1 =

z

the system (4) and (2) transforms into: (17)

do = _ CO2L(c O_ I)= L*(CO), da

(18)

o(o) =

and the domain D is mapped by ~ = 1 [z onto the bounded domain A. the function L*(co) is regular in A, and we can apply Theorem 1 to the system (17) and (18). Thus, there exists a unique analytic solution co(a, ~) which is regular in a, regular and univalent in ~ for I a I < ao, 6o >0, and ~ e A. By (16) this implies that W(a, z),

92

MEIRA LAVIE

which satisfies (4) and (2), is regular in a, regular and univalent in z for z ~ D and ]a [ < a o, unless o~(a, C) = 0. But, since o~(a, 4) is univalent in A (for l a I < ao) it has at most one simple zero Co E A. (For every a the point Co may be different). Hence W(a, z) has at most one simple pole at some point Zo = 1/Co which belongs to D. Note that if L(z) is regular or has at most one simple pole at z = ~ , then L*(0) = 0. Hence ~ = 0 is a fixpoint of ~(a,C), i.e. co(a,0) = 0, and by (16) it follows that W(a, z) has one simple pole at z = ~ , so that z = ~ is a fixpoint of

w(a,z). By Theorem 1, ~o(a,~) satisfies equation (1) under the specified conditions, and it easily follows that the same is true for W(a, z). REFERENCES

1. L. Bieberbach, Theorie der Gewohrdichen Differentialflieehungen, Springer, Berlin, 1953, p. 1-5. 2. P. Erdt~s and E. Jabotinsky, On analytic iteration, J. Analyse Math. 8 (1960--1961), 361376. 3. E. Jabotinsky, On iterational invariants,Technion-IsraelInst. of Tech. Publ. 6 (1954-1955), 68-80. CARNEGIE INSTITUTE OF TECHNOLOGY,

PrrTSnVgOH,PENNSYLV~,~IA,U.S.A. ON LEAVEFROM THE T~rrlNION-ISRAEL INSTITUTE OF TECHNOLOGY,

HAIFA,ISa~L

THE

LINE ANALOG

OF RAMSEY

NUMBERS*

BY

MEHDI BEHZAD AND HEYDAR RADJAVI

ABSTRACT

For positive integers r and s, f'(r, s) is defined as the smallest positive integer p such that every connected (ordinary) graph of order p contains either r mutually adjacent lines or s mutually disjoint lines. It is found that f'(r, s) = ( r - 1 ) ( s - 1 ) + 2 unless r = 2 and s ~ 1, in which case f ' ( 2 , s ) ~ 3 .

By the Ramsey number f(r, s) is meant the smallest positive integer p such that every graph (finite, undirected, with no loops or multiple lines) of order p contains either r mutually adjacent points or s mutually disjoint points. These numbers have been studied extensively by Erd6s [1, 2, 3, and 4], Erd6s and Szekeres [5], Greenwood and Gleason [6], and others, who have found various bounds for f(r, s). The exact values o f f ( r , s) are not, in general, known. In this note we consider the line analog of this problem to which we give a complete solution. First we observe that in the definition, given above, of f(r, s) the class of graphs can be restricted to that of connected ones except for the trivial cases in which r = 2. We now define, for positive integers r and s, f'(r,s) as the smallest positive integer p such that every connected graph of order p contains either r mutually adjacent lines or s mutually disjoint lines. Our main result (Theorem 3) gives the exact value off'(r,s) for every r and s. It is worth mentioning here that if the word "connected" is omitted in the above definition, then f'(r,s) would not exist. For convenience we introduce the symbol ~¢(r,s) to denote the class of all connected graphs which have either r mutually adjacent lines or s mutually disjoint lines. LEMMA 1. Let g(r,s) be the smallest positive integer p such that every tree of order p is in sd(r, s). Then g(r, s) = f'(r, s). Proof. It suffices to show that f'(r, s)< g(r, s). By the definition of f'(r, s) there exists a connected graph G of order f'(r, s) - 1 which is not in d ( r , s). Let Go be a spanning tree of G (a subgraph of G which is a tree and contains all the points of G). But Go cannot belong to ~¢(r, s), implying that g(r, s) > f ' ( r , s) - 1. Received January 9, 1967. * Definitions not given here can be found in [7, 8].

93

94

MEHDI BEHZAD AND HEYDAR RADJAVI

[April

The above lemma permits us to restrict ourselves to trees in what follows. LEMMA 2. In any non-trivial tree there exist two adjacent points a and b, deg a = 1, such that at most one point adjacent with b has degree greater than 1.

Proof. Assuming that the lemma holds for all trees with less than p points, p > 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, the lemma immediately follows. Assuming deg 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<_-;to<;tl<...
• 1/;ti= oo. i=l

We shall discuss the following problems: What are the conditions, necessary and sufficient, on a sequence {#,} (n > 0) in order that it should possess the representation (1.2)

#, =

tardy(t)

fo

n = O, 1, 2,...

where ~(t) is of bounded variation in [0,1]. What are the conditions, necessary and sufficient, on a sequence {#,} (n >-O) in order that it should possess the representation:

1

(1.3)

/~, =

f0

tx"f(t)dt

n = 0,1,2,...

where f(t) belongs to a given class of functions integrable over [0,1]. Hausdorff [3] gave the answer to the first problem in the case 20=0. Endl [2] solved the same problem in the case ;to > 0 and the function ~(t) is nondecreasing in [0,1]. Schoenberg [9] obtains the same solution as Hausdorff [3] in another way and we shall use in this paper some of his results. Roceived March 16, 1966 97

98

D. LEVIATAN

[April

Let A be an infinite matrix of real numbers A = [[a,~[[

n =0,1,2,...

m = 1,2,...

where air = 1 i = 0 , 1 , 2 , . . . . Denote

(i,,'",im) = detll a,~,,l[ 0 < i ,

< ...
Let us assume that (i~, ..., is) > 0 for every 0 < i~ < i2 < .." < iF. For a sequence {/~,} (n > 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 <

Let M(u) be an even, convex continuous function satisfying 1. M(u)/u ~ O(u ~ 0), 2. M(u)/u ~ co (u ~ oo). Denote by LMtO, 1] the class of functions integrable over t 0, I] such that j'o1Mtf(x)] dx < oo. LMtO , 1] is the Orlicz class related to M(u). (See [61). If we take M ( u ) = In I" p > 1, LMt0, 11 is the space fit 0,11 . The Orlicz class L~tt0,11 is not necessarily a linear space (see [6] Theorem 8.2). Denote by M[0,1] the space of all functions essentially hounded in t 0,1]. THEOREM 2.2. Suppose that {qb,(x)} (n __>0) spans the space C[0,1] in the supremum norm. The sequence {/t,} (n > O) possesses the representation: (2.4)

/.t, =

fo'+

n = n = 0,1, 2,...

,(Of(t) dt

where: (i) f(t) ~ LMt0, 1] if, and only if, (2.5)

n~_O m=O sup ~2 [f°'

2,m(t)dt M

]

--H
I I1 Anm(t) dt

(ii) f(t) ~ M[O, 1] if, and only if, (2.6)

sup

I ..I

_= H < o o .

O~_m~_n n > 0 fo1 Anm(t) dt

(0,m + 1,...,n) n)D,_.,d;m(t) for 0_< m < n = 1,2,-..

(2,,,(t) = (m + 1,..., n) (m,...,

and,~,,(t) = ~b,(t), by [91 Theorem 8.1 2,,,(0 => 0 for 0 _< t _< 1, 0 < m < n = 0, 1, 2,..-) THEOREM 2.3. Let {2,} (i >=O) satisfy (1.1) with 20 = 0. The sequence {g,) (n >=O) possesses the representation (1.3) where: (0f(t) ~ L,[0,1] if, and only if, sup I]

n~O m=O

(2.7)

(-1)"-"2,,,+1 ..... ;t,[t*', ... t*"ldt M

[u,,, ...,

ItS, ..., t"-] at

= H
100

(ii) (2.8)

D. L E V I A T A N

[April

f(t) ~ M[0,1] if, and only if, [/~m, "", ~ ]

sup O~m~n n~_O

- H
l[t2% ... , t2- "] dt

fo~

•

By (2.1), and Miintz theorem (see [7] Theorem 2.8.1), Theorem 2.3 in the case 21 = 1 follows from Theorem 2.2. For 2 ~ = i , i = 0 , 1 , 2 , - . , and M(u)=lul p l < p < ° ° , T h e ° r e m 2.3 (i) is Hausdorff's Theorem III [4"] and for 2i = i, i = 0,1, 2,-.., Theorem 2.3 (ii)is Hausdorff's Theorem IV [4]. For 2~ = i,i = 0,1, ... Theorem 2.3 (/) was proved by Berman [1]. 3. Proofs of the Theorems. Proof of Theorem 2.1. We have to prove the theorem only in the case 20 > 0 since for 2o = 0 this is Hausdorff's Theorem VI [3]. First we prove the necessity. Define the sequence{/~,}, {~,} (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

.....

.
Thus we prove (2.2). In order to prove the sufficiency let us define the sequences {/~,}, {J[,} (n > 0) by (3.1), with one exception,/~o is arbitrary. By Hausdorff (7) [3] we get: ( - 1 ) ' - ' , ~ . + 1 . . . . . L [ ~ . , "", ~"] = ~o. m=0

1967]

A GENERALIZED MOMENT PROBLEM

101

(by (3.4)) n-I

(--1)n~[1 ..... )~n[Po, "",/~n] = P o -

]~ ( - - 1 ) " - l - ' ; t , + l . . . . . i n - l [ / / , , " ' , # , - i ] • m----0

Hence by (2.2)

(35)

xl ..... x.[[~o,

.~,31 = I~ol +n.

By (2.2), (3.4) and (3.5) we get for every n > 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,,] <

where H = esssup o$t~I

2,,,(0

I,,,It at < H fo

2,,,(t) dt

If(t)].

Thus we proof necessity. We prove now sufficiency. By (2.6) and since (3.7)

~2 2..,(0 = ~bo(t) = 1

(see [9] p. 607 (8.23)),

ra=O

we get

.=o1~..I __
.=o~'(0

at

=

H.

Hence by Corollary 8.1 of Schoenberg [9], {/~,} (n > 0) possesses the representation

102

[April

D. LEVIATAN

(3.8)

#n = f j $ , ( t ) d ~ t ( t )

n = 0,1,2,...

where et(t) is of bounded variation in [0,1], and if we define ~t,(x) by:

~.(O)=O,~.(x)=

Z ;t.m

O<X_--
t.,,,~x

then there exists a subsequence {n~} ( i > 0 ) such that lim~n,(x)=~(x) for 0<x_<__l. Let x , y , 0 < x < y < 1, there exist r,s satisfying t.,. < x < t.,.+ t ,

t.,~ <=y < t.,~+ x

(r,s depend on n).

Now

< .__X+ laml

hence for every n > 0:

<= H m=r+l

l a*(y) - ~"(x)l

-__ n .

2,,,(Odt ra=r+ 1

We have lim~_.oo{(%,(Y) - %,(x)) = ct(y) - ct(x). Since {¢,(x)} (n > 0) spans C[0,1] we have by [9] Theorem 8.1 and Corollary 8.1 that the solution of the moment problem is unique. By Helly's theorem every sequence {nf} (i >=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 <

m=O

Define functions g~(x) by:

an(0)=0

an(X)=

g

(--1)n-',~.+1 ..... 'Zn[.s,"','n]

tt/gt m-,a ~ x

0<X=
and we get by Schoenberg [9] that for every k > 0

fo

ltakda.(t)=

~] ~'nm ,~/~,, ,,"-"~~ m + ~, ~ .t] m=O

1 ....

as n ~ oo. Using Helly's theorem (see [10] p. 29), since an(X) are of variations uniformly bounded in [0,1] we get limgn,(x)= ~(x) for 0 < x < 1. By HellyBray theorem (see [10] p. 31)

#~ =

f0

1ta~d~( t)

W e conclude the proof as in Theorem 2.2.

k = 0,1,2,.... Q.E.D.

BIBLIOGRAPHY 1. D. L. Berman, Application of interpolatory polynomial operators to solve the moment problem, Ukrain Math. Z 14 (1963), 184-190. 2. K. Endl, On system of linear inequalities in infinitely many variables and generalized Hausdorffmeans, Math. Zcit. 82 (1963), 1-7. 3. F. Hansdortf, Summationsmethoden und momentenfolgen I, II, Math. Zeit. 9 (1921), 74-109, 280-299. 4. , Momentenproblem fiir ein endliches interval. Math. Zeit. 16, (1923), 220-248. 5. A. Jakimovski, The product of summability methods; new classes of transformations and their properties II, Technical (Scientific) Note No. 4 Contract No. AF 61 (052)-187, August 1959. 6. M. A. Krasnosel'skii and Ya. B. Rutickii, Convex functions and Orlicz spaces, Translated by Leo F. Boron. P. Noordhoff L t d . - Groningen- The Netherlands 1961. 7. G. G. Lorentz, Bernstein polynomials, Toronto, 1953. 8. Yu. Medvedev, Generalization of a theorem of F. Riesz, Uspehi Matem. Nauk, 8 (1953) 115-118. 9. I. J. Schoenberg, On finite rowed systems of linear inequalities in infinitely many variables, Trans. Amer. Math. Soc. 34 (1932), 594-619. 10. D. V. Widder, The Laplace transform, Princeton, 1946.

T~-Awv UNIVERSITY, T~L-AVlV, I S ~

ON SHRINKING ARCS IN METRIC SPACES BY

P. H. DOYLE(1) ABSTRACT

By a sin (l/x)-eurve is meant a metric continuum that is a 1-1 continuous image of the disjoint union of an arc and a semi-open interval that has the image of the arc as continuum of convergence. It is shown that if M is a compact metric space, A c M an arc, while M[A is an arc having A[A as an end-point, then M is an arc, a tried, some sin (1/x)-curve, or some sin (l/x)curve with an arc attached at one point, or some sin (1/x)-curve with two ares attached. The case of shrinking finitely many arcs is also considered in an attaching theorem. I f M is a c o m p a c t metric space, 9 an upper semicontinuous decomposition o f M , let X = M [ 9 and suppose r / : M ~ X is the natural map.(2) We will examine the case in which 9 has only finitely m a n y nondegenerate elements that are arcs. F o r X an arc, T h e o r e m 1 describes the topological type o f M where 9 has only one arc d e m e n t . I n T h e o r e m 2 this result is generalized to the case o f finitely m a n y arcs. I n the sequel M [ A will denote the topological space obtained f r o m the topological space M by identifying the points o f the subset A o f M. LEM~A 1. Let M be a compact metric space and A c M a closed subset such that X = M / A is an arc with ~l(A) = 09 as an end-point u n d e r the natural m a p ~l: M ~ X . T h e n M - A n A is a continuum. Proof. The m a p ~/[ M - A is a 1-1 m a p that is open and is thus a h o m e o morphism. I t follows that M - A is topologically the semiopen interval [0,1). Thus M - A is a c o n t i n u u m with M - A - ( M - A ) c A. So without loss o f generality one m a y assume that M = M - A a n d that A = ( M - A) n A. Let g: [0,1) - , M - A be a h o m e o m o r p h i s m o f [0, 1) onto M - A and define Ck = g { [ ( k / k + 1), 1)} t 2 A for k = 0,1,2, ..., n, .... Then Ck is a c o n t i n u u m and n ~o Ck = A = M - A n A is a c o n t i n u u m since Ck ~ Ck +1" Trmom~M 1. M is a compact metric space and A c M is an arc. I f X = M / A is an arc with ~l(A) = ~o as an end-point u n d e r the natural m a p ~l: M ~ X , then M is an arc, a t r i e d , the union S u K o f a sin (I/x) - curve S (0 < x ~ 1) and Received August 9, I966, in revised form January 11, 1967. (1) Prepared under a NASA Research Grant No. NsG-568 at Kent State University. (2) The notation X = M / ~ is not commonly used and an upper semi-continuous decomposition of a space Mis usually denoted by 9 . We prefer however to distinquish between the family 9 and the topological space ~ (denoted here by X = M]9). 104

ON SHRINKING ARCS IN METRIC SPACES

105

the interval of convergence K = {(x,y),x = O, - 1 <- y < 1}, the set S u K with an arc attached at an end-point of the interval of convergence, or the set S U K with two disjoint arcs so attached, one at each of the end-points of the interval of convergence.(s) Proof. By L e m m a 1, C = M - A ~ A is a continuum and since A is an arc, C is a point or an arc. I f C is a point, then M - A = (M - A) u C is the 1-point compactifieation of a half-open arc and is thus an arc. Hence M is a union of two arcs M - A and A with the end-point C of M - A on A. Clearly M is an arc or triod. In case C is an arc, let its end-points be a and b. Since M - A is a semi-arc and C = M - A n A, one can represent M - A as a union of arcs An and B , , : M - A = U ~ = I ( A , , U B n ) where A,,=[an, bn'] Bn = [bn, a , + l ] n = 1,2,.-. and a~, b~, denote the end-points of these arcs. Moreover one can assume that A n ~ B n = ( b n ) , B , N A , + l = ( a n + l ) and a , - o a and b ~ b where al is the end-point o f M - A ( ~ l ( a x ) ~ to). Then representing similarly a sinl/x-curve S(0 < x < 1) one can find a mapping of the interval K onto C obtaining a homeomorphism between S u K and ( M - A ) U C. It follows that M - A is topologically S U K. I f C---A, M is (topologically) the set S U K. Otherwise there are two cases. I f C # A, C either lies in the interior of A or C has an end-point in c o m m o n with A. These two cases yield the last two values of M in Theorem 1. In Theorem 1, there occur five different class counter-images of an arc. We call this set of continua F. Suppose that M is a compact metric space, A c M is an arc while X = M / A is an arc with ~/(A) = 09 an interior point of X. Then X is a union of two arcs X1 and X2; X1 n X2 = 09 is an end-point of each. F r o m Theorem 1 we know that each of the spaces M~ = ~/-1 (X~)(i = 1,2) is a member of F, and M1 n M2 c A. Thus X is obtained by attaching members of F at points of an arc. Using the operation of attaching of two spaces by a function f one can prove the following: THEOREM 2. Let M be a compact metric space a n d ~ an upper semi-continuous decomposition of M whose only non-degenerate elements are a finite number of arcs A1,A2, ...,A,. I f X = M / ~ is an arc then M can be represented as a union M1uM2u...uM n so that M i ~ F , i = 1 , 2 , . . . , n and M 1 U . . . u M i + ~ is obtained from M~ U "" u Mi by attaching Mi+ ~ to M 1 U "" u M~ by a continuous function fi, i = 1,2,n - 1. The question what 1-dimensional continua can be decomposed into arcs and points so that the hyperspace is a preassigned continuum may be of some interest. (3) Here S U K denotes any metric continuum that is a 1-1 continuous image of a semiopen interval S and a disjoint arc K having K as continuum of convergence. We thank H. Davis for this definition.

106

P.H. DOYLE

It should perhaps be pointed out that a relationship exists between the observations made here and the cyclic element theory as presented in [1], [2], [3]. Though the continua we have considered here are not locally connected, Theorem 2 does provide us with a representation of a continuum analogous to the cyclic chains. Links in the chain may or may not be locally connected in our case. The development in [3] of the higher order cyclic element theory is actually carried out for compact finite dimensional metric spaces. REFERENCES 1. D. W. Hall, On a decomposition of true cyclic elements, Trans. Amer. Math. Soc., 47, (1940), 305-321. 2. C. Kuratowski, Topologie II, Warszawa, (1952), 231-251. 3. G. T. Whyburn, Cyclic elements of higher orders, American Journal of Mathematics, 56, (1934), 133-146. KENT STATEUNIVERBITY KeST, Omo

ANALYSE HARMONIQUE DANS QUELQUES ALGI~BRES HOMOGt~NES PAR

Y. KATZNELSON ET P. MALLIAVIN

ABSTRACT

Etude d'alg~bres homog~nes sur le cercle ayant les m&nespropri6t6, de ealcul symbolique et de synth~e speeiale clue l'al#bre de Wiener.

Nous nous placerons sur le cercle T. Nous noterons par C(T) l'alg~bre de Banach des fonctions continues sur T, par A celle des fonctions ayant une sdrie de Fourier absolument convergente. Les alg~bres que nous consid~rerons seront des alg~bres de Banach, semi-simples, autoadjointes, ayant pour spectre T. Nous les identifierons/t des alg~bres de fonctions de C(T). Une telle alg~bre B sera dite homog~ne si l'application de B dans B d6finie par la translation f(x) ~ f ( x + a) est une application isom&rique de B sur B, telle que pour t o u t f e B, ffix6, l'application a o f ( x + a) soit une application continue de T dans B. On dira qu'une alg~bre homog~ne B est fortement homog~ne [1] si de plus quel que soit l'entier l'application de B dans B d6finie par f ( x ) ~ f ( k x ) est de norme 1. Nous allons 6tudier des al#bres homo#nes qui sont comprises entre C(T) et A. Nous pourrons pour des classes g6n6rales de telles alg~bres #ngraliser les r6sultats classiques pour A: seules les fonctions analytiques op~rent, non synthgse

spectrale, tous les automorphismes diffdrentiables de l' algdbre sont donnds par des changements de variables lindaires. Par contre un probl~me ouvert pour A, ~t savoir celui de la dichotomie du calcul symbolique sur les alg~bres de restriction, pourra 8tre d6montr6 non vrai dans le cadre de ces alg~bres homog~nes. Comme une tendance actuelle est de penser que la dichotomie vaut dans A, ce dernier r6sultat montre que s'il enest ainsi sa d6monstration ne pourra 8tre donn6e que par une &ude fine des propri&6s de A qui la distingue des alg6bres homog6nes. La m&hode utilis6e pour construire des alg~bres homog~nes interm6diaires entre C(T) et A sera celle de [1]; nous allons la rappeler ci-dessous. ReceivedJune 24, 1966 I07

108

Y. KATZNELSON ET P. MALLIAVIN

[April

Si E est une partie ferm6e de T, nous noterons par A(E) l'alg6bre des restrictions de A / i E, c'est ~t dire le quotient de A par l'id6al de toutes les fonctions de A s'annulant sur E. Remarquons que l'espace dual (A(E))* de A(E) s'identifie avec les pseudomesures, dent le support est contenue darts E, et qui de plus sent orthogonales ii toutes les fonctions de A s'annulant sur E. Avec [1], nous noterons par AE la plus grande aig6bre homog6ne dent la restriction f i g est A(E). La norme dans AE est d6finie par

IlsllA -- sup II*oslIA,E, (on (*or) (x) -- S(x + a)). act

De m~me on introduit la plus grande alg6bre fortement homog6ne contenant A g est d6finie par

A(E), soit A E. La norme darts

Ilsh. = sup II h,fll keg

(04 (hkf)(x) =f(kx)).

Dans ce travail nous donnerons une condition sur E pour que A r = A, puis une autre condition sur E pour ClUedarts A E la synth~se spectrale soit fausse; enfin une condition sur E permettant de d6crire les automorphismes de A~. Ensuite nous terminerons par deux exemples, celui d'une alg6bre contenant toutes les fonctions ayant un module de continuit6 co donn6 ~t l'avance, et qui n6anmoins poss6de toutes les propri6t6s de A, enfin celui d'une alg6bre fortement homog~ne telle clue la dichtomie au sens fort n'ait pas lieu. I. Conditions sur E. Etant donne une partie E de T nous expeirmerons "l'6paisseur" de E en termes du comportement des transform6es de Fourier d'6Mments de (A(E))*. Etant donn6 p e (A(E))*, e > 0 et trois entiers k, l, N nous noterons par

G,={.ezl [.[ <-N. IP(k.+ OI Si nous nous donnons de plus un entier d nous poserons

HI= ,-I_~V~_,+~, G,. Nous averts alors les 3 conditions suivantes (Q'), (Q"), (Q): (Q') Quels que soient 8 > 0 et N, on peut trouver p e (A(E))* et k tels que Hp Ila. = 1 et que, quel que soit I e Z, G~ poss6de au plus un 616ment. (Q ") Quels que soient r/> 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 <

8

Soit p une pseudo mesure satisfaisant la condition (Q) avec g = 8t//2, N e t J. Posons p,(x) = ~(x - t)p(x)

comme

fo n ,~(x

- t) ~dt = 1

l'int6grale 6tant une int~grale forte ~ vaIeurs dans A, on obtient en multipliant les deux membres par p(x)

fo 2~ pt'-~n dt = p l'int6grale ~tant une int6grale forte ~t valeurs dans A*, comme lip I1, = 1, la fonction Ptque l'on int~gre v&ifie II/~to[It~ > 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

<

2 sup I1R(kx)p(x + t)h*, ir

On a

~, p( l - kn )e~°-P"°t.~(n),

R(kx)p(x + t)(l) =

tim --N

d'ofl

1[R(kx)p(x + t)[[a. <= ~, p(l - kn)ei°-*n'tz~(n) + Ne, n ~GI

D'apr~s I'hypoth~se (Q') la somme comporte un seul terme d'o6

IIhkR ]la: < 1 + rl d'oi~

Il < IIh~PIIA~IIh~RIIA:~(1 + ~) IIhkPIIA~ d'oO, ~/ 6tant arbitraire, l'6galit6 (1). C.Q.F.D. Exemple. Supposons que E eontienne des progressions arithm6tiques arbitrairement longues (ou plus g6n6ralement que, pour tree suite de mk ~ o0, on puisse trouver z~,z2, ...Zmktelle que les 2 mk points xl + x2__+~a + "'" +---~m~ appartiennent ~t E). Alors on salt (cf. I2] d6monstration du th6or~me 7) que E satisfait la condition (Q) (done ~t fortiori h (Q')). Alors pour ces ensembles A B = A. D'autre part, on sait que (of. 1'1]) si E est de mesure nulle An ~ A. L'ensemble parfait de Cantor est ainsi un exemple oh A ~ = A =/=At. IlL Une condition de non synth~se speetrale darts AE. On a: Tr~O~M~ 2. Si E satisfait la condition (Q'), alors A e n e satisfait pas la synth~se spectrale. Preuve. Utilisant une technique connue [3], on voit qu'une condition suffisante pour que Ag ne satisfasse pas la synth~se spectrale est qu'il existe une fonction r~elle f, f ~ A telle que

112

Y. KATZNELSON ET P. MALLIAVIN

[April

A(u)-- II v6rifie

f+~a(u) l u[ du <+oo. On sait que l'on peut trouver une fonction g e A, g r6elle telle que

a(u) = IIe"~ll.,. < exp(- ,'/=). Nous poserons f ( x ) = ~, a.g(k.x) oi~ a n "=

2 -z.-t

et oi~ k, est une suite d'entiers d6termin~s ci-dessous. Posons q

fq(X) = ~, a,,g(k,,x) n=l

c.(u) =

e x p ( i u g ( x ) - inx) dx 2~

fo 2"

• .(u) = Inl>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 <

_q _ , aqu > 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

<

2

20 / S

-- ¥

~n

N-~/2.

1967] ANALYSE HARMONIQUE DANS QUELQUES ALGEBRES HOMOGENES Notons

Nlxl2

AN(x) = Sup 0, N 1

115

, F = e-'"÷*As et # = N ~e~,

(6~ est la mesure de Dirac au point x;/~ est donc une mesure port6e par EN qui attribue h chaque x e E N la masse 1/N F(x)). L'on sait, d'apr6s Herz [5] que

II. II

<: 1] e-i"4'

II

=<

:i N

Comme F(x) est une moyenne des valeurs de e -~"* dans l'intervalle x

2n - - ~ - , x + --~-)

et que

-•x

N e'~e = l u [q~'(x) < l'-0~c " c =

N 1~

'

il r6sulte que 2z~ IF(x)

-

1

e-'~*<~)l <= NO < 1-6

et par cons6quent

[ fe"*a,[ > -g, 1 ce qui entraine

V. Exenlples d'alg6bres homog6nes. 6vident suivant:

Commen~ons par 6noncer le lemme

LEMME. Soit to un module de continuit~ donnd, prenons pour E une suite {an} tendant vers z&o et telle que Z to(~,3 < ~ .

Alors toute fonction admettant to pour module de continuitd appartient gt AE. Ceci &ant des th6or6mes 2 et 3 nous obtenons: THEORf~ME 4. Soit to(h) un module de continuitd. II existe une alg~bre de Banach homog~ne sur le cercle contenant A et toute fonction admettant to pour module de continuitd et qui poss~de les propridtds suivantes: 1) Toute fonction qui op&e dans B est analytique

116

Y. K A T Z N E L S O N ET P. M A L L I A V I N

[April

2) Dans B la synthdse spectrale est fausse 3) Tout automorphisme deux lois diffdrentiable de Best ndcessairement une translation. Demonstration. On prend pour B l'alg6bre Ae, E 6tant une suite tendant vers z6ro tr6s rite mais contenant n6anmoins des progressions arithm6tiques arbitrairement longues. Le probl~me de la dichotomie du calcul symbolique pour les alg6bres homog6nes est la question suivante: Est-il vrai que si Best une alg6bre de Banach homog6ne sur T telle que A c B ~ C(T), alors seules les fonctions analytiques op6rent darts B. Si la r6ponse est positive, m~me en restreignant la question aux alg6bres du type Ae, cela entrainerait la dichotomie du calcul symbolique pour les alg~bres quotient de A. Le dernier r6sultat de cette note montre que si l'on remplace la question ci-dessus par "Est-il vrai que toute fonction qui op~re de A dans B e s t n6cessairement analytique?". La r6ponse est n6gative. TI-I~ORi~ME 5. II existe une alg~bre fortement homog~ne B, satisfaisant C(T) D B D A(T) B ~ C(T) et telle que route fonction deux fois continuement ddrivables opkre de A dans B. D6monstration. D6signons par A~ 0 < 8 < 1 les alg6bres (fortement homog6nes) obtenues par le m6thode d'interpolation de Calderon [6] entre Al = A(T) et Ao = C(T). Posons

N R)--sup

Ile"ll

[IfllA_ga f r6elle

II est facile de voir que N~(R) = eaR. Soit E une suite tendant vers z6ro, r6union des suites Ej = { g l - k~0}{=o et de {0}, avec ~j =(Jr) -~,

~0 = ~ "

On voit que l'on peut choisir ej > O, ej ~ 0 de mani~re /t ce que N*
sup Ilflla_gR f r6dl¢ j=1,2,...

satisfasse N*(R) ~ oo, N*(R) <=R + 1. Soit Bo l'alg6bre des fonctions ~b continues sur E satisfaisant

l[ I[o -- supH avec la norme It $]lo-

< oo

1967] ANALYSE HARMONIQUE DANS QUELQUES ALGEBRES HOMOGENES 117 L'on peut alors d6finir B comme l'alg6bre fortement homog6ne la plus grande, dont la r6striction /t E coincide avec Bo, c'est-/~-dire l'alg6bre des fonctions continues sur T telles que (i) d/(nt + ct)1~~ Bo pour tout n, 0t (ii) ~k(nt + ct)[E depend contnument de ~ (en tant qu'616ment de Bo), (iii) II II. = sup,,, II + )I 11.o < II est clair que A ( T ) c B ~ C(T). De l'inegalit6 concernant N*(R) l'on obtient 11e"SllB = O(n) pour t o u t e f r6elle dans A(T); si F(x) est deux fois continument differentiable que, sans perte de g6n~ralit6, l'on suppose 2n-p&iodique, l'on a F(x) = ~,P(n)e "J" avec ]~l ~(n) l I n [ < oo et par consequent F ( f ) = ~,t~(n)el"I est une serie convergente dans B. L'alg~bre B dont l'existence est assur6e par le th6or~me 5 a la propri6t6 suivante: Si # est une mesure/t support fini port6e par T et si l'on d6note par [/~] l'op6rateur de convolution sur B I/z] : f ~/~*f f E B

If il--masse

alors totale de/~. Ceci r6sulte de [1] th6or~me 2 (oil l'on parle de fonctions qui op~rent dans B mais la m~me d~monstration reste valable pour des fonctions qui op~rent de A dans B. BIBLIOGRAPHIE 1. Y. Katznelson, Calcul symbolique dan~ les alg~bres homog~.es, C.R. Acad. Sci. 2.54(1962), 2700-2702. 2. J.-P. Kahane et Y. Katznclson, Fonctions de la classe A: contribution d deux probl~mes. Israel J. of Math., 1 (1963), 110--131. 3. P. Malliavin,Imposibilitede la synth~se spectrale sur les groupes abeliens non compacts I.H.E.S. No. 2 (1959). 4. Zygmtmd, Trigonometric series. Cambridge Vol. 1, (1959) p. 197, 5. Hcrz, Spectral synthesis for Cantor set. Proc. of the Nat. Acad. Sci. U.S.A. 42 (1956). 6. A. P. Caldcron, Intermediate spaces and interpolation, Studia Math. 24 (1964), 113-190. UNIVERSII~ HEBRAIQUE DE JERUSALEM ET UNrCEP.~TE DE PARIS

INTRINSIC EQUATIONS OF ROTATIONAL GAS FLOWS BY

E. R. SURYANARAYAN ABSTRACT

Considering a vortex line as a C3 curve in Ea, equations governing the flow of a steady, compressible gas are expressed in the intrinsic form. These intrinsic relations are applied to derive some geometric properties of rotational motions, and to study a class of flows whose vortex lines form a family of helices on right circular cylinders. 1. Introduction. Coburn [1], Kanwal [2], Wasserman [3] and Truesdell [4] have derived the intrinsic form o f the equations governing the steady motion of a gas, by considering a stream line as a space curve in in E a, a three-dimensional Eucldean space. In this paper we derive the equations of motion, continuity and entropy of a rotational flow in the intrinsic form by considering a vortex line as a C 3 curve in E a. It is assumed the gas is non-viscous and non-heat-conducting. By expressing the equations of motion along the tangent, principal normal and the binormal vectors of the vortex line, it is shown that for a circulation preserving complexdemellar flow the Lamb surfaces and the surfaces normal to the stream lines intersect orthogonally along the vortex lines, and therefore, if the vortex lines are lines of curvature on either the Lamb surface or the surface orthogonal to the stream lines, then they are lines of curvature on the other also. It is found that in a circulation preserving motion, a vortex line is a geodesic on the Lamb surface if and only if the velocity component along the principal normal vector is zero. A necessary and sufficient condition is determined for the Lamb surfaces to be a family of parallel surfaces. It is shown that if the vortex line admits normal surfaces then these surfaces are minimal if and only if the magnitude of the vorticity vector does not vary along the vortex line. The intrinsic equations are used to derive the flow equations when the vortexlines are right circular helices.A class of solutions o f these equations are obtained in the case when the binormals of these helices form a normal congruence. 2. The basic equations. Let x~(j = 1,2,3) denote a Cartesian orthogonal coordinate system in E a and let us denote the partial derivative by the symbolism t~j - axJ" Received December 8, 1966

118

INTRINSIC EQUATIONS OF ROTATIONAL GAS FLOWS

119

In E 3 covariant and contravariant components are equivalent. However, in order to use the summation convention, we shall write the indices in covariant and contravariant positions. Let g~j denote the fundamental tensor of E 3. The equations governing the flow of a stationary gas, neglecting viscosity and thermal conductivity, are (2.1)

Oj(pu j) = O,

(2.2)

puJOju~ + OiP = O,

(2.3)

uJ Ojtl = O, p = P(p) SOl),

(2.4)

where, uj are the components of the velocity, p is the density, p is the pressure, and q denotes the specific entropy. For a polytropic gas

P(p) = p'/~,

(2.5)

being the adiabatic exponent. If e *jk denotes the permutation tensor, then the vorticity vector is defined by ¢0 i = eiJkOjuk.

(2.6) The Lamb vector 2~ is defined by (2.7)

~i = eiyk ¢Ojuk.

The equations of motion (2.2) can be written in terms of the Lamb vector [5] (2.8)

2~ = TOyl - O~H

where, T is the absolute temperature and H is the stagnation enthalpy defined by (2.9)

1 u2

H = ~

+ I.

Here u is the magnitude of us and I is the specific enthalpy. A consequence of the relation (2.6) is (2.10) 3. The basic decomposition. then we may write (3.1)

0j~J = 0. If fl is a unit vector along the direction of aft,

off = o~sj

where a~ is the magnitude of ~o( Let n j and b Jbe the unit vectors along the principal normal vector and the binormal vector respectively, of the vortex line. The velocity vector can be expressed in terms of s j, n j and bJ:

120

E.R. SURYANARAYAN

(3.2)

U 1-- O~lSJ+ 0~2nj + ~t3bJ.

[April

Substituting for coJ from (3.1) and for uJfrom (3.2) into (2.8) and using the crossproduct relations of s/, n j ans b j, (2.8) becomes (3.3)

o)(%bt - %n3 = Td;1 - a~H.

Decomposing the right hand side of the above relation along st, n~ and bt and equating their coefficients respectively, we get dH d--7.= 0,

(3.4) dtl T dn

(3.5)

dH dn = - °ta°~'

drl T db

(3.6)

dH db = °t2t°'

where, d/ds, d/dn and d/db denote the directional derivatives operator along the directions s j, n I and b~ respectively. The equation of continuity (2.1), by use of (2.3), (2.4) and (3.2), becomes (3.7)

(°q--a~' + Ix2d-n' + %-d-b-'t P + P (°qAt + ct2A2 + %Aa + d°q + dn + dct3~db] = O, where (3.8)

At ~- cOisl,

A2 = 0in ~,

A3 = a~bt.

The equation (2.3) can be written (

d d dl cq-d~- + ~2 ~ - + =3 ~ - r/=O.

Substituting into (2.6) for coi from (3.1) and for u s from (3.2), and equating the coefficients of s ~ n Sand b ,~we find after a lengthy but direct computation that d%

(3.9)

oft~'~t + %d - %t + d ~

(3.1o)

%f22

(3.11)

dot2 %f~3 - %e + kcq -t ds

d% + %q - ~

+

d~x2

~

= co,

d~l ----0 db -'

doq d---n = O,

where f~t, f~2, ~s are rotation coefficients [6] defined by (3.12)

f~t . ~bk'~- .

nk. ~'-,dsk. ~2

bk ~

s ._dnk d_~

f~a = nt-d~"

st db~dn

1967]

INTRINSIC EQUATIONS OF ROTATIONAL GAS FLOWS

121

k is the curvature of the vortex lines, and (3.13)

dnk. k d=-d-~b,

e=

_~

dbk k s~, q=--d-Gs,

dbk t=-d-~n.

The relation (2.10), when substituted for co~from (3.1), yields (3.14)

d logco) = - A1. -~(

Let us now draw some conclusions from our calculations. The relation (3.4) shows that the surfaces H = constant contain the vortex lines. This is a well known result [5]. Relations (3.5) and (3.6) show that the rate of change of H along the principal normal direction is completely determined by the rate of change of ~/along this direction, and similarly for the rate of change of H along the binormal direction. For a circulation preserving motion, there exists an acceleration potential ~b and therefore 14] 1 2). (3.15) = + That is, ~b satisfies the relation

The surfaces ~b+(1/2)u 2 =constant are the Lamb surfaces. From (3.15) it is clear that the Lamb surfaces contain both stream lines and vortex lines. If the motion is complex-lamellar, that is, if there exists a family of o01 surfaces orthogonal to the stream lines, then these surfaces contain vortex lines. And these two families of surfaces, the Lamb surfaces and the surfaces orthogonal to the stream lines, intersect orthogonally. Therefore we have: THEOREM. In a circulation preserving complex-lamellar flow, the Lamb surfaces and the surfaces orthogonal to the stream lines intersect orthogonally along the vortex lines. This result is more general than the one in the paper [7]. Since the Lamb surfaces and the surfaces orthogonal to the stream lines are orthogonal we apply a classical theorem due to Joachimsthal [8] and obtain the following statement: In a circulation preserving complex-lamelIar motion if the vortex line is a line of curvature on either the Lamb surface or the surface orthogonal to the stream line, then it is a line of curvature on the other also. From the relation (2.8), (3.3) and (3.5) we find that the normal to the Lamb surface is along the principal normal of the vortex line if and only if ~2 = 0. Therefore, a vortex line is a geodesic on the Lamb surface in a circulation preserving motion, if and only if the component of the velocity vector along the principal normal vector of the vortex line is zero.

122

E.R. SURYANARAYAN

[April

From (2.8), (3.3) and (3.15) we find that

where d/d~ is the directional derivative along the Lamb vector. Therefore, a necessary and sufficient condition that the surfaces, containing ui and coi, be parallel surfaces is that 0,

+

= a2

where a is a constant along each surface of the family. A similar result has been obtained by Coburn [1]. The above condition is also equivalent to o~2 =

a --cosoc, ~0

a . o~3 = - - s l n c x , 60

being a parameter. The relation (3.14) shows that the rate of change of - l o g o along the vortex lines equals the divergence of the vector sf. I f A1 = a~sf = O, then o~ does not vary along the vortex lines, and conversely. The condition A t = 0 is a geometrical condition on the congruence of vortex lines. It means, roughly, that the vortex lines do not converge (or diverge). A motion for which the vortex lines possess a family of normal surfaces satisfies f~x = 0, and A t is then the mean curvature of these normal surfaces. In particular ~o does not vary along the vortex lines if and only if the mean curvature A t vanishes, that is, if and only if the family of normal surfaces constitute a family of minimal surfaces. From the previous results in this section we find that, in a circulation preserving complex lamellar motion with vortex lines possessing normal surfaces, the velocity vector, the vorticity vector and the Lamb vector are mutually orthogonal at every point of the flow, that is, the Lamb surfaces, the surfaces orthogonal to the stream lines and the surfaces orthogonal to the the vortex lines are mutually orthogonal. This result is known [9]. For a screw motion, that is, for a motion with co~parallel to u s, a 2 = eta = 0, cq = u . Therefore (3.10) and (3.11) imply that u does not vary along the binormal of the vortex lines and that the variation of logu along the principal normal equals the curvature of the vortex lines. 4. Flows whose vortex lines are right circular helices. Here we shall consider the case where the vortex lines form a family of right circular helices. We introduce the cylindrical coordinates r, 0 and z and write

(4.1)

s~ -- 0~sin/~ + z~cos/~,

(4.2)

n~ =

(4.3)

b~ = - 0~cosfl + zz sinfl.

-

r~,

1967]

INTRINSIC EQUATIONS OF ROTATIONAL GAS FLOWS

123

Here 0~ and z~ are the unit vectors along the increasing 0 and z directions respectively and r~ is the unit vector along the radius of the cylinder, fl is the angle of the helices and is in general a function of r. Since the vortex lines form geodesics on cylinders, we have c~2 = 0. (Section 3). Since helices form a congruence of parallel curves on cylinders, At = 0 [10]. After some computation we find that: (4.4)

At

A3 =

d = e = q = 0,

sin 2 fl k = ~ /",

(4.5) (4.6)

=

f~l =

cos 2 fl t = ~ g,

dfl + sinpcosp tO2 = O, tO3 = r '

sin/~ cos# + _ r dr

and (4.7)

d sin p O 0 d d'--~= r O0 b cosfl ~--~, ~-~ =

0 d cosfl 0 Or'db= r O0

sin fl d-~

Substituting these relations into (3.4)-(3.7), (3.9)--(3.14) we find that (4.8)

dH ds = O,

(4.9)

Tan 0H Or + Or" = --aa¢°

(4.10)

drl T-~

(4.11)

o q ~ - + cq-d-~ P + P ~ ds + db ] = O,

(4.12)

r

\d-r

(4.15)

Or

d% dat ds +-d-b-" = 0,

(4.13) (4.14)

dH db = 0

%

sinflcosfl r

+

dfl) Trr

+

sin 2 r

-fla 1 +

d¢o

d----T = 0.

The entropy condition (2.3) now becomes (4.16)

d (a1~+~3

d) ri=0.

0al W

= 0,

124

E.R. SURYANARAYAN

[April

In the above relations d/ds and d]db are given by (4.7). The above equations constitute the flow equations for a motion whose vortex lines are right circular helices. 5. A special flow. We shall now find a class of solutions of the set of equation s (4.8)-(4.16). We shall assume that the flow is isentropic that is, r / - constant, and the gas is polytropic. Further let

dfl (5.2)

cosp

dr

(5.1)

sinp

=

0

r

~t3 = 0q(r), ~t1 = al(r), H = H(r), co = oJ(r), P = P(r).

The condition (5.1) implies that f~a = 0 and therefore b~ forms a normal congruence and tanp = r/l,

(5.3)

where ~ is an arbitrary constant. The equations O.8)-(4.16), by virtue of (5.1)-(5.3), now reduce to (5.4)

(5.5)

--

dH d"-'r-= ¢0~3'

cos2p

-~s

r

(2,)

' ' +

(5.6)

~

d~ 3 -

dr

62 = 0 ,

(5.7)

d~ -dr -

r ~ 2 + r 2~1.

Integrating equation (5.7), we get (5.8)

~

-- A~(~ 2 + r2)-~ ,

where A is an arbitrary constant. (5.3),(5.7) and (5.8)imply that ~2 (5.9)

2A~2

-~3 r({2+r 2) | (~2+r 2)

d~ 3

d~ --t°"

In particular if (5.10)

~3

=

-

Ar(~ 2

+ r2)-~ ,

the equation (5.9) reduces to (5.11)

co = 4A{2({ 2 + r2)-~ .

Substituting for ~3 from (5.10) and for to from (5.11) into (5.4) and integrating, we obtain

1967]

INTRINSICEQUATIONS OF ROTATIONAL GAS FLOWS

(5.12)

H = B2

125

2A2~2 (~2 + r2) •

Here B is an arbitrary constant. Since we have assumed the gas to be polytropic, the specific enthalpy [5] I = ~'dp Jo 7

(5.13)

= fo" Tdp =

s(r ~_ 1)pC,_1)/,,

where S = S(t/) --- constant. From (5.8) and (5.10) the magnitude of the velocity is found to be constant and is equal to A. Therefore from (2.9), (5.12) and (5.13), we find that the pressure is given by (5.14)

2A2~2 ~2+r2

B2

p(~-I)/~

Y

½A2 = S ( 7 -

1)

From (3.2), (5.8), (5.9), (4.1) and (4.3), the velocity vector becomes (5.15)

u~

A ---- ~2 +

r2

[2~rO~+ (~2_ r2)zi].

For an isentropic motion of a polytropic gas, the equations of motion (2.2) and continuity (2.1) assume the form

Spll~uJ~lui + ~ip =

0,

a~(f/'u j) = O. Transforming the above equations into polar coordinates, we find that the components of velocity and pressure satisfy the transformed equations. From (5.15), we find that the stream lines are also right circular helices, making an angle 6 with the z axis, where cos6

=

~2 _

r2

~2 + r ="

As r changes from 0 to infinity t~ changes from 0 to n. A t r = ~, 6 = n/2. The vorticity vector, from (3.1), (4.1) and (5.11), becomes

09i --

4A~2

(~2 + r2)~ [rO~+ ~zJ.

The angle/~ of the vortex lines changes from 0 to n/2 as r changes from 0 to infinity. We find that the angle between u ~ and tot is also ft.

Since [1] C2

I=

7-1

where c is the local sound speed, the relation (5.13) and (5.14) yield

126

E.R.SURYANARAYAN =

7-1

Here M = u/c is the Mach number. For real M the constant B must satisfy B2 >

c2

7-1"

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

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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<° )l

=

f.

I

1 ~[x~) I

dP

we have I1 - x II ~ 0 if and only if x. ~ x in probability. So considered, Theorem 1 states that M has the circle property if and only if it is isomorphic either to an Euclidean space (in the case of finitely many atoms) or to the F-space (s) of all complex sequences c = (Ca, c2,'") with the norm °°

1

]c,I

Ilcll = z .=, z. l+lc.I (in the case of countably many atoms). Now let us consider the general case (1). Ordinary convergence (0) of a series ~,x., x . ~ X , is defined as usual. We follow A. Dvoretzky [5] and call ]Ex, absolutely convergent (A) if

E pv(x.) <

n

for every neighborhood V of 0 ~ X, where pv(X) is the Minkowski functional of V at x, i.e.

pv(x) = i n f { ~ [ z > 0, x ~ x v } . In a linear metric space with norm IIx [I, the series ]~x. is said to converge metrically (M) if ]~1[x. ]l < oo. In such a space we have M =~ A, in Banach spaces M ~.A, in F-spaces M =~ O.

130

LUDWIG ARNOLD

[April

In general, in an arbitrary linear topological space A does not entail O, and 0 does not entail A. But, for a power series (1) we have the following simple facts: LEM3~A 1. Let X be a linear topological space and (1) a power series in X. (a) There always exists a circle of convergence for A. (b) O at z I (or only {anz~} bounded)=:, A at every z with [z[ < [zt[.

Proof. (a) For each linear topological space the family W of circled 3 neighborhoods of 0 is a local base (see [6], p. 35), so that it is sufficient to consider W. For every U ~ W

= I [ p (x) v

c

see [6], p. 15). Therefore

Xpu(a.z") - X pu(an) l z In converges in a circle with radius (lim sup ~ / ~ -

t, and altogether we have A for

[ z [ < r(A) = inf (lim sup ~ / ~ ) -

',

UeW

I

and absolute divergence of (1) V I z > 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 =<~. Thus, for Izl < l z , I ~(z~)") ~,Pv(an zn) = ~,Pv ( a.z

1 ~, Z~ n < oo, = Y~pu(a.z ~lZ~]" < -~

q.e.d. We denote by r(A) the radius of the A-circle, and if X has the circle property (for 0), we denote the corresponding radius by r(O). LEMMA 2. Let X = II,~TX, be the Cartesian product of linear topological spaces X t each having the circle property with r(°(O) = r t ° ( A ) - - r, V t ¢ T. Then X also has the circle property with r(O) = r(A) = inf rr teT

Proof. These statements follow at once from the fact that a series :~:x, in X, x, = (..., x , (o,...), converges O(A) if and only if ]~x~t) converges O(A)Vt e T, q.e.d. Lemma 2 already covers the atomic case of Theorem 1. Lemma l(b) states that one part of the classical situation can be found in every linear topological (3) A set V in a linear space is circled if aV c V whenever I a I ~ 1.

1967]

131

RANDOM POWER SERIES

space. The other part, A =~ O, would, together with Lemma l(b), imply the circle property and r(O) = r(A). But for this we need additional conditions. For instance, every finite-dimensional space has the circle property with r(O)= r(A). Other classes will be described below. 4. Complete locally convex spaces. TI-mOR£M 2. I f X is a complete locally convex space, then A =~O, so that X has the circle property and r(O) = r(A) for every power series (1).

Proof. A =~ O follows easily from the fact that the Minkowski functional of a circled convex body is a pseudonorm and X is complete. The rest follows from Lemma 1, q.e.d. It should be mentioned that it is possible to develop a satisfactory theory of holomorphic functions defined in a domain G c C with values in a complete locally convex space (see also [7]). 5. Complete locally bounded spaces. Each locally bounded Ispace (i.e. a space possesing an open bounded set) can be endowed with a p-homogeneous norm II x 11 0.e. II~x II -- I~ I ~11x II ~ ~ C where 0 < p < 1) reproducing the original topology (see [8]). Conversely, a linear metric space with a p-homogeneous norm is locally bounded. TrmOREM 3. If X is a complete locally bounded space and if r(M) denotes the radius of M-convergence of (1) with respect to a (always existing) p-homogeneous norm LII[, X has the circle property with r(O) = r(A) = r(M) = (lim sup ~ / ~ -

l/p.

Proof. Clearly, by the p-homogenity of II" II,we have r(M)= (lira sup ~/II a,, II)-'/~. Taking into account the relations among O, A, and M stated in section 3 it remains to prove r(A) = r(M). This follows from the fact that for a p-homogeneous norm IIx, II1,~< oo is a necessary and sufficient condition for A-convergence of ]~x,, q.e.d. Consider an arbitrary F-space X with norm l~ll-111Without loss of the generality, we can always assume that II~x II---
~.x

(~ ~ o)

IIxil

x#O

has the properties ~b(O)-- O, ~b(1)= 1, ~b(0~+ fl) ~ ~(u) + ~b(fl), ~b(ufl)~ qS(00~(fl), and is increasing with ~. Clearly, since ~(~2) =<~(~)2, we have either ~b( + O) = 1 (and hence ~b(~)=l ¥~e(O,1]) or ~b(+O)=O. If [[. 1[ is p-homogeneous, ~(~) -- ~P. For (s) and M and in general for a bounded norm we have ~( + O) = 1,

132

LUDWIG

ARNOLD

but the same is true also for the unbounded norm real axis. However, we have TI-mOREM 4.

[April

Ilxll = l o g ( I

+

Ixl) on

the

If for an F-space X

~(+o) =o then X has the circle property with r(O) = r(A) = r(M).

Proof. Since [10txII < ~(~t)II x II, x has bounded spheres and is therefore locally bounded whence r(O)= r(A). For r(O)= r(M) it is sufficient to show that 0 at zt entails M in Izl < [zl I" Indeed, 0 at zl implies [I a,z~ II ~ O so that I[ a,z~ [I ~ c Y n. Now let [z[ < [ z tl. We have

zo IIa.z"ll = ~o IIa.z

II__
The right hand side is bounded for N ~ oo for every I z I < [zt ] if and only if ~ ( = " ) < oo v~ e [o, 1]. Applying the integral criterion this is equivalent to

fo 1

,~(=)

as <

oo.

But there exists an % ~ (0,1) with ~(%) = d < 1, therefore ~(0t~ < d" and

f. 1

dO

dq = _ ~

n=0

d~ =
O~

Z

< o%

n

q.e.d. Theorems 3 and 4 apply, for instance, to Lv, Iv, and H v (0 < p < 1). In a locally convex F-space, we have in general only r(O) = r(A) >=r(M) where the inequality may be strict as examples in (s) show. We note that local convexity or local boundedness are only sufficient conditions for the circle property. For instance, X = (Lp)°° (0 < p < 1) is neither locally convex nor locally bounded, but has the circle property by Theorem 3 and Lemma 2. 6. F-spaces without circle property. We simulate in a general F-space what led to the construction of the counterexample in the non-atomic case of Theorem 1: THEOREM 5. Let X be an F-space. I f there exist a constant c > 0 and for every 8 > 0 a finite number n = n(O of elements x~, ...,x, e X with sup II=x,

for i = 1, ..., n

~teC

and

IIz x, II>__c 1

19671

RANDOM POWER SERIES

133

then for every zl, z2 (0 < [zl[ < [z2]) there exists a power series Y~a~z~ converging 0 at z = zz (to OEX) but diverging 0 at z = z 1.

ProoL Let 8k--*0 and let xt¢t), °'°' J~'k ~t~) be the elements fulfilling the above conditions for e=ek, set he=O, hi = n l , h t = n l +.." + n t , s o = O c X , and for hk-1 < J ~ hk, k = 1, 2, ... s, =

iz2\ J ~

Then the power series ~a,,z" with ao = 0 and

(n _>--1)

a, = z~"(s, - s , - t )

has the required property. Indeed

(3)

Y, a : " =

sN +

1- z

S~z~'

0

0

so that for z = zz

Z~

N

11

a,z"

II IIsN II 8,

for hi-1 < N _ hk. Thus, Y a : " converges O at z = z 2 because ek -* 0. On the other hand, if ~a,z~would converge O, so also, by (3) and SN(Zl/Z2)M--~0 would Ys,z~z2-? This is impossible since by definition of the s,'s

h•

hk - + 1

snz~zi"

=

n~ X (k) 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