Pacific Journal of Mathematics

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(1)Pacific Journal of Mathematics IN THIS ISSUE— Larry Armijo, Minimization of functions having Lipschitz continuous first partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Edward Martin Bolger and William Leonard Harkness, Some characterizations of exponential-type distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James Russell Brown, Approximation theorems for Markov operators . . . . . . . . . . . . . . . . Doyle Otis Cutler, Quasi-isomorphism for infinite Abelian p-groups . . . . . . . . . . . . . . . . . Charles M. Glennie, Some identities valid in special Jordan algebras but not valid in all Jordan algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas William Hungerford, A description of Multi (A1 , · · · , An ) by generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James Henry Jordan, The distribution of cubic and quintic non-residues . . . . . . . . . . . . . . Junius Colby Kegley, Convexity with respect to Euler-Lagrange differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tilla Weinstein, On the determination of conformal imbedding . . . . . . . . . . . . . . . . . . . . . . Paul Jacob Koosis, On the spectral analysis of bounded functions . . . . . . . . . . . . . . . . . . . Jean-Pierre Kahane, On the construction of certain bounded continuous functions . . . . . V. V. Menon, A theorem on partitions of mass-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . Ronald C. Mullin, The enumeration of Hamiltonian polygons in triangular maps . . . . . Eugene Elliot Robkin and F. A. Valentine, Families of parallels associated with sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Melvin Rosenfeld, Commutative F-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Seidenberg, Derivations and integral closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Verblunsky, On the stability of the set of exponents of a Cauchy exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Herbert Walum, Some averages of character sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Vol. 16, No. 1. 1 5 13 25 47 61 77 87 113 121 129 133 139 147 159 167 175 189. November, 1966.

(2) PACIFIC JOURNAL OF MATHEMATICS EDITORS H.. *J. DUGUNDJI University of Southern California Los Angeles, California 90007. SAMELSON. Stanford University Stanford, California R. M. BLUMENTHAL University of Washington Seattle, Washington 98105. RICHARD ARENS. University of California Los Angeles, California 90024. ASSOCIATE EDITORS E. F. BECKENBACH. B. H. NEUMANN. F. WOLF. K. YOSIDA. SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY UNIVERSITY OF SOUTHERN CALIFORNIA. STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON * * * AMERICAN MATHEMATICAL SOCIETY CHEVRON RESEARCH CORPORATION TRW SYSTEMS NAVAL ORDNANCE TEST STATION. Printed in Japan by International Academic Printing Co., Ltd., Tokyo Japan.

(3) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. MINIMIZATION OF FUNCTIONS HAVING LIPSCHITZ CONTINUOUS FIRST PARTIAL DERIVATIVES LARRY ARMIJO. A general convergence theorem for the gradient method is proved under hypotheses which are given below. It is then shown that the usual steepest descent and modified steepest descent algorithms converge under the some hypotheses. The modified steepest descent algorithm allows for the possibility of variable stepsize. For a comparison of our results with results previously obtained, the reader is referred to the discussion at the end of this paper. Principal conditions* Let / be a real-valued function defined and continuous everywhere on En (real Euclidean w-space) and bounded below En. For fixed x0 e En define S(x0) = {x : f(x) ^ f(x0)}. The function^ / satisfies: condition I if there exists a unique point x* e En such that f(x*) = inf/(a); Condition II at x0 if fe C1 on S(x0) and Vf(x) = 0 xEE. n. for x e S(x0) if and only if x = x*; Condition III at x0 if / e C1 on S(x0) and Ff is Lipschitz continuous on S(x0), i.e., there exists a Lipschitz constant K> 0 such that |Ff(y) — Ff(x) \ S K\y — x\ for every pair x, yeS(x0); Condition IV at x0 if feC1 on S(x0) and if r > 0 implies that m(r) > 0 where ra(r) = inf | Ff(x) | , Sr(xQ) = Sr Π S(x0), Sr = χesr(χ0). {x : I x — x* I ^ r } , and #* is any point for which f(x*) — inf f(x). (If n. xEE. Sr(x0) is void, we define m(r) = oo.) It follows immediately from the definitions of Conditions I through IV that Condition IV implies Conditions I and II, and if S(x0) is bounded, then Condition IV is equivalent to Conditions I and I I . 2. T h e convergence theorem* In the convergence theorem and its corollaries, we will assume that / is a real-valued function defined and continuous everywhere on En, bounded below on En, and that Conditions III and IV hold at x0. THEOREM.. (1). // 0 < δ g 1/41SΓ, then for any xeS(x0),. S*(a, δ) = {xλ: xλ = x- Wf{x),. the set. λ > 0, f(xλ) - f(x) ^ - δ \Ff(x)\2}. Received January 30, 1964. The research for this paper was supported in part by General Dynamics/Astronautics, San Diego, California, Rice University, Houston, Texas, and the Martin Company, Denver, Colorado. The author is currently employed by the National Engineering Science Company, Houston, Texas..

(4) 2. LARRY ARMIJO. is a nonempty subset of S(x0) and any sequence {xk}ΐ=0 such that xk+1 e S*(xk, d), k = 0, 1, 2, •••, converges to the point x* which minimizes f. Proof. If xe S(x0), xλ = x - Wf(x) and 0 ^ λ ^ 1/Z", Condition III and the mean value theorem imply the inequality f(xλ) — f(x) ^ 2 2 - (λ - X K) I Pf(x) | which in turn implies that xλ e S*(x, δ) for λx ^ λ ^ λ 2 ,. λ<. so that S*(x, d) is a nonempty subset of S(xQ). If {^fc)Γ=o is any sequence for which xk+1e S*(xk, δ), k = 0 , 1 , 2, •••, then (1) implies that sequence {f(xk)}~=0, which is bounded below, is monotone nonincreasing and hence that | Vf(xk) \ —* 0 as k —> oo. The remainder of the theorem follows from Condition IV. COROLLARY. 1. (The Steepest Descent Algorithm). xk+1. = x. k. If. - -Lj7/(a? 4 ), k = 0, 1, 2, Δίί.. then the sequence {xk}^=Q converges to the point x* which minimizes. f.. Proof. It follows from the proof of the convergence theorem that the sequence {xk}^=0 defined in the statement of Corollary 1 is such that xk+1 e S*(xk, 1/4Z), k = 0,1, 2, . . COROLLARY 2. (The Modified Steepest Descent Algorithm) If a is an arbitrarily assigned positive number, am = a/2m~\ m = 1, 2, , and xk+1 = xk — ccmjPf(xk) where mk is the smallest positive integer for which. ( 2). f(xk - am/f(xk)). - f(xk) ^ - \amh. 2. \ Ff(xk) | ,. k = 0, 1, 2, •••, then the sequence {xk}ΐ=0 converges to the point x* which minimizes f. Proof. It follows from the proof of the convergence theorem that if x e S(xQ) and xλ=x - \Pf(x), then f(xλ) - f(x) ^ - (1/2) λ | Vf(x) | 2 for 0 ^ λ ^ 1J2K. If a ^ l/2iΓ, then for the sequence {xk}ΐ=0 in the statement of Corollary 2, mk = 1 and xk+1e S*(xk, (l/2)a), k = 0,1, 2, . If a > 1/2Z", then the integers m^ exist and amfc > l/4iΓ so that.

(5) MINIMIZATION OF FUNCTIONS. 3. 3* Discussion* The convergence theorem proves convergence under hypotheses which are more restrictive than those imposed by Curry [1] but less restrictive than those imposed by Goldstein [2]. However, both the algorithms which we have considered would be considerably easier to apply than the algorithm proposed by Curry since his algorithm requires the minimization of a function of one variable at each step. The method of Goldstein requires the assumption that feC2 on S(x0) and that S(x0) be bounded. It also requires knowledge of a bound for the norm of the Hessian matrix of / on S(x0), but yields an estimate for the ultimate rate of convergence of the gradient method. It should be pointed out that the modified steepest descent algorithm of Corollary 2 allows for the possibility of variable stepsize and does not require knowledge of the value of the Lipschitz constant K. The author is indebted to the referee for his comments and suggestions. REFERENCES 1. H. B. Curry, The method of steepest descent for nonlinear minimization problems, Quart. Appl. Math. 2 (1944), 258-263. '2. A. A. Goldstein, Cauchy's method of minimization, Numer. Math. 4 (2), (1962), 146-150..

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(7) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. SOME CHARACTERIZATIONS OF EXPONENTIAL-TYPE DISTRIBUTIONS E. M. BOLGER1 and W. L. HARKNESS. Let / = {fix; δ) = exp [xδ + q(δ)], δ e (α, b)} be a family of exponential-type probability density-functions (exp. p.d.f.'s) with respect to a σ-finite measure μ. Let M(t; δ), a — δ < t < b — δ9 denote the moment generating function (m.g.f.) corresponding to fix; δ) 6 /, and let c(t; δ) = In M(t; δ) = 2Γ=i h(δ)tk/kl be the cumulative generating function. The main results pertain to characterizations of certain exp. p.d.f.'s in terms of the cumulants 2k(δ). First, it is shown that if M(t; δQ) is the m.g.f., respectively, of a degenerate, Poisson, or normal law for some δQ £ (α, 6), then Mit; δ) is the m.g.f. of the given law for all δ e (a, 6), and that infinite divisibility (inf. div) of M(t; δQ) for some δ0 implies inf. div. for all δ. Further, it is shown that if φ(t) is a nondegenerate, inf. div. characteristic function (ch. f.) with finite fourth cumulant λi, then λ± — 0 if and only if ψit) is the ch.f. of a normal law, while if ^4 = CΛJ = a2λ2 Φ 0, then φ(t) is the ch.f. of a Poisson law. Combining these results, it follows that if M(t; δ0) is inf. div., and nondegenerate, with Λ4(d0) = 0, then M(t; δ) is the m.g.f. of a normal law for all δ e (α, 6). A similar result characterizes the Poisson law. Finally, it is proved that the normal law is the unique exp. p.d.f. which is symmetric.. An exponential-type family of distributions is defined by probability densities of the form (1). f(y; δ) = exp [yδ + q(δ)] ,. a < δ <b. with respect to a σ-ίinite measure μ over a Euclidean sample space (X, SI). It is known ([1], p. 51) that the set of parameter points d such that \exp [δy]dμ(y) < 00, is an interval (finite or not). The binomial, Poisson, normal, gamma, and negative binomial disiributions provide familiar examples of exponential-type distributions. A few structural properties for this family are considered. Section 2 contains some useful lemmas which are applied in § 3 to obtain some characterizations of the Poisson and normal distributions. 2* Some lemmas. Patil [3] has shown that a collection of d.f.'s {F(x; δ): δe (a, b)} is of exponential-type if and only if the Received March 12, 1964 and in revised form July 27, 1964. 1 Now at Bucknell University..

(8) 6. E. M. BOLGER AND W. L. HARKNESS. cumulants, λfc(δ), exist for all k and satisfy (2). χk(δ) = ^=M for ft = 2 , 8 , 4 , . dδ Further, he has shown [3, equation (12)] that M(t; δ) is the moment generating function of an exponential d.f. if and only if M(t; δ) = exp {q(δ) — q(δ + t)}. Lehmann ([1], p. 52) has shown that e~9{8) is an analytic function of δ for a < Re δ < b. It follows that q(δ) is analytic for a < Reδ <b. Then Xk(δ) is analytic for a<Reδ<b and k ^ 1. Hence, if δoe (α, 6), there is a neighborhood Δ of δ0 such that λ. (β)=. £ λ i + f e (g 0 )(g - go) k=o. ΐov δeΔ,. kl. LEMMA 1. If M(t; δQ) is degenerate for some δ0 e (α, δ), then M(t; δ) is degenerate for all δe (α, b).. Proof.. M(t; δ0) degenerate implies λj(δ0) = 0 for j ^ 2. Write US) - Σ. λ2+i(go)(g. "" g o ) i. for δe Δ .. Thus, λ2(δ) = 0 for δe Δ. Since λ2(δ) is analytic for a < Reδ < 6, we have λ2(<5) = 0 for δ e (a, b) and the conclusion follows. COROLLARY. If λ2(<?0) is different from zero for at least one δQe (α, 6), then λ2(δ) is different from zero for all δe (α, 6). LEMMA 2. If M(t; δ0) is the m.g.f. of a Poisson type distribution for some δQe {a, b), then M(t; δ) is the m.g.f. of a Poisson type distribution for all δe (α, b).. Proof.. By assumption.. M(t; «o) = exp and 2. λy(δ0) - c^- λ2(δ0). for j ^ 2 .. \j(δ) = c>'~2X2(δ). for j ^ 2. If it can be shown that ( 3). and all δ e (α, 6), then the Lemma will follow.. The proof of (3) is by.

(9) SOME CHARACTERIZATIONS OF EXPONENTIAL-TYPE DISTRIBUTION. 7. induction on j . Let h(δ) = \(δ) — cX2(δ). Now h(δ) is analytic for a < Reδ <b. Furthermore, h(δ0) — 0, and h^(δ0) = Xs+k(δ0) - cX2+k(δ0) = ck+1X2(δ0) - cckX2(δ0) = 0. It follows that h(δ) = 0 for δe (α, δ). So XB(δ) = cλ2(δ). Now, assume y 2 Xj(δ) = c ~ λ2(δ). Differentiation of both sides yields. This completes the proof of (3). It follows that M(t; δ) = exp LEMMA 3. If Λf(t; δ0) is normal for some δoe(a,b), is normal for all δe (α, 6).. then. M(t;δ). Proof. Since j|f(£; δ0) is normal, X2(δ0) Φ 0 and \j(δ0) = 0 for i ^ 3. Write for δ e A,. Then λ3(δ) Ξ 0 for δ e (α, δ). Because of (2) it follows that λ, (<5) = 0 for j ^ 3. Finally, X2(δ0) Φ 0 implies X2(δ) φ 0 for any δe (α, b). LEMMA 4. JΓf M(ί; δ0) is infinitely divisible for some then M(t; δ) is infinitely divisible for all δe {a, b).. δoe(a,b),. Proof. If λ2(<50) = 0, the result follows from Lemma 1. So assume λ2(<?) Φ 0 for any δe{a,b). Now, (Lukacs [2]), there exists a distribution G(x; δ0) such that λ 2 (δ 0 + t)/X2(δ0) = \extdG(x; δQ). for te (a — δQ, b — δ0). Let δx be an arbitrary element of (α, δ). If te(a-δl9bδj, t h e n t + δ, e (α, δ) a n d t + δ, - δoe(a - δOfb - δ0). H e n c e , f o r t e (a — δ19 b — δλ) \(δi + t) _ λ 2 [δ 0 + (£1 + δλ — δ0)] r x. dG1(x] δQ).

(10) E. M. BOLGER AND W. L. HARKNESS. where dG^x; δ0) = (λ2(δ0)/λ2(δO)e(δl~So)*dG(α?; δ0). It is easy to see that G^x; δ0) is a distribution function. Thus,. is a moment generating function for t e (a — δί9 b — δ±). Hence, M(t; δλ) is infinitely divisible. Since δλ is an arbitrary element of (α, 6), M(t;δ) is infinitely divisible for all δe(a, b). In the following two lemmas, we assume that f(t) is a nondegenerate, infinitely divisible characteristic function (ch. f.) and φ(t) — \ogf(t) has four derivatives at ί = 0. Let 7 - 1 2 3 4. x. 3. dt From the results of Shapiro [4], it is easily deduced that — (llX2)(d2φ(t)/dt2) is the characteristic function of a d.f. with mean λ3/λ2 and variance (λ 2 λ 4 — λ^/λ j . LEMMA. a normal. 5. If λ4 = 0, then f(t) is the characteristic function of distribution. 2. 2. Proof. —(l/X2)(d φ(t)/dt ) is a characteristic function of a distribution with mean λ3/λ2 and variance (λ2λ4 — λj)/λj. Thus λ4 = 0 implies 2 2 λ3 = 0 since the variance is nonnegative. Therefore, —(l/X2)(d φ(t)/dt ) is the ch. f. of a degenerate distribution with mean 0. Hence, l. 2. λ2. dt. 2. and, it follows that φ(t) = iXjb - (λ2t /2) for all t. Note that the single assumption that λ4 = 0 does not suffice to ensure normality since the binomial distribution, while not infinitely divisible, with pq = 1/6 has λ4 = 0. 2. LEMMA 6.. If λ4 = αλ3 = α λ2 Φ 0, and f(t) is infinitely divisible, then f(t) is the characteristic function of a Poisson type distribution. Proof. ~(l/X2)(d2φ(t)/dt2) is the ch.f. of a distribution with mean λ3/λ2 = a and variance (λ2λ4 - Xl)/X22 = {a2X\ - a2X\)IX\ = 0. So, — (l/X2)(d2φ(t)/dt2) is a ch.f. of a degenerate distribution with mean a. That is, __ 1 d2φ(t). λ2. It follows that. dt2. _. Qiat.

(11) SOME CHARACTERIZATIONS OF EXPONENTIAL-TYPE DISTRIBUTION. 9. REMARK. ,(ί) = ^ 2( β * . - l ) + i ί λ 1 - ^ a \ a. 1. I t is not sufficient to assume infinite divisibility and. λ 3 = λ 4 Φ 0. u. Let φ(t) = X(e - 1 ) + i λ ί - (f/2). Then λ 3 = λ 4 = λ Φ 0. <p(f) is t h e ch.f. of t h e composition of normal and Poisson distributions. EXAMPLE.. REMARK. 2. I t is not sufficient to assume infinite divisibility and. λ 2 = λ 3 Φ 0. EXAMPLE. REMARK. 2ίt. L e t φ(t) = e. 2. - 1 - 2ί . Then λ 2 = λ 3 = 8.. 3. I t is not sufficient to assume λ 2 = λ 3 = λ 4. Φ. 0.. Let x0 = (1 + i/Ϊ3)/2 and ^ = 1 — sc0. Let p 0 = (α?o — l)/(2a?0 — 1) and ^ = 1 — p 0 . I t is easy to see t h a t 0<p09p1<l. ίx i t Let flfi(t) = e ^p0 + e ^ p1 and g2(t) = 1. Then, if EXAMPLE.. it follows by direct computation t h a t λ 2 = λ 3 = λ 4 = 1. Here, git) is obviously not an infinitely divisible ch.f.. 3* Characterization of the normal aud Poisson distributions* THEOREM 1. If M(t; δ0) is infinitely divisible and nondegenerate, and if λ4(δ0) = 0, then M(t; δ) is the m.g.f. of a normal distribution, for all δe (a, b).. Proof. By Lemma 5, M(t; δ0) is the m.g.f. of a normal distribution. Then by Lemma 3, the conclusion holds for all δ e (a, 6). The family of normal distributions has the property that all its members are symmetric distributions. This means that all central moments of odd order vanish; in particular, the third central moment μ9 = λ3, must vanish. The next theorem, which follows easily from equation (2) and Lemma 3, implies that the normal law is the unique exponential-type distribution which is symmetric. 2. Let / = IF(X; δ) = Γ eyS+q{8) dμ(y); δe(a, b)\ be a family of exponential-type distributions, and assume that λ3(δ) = 0 THEOREM.

(12) 10. E. M. BOLGER AND W. L. HARKNESS. for all δe(a,b) and λ2(δ0) > 0 for some δoe(a, b). family of normal distributions.. Then y is a. The following question now arises: If, for some δ0 e (α, δ), M(t; δ0) is infinitely divisible and λ3(<50) = 0, must M(t; δ) be normal? The answer is no. EXAMPLE.. Let N(t) = e-t+t"12 for — <χ> < t < oo,. Pit) = and JVΊ(ί) = e P(<) . Then, (Lukacs [2]), iVΊίί) is an infinitely moment generating function. Clearly,. is an exponential-type moment generating function. that M(t; μ) is infinitely divisible. Now =. divisible. It is easy to see. d*logM(t;μ) dt*. __ dΨ(t + μ) t=o dt" = ( — 1 + μ)i. ί=0. _ dN(t + μ) dt. ί=0. __ dN(μ) dμ. so that λ 3 (l) = 0. However, \z(μ) is not identically zero so that M(t; μ) is not the m.g.f. of a normal distribution for any value of μ. [For M(t; μ0) normal would imply M(t; μ) normal for all μ which, in turn, would imply λ8(/ι) = 0.] THEOREM 3. If M(t; δ0) is infinitely divisible for some δoe (α, δ), and if λ4(δ0) = cλ3(δ0) = c2λ2(<50) Φ 0, then M(t; δ) is the m.g.f. of a Poisson type distribution for all δe {a, b).. Proof.. This follows directly from Lemmas 2 and 6.. THEOREM 4. If λ3(<5) = cλ2(<5) for all δ e (α, δ) where \2(δ) and λ3(<5) are cumulants of an exponential-type distribution, then M(t; δ) is the m.g.f. of a Poisson type distribution.. Proof.. First we show by induction that λ i+2 (δ) - c%(δ) .. By assumption, this is true for j = 1. Assume now that λ i+2 (S) =.

(13) SOME CHARACTERIZATIONS OF EXPONENTIAL-TYPE DISTRIBUTION. cjλ2(d).. 11. Differentiating both sides, we get λ i+3 (δ) = c%(δ) =. Then, log M(t; δ) = *#>(*« - 1) + c2 Let δ0, δλ e (α, 6). Many of the preceding results would be trivial if there existed constants c, d with c Φ 0 such that REMARK.. M(t; d0) = e d ί i. However, that this is not always the case is shown by taking M(t; δ) - e e S ( e i - 1 } ,. ί, § e ( - c o , oo) .. REFERENCES 1. E. L. Lehmann, Testing Statistical Hypotheses, John Wiley, New York, 1959. 2. Eugene Lukacs, Characteristic functions, Hafner, New York, I960. 3. G. P. Patil, A characterization of the exponential-type distribution, Biometrika 50 (1963), 205-207. 4. J. M. Shapiro, A condition for existence of moments of infinitely divisible distributions, Canad. J. Math. 8 (1956), 69-71. THE PENNSYLVANIA STATE UNIVERSITY.

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(15) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. APPROXIMATION THEOREMS FOR MARKOV OPERATORS JAMES R. BROWN. Let (X, ^ ,m) be a totally (/-finite measure space. A Markov operator (with invariant measure m) is a positive r. operator T on L^X, v ^ , m) such that Tl = 1 and f Tfdm = 1 fdm for all feLi(X,. ^ , m) n Loo(X, ^~, ra). If φ is an in-. vertible measure-preserving transformation of (X, ^", m), then φ determines a Markov operator Tφ by the formula Tφf(x) = f(φx). The set Λf of all Markov operators is convex and each Tφ is an extreme point. In case (X, ^~,m) is a finite, homogeneous, nonatomic measure space, M may be identified with the set of all doubly stochastic measures on the product space (X X X, J^~ X J^", m x m). The main result of the present paper is that M is compact in the weak operator topology of operators on L2(X,j^,m) and that the set Φ of operators ΎΨ is dense in M. It follows that M is the closed convex hull of Φ in the strong operator topology. We shall further show that Φ is closed in the uniform operator topology and that the closure of Φ in the strong operator topology is the set Φx of all (not necessarily invertible) measure-preserving transformations of. We shall denote LP(X, <^~, m), 1 g p ^ ©o, more briefly by Lp. The operators Tφ arising from invertible measure-preserving transformations are in certain respects the most pleasant of all Markov operators (in particular, they are unitary). Therefore, it seems worthwhile to determine what role they play in the structure of M. Interest in special cases of this problem is indicated by the attention which has been devoted to the solution of Birkhoff's Problem 111, which is concerned with the case of a denumerable space X with a measure m uniformly distributed on the points of Xo Thus J. R. Isbell [4], B. A. Rattray and J. E. L, Peck [12] and D. G. Kendall [6] have given approximation theorems for doubly stochastic matrices in terms of convex combinations of permutation matrices. A second type of solution has been given by Kendall [6] and by Isbell [5]. They have shown that the set of permutation matrices coincides with the set of extreme points of M. Still a third type of solution has been offered by P. Revesz [13], who has shown (essentially) that every Received September 21, 1964. This paper contains a portion of the author's dissertation presented for the degree of Doctor of Philosophy at Yale University. 13.

(16) 14. JAMES R. BROWN. doubly stochastic matrix is an integral over the set of permutation matrices. In the case of the real line X with Lebesgue measure m, Peck [11] has given a solution of the first type above for the set M of what he calls doubly stochastic measures. He also alludes to the corresponding result for the unit interval with Lebesgue measure. In § 3 we shall show that, for finite measure spaces,' the set of doubly stochastic measures can be identified with the set of Markov operators. This is not the case for a nonfinite measure space. However, approximation theorems for Markov operators always imply the corresponding approximation theorems for doubly stochastic measures or matrices. In this paper we shall restrict our attention to finite measure spaces. The σ-finite case is more complicated and will be considered at a later time. In § 4 we give a solution of the above-mentioned problem (Theorem 1) for the case of a nonatomic, finite measure space. This is an approximation theorem like that of Peck [11], but is a stronger result in that the convex closure of the set Φ of invertible measure-preserving transformations of (X, ^~, m) is replaced by the closure of Φ. The topology on M is the weak operator topology for operators on L2. Simple examples can be constructed to show that the second and third types of solutions mentioned above for doubly stochastic matrices do not extend to doubly stochastic measures on a nonatomic measure space. The Choquet representation theorem can be invoked to give solutions of the third type as integrals over the set of extreme points of M. These extreme points have recently been characterized by Jo Lindenstrauss [7], but this approach will not be pursued here. In § 5 we consider the closure and the convex closure of Φ in the strong operator topology and the uniform operator topology. The results of that section are obtained directly from Theorem 1 and an interesting geometric characterization (Theorem 5) of the operators Tψ arising from measure-preserving transformations. The author would like to express his gratitude to Prof. S. Kakutani and Prof. R. M. Blumenthal for valuable discussions of the problem considered in this paper. In particular, acknowledgement is made of certain observations of Prof. Blumenthal which led to considerable simplification in the proof of Theorem 1. 2* Markov operators* A Markov process with discrete time parameter and state space (X, ^) is determined (cf. [1], p. 190) by a stochastic transition function P(x, B), i.e. a nonnegative function of xe X, Be &~ such that (i). P(x, B) is a probability measure in B for each fixed xe X;.

(17) APPROXIMATION THEOREMS FOR MARKOV OPERATORS. 15. (ii) P(x, B) is a measurable function of x for eachfixedB e / ' . We assume, in addition, that m is an invariant measure for P(x, B) in the sense that (iii). ί P(x, B)m{dx) = m(B) ,. Under these conditions P(x, B) defines a bounded linear operator T on Loo by the formula (2). Tf(x)=^f(y)P(x,dy).. The operator T clearly has the properties: (3). /^0-Γ/^O. (4) (5). Γl=l f Tf(x)m(dx) = \ f(x)m(dx) , Jx. Jx. /eLw.. For instance, equation (5) follows from condition (iii) when / is a characteristic function of a measurable set and more generally by an approximation argument. Likewise, (4) follows from (i). We shall refer to any linear operator T on L^ which satisfies (3), (4) and (5) as a Markov operator with invariant measure m, or simply a Markov operator. It follows from (3) and (4) that T is a positive operator on L^ with || T|!oo = 1 and from (5) that T may be uniquely extended to a positive operator on hx with || T\\x — 1. This extension is again given by (2). According to the Riesz convexity theorem, T is a contraction operator on Lp for each p, 1 ^ p fg °o# That is, T is a positive operator with || T\\p ^ 1. The adjoint Γ* of Γ defined by ( 6). (Γ*/ f g) = (/, Tg) = [ f(x)Tg(x)m(dx). for fe Lp, ge Lq, 1 ^ p, q ^ oo, l/j> + 1/g = 1, is also a Markov operator. Indeed, equation (5) is equivalent to (5*). Γ*l= 1.. Thus equations (3), (4) and (5*) may be taken as the definition of a Markov operator. In general, a Markov operator can not be defined in terms of a stochastic transition function. However, under suitable separability restrictions on (X, ^ , m) there will always exist a stochastic transition.

(18) 16. JAMES R. BROWN. function P{x,B) such that (2) holds almost everywhere for each fe L«> (cf. [1], pp. 29 ff). A measurable transformation (function, mapping) φ from (X, ^ ) into itself is said to be measure-preserving if (7). miφ-'B) = m(B) ,. B e jF~.. It follows that φ is essentially onto, he. the complement of its range has measure zero. As usual we shall identify functions which are equal almost everywhere. Hence we may assume that φ is onto. If it is also one-to-one and if φ"1 is measurable, we shall say that φ is an invertible measure-preserving transformation. It follows that φ"1 is also measure-preserving. The correct notion for our purposes is actually that of a measurepreserving set function from JΓ into ^ . However, for the measure spaces that we shall be considering in this paper, namely, products of the unit interval with the product Lebesgue measure, every measurepreserving set transformation ψ is given by a measure-preserving point transformation φy ψ{B) — φ~\B). Let us denote by Φ the set of all invertible measure-preserving transformations of (X, J^, m) (or rather all equivalence classes of such transformations modulo null transformations) and by M the set of all Markov operators. Then Φ may be identified with a subset of M by the correspondence φ—>Tφ where (9). Tφf(x)=f(φx),. feLp.. We shall show in § 4 that, subject to a homogeneity condition on (X, ^ ~ , m), Φ is dense in M in the weak operator topology. 3. Doubly stochastic measures. It is well known that any finite measure space (X, j^~, m) with m(X) = 1 which is nonatomic and for which there exists a countable class ^ of measurable sets that generates ^ is measure-theoretically equivalent to the unit interval. That is, there exists a one-to-one mapping (modulo sets of measure zero) of ^~ onto the class of Borel subsets of the unit interval which preserves set operations and the measure ([3], p. 173). More generally, if m is any finite measure, it can be shown [8] that X is essentially a countable union of measurable sets Xn which are measure-theoretically equivalent either to a single point or to a product of intervals with the product Lebesgue measure. The measure spaces (Xn, ^ Γi Xn,m) are homogeneous in the sense that (i) there exists a class ^~n of measurable subsets of Xn which generates ^~ Π Xn and (ii) for each subset Y of Xn of positive measure, the σ-algebra.

(19) APPROXIMATION THEOREMS FOR MARKOV OPERATORS. 17. ^ Π Y is not generated by any class of smaller cardinality than that of ^l. The cardinality of J^ is called the character of Xn and determines the number of copies of the unit interval which go into its representation as a product measure space. It is clear that, except for the atoms (character 1), each of the spaces Xn may be assumed to be of different character, hence not measure-theoretically equivalent to each other. In particular, any invertible measure-preserving transformation of X must leave invariant each of the nonatomic Xn as well as their union. Now suppose that X — Xx (J X2 where X1 and X2 are disjoint and Xx is either one of the nonatomic Xn of the preceding paragraph or the union of all of the atoms. Consider the Markov operator T defined on Lp by. Tf(x) = ( f(y)m(dy) . JX. For this operator we have (Xzv TχX2) = ^ χZl(x)TχZΛ(x)m(dx). = m(X0m(X 2 ) .. (Here and in the remainder of the paper χA denotes the characteristic function of the set A.) On the other hand, if φ is any measure preserving transformation of X, then we know that (Xxv TφX,2) = m(Xx Π qr'Xt) = 0 . Thus T cannot be a limit of convex combinations of elements of Φ in any operator topology. We therefore assume that (X, «^r, m) is nonatomic and homogeneous. As noted above, this implies that (X, j ^ ~ , m) is essentially a product of unit intervals with the product Lebesgue measure. As such it has a natural topology, the product topology, in which X is compact, ^ coincides with the class of Borel sets of X and m is a regular Borel measure. Now let T be a Markov operator on !/«,. We shall denote the product measure space (X x X, ^ x ^) by (X 2 , ^ ^ 2 ) . We shall further denote the algebra of finite unions of measurable rectangles A x 5 , A, Be ^ , by ^ 2 . For each such rectangle we define (10). λ(A x B) - (χΛ, Tχΰ) .. Since λ is additive in A and B individually, it follows that it can be uniquely extended, by additivity, to a finitely additive, nonnegative set function on ^ 2 . Moreover, (11). λ(A x X) = λ(X x A) = m(A) ,.

(20) 18. JAMES R. BROWN. We shall show that λ is count ably additive. This is a special case of a theorem of E. Marczewski and C. Ryll-Nardzewski [9, 10] on nondirect products of compact measures. For completeness and because the special case is much simpler than the general case, we include a proof. 2 Any nonnegative, finitely additive set function λ on J^l which satisfies (11) will be said to be doubly stochastic. Let X be a doubly stochastic set function on ^l2. Then X is countably additive and regular, hence it has a unique extension to a doubly stochastic measure on LEMMA.. Proof. If A and B are Borel subsets of X, then by the regularity of m there exist compact sets At and Bι such that A1aA, BίaB1 m(A — Aj) < ε and m(B — B^) < ε, where ε is any preassigned positive number. It follows that At x B1cz A x B, A1 x Bt is compact, and X(A x B - A, x Bx) ^ X(A x (B - B,)) + X((A - A,) x B) ^ m{B - Bx) + m(A - A,) < 2ε . Thus any set in j^2 can be approximated from the inside by compact sets, i.e. λ is regular. It follows by Alexandroff's theorem ([2], p. 138) that X is countably additive. The existence and uniqueness of the extension then follow from the Hahn extension theorem. COROLLARY. The relation (10) determines a one-to-one correspondence between the set M of Markov operators on L«, and the set of all doubly stochastic measures on. Proof. We have shown that each Markov operator T determines a unique doubly stochastic measure λ satisfying (10). Conversely, suppose that λ is a doubly stochastic measure. Let g e L^ and let / be a simple function on (X, ^~, m). Set (12). G(f) = \f(x)g(y)\(dx,dy).. Then \G(f)\^\\g\u\\f(x)\\(dx,dy). Thus (12) defines for each # e LM a bounded linear functional G on a dense subset of Lx. It follows that there exists a function Tg e L«, such that.

(21) APPROXIMATION THEOREMS FOR MARKOV OPERATORS. <13). 19. (/, Tg) = j f(x)g(y)X(dx, dy). for all fe Llo If g ^ 0, then G is a positive functional so that Tg Ξ> 0. Thus T is a positive linear operator on L«, Clearly, Γ l = 1. Moreover, I Tg(x)m(dx) = I g(y)X(dx, dy) . Γ Γ Since λ is doubly stochastic, it follows that I Tg — \ g for all simple functions g and hence for all g e L^, Thus T is a Markov operator. Since (10) is clearly a special case of (13), the proof is complete. Suppose that T — Tφ is determined by a measure-preserving transformation φ e Φ. Then according to (10) and (9) ψ determines the doubly stochastic measure λ^ defined by (14). \(A. x B) = m(A Π φ~ιB) .. Thus λ^ is concentrated on the graph of φ« Such measures are sometimes referred to as permutation measures . 4* Weak approximation* the main result of this paper.. We are now in a position to prove. THEOREM 1. M is a compact convex set of operators and Φ is dense in M in the weak operator topology of Lp, 1 o / / (X, j ^ ~ , m) is a separable measure space, then M is metrizable.. ProofΌ The convexity of M is clear. Let us show that M is compact. Suppose that T belongs to the closure of M in the weak operator topology of bounded linear operators on Lp. Since L^czLpftLq, where 1/p + 1/q = 1, and since (/, Tg) is a continuous function of T for each fixed feLp1geLq, it follows that T has the properties (3)~ (5) of § 2. For instance, (3) is equivalent to. It follows from (3) and (4) that T maps L^ into itself. Thus T is a Markov operator and M is closed. Since Lp is reflexive, the closed unit sphere in the space of bounded linear operators on Lp is compact ([2], p. 512). Since || T\\p ^ 1 for each TeM, it follows that M is compact. Note that each of the weak operator topologies corresponding to different values of p is stronger than the weak topology on M determined by the functionals (/, Tg) for f^geL^o Since the latter is, nevertheless, a Hausdorff topology and since M is compact in each of.

(22) 20. JAMES R. BROWN. the weak operator topologies, it follows that they all coincide and hence there is no ambiguity in referring to the weak operator topology on M. Now suppose that (X, J ^ , m) is separable. Then Lp and Lq are separable in their norm topologies. Let {/„} and {gn} be dense sequences in Lp and Lg, respectively. The series τ ). =. v v. 1 .. \(fn,(T-S)gm)\. is uniformly convergent in S and T and thus defines a continuous (weak operator topology) metric on Mo It follows that the resulting metric topology is weaker than the weak operator topology and hence, by the compactness of M, that the two topologies coincide. It only remains to show that Φ is dense in M. A basis for the weak operator topology of M is given by all sets of the form (15). {T: \(fk, Tgk)-(fk,. Togk)\ < ε, k = 1,. , n}. where fk and gk run through a dense subset of L 2 , Toe M and ε > 0. In particular, we may take fk and gk to be continuous. (Recall that X has a natural topology for which X is compact and m is regular.) In this case they are bounded and, by the arbitrariness of ε in (15), we may assume that they are bounded by l β Let ToeMB We shall show that there exists a measure-preserving transformation φ such that Tφ belongs to the set (15). For A.Be^ we introduce the notation λo(A xB) = (χA, ToχB) Xφ(A x B) = (χA, TφχB) - m(A n φ~ιB) ,. φe Φ .. According to the lemma of § 3, λ0 and λ^ may be extended to doubly stochastic measures on (X 2 , ^βr2)o Let us set hk(x, y) — fk(x)gk(y), k = 1, , n. According to (13), we have (Λ, Togk) - \ hkd\Q and (fk, Tψgk) = 1. hkdXφ . 2. Since each hk is uniformly continuous on X , it follows that there exist disjoint sets Xu — , Xse ^ such that X=\J*i=1Xi and the oscillation of each hk is less than ε/3 on each rectangle X{ x X3 , i, j — 1, •••, s. Since λ0 is doubly stochastic, we can choose, for each i — 1, , s, disjoint measurable subsets Xi3 of X{ such that m(Xi3) =.

(23) APPROXIMATION THEOREMS FOR MARKOV OPERATORS. 21. λo(Xί x Xj), j — 1, , So (Recall that X is nonatomic.) Similarly, for each i = l, ,s we can choose disjoint measurable subsets Yi3 of Xj such that m(Yiά) = λ o (X ί x X3 ), i = 1, , s. For each i and j" there exists an invertible measure-preserving transformation φi3 of X^ onto Yi3 since X^ and Y^ are both homogeneous with the same character and the same measure. Define φ on X by. We shall show that Tφ belongs to the set (15). Since φ maps each Xir onto YίraXr follows that. and since Xt = U ^ J i r , it. XyiXi x Xj) = m(X, Π φ-'Xj) - m(X { i ) = λo(X, x Xs) for each i,j = l, ,s. Recalling that the oscillation of each of the functions hk is less than ε/3 on each of the sets Xi x X3 and that hk(x, y) I ^ 1, we have ( Λ , Γffflffc) - ( Λ , Γ o g /C ) I hkdX0. i. -u i \ f y v y ^ i. I ί\jφ\u^Li A. -AJ7. Λ. (y. v y ^ it. /\;Q\-Λ.^ A. y\.j). |j. and the proof of Theorem 1 is complete. 5, Strong and uniform approximation* Since convex sets have the same closure in the weak operator and the strong operator topologies ([2], p. 447), we immediately obtain the following approximation theorem. THEOREM 2. M is the closed convex hull operator topology.. of Φ in the strong. It is natural to ask whether Theorems 1 and 2 can be strengthened to give M as the closure of Φ in the strong operator topology. The answer, at least on L 2 , is negative as we shall now show. Henceforth, all operator topologies will refer to operators on L 2 . Let us denote by Φ1 the set of all (not necessarily invertible) measure-preserving transformations of (X, J?~, m). Again we identify Φ1 with a subset of M by setting Tψf(x) = f(φx), feL2. It follows that ΦdΦ.dM..

(24) 22. JAMES R. BROWN THEOREM. 3.. Φγ is the closure of Φ in the strong operator topology*. THEOREM. 4.. Φ is closed in the uniform. operator topology.. Theorems 3 and 4 follow easily from Theorem 2 and a pair of simple algebraic propositions which we give as Theorem 5 below. Note that nothing is said about the closed convex hull of Φ in the uniform operator topology. This is apparently still an open question. 5. Let T be a bounded linear operator on L2. Then TeΦx if and only if T is doubly stochastic and isometric; TeΦ if and only if T is doubly stochastic and unitary. THEOREM. Proof. It is well known that every T e Φλ is isometric while every TeΦ is unitary. Moreover, T is unitary if and only if both T and T* are isometric. Thus the second part of the theorem follows from the first. It only remains to show that every TeM which is isometric is induced by a measure-preserving transformation φ of X. Suppose that T e M is isometric and let A and B be measurable subsets of X. Then \χ (TχΛ)(TχB)dm (16). = (TχA, Tχΰ) = (χA, χB) =(XΛ^l) = (. =. (TχΛnB9l). TχArίSdm.. However, since T is positive and 0 ^ χA ^ 1, we have that 0 ^ TχA <g Γ l = 1 and so (17). 0 ^ (TχAγ. ^TχA^lo. It follows from (17) and (16) with A = B that (TχA)2 = TχA and hence that TχA is (essentially) the characteristic function of some measurable set. Let us denote this set by ψ{A)o Using the positivity of T again we have, moreover, that TχAΓB ^ min{TχA,TχB} so that (18). 0 ^ TχA,B = (TχA,Bf. rg (TχA)(TχB). From (16) and (18) we see that TχAnB =. (TχA)(TχB). or. n B) = ψ(A) n ψ(B). ..

(25) APPROXIMATION THEOREMS FOR MARKOV OPERATORS. 23. for all measurable sets A and B. Thus ψ preserves finite intersections. From the relations χAϋB = χA + χB - χ 4njB and χ^, = 1 — χA and the fact that Tl = 1, it follows that ψ* preserves finite unions and complements as well. From the relation m(f (A)) =. (TJU,. Tl) = (χ^1) - m(A). it follows that ψ preserves countable unions and intersections as well as the measure. It follows that there exists a measure-preserving transformation φ of (X, jr, m) Such that ψ(A) == ^(-A) for all A e jrm Thus TχA(x) = Xir(A)(x) = %A{φ%)- It follows that Γ coincides with 2^ on all simple functions and hence on all of L2. This completes the proof of Theorem 5. It follows immediately from Theorem 5 that Φ is closed in the uniform operator topology and that Φλ is closed in the strong operator topology. According to Theorem 2, every element TeΦ1 is the limit in the weak operator topology of a convergent net Ta of elements of Φ. It follows that || ( Γ - Γ β )/||ϊ = 2(/,/) - (Tf, Taf)-(Taf,. Tf)~>0. for each feL2o Thus Ta-+T and Φ is dense in Φx in the strong operator topology. This completes the proofs of Theorems 3 and 4. BIBLIOGRAPHY 1. J . L. Doob, Stochastic Processes, N e w York, 1953. 2. N. D u n f o r d a n d J . T. S c h w a r t z , Linear Operators, P a r t I, N e w York, 1958. 3. P . R. Halmos, Measure Theory, P r i n c e t o n , 1950. 4. J . R. Isbell, Birkhoff's problem 111, P r o c . A m e r . M a t h . Soc. 6 (1955), 217-218. 5# doubly stochastic matrices, Canad. M a t h . Bull. 5 (1962), 1-4. 1 Infinite 6. D. G. K e n d a l l , On infinite doubly stochastic matrices and Birhhoff's problem 111, J. London M a t h . Soc. 3 5 (1960), 81-84. 7. J . L i n d e n s t r a u s s , A remark on extreme doubly stochastic measures (unpublished). 8. D. M a h a r a m , On homogeneous measure algebras, P r o c . N a t . Acad. Sci. U S A 2 8 (1942), 108-111. 9. E. Marczewski, On compact measures, F u n d . M a t h . 4 0 (1953), 113-124. 10. a n d C. Ryll-Nardzewski, Remarks on the compactness and non direct products of measures, F u n d . M a t h . 4 0 (1953), 165-170. 11. J . E. L. Peck, Doubly stochastic measures, M i c h i g a n M a t h . J . 6 (1959), 217-220. 12. B. A. R a t t r a y a n d J . E. L. Peck, Infinite stochastic matrices, T r a n s . Roy. Soc. Canada, Sec. I l l , (3) 4 9 (1955), 55-57. 13. P . Revesz, A probabilistic solution of problem 111 of G. Birkhoff, Acta Math. Acad. Scien. H u n g a r i c a e 1 3 (1962), 187-198. OREGON S T A T E U N I V E R S I T Y.

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(27) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. QUASI-ISOMORPHISM FOR INFINITE ABELIAN ^-GROUPS DOYLE 0.. CUTLER. This paper is concerned with the investigation of two closely related questions. The first question is: What relationships exist between G and nG where G is an Abelian group and n is a positive integer? It is shown that if G' and W are Abelian groups, n is a positive integer and nG' = nH', then G = H where G' = S 0 G and H' = T®Ή such that S and T are maximal ^-bounded summands of Gf and Hf, respectively. A corollary of this is: Every automorphism of nG can be extended to an automorphism of G. We define two primary Abelian groups G and H to be quasi-isomorphic if and only if there exists positive integers m and n and subgroups S and T of G and H, respectively, such that pnG c S, pmH c T and S = T, the second question is: What does quasi-isomorphism have to say about primary Abelian groups? It is shown that if two Abelian p-groups G and H are quasi-isomorphic then G is a direct sum of cyclic groups if and only if H is a direct sum of cyclic groups, G is closed if and only if H is closed, and G is a 2*-group if and only if H is a i^-group. In this paper the word "group" will mean "Abelian group/ 7 and we shall use the notation in [5] except that a direct sum of groups A and B will be denoted by A 0 5 . Also if aeA then HpA(ά) will denote the p-height of a in A. (If it is clear which group or which prime is referred to then either sub- or super-script may be dropped or both.) At a symposium on Abelian groups held at New Mexico State University, L. Fuchs asked the question: What does quasi-isomorphism (see Definition 3.2) have to say about primary Abelian groups? A question posed by John M. Irwin that arises in the investigation of this question is: What relationships exist between G and nG where G is an Abelian groups and n is a positive integer? The purpose of this paper is to investigate these two questions,, First, we will begin by considering to what extent nG determines G where G is a group and n is a positive integer. It will be shown that if Gf and H' are groups, n is a positive integer, and nG' ~ nH', f then G~H where G = S φ G and H' = T 0 H such that S and T Received July 24, 1964. The results in this paper were part of a doctor's thesis completed in May 1964 under Professor John M. Irwin at New Mexico State University. 25.

(28) 26. DOYLE 0. CUTLER. are maximal ^-bounded summands of G' and H', respectively. A corollary to this is: Every automorphism of nG can be extended to an automorphism of Gβ In looking at quasi-isomorphism of primary Abelian groups, it is shown that if two Abelian p-groups G and H are quasi-isomorphic then G is a direct sum of cyclic groups if and only if H is a direct sum of cyclic groups, G is closed if and only if H is closed, and G is a J^-group if and only if H is a J^-group. Other related results are also obtained. 1*. An isomorphism theorem* Let G be a p-group and Br be a basic subgroup of Then there exists a basic subgroup B of G such that pnB = B\. LEMMA 1.1. n. p G.. Proof. Write B' = χ λ e ^ {xλ}. Now there exists yλ e G, for all λ e J , such that pnyλ = %λ. Let B* = J^λeA{yλ}. Now B* is pure in G and [yλ : λ e A] is a maximal pure independent set with respect to the property of J3* having no cyclic summand of order ^-p*. To see the purity of B* first notice that (B*)[p] = (B')[p\. Let xe(B*)[p]. Now HB\{x) — n Λ- HpnG(x). and if HpnG(x). — m, HG(x). = m + n and HB*n(x) =. m + n\ Hence HG(x) — HB*n(x), and Bn is pure in G. That B* is maximal pure as above is clear. Thus Bt can be extended to a basic subgroup B of G, and B = Sn 0 5 * where p w S Λ = 0 (see p. 97 of [5]). Hence pnB = B'. Using the above notation note that G — Sn@Gn where Gn = JBJ + pwG and Bt is basic in G%β This follows from a theorem of Baer's (Theorem 29β3 in [5]). We shall continue using this notation and refer to this decomposition as Baer's decomposition. From a theorem of Szele's [Theorem 29O4 in [5]) it follows that Sn is a maximal ^-bounded summand of G« From this it is easy to show that, if Hf is a group, then H' = Γ φ ί ί such that T is a maximal ^-bounded summand of H'. Let G and H be p-groups such that pnG = pnH (under an isomorphism φ) for some positive integer n. Then Gn = Hn according to Baer's decomposition. THEOREM 1.2.. Proof. We may assume that G and H are reduced by Test Problem I and Exercise 9 in [9]. Now pnGn = pnG ^ pnH = pnHn. Let pn(BGJ be a basic subgroup of pnG such that BG% is a basic subgroup of Gn, and let Bπ% be a basic subgroup of iJ % such that φ(pnBGJ — pnBH% a n basic subgroup of p H. This is possible by the above lemma and the n n fact that p G ~ p H under φ. From the proof of the above lemma it is easily seen that there exists an isomorphism φ: BG —•* BH such that.

(29) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. φ 1 pnBGn = Φ I pnBGn.. We may write Gn = BGn + pnGn and Hn=Bffn. 27. + pnHn.. Define ψ : G n — Hn: gn = & + 2> where 6 e I?0 Λ and g%l e. ffni. -> 0(6) + 0(2> flrni). Gn.. n. Suppose gn = b + p g%l = V + p ^ % 2 , where 6, V e BG% and gnil g%2 e Gn.. Then & - &' = pn{g%2 - gn) implies b - 6' e p % £ ^ , and 0(6) -0(6')=0(6 - V) = Φ(Pn9n2 ~ VnQn) = ^(P w ^ 2 ) ~ Φivn9n). Therefore n. n. 0(6) + Φ(v gni) = Φ(V) + φ{p gn) . Hence ψ is well defined. Clearly f is a homomorphism. Suppose gn e Gn and f{gn) = 0β Now gn = b + p ^ % l where 6 e BGn n w and flfni G Gw, and f(gn) = ψ(b + p gn) = 0(6) + 0(p ff1l) = 0β Hence 0(6) = n n n -φ(p 9n) and b e p Gn. Thus φ(b) = 0(6) = φ{-p g%) and 0(6 + Λ ) = 0. Therefore gn = b + p w # %1 = 0 and ψ is one-to-one. If hn e Hn then An = b + j>wΛΛl for some beBHn and hnie Hno Since 0 and 0 are onto, ψ is onto. 1.3. Let G be a p-group. Let B and Bf be basic subgroups of G such that Gn = Bt + p"G α^ίί G'n = (B')ί + 2>ΛG. COROLLARY. 1.4. Let G and H be torsion groups such that nG = nH for some positive integer n. Then G 0 Bλ = H 0 J32 where Bx and B2 are groups of bounded order bounded by no COROLLARY. l β 5. Let Gf and Hr be torsion groups and n a positive integer. Write G' = S 0 G and Hf = TQ) H where S and T are maximal n bounded summands of Gf and Hf, respectively. Suppose that nG1 = nHf. Then G = H. COROLLARY. 1.6. Let G and H be p-groups such that pmG — pnHy Then pn~mHm = Gm.. COROLLARY. n>. m.. Proof. Now pm(pn-mH) ~ pmG implies (pn~mH)m = Gm by 1.2. By Baer's Theorem H=Sm@Hm where Hm = B* + pnH. Thus pn~mH = Pn~mSm 0 (pn~mB* + pn~m(pmH)). Also p«-mH = pn~mSm 0 (pn-mBZ +. pm(pn~mH)). since pn~m(Sm 0 Bt) is a basic subgroup of pn~mH. pn~m{Hm) and we have that Gm = pn~mHm.. Thus (pn~mH)m. A generalization of Theorem 1.2 is the following:. ^.

(30) 28. DOYLE 0. CUTLER. 1.7. Let Gr and Hr be groups such that nGf ~ nH\ for some positive integer n. Write G' — S φ G and H' = TQ)H such that S and T are maximal n-bounded summands of Gf and H\ respectively. Then G = H* THEOREM. Proof. If Gf and Hf are torsion groups, we are done by 1.5. Thus suppose that Gr and Hf are not torsion groups β It suffices to prove the theorem for n = py a fixed but arbitrary prime. To see this write n = p^ pi2 pkmm such that p/s are distinct primes. Let ^ be chosen such that Pin{ — n. Observe that if p^n^G) ~ PiiriiH) implies that n{G = n{H, then by finite induction G = H. Suppose that pGf = pH' under the isomorphism σ. Write G' = and Hf — T 0 H such that S and T are maximal p-bounded summands of G' and if', respectively. Let Gt and Ht be the torsion subgroups of G and H, respectively. Then pGt ~ pHt under the isomorphism σ I pGt. By 1.5, Gt = Ht under an isomorphism σt such that σt \ pGt — σ I pGt. Define a map + pH: φ(t + g) = σt(t) + σ(g). Φ: Gt + pG^Ht. where t e Gt and g e pG. Note that φ is an isomorphism since σ I Gt Π pG = σt \ Gt Π pG and σ(Gt Π pG) = Ht Π p i ϊ . (The proof of this is similar to that in Theorem 1.2.) Next we will show that there exists a pair (£>', φ') such that S' = {Gt, pG, x} with xe G and a;?G ( + pG, Φf is an isomorphism from S' to a subgroup of iϊ, and ^ | Gt + pG = ^. To this end let x e G such that xί Gt + pG. Then aj is torsion free. Let ye H such that py = Define Φ'\ {Gt, pG, x] — {Ht, pH, y}\ Φ\t + g + wa?) = 0(ί + 9) + ny such that (w, p) = 1 or w = 0, t e Gt and geG. Note that Hp(nx + £) = 0 for all ί e G t and (w, p) = 1. If there exists zeG such that pz— nx + t then nx — pz — teGt + pG, a contradiction to the choice of x. Thus iP(£ + g + WE) = 0 for (n, p) = 1, t e Gt and βr e pG. Suppose that flf' + nx = g" + mx with flf', ^" e Gt + pG. Then ^'(^' + nx) = Φ(gf) + ny and ^'(^" + ma;) = Φ{g") + m^/. Now (n — m)x = ^" — g' e pG since iP(&£ + 0 = 0, (&, p) = 1, for all t e Gt. For if p does not divide (n — m), then ίP(α; + ί) > 0 for some t e Gt since g" — gf' = g — t such that g e pG and t e Gt. Thus n — m — pnly and r. m. Φ(g" - Q ) = Φ((n ~ )x) = Φin.px) = nλpy = (n - m)y . Hence ψ{gr) + ny — φ{g") + my, and the map is well defined. Now φ' is clearly a homomorphism onto {Ht, pH, y). If φ'(z) — 0 for some.

(31) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. 29. z e {Gt, pGy x) then pz — 0 since φ' \ pG is one-to-one. Thus z e Gta But f r φ I Gt is one-to-one and hence z — 0. Thus we have an extension (S', φ ) of (Gt + pG, φ). Next let S/' be the set of all pairs (Sa, φa) such that Sa is a subgroup of G containing Gt + pG and ^ α is an extension of φo Partially order 6f as follows: If (Sa, φa), (Sβ9 φβ)e^ then (Sa, φa) ^ (Sβ, φβ) if and only if Sa ZD Sβ and φa is an extension of ψβo Now y ^ 0 as shown above, and every chain ^f has a least upper bound in Sf. To see this let ίT = [(S α , φa): a e A\. Let Sc = (J«e i & and φc be defined by 0c(sα) = φβ(sa) for /9 ^ a,saeSa. Clearly (Sc, φc) is the least upper bound and (Sc, Φc) e 6^. Therefore by Zorn's Lemma there exists a maximal element (SM, φM)o Now SM — G9 for otherwise there exists xeG such that x g SM, and we may extend φM to {SM, x] as before. Thus we have an isomorphism φM from G into ίί β If ye H and x e G such that px — Φ~ι(py) then φM{x) — y — te Ht which implies that y = φm(x) — te H9 Thus φM is onto and G and H are isomorphic. COROLLARY l o 8 β Let G' be a group and n a positive integer. every automorphism of nGf can be extended to an automorphism. Then of G\. This is actually a corollary to the proof of Theorem 1.1. For if we write G' = S 0 G, where S is a maximal ^-bounded summand of G\ every automorphism of nGf can be extended to an automorphism of G, as the proof of Theorem 1.7 indicateso This together with any automorphism of S gives the desired automorphism of GO COROLLARY L9 β Let G and H be groups and n a positive integer. Suppose that nG = nH and the maximal n-bounded summands of G and H are isomorphic. Then G ~ H.. COROLLARY 1.10. Let G' be a group and n a positive integer. The only pure subgroups between G' and nG' are groups of the form S ® G where S is a pure subgroup of Gf bounded by n and Gf = S ' ® G such that Sr is a maximal n-bounded summand of Gf. Proof. Let H' be a pure subgroup of G' such that G ' ^ H' Z) nG'. Then G'/H' = T a group of bounded order bounded by n. By Theorem 5 in [9], G' = H'@ T' such that T' = T. Thus nG' = nH' 0 nT' = nH'. By 1.7, H=G where H' = K@H with K a maximal ^-bounded summand of H'. Since H is pure in G' and G'/H is bounded by n, H is a summand of G' and G' = S' 0 if. Therefore H' ~ SφG where S is a pure subgroup of G' bounded by n..

(32) 30. DOYLE 0. CUTLER. 1.11. Let G be a group and n a positive integer. The pure subgroups between G and nG are all isomorphic up to summands of bounded order, bounded by n. COROLLARY. 1.12* Let G be a group and suppose that G — S φ G ; = T 0 G" where S and T are maximal n-bounded summands of G. Then Gf ~ G". COROLLARY. 2. Some properties of G and nG. In this section we will be concerned mainly with the question: If P is a property of a group, does G have property P if and only if nG has property P where G is a group and n is a positive integer? This question is of interest in itself, but it is also of interest in looking at the question of Fuchs: What does quasi-isomorphism have to say about primary Abelian groups? We shall begin by proving a decomposition theorem. 2 β l β Let G be a group and n a positive integer that nG = Σ λ 6 Λ G(λ. Then G — Σ λ e ^ Gλ such that nGλ = G'λ. THEOREM. such. Proof. Let a e A. Set Ha = G/ΣΛ* Gfλ, where A' = A - a, for all a e A, and observe that nHa ~ G^. Now Hλ = S* φ {Hλ)n where S^ is a maximal ^-bounded summand of Hλ. Set G = 2 ^ (Hλ)n φ Sn (external direct sum) where Sn is a maximal ^-bounded summand of G. Now nG ~ nG. By 1.9, G = G. Therefore G = ΣΛGX where Gλ s (Hλ)n for all λ G A except /S e J , and Gβ = (fl"β)w 0 £„« Also ^ G λ = Gί. Now nG = ΣΛ nGλ = ΣG'λ. Let ^ be an automorphism of nG such that ψ(nGλ) = G^. By 1.8, we can extend φ to an automorphism of G, say f. Thus we have G = Σf(Gλ) such that nψ(Gλ) = THEOREM 2.2. Le£ G be a group. Let H be a pure subgroup of nG, n a positive integer. Then there exists a pure subgroup K of G such that nK = H and K[p] — H[p] for each prime p.. We will first prove two lemmas. 2.3. Let G be a p-group. Let Hr be a pure subgroup of pnG. Then there exists a pure subgroup H of G such that pnH — Hf and H[p] = H'[p\. LEMMA. f. Proof. Let X=[ye H : HH,(y) = 0]. For each yeX n such that p xy — y. Let H = {[xy: yeX], H'}. Now pnH = {[pnxy: y e X], p"H'} c H' .. let. xyeG.

(33) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. 31. If x e Hf then either Hn,{x) = 00 or HHI{x) < co. If HH,(x) = 00 then x e pnH' and x e pnίl. If HHf(x) = k < 00 then there exists ze H' such k n that p z = x, and there exists xze X such that p xz — z. Thus p***^ = x and a? 6 pnH. Therefore Hf c pΛίf, and pnH = H'. Now write H — Sn 0 Hn according to Baer's decomposition. Let H = Hn. Clearly pnH = H'. Also if|i>] = H'[p] and JET is pure in G. Now H[p] = H'[p\ since if = Hn and thus if[p] = (pΉ)[p] = Hr[p\. To see that H is pure in G suppose not; i.e., suppose that there exists ye H[p] such that HH(y) < Hβ(y). Since H[p] = iP[p] and if' c if we may assume that ##(2/) = & < co. Now HH,{y) — k — n, (k > n). Thus HH(y) — H%(y) — k, a contradiction. Hence if is pure. LEMMA 2.4. Let G be a torsion group. Let H be a pure subgroup of nG, n a positive integer. Then there exists a pure subgroup K of G such that nK — H and K[p] = H[p] for each prime p.. Proof. Write n = pi1 ptm, plf , pm distinct primes. Now by Theorem 1 in [9], G = ΣP GV, p a prime and Gp the p-primary component of G. Likewise H = ^ p Hp where Hp is a pure subgroup of nGp. (Note that if (p, pt) = 1 for i = 1, , m then nGp — Gpo) If (p, p{) = 1, i = 1, , m, let iΓ^ = Hp. If (p, p^) ^ 1 for some i — 1, , m, let Zp be a pure subgroup of Gp such that nKv = ίί^ and if P [p] = Bp[iP]. Such a Kp exists by 2.3. Now define K = ^PKP. Clearly K is pure, K[p] = H[p] for all primes p and nK = H. Now we are ready to prove Theorem 2.2. Proof. If H is a torsion group we are done by 2.4. Thus suppose that H is not a torsion group. Let Ge and Ht be the torsion subgroups of G and if, respectively. Then Ht is pure in nGt — (nG)t, and by 2.4 there exists Kt, a pure subgroup of Gt, such that nKt — Ht and Kt[p] — Ht[p] for all primes p. Now let U = [xe H:x is torsion free]. Let V=[xeU: H${x) = 0 for i = 1, , m]. For each a; e F let ^ e G such that fl#β = x. Let T7 = [T/,: X e V]. Define J£ = {Kt, H, {W}}. Now K = Sn 0 iΓ where S^ is a maximal ^-bounded summand of Z , and K is the desired group. To see this let Kt be the torsion subgroup of K. Now. Thus ^iΓί = nKt —' Ht and Kt[p] — Ht[p] for all primes p. Hence Kt is pure in Gt. Thus to check the purity of K, we need only check torsion free elements in K. Let x e K such that o(x) = oo, and suppose that there exists yeG such that pry — x. (We need only check.

(34) 32. DOYLE 0. CUTLER. divisibility for a power of an arbitrary prime p by p. 76 in [5].) We will show that there exists z' e K such that przf = x9 Now there exist 1 1 nonnegative integers jι^k1,^-,jmS km such that n, = pi pt and w, is the least positive integer such that njx e Ho We consider two cases. First suppose that (p, p{) = 1 for all i = 1, , m. Then there r r exists ze K such t h a t p nάz — ΠjX. Thus p z — x ~ kte Kt and HEβt). ^ min (HS(pkz), Hnθ(x)) ^ fc .. Thus since Kt is pure in Gt there exists fcj 6 Kt such t h a t prfcί = kt. Hence pr(z — ftj) = x and 2' = z — k[ e if. Next suppose t h a t (p, p{) Φ 1 for some i = 1, , m, say i = v Then H&(nάx) ^ j i f l + r . lί x$H then H'βinjX) g ά,o and hence there exists 2 e if such t h a t p % 2 = wΛc. Thus prz — x = kte Kt, and as before there exists k[e Kt such t h a t pr(z - Jkί) = x and 2' = 2 = k[ e Ko It xeH, then H§(x) = Hpπ{x) + fcίo < r, and hence again there exists zf e K such that prz' = χo Thus if is a pure subgroup of G with the desired properties. It is easily seen that another proof of Theorem 2.1 can be obtained by using Theorem 2O2. One might note at this point that if n is a positive integer and K is pure in G then nK is pure in nG. Also Kι — f\nn\ K is pure in G1 — f\nn\G (see p. 452 in [7]). We have just shown that if H is pure in nG then there exists K in G such that K is pure in G and nK — Ho The question then arises: If H is a pure subgroup of G1, does there exist a pure subgroup K of G such that K1 — HI The answer to this question is in the affirmative as will be seen in the next theorerru First we need a lemma. 2O5O Let G be a group and K a pure subgroup of G. EQ be the divisible hull of G and Eκ a Ea the divisible hull of Then EκnG = K.. LEMMA. Let K.. Proof. Clearly EκΓ\Gz)K. Let 0 Φ x e Eκ Π G. Then {x} Π K Φ 0 (see Lemma 20β3, p. 66 in [5]). Let n be the smallest integer such that nx e K. Now since K is pure, there exists ye K such that ny = nx. Observe that y e Eκ and thus x — ye Έκ. We shall show that {x — y) Π K = 0, thus {x — y} — 0 and thus x — y e K. Suppose there exists m such that 0 Φ m(x — y)e Ko Then clearly (m, n) — i < n and there exist integers s and t such that ms + nt = i. Now mcc e if since mx — my9 my e Ko Also msx + ?ιsίc = ix, and since msx, ^sx e K, we have iίc e K. But i < n, and this contradicts the fact that n was the smallest integer such that nx e Ko Thus {x — y] Π if — 0, and we have x G if. Therefore EKΠG = if. In [6] a high subgroup is defined to be a subgroup H of a group G.

(35) QUASI-ISOMORPHISM FOR INFINITE ABELIAN ^-GROUPS. 33. maximal with respect to disjointness from G1. 1. 2.6. Let G be a group with G = f\nn\G Φ 0. Let H be a high subgroup of G. Let K be a pure subgroup of G1. Then there exists a pure subgroup M of G such that M1 = K and H is high in M. THEOREM. Proof. Let φ be the natural homomorphism from G to G/H. Then 1 ι φ I G is an isomorphism. Thus φ(K) is pure in φ(G ). Let E be the ι divisible hull of Φ{G ) in G/H (note that E = G/H), and let D be the divisible hull of Φ{K) contained in E. Then D Π ΦiG1) = Φ{K) by 2.5. Now define M = Φ~\D). Observe that H is pure in M, M/H = D, and M/Ha G/H as a pure subgroup. Thus by Lemma 2 in [9], M is pure in G. Now G1 Z) ilί 1 =) i ί by construction, and M1 c K since 0(AF)c:Z) Π ^(G1) = Φ(K). Hence ifcf1 = K. Also £Γ is high in M since flTl M 1 = 0 and {H, x} Π M1 Φ 0 for any as e M\iϊ. The latter statement is true since if there exists an x e M\H such that {H9 x) Π M1 — 0, this would imply that {H, x} Π G1 = 0 which would contradict the hypothesis that H is high in G. We will now show that several properties are possessed by a group G if and only if they are possessed by pnGo We will only consider primary groups. THEOREM 2β7. Let G be a reduced p-group. and only if pnG is closed.. Then G is closed if. Proof. Suppose G is closed. Then G — B for some basic subgroup B of G. Now B = Σί=i Bn9 pnΠBn = ΠpnB9 and thus pn(B) = ψB. Therefore pnG is closed. n. n. Suppose p G is closed. If B is a basic subgroup of G then p B is a basic subgroup of pnG. Let B be a closed subgroup with basic subgroup B and identify G with its pure subgroup between B and B (see p. 112 in [5])β Now p % 5 = p%G since p%G and pnB are closed and have the same basic subgroup B. Also (B)n ~ Gn by Theorem 1.2. Thus G ~ B since they contain the same basic subgroup. Therefore G is closed. DEFINITION 2.8. A -Γ-group is any group G all of whose high subgroups are direct sums of cyclic groups, (see [6]).. 2.9. Let G be a reduced p-group. n if and only if p G is a Σ-group. THEOREM. Then G is a Σ-group.

(36) 34. DOYLE 0. CUTLER. Proof. Theorem 11, p. 1370 in [8], states that "Every subgroup L of a torsion I'-group G with L 1 = L Π G1 is a I^-group". Thus if G n n 1 is a J?-group then p G is a I'-group since {p Gf = G . If pwG is a Jf-group and H is high in G then p " i ϊ is high in pnG (see p. 1368 in [8]). Thus pnH is a direct sum of cyclic groups, and hence H is a direct sum of cyclic groups. Therefore G is a J-group. DEFINITION 2.10. A group G is a direct sum of countable groups if and only if G = Σλ€Λ Gλ such that | Gλ | g Ko for all XeA. THEOREM 2.11. A group G is a direct sum of countable if and only if pnG is a direct sum of countable groups.. Proof.. groups. If G is a direct sum of countable groups then clearly pnG is.. Now suppose that pnG is a direct sum of countable groups. Write G = Sn © Gn according to Baer's decomposition. Then pnG = pnGn = Σ λ e * G ' λ such that | Gfλ\ S Ko By Theorem 2.1, we may write Gn = Σ λ e , Gλ such that pnGλ - G'λ. Now Gλ[p] = G'[p] and thus r(Gλ) = r(Gλ[p]) = r(G'λ[p]) = r(G'k). Also | G λ | = r(G λ ) Ko and since r(Gx) = r(G'λ) ^ ^o we have that | Gλ | g ^ 0 (see pp. 32-33 in [5]). Thus Gn is a direct sum of countable groups. Since Sn is bounded, Sn is a direct sum of cyclic groups and hence a direct sum of countable groups. 2.12. A p-group G is a direct sum of closed groups if and only if pnG is a direct sum of closed groups. THEOREM. Proof. Suppose that G is a direct sum of closed groups. Then G — ΣλeΛ G λ such that G λ is closed for each XeA and pnG = χ λ 6 i l pnGλ. By 2.7, pnGλ is closed for each XeA. Thus pnG is a direct sum of closed groups. Now suppose that pnG is a direct sum of closed groups. Then P G = ΣλeΛG'λ such that G'λ is closed for all λ e Λ . By 2.1, G = ΣxβΛ Gλ such that pnGλ = G'λ. Thus by 2.7, G is closed for each λ e A. n. 2.13. A group G is essentially indecomposable if G = implies that A or B is finite.. DEFINITION. A0S. THEOREM 2.14. Suppose that the first n Ulm invariants of a p-group G are finite, n a positive integer. Then G is essentially indecomposable if and only if pnG is essentially indecomposable..

(37) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. 35. Proof. The proof follows immediately from 2.1. DEFINITION 4.14. A group G is "Fuchs 5" if and only if every infinite (pure) subgroup of G is contained in a summand of the same cardinality.. 2.15, A p-group G is a Fuchs 5 group if and only if p G is a Fuchs 5 group. THEOREM. n. Proof. First assume pnG is Fuchs 5. Let H be an infinite unbounded pure subgroup of G. Then pnH is a pure subgroup of pnG. Since pnG is Fuchs 5, pnH is contained in a subgroup K' of pnG such that Kf is a summand of pnG and | pnH \ = | K'\. Thus pnG = K' © U. Now Gn = KQ) L (where G = S% © Gn according to Baer's decomposition) such that p*K = K' and pnL = 1/ (by 2.1). Write H = S* 0 H w according to Baer's decomposition0 Then pnHn is a pure subgroup of pnK which is a pure subgroup of p"G. Let BII% be a basic subgroup of Hn. Then p^-B^ is a basic subgroup of pnHn and can be extended to a basic subgroup of pnK, say p%l? = p%l?' 0 pnBHn where £>%.B' = Σ{xi}. For each cc^ there exists yi e K such that pny{ = x{ and Ή.κ{y^) — 0. Define £ ' = Σ{y^* Define groups B = B' © BΠn and M = {B, pnK}. Notice that S is a direct sum of cyclic groups, B is pure in M and M/B ~ (B + pnK)/B ~ pnK/(B ΓΊ pnK) ^ pnK/pnB a divisible group since p%j3 is a basic subgroup of pnKa Hence B is a basic subgroup of M. Now M ~ K, Hn c Λf, and G = S^ 0 i l ί φ Lo To see that ΛΓ ~ K notice that p%J5 is a basic subgroup of pnK and thus p"Λf ~ {pnB, p2nK} = pnK. By Theorem L2, if p%M = pnK then MΛ ^ iΓa (where M = Sf 0 M, and 1 ? = Si©uT n according to Baer's decomposition), and since M — Mn and K = Kn, M = KB (M = Mn since B is a basic subgroup of ΛΓ and JB is isomorphic to a basic subgroup of K.) Also Hn c Λf since n n Hn = {B^, r i ϊ . l , S f f n c δ and p Hnczp K. To see that G = S;?©M©L we first observe that M[p] = (pnM)[p] = (p w iί)[p] = ur[rf and thus Λf Π L = 0. Thus ikf 0 L is a direct sum and hence S° 0 (If 0 L) is a direct sum since M®L~K@L. Clearly S ^ φ f φ L c G . Now n Gu = i i 0 L = {BK, p K}@L where BK = B'@B" such that β " = r n n Λ n I'ίWί} and J5Hn=2 {2i} with p w~p zi (i.e., p β " = p B3J. Thus to show that GaS^M®L, it suffices to show that each wi e S° © I φ L. Now p ^ i = p X and thus w{ — 2< = s + fc + ί G S * φ j K " © L with % se Sn,ke K,te L, and o(/b), o(t) ^ p . But J5" may as well have been chosen such that its ίth generator was Wi — k, and thus we may assume that Wi - zi = ί + s e L © Sf. Hence w* = ^ + t + s e S° © l ί φ L. Therefore G c S i © M © L and G = S%σ φ M © L. Now I pM3"n I = I p AΓI, (pnH)[p] = iϊ n [p] and (pnΛf)[p] = Λf [p]. Thus = \M\. Each GTO is an absolute summand of G and we may.

(38) 36. DOYLE 0. CUTLER. write G = S£(BGn such that S? is a summand of S?. Thus H = S* 0 iJ w c Sf © Λf, a summand of G and clearly | S f © Λf| = |£Γ|. Therefore G is Fuchs 5β Next assume G is Fuchs 5. Let H be a subgroup of pnG. Then i ϊ is a subgroup of G and is contained in a subgroup K oί G such that JSL is a summand of G and | £Γ| = | K|. Thus G = ϋΓ 0 L and pnG = pnK@ pnL. Now Ha pnG and hence ί f c pnG ί l ϊ = £>%if by the purity of X in G. Thus \H\^\p*K\^\K\ = \H\. Therefore p«G is Fuchs 5. DEFINITION 2.16. A group G is a Crawley group if G contains no proper isomorphic subgroups.. The existence of such groups has been shown by Peter Crawley in [4]. For he has constructed a group C between B — X°Li (Cp1) and B = T(Y[Γ=i Cip1)) (the torsion subgroup of the complete direct sum of the C(p%) for i — 1, •••) which has no proper isomorphic subgroups. This group can be chosen such that r(B/C) — 1. This fact was first observed by R. A. Beaumont and R. S. Pierce in [3]. This group is also essentially indecomposable. The fact that if r(B/C) = 1 then C is essentially indecomposable was first proved (as follows) by John M. Irwin: Suppose C = H@K. Then B = 2 ? 1 φ ϊ ? 2 where HaB1 and KaB2. Thus either H = Bι or K = B2 since rφlC) = 1. Suppose H = B,. Then there exists a copy of BcB^ and thus a copy of C in j?1# It seems that this class of groups will be quite important in the study of ^-groups. 2.17. If C is a Crawley group then pnC is a Crawley group. If p C is a Crawley group and C is essentially indecomposable then C is a Crawley group. THEOREM n. Proof. Suppose first that C is a Crawley group. Suppose that there exists a group L §Ξ pnC such that L ~ pnC. Let U — [x e L: HL{x) = 0]. Then for each xe U there exists yeC such that pny = x. For each x let yx G C such that pnyx = x. Let V = [# β : a? e U7]β Define CL = {F, L}. Now p w C z = L. Let (C z ) n be a summand of C z according to Baer's decomposition and let Gr = S n 0 (CZ)Λ where B = Sn@BZ,B a basic subgroup of C and £ w a maximal p71-bounded summand of B. Now C £ C and p w C = L ^ p % C. Thus by 1.9, C = C. But this contradicts the fact that C is Crawley. Suppose pnC is Crawley. Then if C is essentially indecomposable then C is Crawley. For if there exists L gΞ C such that C = L, then p"C = p % L gi p^C which would contradict the fact that pnC is Crawley. COROLLARY. 2β18.. There are at least fc$0 nonisomorphic. Crawley.

(39) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. 37. groups C between B and B. Proof. Observe that B ~ pnB and B ~ pnB for all n. Let ψ be an n isomorphism between B and p B. Let C be a Crawley subgroup of B. n Then p C is a Crawley subgroup of pnBy and pnC is not isomorphic to C. n Thus φ(p C) is a Crawley subgroup of 5 which is not isomorphic to C. 3* Quasi-isomorphic p-groups* In a recent paper by R. A. Beaumont and Ro So Pierce (see [2]), it was shown that two countable primary groups are quasi-isomorphic if and only if their basic subgroups are quasi-isomorphic and their subgroup of elements of infinite height are isomorphic. Also if two primary groups are quasi-isomorphic their basic subgroups are quasi-isomorphic. In [3] they have considered quasi-isomorphism in relation to direct sums of cyclic groups. We will extend these results to closed p-groups. In considering quasi-isomorphism, it is of interest to investigate what properties of primary Abelian groups are preserved under quasiisomorphisnio It will be shown that if G and H are quasi-isomorphic primary groups, then the statement that G has property P if and only if H has property P is equivalent to (1) property P is preserved under isomorphism, (2) G has property P if and only if pnG has property P and ( 3 ) groups between G and pnG have property P if G does. This reduces this problem to considering G, pnG, and groups between G and pnG. DEFINITION 3.2. Let G and H be p-groups. Then G ^ H (quasiisomorphic) if there are subgroups SczG, TaH and positive integers m and n such that f G c S , pnHa T, and S ~ T.. The following theorem (among other things) shows that if two p-groups are quasi-isomorphic, then their subgroups of elements of infinite height are isomorphic. THEOREM 3.2. Let G and H he p-groups. If G and H are quasiisomorphic, then G/G1 is quasi-isomorphic to H/H1 and G1 = H1.. Proof. Now G and H quasi-isomorphic implies that for some positive integers m and n there exists subgroups S and T of G and if, respectively, such that G =) S z> pmG, H^TZD pnH and S ~ T. Now clearly G1 = S\ H1 ~ T1 and S1 ~ T1. Thus G1 ~ H\ Now S/S1 = Ύ\T\ Thus pm{GIGx) ~ pmGjGι c S/G1 c G/G1 and p^H/H1) ~ pnH/H1d T/H'aH/H1. Hence H/H1 is quasi-isomorphic to G/G1. The converse of the above theorem is not true as can be seen from.

(40) 38. DOYLE 0. CUTLER. an example on p p . 134-135 in [ 5 ] . PROPOSITION 3.3. L e t p be a property of p-groups. Then t h e following statements a r e equivalent: ( 1 ) L e t G and H be quasi-isomorphic p-groups. Then G h a s P if and only if H h a s P. ( 2 ) Property P is such t h a t for a n y p-group L : n (a) L h a s property P if and only if p L h a s property P for all positive integers n. (b) Whenever L has property P and S is a subgroup of L such n that Lz) Sz)p L then S has property P, and (c) Property P is preserved under isomorphism.. Proof. First 2 implies 1: If G and H are quasi-isomorphic then for some positive integers m and n there exist subgroups S and T of G m n and if, respectively, such that Gz> Sz>p G, Hz) Tz>p H, and S = T. If G has property P then S has property P by (b). Thus T has property P by (c), and since pnS(zpnHcSypnH has property P by (b), and H has property P by (a). By symmetry we have 2 implies 1. Next 1 implies 2: Since G is quasi-isomorphic to pnG, we have (a). Also G is quasi-isomorphic to any subgroup S such that G Z) S Z) pwG, thus we have (b). Clearly (c) holds. Using the above proposition, we can show that if G and H are quasi-isomorphic p-groups, then G is a direct sum of cyclic groups if and only if H is a direct sum of cyclic groups, G is a closed p-group if and only if if is a closed p-group, and G is a ^-group if and only if if is a IZ-group. 3.4. Let G and H he quasi-isomorphic p-groups. Then G is a direct sum of cyclic groups if and only if H is a direct sum of cyclic groups. THEOREM. Proof. Now G is a direct sum of cyclic groups if and only if pnG is, and any group between G and pnG is a direct sum of cyclic groups (see p. 46 in [5]). Then by Proposition 3.3 the theorem is proved. LEMMA 3.5. Let G be a reduced p-group. If G is a Σ-group and S is a subgroup of G such that Gz) Sz) pnG, then S is a Σ-group.. Proof. Now S1 = G\ Thus, apply the theorem stated in the proof of Theorem 2.9. 3.6. Let G and H be quasi-isomorphic p-groups. Then G is a Σ-group if and only if H is a Σ-group. THEOREM.

(41) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. Proof.. Apply Theorem 2.9, Lemma 3.5 and Proposition 3.3.. LEMMA. 3.7. Let G be a closed p-group and G D S D p n G .. 39. Then. S is closed. Proof. Let gu , gn, — be a Cauchy sequence in S. Then β β 9u ' * > 9ni is a Cauchy sequence in G. Since G is closed this sequence converges to some g eGo Also g — gme pnG c S for m > n, and since gme S we have # e S. THEOREM 3.8. Let G and H be quasi-isomorphic p-groups. G is closed if and only if H is closed.. Then. Proof. An application of Proposition 3.3, Theorem 2.7 and Lemma 3.7 proves the theorem. An important problem along these lines that seems to be a very difficult one is the following: If G and H are quasi-isomorphic p-groups, then is it true that G is a direct sum of countable groups if and only if H is a direct sum of countable groups? By Proposition 3.3 and Theorem 2.11 this problem is reduced to the following: If G is a direct sum of countable p-groups and S is a subgroup of G such that pnG c S, is S a direct sum of countable groups? We are able to answer two special cases of this question in the following two theorems. THEOREM 3.9» Let G be a direct sum of countable p-groups. Let Kbe a subgroup of G such that G D Ki)pnG and KjpnG is countable. Then K is a direct sum of countable groups.. Proof. Write K/p"G = Σ i e i ί^i + VnG} and pnG = χ λ 6 , pnGλ (where G=ΣGλ) such that | Gλ | = « 0 . Then K = {{kj}jei, {pnG}λeΛ} = {{ks}dei9 pnG}. Now kj G G and hence kj = Σ?=1 gλi such that gλι e G λ ( . Let Λf = [λ e A: for some gλeGκ,gλ is a representative in some kj]. Let A" — A\Ar. Then ίΓ={{fci}i 6 z,{p w G λ } λ6 ^}ΘΣλ6^"3> n G λ . Thus ΛΓ is a direct sum of countable groups. THEOREM 3.10. Let G be a direct sum of countable p-groups, and let K be a subgroup of countable index. Then K is a direct sum of countable p-groups.. Proof. Write G = X λ € ^ Gλ. Now Gλ c K for all but a countable number of λ e A since K is of countable index in G Let A! — [XeA:GλdK]. Let K, = ΣλeΛ, Gλ and ίΓ2 = {[fce JSΓ: fceχλe^ Gλ]}..

(42) 40. DOYLE 0. CUTLER. Then K — Kx 0 K2 such that K2 is countable and Kx is a direct sum of countable groups. The following theorem extends Beaumont and Pierce's results in [1] and [2] to closed p-groups. THEOREM 3.11. Let B and C be closed p-groups with basic subgroups B and C, respectively. Then B = C if and only if B = C. Proof. If B ^ C then B ςk C by [2]. Thus suppose that B ± C. Then there exist subgroups S and T of B and C, respectively, such that S D pw.B, T z> p%C and S = T. Thus BZDSZ) pnB, Ci)f Z)pnC and S = f since closed subgroups are completely determined by their basic subgroups (see pβ 115 in [5]). Thus B = C. 4* Special cases of quasi-isomorphisiru In this section we will impose some restrictions on the definition of quasi-isomorphism, and in some cases we will be able to determine by just how much two quasiisomorphic groups with these restrictions differ. 4.1. G = H (S.B. quasi-isomorphic, i.e., in the sense of Schroeder-Bernstein) if there exist subgroups SaG,T(zH, and positive integers m and n such that G= T, H = S, pmGd S, and pnH(Z T. DEFINITION. DEFINITION 4.2. Two p-groups G and H are purely quasi-isomorphic if for some positive integers m and n there exist pure subgroups S and T of G and H, respectively, such that Gz>Sz)pmG, HZD Tz)pnH and S~ T.. 4.3. G = @H (summand quasi-isomorphic) if there are subgroups SaG, Ta H, and positive integers m and n such that S = T, ^ G ^ O , and pnH, = 0. DEFINITION. DEFINITION 4.4. G ύ H (strongly quasi-isomorphic) if there are subgroups SaG,Tc:H such that S = T, [G : S] < oo, and [H: T] < oo. DEFINITION 4.5. G M ®H if there exists subgroups S c G , Γ c i ϊ such that, G = Sξ&GuH= TφHλ and G, and J^ are finite.. 4.6. Two groups are strongly S.B. quasi-isomorphic if there exist subgroups S and T of G and iϊ, respectively, such that [G:S] < oo,[H: T] < oo, S ~ H and Γ s G. DEFINITION. DEFINITION. 4.7. Two p-groups G and H are purely S.B. quasi-.

(43) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. 41. isomorphic if for some positive integers m and n there exists pure m subgroups S and T of G and ϋ , respectively, such that G ZD S D p G, n Hz>T^p H,G^ T and H ~ S. Each of the above definitions yields an equivalence relation. Definitions 4.2 and 4.3 are equivalent. It will be shown that Definitions 4.4 and 4.5 are equivalent, that Definition 4.6 is equivalent to Gn = Hn for some positive integer n according to Baer's decomposition and that Definition 4.7 implies the groups are isomorphic0 With this it will be clear that each relation (excluding Definition 4.1) is no weaker than the preceding one. Examples will be given to show that Definition 4O1 and 4.2 are stronger than Definition 3.1, and except for equivalent definitions, each relation from Definition 4O2 to Definition 4.7 is stronger than the preceding ones. PROPOSITION. 4.8. G M H if and only if G 'ύ ®U.. Proof. That G £k 0 i ϊ implies G ϊk H is clear. Thus suppose G'ύ H. Let S x c G , T x c H such that S1 ~ 2\ and G/S, and HjT1 are finite. A lemma of R. S. Pierce says: Let SΊ be a subgroup of a reduced p-group G of unbounded order such that the index [G: Sλ] < oo. Then there exists S2 c S1 such that [G: S2] < °° and G = S2 0 L. Thus there exist subgroups S2a Su and T2 c T1 such that S2 is a direct summand of finite index in G and T2 is a direct summand of finite index in H. Let φ be an isomorphism of S1 onto Tx. Then S2 Π ^(T^) has finite index in S lβ Again by the above lemma, there exists a subgroup S of S2 Π ^ ( T i ) such that S ^ S φ L where L is finite. Now S2 = S 0 ( S 2 ί l L ) , where S2 n L is finite. Thus G = S 0 ( S 2 n L ) 0 I , where ( S 2 Π L ) 0 Λ f is finite. Let T = Φ(S). Then Γ 1 = Γ © ί δ ( L ) , where T c T2. Consequently Γ2 = T 0 (^(L) Π Γ2), and J ϊ = Γ φ ( ^ L ) Π T2)@N, where (^(L) n T2) 0 ΛΓ is finite. Since S ~ T, it follows that G ^ ©Jff. The following theorem shows that if two groups are strongly S.B. quasi-isomorphic then they only differ (up to isomorphism) by summands of bounded order. 4.9. Let G and H be strongly SOB. quasi-isomorphic p-groupSo Then there exists a positive integer n such that Gn = Hn according to Baer's decomposition. THEOREM. Proof. Now G and H being strongly S.B. quasi-isomorphic implies that there exist subgroups S and T of G and H, respectively, such that [G: S] < oo, [H: T] < oo, G ~ T and H = S. By the lemma stated in the proof of 4.8, there exist subgroups S1 c S and ϊ\ c T such that Sx and 2\ are pure in G and H, respectively, [G: SJ < oo and [H: ΓJ < ©o. Thus G = S, 0 S, where Sλ ~ G/S, and S, is finite, and H=T1QTί.

(44) 42. DOYLE 0. CUTLER. where Tλ = H/T1 and 2\ is finite. Choose n such that p*φύ = p^T,) = 0. n Now p G = p ^ and p».ff = p»2\. Thus by Theorem 1.2, Gn = (SO and fl"n s (TO,. Also pnS = p Sx and p Γ = p Γx, thus S. = (SX and ΓΛ = (I^)*. Hence by the hypothesis we have that Gn~Tn~ (T,)n ^ Hn. COROLLARY 4.10. IfG and H (as in Theorem 4.13) have isomorphie basic subgroups, then G ~ H.. In the next theorem we will show that if G and H satisfy Definition 4.7, then G ~ H. First we need a lemma. LEMMA 4.11. Let G and H be p-groups such that G ~ G 0 B2 where Bλ and B2 are groups of bounded order. Then G = H.. Proof. Now G = G © B10 B2 where B1 0 B2 is of bounded order. Thus if we write G — Sn@Gn according to Baer's decomposition, where n p (B, 0 B2) = 0, it is clear that Sn = Sn 0 B, 0 B2 = Sn 0 B2 by Ulm's Theorem. Thus H = G 0 B2 ~ Gn 0 (Sn 0 B2) ~ Gn 0 Sn = G. THEOREM 4.12. If G and H are purely S.B. quasi-isomorphic p-groups, then G ~ H.. Proof. If S is a pure subgroup of G such that G/S is bounded, then S is a summand of G by Theorem 5 in [9]. Thus G = S 0 Bx where Bx is of bounded order. Also H — Γ 0 S 2 where B2 is of bounded order. Thus G ~ H® B± and H = G 0 B2 and by 4.11, G ~ H. Now to see that Definitions 4.1 and 4.2 are stronger than Definition 3.1, let G = C(p2) and H = C(p). Then G & H but G and i ί do not satisfy either Definitions 4.1 or 4.2. To see that Definition 4.4 is stronger than Definition 4.3, let G = ΣΓ=i W ) and H - G 0 Σ^ o C(^)» N o w c l e a r 1 ^ G = ® ^ a n d b ^ U l m ? s Theorem and 4β8, it is clear that G and H are not strongly quasiisomorphic. Next we show that Definition 4.6 is stronger than Definition 4.5. Let G = C(p) 0 C(p) and H = C(p). Now clearly G ^ ©if, but G and if do not satisfy Definition 4.6. Finally to see that Definition 4.7 is stronger than Definition 4.6, let G = Σ * o C(p2) and H = G 0 C(p). Then clearly G and i ί satisfy Definition 4.6 but not Definition 4.7. 5* Some related problems. Given a p-group G and a subgroup n if containing p G, n a positive integer, the question arises: If B is a basic subgroup of G, does there exist a basic subgroup Bf of if such.

(45) QUASI-ISOMORPHISM FOR INFINITE ABELIAN ^-GROUPS. 43. that Bz)Br ZDpnBςl We will give an answer to a very special case of this question in a corollary to the following theorem. THEOREM 5.1. Let G be a p-group and H a subgroup of G conn taining p G. If N is a high subgroup of G, then there exists a high subgroup M of H such that Nz) MzDpnN. n. Proof. Let N be a high subgroup of G. Now p N is a high subgroup of pnG (see p. 1380 in [6]), pnN(Z H and pnN f] H1 = 0. Thus let M be the maximal subgroup of H such that NZD MZD pnN and MΠ H1 — 0. Then M is high in if. To see this suppose not, iβe., suppose there exists xe H, x$ M, such that {x, M) Π H1 — 0. Since JV is high in G, {x, N} Π H1 Φ 0. There exists yeN such that pkx + y = hx e H1(h1 Φ 0). Now pkx, h±e H and hence yeH. Since. yeHf)N,yeM.. 1. Hence {&, Af} Π i ϊ Φ 0, a contradiction.. COROLLARY 5.2. Let G be a p-group and H a subgroup of G containing pnG. If G is a Σ-group and B is a basic subgroup of G which is also a high subgroup of G, then there exists a basic subgroup Bf of H such that B Z) Bf ZD pnB. Here Bf is a high subgroup of H*. The general question seems to be a little more elusive,, The following two theorems are results related to this problem. THEOREM 5.3. Let G be a reduced p-group and H a subgroup of G such that H z> pnG. Let Br be a basic subgroup of H. Then there exists a basic subgroup B of G such that B 3 Br.. Proof. We may write B' = Uϊ=i Sm, Sm = χr=i Bi9 and Br = ^ B, m such that B{ - ΣC(p% Now H=Sm@Hm where Hm = (ΣΓ=m,i Bi9 p H). Thus S m Π Hm = 0. Since Hm[p] = (pmH)[p\, Sm Π pmH = 0. Also since n n+m n+m p G c H, Sm Π p G = 0. Thus Sm is contained in a maximal p bounded summand of G, and therefore the height in G of any element f of Sm is bounded by n + m. By Kovacs' Theorem (p. 99 in [5]), B can be extended to a basic subgroup B of G. Hence B! c ΰ a basic subgroup of G. 5.4. Let G be a reduced p-group. Write G — SnQ) Gn according to Baer's decomposition. Let T be a subgroup of Gn such that T ZD pnG. Then there exists a basic subgroup Bτ of T such that BT = ΣΓ=«+i Li with H&n(x) = i — 1 for all x e L^p]. THEOREM. Proof. To prove the theorem we will construct a basic subgroup Bτ of T such that I? Γ = \JT=n+i Si where if x e Si+1[p] then HQ (x) ^ i.

(46) 44. DOYLE 0. CUTLER. and Si+1 = Si 0 L < + 1 such that if x e Li+1[p], HGn{x) = i . will have the desired properties.. Note that Bτ. Let ^ + 1 = [L: L is a summand of T and if x e L[p], HQ{x) — ri\. If ^ + 1 = 0 let Sn+1 = 0 and if ^n+1 Φ 0 let Sn+1 be a maximal element of S^n+1. (Clearly such a maximal element exists by Zorn's Lemma.) Assuming that Si has been defined, define Si+1 as follows: Let S^i+1 ~ [L: L is a summand of T; S ^ c L ; if x e L, HG%{x) g i; and L — S< 0 £ such that if j/e £[#>] then HGn(y) = i ] . Let S ί + 1 be a maximal element of t5t+i which we will now show exists. Clearly some S^ Φ 0. Thus partially order ^ t + 1 by set inclusion and let Lγ c L 2 be a chain in J?t+i. Put L = (JΓ=i •ί'i Then L is pure in Γ since each Lj is pure in T. Also if X G L , then cc e L, for some j which implies that HGn(x) ^ i. Hence L is of bounded order (with order bound ^pί+1) and thus L is a summand of Tβ Clearly LZD SίΛ Note that Z^ is a summand of Lj+1 for all i = 1, 2, . Now I/y = S w + 1 0 £y such that for x e Lj[p], HQn{x) = i, and L y + 1 = Sn+1 0 L y + 1 such that for x e Li+1[p], HQn(x) = i. To show that L — UΓ=i L3 e S^+1 we need only show that Lj+1 can be chosen such that it contains Lj. Write L3 = Σ{xa} such that o(xa) = na + 1. Now pW£*xα e Lj+1[p] since otherwise H&n(pn<*xa) < i. If ίcα e £ ί + 1 we are done. So suppose that xa $ Lj+1. Since xa e L J + 1 , ccα = s + ya where s e S n + 1 , ί/α e £ J = 1 and pn»xa = p%Ω>i/α. By the purity of L3 in Lj+1, Lj+1 — {7/α} + K, and thus we may rewrite Lj+1 as L i + 1 = Sn+1 0 ({s + ^/α} 0 K) where our new Lj+1 — {s + ya} © K and hence contains xa ~ s + ya. Since we may do this for any xa e Ljf we can choose Lj+ί to contain £ i β Therefore L — (JΓ=i J^i = S< 0 UΓ=i ^^ a n d J^ has the desired properties. Next set Bτ = Ui°=^+i S<, and note that Bτ is pure in T and J5 r is a direct sum of cyclic groups (see Theorem 11.1 in [5]). We will now show that TjBτ is divisible which will imply that Bτ is a basic subgroup of T. To do this we will show that for all x e T[p] such that x ί Bτ[p], Hτ}Bτ(x + Bτ) — oo. This is sufficient since, by Lemma 1 in [9], if x + Bτ e (T/Bτ)[p], there exists x' e T[p] such that x' + Bτ = x + Bτ. Let x e T[p] such that x g Bτ[p], and suppose that HG%{x) = m. Suppose that ί/"Γ(ίc) = fc, and let z e T such that pkz — x. Now a; 6 S m + 1 which implies that either S m+1 0{z} is not pure in T or that HGn(x-\-y)>m for some yεSm+1[p] (by the maximality of S m + 1 ). If S m+1 0{2;} is not pure in T then there exists y1 e Sm+1[p] such that iϊr(^i + ») = h > k. Let zxe Tsuch that p^zx = y1JΓx* Now H^ix + y^^m, and x + Vii S w + 1 so that again either S m + 1 0 {^} is not pure in T or HGn(x + y1 + j/) > m for some j/6S m + 1 [p]. If S w + 1 © {sj is not pure in Γ then there exists V* G S»+i such that £ΓΓ(^ + 2/i + 1/2) = fe > ki > k* Thus in either case there exists ye Sm+1[p] such that H&n(x + y) — mι> m. Now clearly τ + 7/ ί Sm +1 , and by a similar argument there exist a yι e Sm + 1 such.

(47) QUASI-ISOMORPHISM FOR INFINITE ABELIAN p-GROUPS. 45. 1. that HG%(x -i- y + y ) — m2 > mlΛ Continuing by induction we see that the height of x + Bτ e T/Bτ is infinite. It seems that an answer to the general question (asked at the beginning of this section) would give useful information about the structure of infinite p-groups. Another question that arises is the following: If G and H are p-groups such that G έ H and G/G1 ~ H/H\ then is it true that G~HΊ It would be interesting to know the answer to this question at least in the case G/G1 is a direct sum of cyclic groups. It would also be of interest to know: If G is a direct sum of closed p-groups, is a group S between G and pnG also a direct sum of closed p-groups? BIBLIOGRAPHY 1. R. A. Beaumont, and R. S. Pierce, Quasi-isomorphism of Direct Sums of Cyclic Groups, (to appear). 2. , Quasi-isomorphism of p-Groups, Proceedings of the Colloquium on Abelian Groups, Publishing House of the Hungarian Academy of Sciences, (to appear). 3. , Some Invariants of p-Groups, (to appear). 4. Peter Crawley, An infinite primary Abelian group without proper isomorphic subgroups, Bull. Amer. Math. Soc. 6 8 (1962), 463-467. 5. L. Fuchs, Abelian Groups, New York, Pergamon Press, 1960. 6. John M. Irwin, High subgroups of Abelian torsion groups, Pacific J. Math. 11 (1961), 1375-1384. 7. John M. Irwin, and E. A. Walker, On isotype subgroups of Abelian groups, Bull. Soc. Math. France, Vol. 89, (1961), 451-460. 8. N-high subgroups of Abelian groups, Pacific J. Math. 1 1 , (1961), 1 On 1363-1374. 9. Irving Kapϊansky, Infinite Abelian Groups, Ann Arbor, The University of Michigan Press, 1954. TEXAS CHRISTIAN UNIVERSITY AND NEW MEXICO STATE UNIVERSITY.

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(49) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. SOME IDENTITIES VALID IN SPECIAL JORDAN ALGEBRAS BUT NOT VALID IN ALL JORDAN ALGEBRAS C M . GLENNIB A Jordan algebra is defined by the identities: (1). z. %-y = y x,(x-y)'y. 2. = (x-y )-y .. The algebra Aj obtained from an associative algebra A on replacing the product xy by x-y — l/2(xy -f yx) is easily seen to be a Jordan algebra. Any subalgebra of a Jordan algebra of this type is called special. It is known from work of Albert and Paige that the kernel of the natural homomorphism from the free Jordan algebra on three generators to the free special Jordan algebra on three generators is nonzero and consequently that there exist three-variable relations which hold identically in any homomorphic image of a special Jordan algebra but which are not consequences of the defining identities (1). Such a relation we shall call an ^-identity. It is the purpose of this paper to establish that the minimum possible degree for an S-identity is 8 and to give an example of an S-identity of degree 8. In the final section we use an S-identity to give a short proof of the main theorem of Albert and Paige in a slightly strengthened form. NOTATION. The product in a Jordan algebra will be denoted by a dot, thus α δ, and {ahc} will denote the Jordan triple product (2). {abc} = α (δ c) — b (c a) + c (a-b) .. Unbracketed products ax a2 an will denote left-normed products i.e. ( (((V(v) α3) α j . When working in a special Jordan algebra we shall use juxtaposition, thus ah, to denote the product in the underlying associative algebra., Then a b = l/2(ab + ba) and 2{abc} = abc + cba. The free (respectively free special) Jordan algebra on n generators, taken as xu , xn or as x, y, z if n = 3, will be denoted {n) U) by J (respectively J 0 ) and the kernel of the natural homomorphism {n) u) vn (written as v for n = 3) of J onto J o by Kn. The subspace of {n) J spanned by the monomials of degree n linear in each of the generators will be denoted by Ln. The underlying associative algebra for J0(?λ) is the free associative algebra on n generators: we shall denote this by A{n). Throughout the paper we work over some fixed, but arbitrary, field of characteristic not two. Received August 13, 1964. This paper is a revised version of part of the author's 1963 Yale Ph.D. dissertation. 47.

(50) 48. C. M. GLENNIE. I*. The following theorem has been proved by MacDonald [4]:. THEOREM 1. (MacDonald). K3 contains which is linear in one of the generators.. no. (nonzero). element. We have at once the following corollaries: 1.. COROLLARY. than. Kz contains no (nonzero) element of degree less. six.. 2. An element u in J ( 3 ) linear in one generator, or of degree less than six, can be unambiguously represented by the expansion of uv in A (3) . COROLLARY. In this section we shall strengthen Corollary 1 to the following theorem, which I understand has previously been proved by J . Blattner; THEOREM. 2.. K3 contains no (nonzero) element of degree less than. eight. Proof. Let L be the subspace of J ( 3 ) spanned by the elements of degree two in x, two in y and two in z\ M t h e subspace of J ( 3 ) spanned by the elements of degree two in x, two in y and three in z. It is sufficient to show that (i) the restriction of v to L is one-to-one and (ii) the restriction of v to M is one-to-one. For (i) we display a set of elements which span L but whose images are linearly independent in Lv. For (ii) we prove a Lemma which implies that if (ii) does not hold, then (i) does not hold. Let Rh denote the mapping a—>c& 6 in a Jordan algebra. is well-known that: (3 ). Ra.b-c. (4 ). *..... =. (5 ). R..tR. -\-Rt t.cRa + Rc .A. = Ra- bRc. + Rb.c Ra + R . £ί e a. — Uoβ R. b. c. i 73. ~D 1 JΛ/ J.. 73. b. ϊb.c + Rbl ϊca + R• Jx . — JXJRJt,b c a b 0. =.RaRb.c. RbRcRa. ROW.. » + RcRab. + I. So L is spanned by elements of the forms RcRdReRt. ( v ). aRb RcRdRe.f. . RdReRf. (vi). Rc.dReRf. (vii). aRb aRb. (i). aRi. Ί. (ϋ). aRi. b c. (iii). aRi. Ί. (iv). aRi >RCRdeRf. (viii) aRb. .Λ 1-eRf .cR< iRe f Red Ref. Then it.

(51) SOME IDENTITIES VALID IN SPECIAL JORDAN ALGEBRAS. 49. where two of α, 6, c, d, e, f represent x, two represent y and two represent z. Consider those of type (i). We have (16 aRhRcRdRe)v. = {abode + edcba) + (bacde + edcab) + (cabde + edbac) + (cbade + edabc) + (dabce + ecbad) + (dbace + ecabd) + (dcabe + ebacd) + (dcbae + eabcd) - 17,+ C/2+ . . . + ί7δ(say). (where t/x = α&cdβ + βrfc&α, etc) . Cohn has shown [2] that reversible elements in A{3) are in Jo(3) so that each Ui is in J"0(3). Since v is an epimorphism there exist ^ e P 1 (i = 1, , 8) for which ^ v = Z7i# Then (16 aRhRcRdRe)v. ^. Thus =^. lQaRbRcRdRe. (Theorem 1, Corollary 1). and 16 aRhRcRdReRf. = (Σu,)-/ =. Σ(urf). By Theorem 1, Corollary 2, we can use J7< to represent ut without introducing ambiguity. Thus instead of u^f we can write U^f i.e. (abode + edcba)«f, an element in Ji3) but with notation for the part in brackets borrowed from J o (3) . Treating elements of types (ii)-(viii) similarly we see that with this notational convention L is spanned by elements of the forms (abode + edcba) •/ and (abed + dcba) (e f). The following elements then, together with those obtained by permuting x, y and z, span L: T-elements. 2(a). (b).. x (xyzyz + z ^ / x ) = 2x {x{yzy}z} ="2x-{xy{zyz}} . x (xzyzy + yzyzx) = 2a>{φ?/φ} = 2a;-{. 3.. 2x°(xyz2y + ^Λ/x) = 4α? - (α; {i/«2i/}). 4.. 2z°(yzyx2 + x%^) = 4s ({3/33/} sc2). 5.. 2x-(yxzyz + zyzxy) = 4x-{yx{zyz}}. 6.. 2y (2^ 2 2 + ^ y ^ ) = 4«/ {2(2/ ^2)^}. 7.. 2α (2/ ff2 + 3 a?2/ ) = 4a; *{y xz }. 8(a).. x-(yzxyz + zτ/^τ/) = x-f(x,y,z). (b).. y (zxyzx + xzyxz) = y f(y, z, x). 2. 2. 2. 2. 2. 2.

(52) C. M. GLENNIE. 50. (c).. z (xyzxy + yxzyx) = z f(z, x, y) where fix, y, z) = 4{(y-z)x(y z)}. 9. 10(a). (b).. {y{zxz}y} - {z{yxy}z}. 2y (xzyzx) = 2y-{x{zyz}x} aj2 (2/V + z2τ/2) = 2x*-(y*-z*) 2. 2. 2. τ/ (zV + xV) = 2 / . (z . x ). (c). {zyz}) = 2x2-({yzy}-z). 11.. x2-(yzyz +. 12.. 2x2-(yz2y) =. 13.. (x 7/) (£7/z2 + z2yx) =. 14.. (x-y)-(xzyz + 27/20?) = 2{zyz}RzRx.y. 15.. ix-y) ixz2y + yz2x) = 2ix-y)R[xz2y}. 16.. (x-y)-(zxyz. T16 is clearly redundant, while use of formulae (3), (5) and (3) respectively shows that T13, T14 and T15 are also redundant. So the set T (namely T1-T12 together with those elements obtained from T1-T12 by permuting x, y and z) spans L. We now display a set U of Jordan elements. Each Z7-element may be considered as an element in J 0 (3) : as such its expansion in A(3) appears as the corresponding Velement. Alternatively the [/-element may be considered as an element in L: its expression as a linear combination of T-elements appears as the corresponding W-element. For each integer r the validity of the relation Ur = Wr can be checked by appealing to MacDonald's theorem. For example, in the case of r = 7, U7 = W7 is valid in J0(3) and linear. r 1. [/-elements 2xi {yziy}. 2. 2{x{yz*y}x}. 3. 2. 2. 2. 2. W-elements 2. x yz y + yz yx 2. 2 2 2. xyz + zyx 2. 2 2. T12 T3 - T12. 2xyz yx 2 2 2. 2{xψz } 2. F-elements. TlOa - T10b + TlOc. 2 2. 4. 2{x(y 'Z )x}. xy z x + xz y x. 5. 2{x(y-{zyz})x}. xyzyzx + xzyzyx. T l - • W3 T2a + T 2 b - T i l. 6. 2{x*{yzy}z}. x2yzyz + zyzyx2. T2a - T 2 b + T i l. 7. 2. 2{zx {yzy}}. 2. 2. zx yzy + yzyx z 2. 8. 2. T 4 - -W6. 9. 2y {x{zyz}x}. yxzyzx + xzyzxy. T6 --W7 T9. 10. 2{xyx} {zyz}. xyxzyz + zyzxyx. T5 --W9. yzx yz + zyx zy. 11 x f(x,y,z). -yf(y,z,x) xyzxyz + zyxzyx + z-f(z,x,y). T8a - T 8 b + T8c.

(53) SOME IDENTITIES VALID IN SPECIAL JORDAN ALGEBRAS. 51. (3). in {yzy}, so U7 — Wl is valid in J o Suppose now that the sets of [7, V and PF-elements have been augmented by adjoining all elements •obtained from those displayed by permuting x, y and z. The column headed π shows the number of distinct elements obtained for each value of r . It is then easy to check that each T-element is a linear combination of W-elements, so the "PF-elements span L. But their images under v are the F-elements which are clearly linearly independent. So v\L is one-to-one. To complete the proof of the theorem we now prove the following lemma: LEMMA 1. Let n be an odd (positive) integer and u an element in Knf]Ln which is expressible in the form u = ΣΓ=i#ίβ2/i Then y{ e Kn for each i = 1, , n.. Proof. For convenience we denote vn by vo For n = 1 there is nothing to prove. Assume n > 1 and let the coefficient of fft+iffί-ί-2 β β. ^A. β. Xi-i (%2β. β. v* if i = 1). in y-p be μio Then the coefficient of x%+1xτ+2 xnx1 x{ in 2uv is P% + Pi+i Since distinct monomials in A{n) are linearly independent we have μi + /Vi = 0, i - 1, , n — 1 and μn + μγ = 0, whence (n being odd) μ% — 0, i ~ 1, «>, n. In particular μ1 — 0, i a e. the coefficient of #2 xn in 2/J.I; is zero. It follows by considering suitable renumberings of x2, , xn that 2/i^ = 0, i.e. yx e Kn9 Similarly yί e Kn for i = 2, , w. COROLLARY. Leέ % be an element in K3 which is homogeneous Of odd degree such that u — x*a + y b + z°c. Then α, δ, ce K3.. Proof. Suppose u — x°a + y°b + z°c is of degree p in x, q in y and r in 2 with p + # + r = n (an odd integer). Let xu * >, xn be n symbols of which p denote x, q denote y and r denote z. For convenience we denote vz by v. For n — 1 there is nothing to prove. We now proceed almost word for word as in the proof of the lemma. Assume n > 1 and let μi be the coefficient of xi+1 - xnx1 x^ (x2 β β xn if i — 1, a?! #„_! if i = ^) and so also of ^_i xλxn ceί+1 (a;w x2 if ί = 1, x%_λ #! if i — n) in the expansion in A(3) of α^ if ^^ — x, of όv it Xi — y and of cv if ^ = z. Then the coefficient of xi+1 ° # A aji (xx >' ° xn if ΐ = ?ι) in the expansion in A{3) of 2uv is j«i + μi+1 (μn + μλ if i = n). Since distinct monomials in A{3) are linearly independent we have μi + //i+1 = 0, i = 1 , , w — 1 and μn + μ^ — 0. Whence (^ being odd) ^ = 0, i = 1, , w. Since the argument goes through for any distribution of £> x's, q y's and r 2;'s amongst xu , xn {2>) the coefficient of each monomial in the expansion in A of av is zero, i.e. a e iΓ3. Similarly for b and c..

(54) 52. C. M. GLENNIE. It is now sufficient for the proof of Theorem 2 to show that each element in ikfis of the form x α + 7/ 6 + £ c. Let N be the subspace of J ( 3 ) spanned by elements of this form. We shall write a = b to denote a — b e N: thus we wish to show that m = 0 for each me M. Now M is spanned by elements of the forms a°(bcdefg + gfedcb) and (a-b)*(cdefg. + gfedc). where. two of a,b,c,d,e,. f, g represent x, t w o. represent y and three represent z. It is sufficient to show that each element of the form (a-b)-(cdefg + gfedc) is in N> or by formulae (3) and (4) that each of the following is in N: (1). aRbRcRdReRf.g. ( 2 ) aRhRcRd.eRf.g (3). = cRa.bRdReRf.g =. cRa.bRd.eRf.g. aRbRc.dReRf.g. For types (1) and (2) let t — α°δ c. Then we have for (1): (f-g)Rt.d er and for (2): (f g)Rd.e.t. Since Rt — Ra.b.c it follows by two applications* of formula (3) in each case that elements of types (1) and (2) are in N. Since any element in M can be written as zP where P is an operator generated by the right multiplications Ru,ueJ{3), it will be sufficient in the case of elements of type (3) to consider a = z. The possibilities, modulo interchange of x and y, are: ( i ) zRzRx.xRzRy.y. = xRxRz.zRzRy.y. = xRxRzRz.zRy.y. ( ii ) zRzRx.xRyRy.z. = -hzRzRx.xRzRy.y. = 0. by ( i ). (iii). zRxRz.xRzRy.y. == -\zRzRx.xRzRy.y. = 0. by ( i). (iv ) zRxRz.xRyRz.y. = -\zRxRz.xRzRy.y. = 0. by (iii). ( v ) zRzRx.yRzRx.y. - xRyRz.zRzRx.y. = xRyRzRz.zRx.y. = 0. = 0. (type 2). (type 2). ( vi). zRzRx.yRxRz.y. = -hzRzRx.xRyRz.y. = 0. by (ii). (vii). zRxRz.yRzRx.y. = -hzRzRx.yRzRx.y. = 0. by ( v ). (viii) zRxRz.yRxRy.z. = -hzRzRx.yRxRy.z. ^0. by (vi). (ix ) zRyRx.xRzRz.y. = xRxRz.yR2Rz.y. ( x ) zRyRx.xRyRz.z. = -2zRyRx.xRzRy.z. =0. by (ix). ( xi). zRxRx.yRzRz.y. = -izRyRx.xRzRz.y. ~Q. by (ix). (xii). zRxRx.yRyRz.z. = -ϊzRyRx.xRyRz.z. = 0. by (x). (xiii). zRxRz.zRxRy.y. = xRzRz.zRxRy.y. = xRz.zRzRxRy.y. = 0. (type 1). (xiv). zRxRz.zRyRx.y. = xRzRz.zRyRx.y. = xRz.zRzRyRx.y. = 0. (type 1). = ~ϊxRxRz.zRyRz.y. = 0. by (ii). This completes the proof of Theorem 2. It is possible to avoid the use of MacDonakΓs theorem in the proof of Theorem 2 by using the following result, tabulating bases for each subspace spanned by homogeneous elements of degree six and applying.

(55) SOME IDENTITIES VALID IN SPECIAL JORDAN ALGEBRAS. the Corollary to Lemma 1 to each subspace spanned by elements of degree seven. This process is straightforward, tedious, and is in any case largely a special case of theorem. We include Theorem 3, however, as it would of independent interest, in providing easy verification five-variable identities linear in each variable. THEOREM 3.. Kn n Ln = {0} for. n ^. 53. homogeneous if somewhat MacDonald's appear to be of proposed. 5.. Proof. The cases n — 1, 2, 3 follow at once from the case n — 4 with which we begin, taking the generators of J{i) as x,y,z,t. Let Rb, Sbc, Ubc denote the mappings α —>α»6, α-+{αδc}, α—>{bac} respectively. Then. Sbc = Rb.c. + RbRc. - RΰRb,. and Ubc - RbRe. + RcRh. - Rb.e.. Since. L 4 is spanned by the elements tRxRyRz1 tRxRy.z1 tRx.yRz and all others obtained from these by permuting x, y and z and Rb.c = RbRc + RcRb— Ubc, 2RbRc — Sbc + Ubc1 we have that (again to within permutations of x, y and z) L4 is spanned by tRxSyz1tRxUyz, tUxyRz. Now let ueK^OL^ and suppose that u - Σt(axyzRxSyz. + βxRxUyt. +. ΊzUxyRz). where the summation is over permutations of x, y and 2. Since ue KA and distinct monomials in A{i) are linearly independent we have ( 1 ) axyz = Q (coefficient of txyz in A{4)) and similarly each acoefficient is zero, and ( 2) βy + ΊZ = 0 (coefficient of $£7/2 in A(4)) and similarly for each pair of distinct subscripts. So βx — βy ~ βz — —ηx— —yy— —yz and u is a scalar multiple of t(RxUyz + RyUzx + # z £/ x y - L ^ β , - UyzRx -. UzxRy). which is zero by (5). So K4 Π LA = {0}. The result for n — 5 now follows by Lemma 1 and the fact, already noted in the proof of Theorem 2, that L 5 is spanned by the elements α ί> where a is a generator and b is linear in each of the other generators. 2* In order to establish the existence of an S-identity of degree 8 we now examine the situation discussed by Albert and Paige in the paper [1] mentioned in the introduction. Let D be an algebra with an identity element 1 and an involution d—>d. In the algebra Dn of n x n matrices with entries in D we can define an involution M—+M' by taking ( M % = (ilί,-,-), i.e. Mr is the conjugate transpose of M. Further, we can define an involution M—> M * in Dn by choosing a diagonal matrix Γ = diag{Yi, , τ»}.

(56) 54. C. M. GLENNIE. where the 7^ are self-ad joint (7* = 7*), in the nucleus of D and have inverses, and defining M* = Γ^M'Γ. Such an involution is called a canonical involution in Dn. The particular case in which Γ is the identity matrix reduces to the first involution defined and this is called a standard involution. It is clear that the subset of Dn of matrices self-adjoint under a canonical involution (i.e. M* = M) is closed under the product A B = 1/2(AB + BA) where AB is the usual matrix product and forms an algebra relative to this product and the usual addition and scalar multiplication. We denote this algebra by H(DnJ Γ) or simply H{Dn) if Γ is the identity matrix. With this notation the main theorem proved by Albert and Paige can be stated as: 4. {Albert and Paige). If H(Dό) is the homomorphic image of a special Jordan algebra then D is associative. THEOREM. Our first step will be to obtain a three-variable relation, S(x, y, z) — (3) 0, which will be easily seen to hold in J0 and so in any homomorphic image of a special Jordan algebra. Substitution of suitable elements x, y, z from H(D3) will immediately show that D is associative, giving an independent proof of the Albert-Paige result and simultaneously showing that S(x, y, z) — 0 is not valid in every Jordan algebra, since an example is known (with D as the eight-dimensional Cayley algebra) of a Jordan algebra H(D3) in which D is not associative. The homogeneous part of S(x, y, z) — 0 of degree 3 in x, 2 in y and 3 in z then gives the required S-identity of degree 8. Lemmas 2 and 3 are essentially due to Albert and Paige. LEMMA 2. Let θ be a homomorphism from a special Jordan algebra H, embedded in an associative algebra U, onto a Jordan algebra J such that (1) H is generated by elements X, Y, Z and I (I an identity in U) and β ( 2 ) H contains elements Eu , Ek (k ^ 3) such that E{Ej — EjEi in U and such that ely - , ek {ei — Eβ) form a set of orthogonal idempotents in J whose sum is the identity f'= Iθ of J. Then, for a, β in the set 1, , k and A a monomial in U generated by X, Y, Z and I we have (FaAFβ + FβA*Fa)θ e Jaβ where Fa = E%> Fβ = E%, A* is the reverse of A and Jaβ is the a, β component of J in the Pierce decomposition determined by the e^s.. Proof. Let B - EaAEβ + EβA*Ea, C = A + A*. Then FaAFβ + FβA*Fa = EaBEβ + EβBEa - (EaEβ)C(EaEβ) = 2{EaBEβ) - {(Ea.Eβ)C(Ea Eβ)}.

(57) SOME IDENTITIES VALID IN SPECIAL JORDAN ALGEBRAS. 55. So (FaAFβ + FβA*Fa)θ. = 2{ea(BΘ)eβ} -. {(ea-eβ)(CΘ)(ea-eβ)}eJaβ. LEMMA 2'. Wϊί/& iϊ, J, 0 and condition (1) (6uί no£ condition (2)) as m Lemma 2 suppose that Eλ — 1/2(X2 + X), JS72 = I — X 2 , E3 = 1/2(X2 - X) and X# = a?, 10 = /, £^0 = e1? Eφ = e2, # 3 0 = e8. Tfcen i / (2)' α;3 = x, we have that (a) eu e2, e3 are orthogonal idempotents with sum f and (b) (EaAEβ + EβA*Ea)θ e Jaa + Λ^ + Jββ.. Proof, (a) This follows immediately from the definitions of e1} e2, e3 and condition (2)'. (b) Let B = XA(I - X) + (/ - X)A*X. Then. + E2)B(2E1 = {(2ex + e2){BΘ){2e, + e2)} e J u + J 1 2 + J 2 2 Similarly for other choices of a and /3. LEMMA. 3.. With notation as in Lemma 2': 2[(EaAEβ + EβA*Ea)*(EβDEy = [EaAEβEβDEy. +. + EyD*Eβ)]θ. EyD*EβEβA*Ea]θ. where D is a monomial in U generated by X, Y, ^ and 7, and ar, /S, 7 are distinct integers chosen from 1, 2, 3.. Proo/. 2[(£7 β A^ + EβA*Ea)*(EβDEy + EyD*Eβ)]θ = (EaAEβEβDEy + EyΌ*EβEβA*Ea)θ + (EaAEβEyD*Eβ + EβDEyEβA*Ea)θ + (EβA*EaEβDEy + EyD*EβEaAEβ)θ + {EβA*EaEyD*Eβ + EβDEyEaAEβ)θ . Now, since a, /5 and 7 are distinct, / α β Jβ Y g / α γ β So, by Lemma 2', the left-hand-side is in J α γ β The result now follows from Lemma 2' and the disjointness of the Peirce decomposition. COROLLARY.. . (E2AE2E2ZE3 Equation (6) suggests the following relation in U:.

(58) 56. C M . GLENNIE. + (E2ZEJΞ2CE2E2ZEZ. E,ZE2E2C*E2)] + E.ZE.E.C^E.E.ZE,). E2. -. (E1ZE2E2CE2E3ZE2. -. (E2C*E2E2ZE1E2ZES. -. (E2C*E2E2ZE1EzZE2. where, for reasons which will appear later, we take C = YXZY and C* = ΓZXΓ. In turn, (7) suggests the following relation in J<3), (this is the relation referred to previously as S(x, y, z) — 0) 4:{e1ze2}*p1 + {(e2 + 2e3)q1(e2 + 2e3)} -. {(2β! 2. - {(2ex + e2)q2(2e1 + β2)}. {(e2 + 2e 3 )r 2 (e 2 + 2e 3 )} - {e2s2e2} where e1 = l/2(^ 2 + x), e2 = 1 — x\ e3 = l/2(x2 — x) and p l y 2qu 2ru slt p2, 2q2, 2r 2 , s2 are Jordan elements in J 0 (3) equal respectively in A (3) to e2yxzye2e2zeB + eBze2e2yzxye2 , (1 + x)ze1e2yxzye2e2zx xze2e^e2e2yzxy(l. + xze2e2yzxye2exz{l. + x) ,. — a?) + (1 ~ x)yxzye2e2zeze2zx. ,. e1ze2e2yxzye2 + e2yzxye2e2ze1 , #3e2e22/2C32/e2β33(l — x) + ( 1 — x)zese2yzxye2e2zx. (1 + x)yzxye2e2ze1e2zx. + xze^ze^yxzyiX. ,. + cc) ,. zeBeλze2e2yxzy .. Now, (8) is an S-identity. By construction it holds in J0(3) and we may see that it does not hold in ίί(C3), where C is the eight-dimensional Cayley algebra, by substituting. where w, v and w are arbitrary elements in C, and examining the 1, 3 element on each side of (8). The calculation is quite simple: by choice of x, the only nonzero contribution on each side arises from the first term. Further, px and p2 are of degree two in z and so may be evaluated as though C were associative, that is by substituting directly.

(59) SOME IDENTITIES VALID IN SPECIAL JORDAN ALGEBRAS. 57. into their equivalent associative forms displayed above. The result is u[(v — v)w] on the left and [u(v — v)]w on the right. Since self-adjoint elements in C are in any case in the nucleus we have u[(v + v)w] = [u(v + v)]w Whence u(vw) — (uv)w. But C is not associative. So (8) does not hold in the Jordan algebra H(C3) and is thus an S-identity. 6 The relation (8) can be written as Σi =3fi(x9y9z) — 0, where fi(x9 y, z) is a Jordan polynomial of degree i in x. Now fi(x, y9 z) can (3) be expanded in A as a linear combination of monomials in x, y, z of degree i in x. Since A{3) is free, fι{x, y, z) = 0 for each i. We consider the case i = 3. The parts of the terms of (8) which are of degree 3 in « are equal respectively in A[Z) to:. (a) (b) (c) (d) (e) (f) (g) (h). —4(x z) (yxzyzx + xzyzxy) zxyxzyzx + xzyzxyxz xzxzyzxy + yxzyzxzx zxxzyzxy + yxzyzxxz — 4(2β a?) (xzyxzy + yzxyzx) xzyxzyxz + zxyzxyzx yzxyzxzx + xzxzyxzy yzxyzxxz + zxxzyxzy. W e now make the following choices for Jordan expressions of the above: (a) + (c) + (d): -£{(x*z)y{x{zyz}x}} (e) + (f) + (g) + (h): -2{φMa>sMφ} and obtain the following relation which clearly holds identically in J 0 (3) : (9). 4{{z{xyx}z}y(z x)} - 2{z{x{y(x s)i/}φ}. Substitution in (9) of the same elements as were substituted in (8) shows that (9) is an S-identity. 3.. In H{D3) let la. p. q\. g = ip. β. r. \q. f. 7/. Then we have. /•. ,. x = (1. 1.

(60) 58. C. M. GLENNIE. jβ. V. {xgx} = \p. a. Λ ). and. {ygy} = [. 7 r ), r. while ##?/ (ordinary matrix multiplication) is equal to r. β\. q p\ . With these results in mind, (8) suggests the following candidate for an S-identity: 2{xzx}-{x{zy2z}y} - {x{z{x{yzy}y}z}x}. (10). = 2{x{zx2z}y}'{yzy} - {y{z{y{xzx}x}z}y} We verify that (10) is an S-identity by using it to prove the AlbertPaige Theorem in a slightly strengthened form. (Albert and Paige mention that their method will give the stronger result but do not give the details.) 4(a). If H(Dny Γ),n ^ 3, is the homomorphic image of a special Jordan algebra then D is associative. [Theorem 4(a) is also a stronger form of a theorem due to Jacobson [3] viz: If H(Dn, Γ), n ^ 3, is a special Jordan algebra then D is associative.] THEOREM. Proof of Theorem 4(a). It is sufficient to prove the result for n = 3. Since H(D3, Γ) is the homomorphic image of a special Jordan algebra the relation (10), which clearly holds in J0(3\ holds in H(D3, Γ). Now suppose that la Γ=. . β. \. " 7/. and let. y = .. where u, v and w are arbitrary elements in D. gives, in the first row, third column:. left hand side: βuaβiavyβywβy) right hand side: (βuaβavyβ)ywβy. Substitution in (10).

(61) SOME IDENTITIES VALID IN SPECIAL JORDAN ALGEBRAS. 59. Since u, v and w are arbitrary and a, β and 7 are in the nucleus of D with inverses the result follows at once. REMARK. It can be shown by using the corollary to Lemma 1 that the S-identity (10) is generated by S-identities of degree 8. We do not give the details here as we hope to embody them in a later paper. REFERENCES 1. A. A. Albert and L. J. Paige, On a homomorphism property of certain Jordan algebras, Trans. Amer. Math. Soc. 9 3 (1959), 20-29. 2. P. M. Cohn, On homomorphic images of special Jordan algebras, Canadian J. of Math. 6 (1954), 253-264. 3. N. Jacobson, Structure of alternative and Jordan bi-modules, Osaka Math. J. 6 (1954), 1-71. 4. I. G. MacDonald, Jordan algebras with three generators, Proc. London Math. Soc. series 3, 10 (1960), 395-408..

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(63) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. A DESCRIPTION OF MULT, (A1, ••, An) BY GENERATORS AND RELATIONS THOMAS W.. HUNGERFORD 2. If R is a ring (with unit) and A^RtRA Ri > -, RAI'RA71 are #-(bi)modules, then Multf'"(A1, , An) is defined to be the ^th left derived functor of the multiple tensor product A1 (g) ® An 1 n r (® = ®Λ); i.e., HiiK (g) ® K )9 where each K is a projective resolution of Ar. The purpose of this paper is to give a description of Multf»w(AS •••, Aπ) in terms of generators and relations, analogous to that given by MacLane in the case n=2 [and Mult; = Torf (A1, A2)]. Throughout this paper R is a ring with unit, all modules are unitary, and ® means (g)Λ. If A^, ^A^, • • ,ΛA£~\ ^A* are iϋ-modules (or bimodules, as indicated), then Multf w(A\ -.., An) is defined to be the itla left derived functor of the multiple tensor product A1 (g) 0 A%; iβeβ. ίW. 1. Θ. <g) κ«),. where each i£ r is a protective resolution of A r . When no confusion can arise we shall often write Mult; or Mult? in place of Multf>\ Note that for n = 2, Mult^ is simply the functor Torf (A1, A2). A description of Multf'% (A1, •• , An) is given in [1]. MacLane [2] has described Torf (A1, A2) in terms of generators and relations. The purpose of this paper is to extend this description to the functors Multf >n (A\ , An)c The first difficulty in doing this is to formulate the proper definition of the generators and defining relations. Once this is done, however, most of the proofs are analogous to (though usually considerably more complicated than) the proofs given for Torf(A\ A2). A notable exception to this is Theorem 3,1, in which the results for n — 2 are used as the first step in an inductive procedure, which is much simpler than a direct proof. Unfortunately, this technique apparently cannot be applied in the proof of the crucial Theorem 3.6, where we must resort to a long and somewhat involved procedure. Throughout this paper we shall often use the term iϋ-module for left-iϋ-modules, right i?-modules, or iϋ-bimodules, the specific meaning being indicated by the context. Received July 2, 1964. 61.

(64) 62. THOMAS W. HUNGERFORD. 2* Definition a n d basic properties* consider chain complexes E of length i 777.. 777 ,. d. 777. i l / . jβ/ 0 <. hlγ. ,. 3. ,. « « <. <. ^. For a fixed ί ^ 0, we. 777. hi I ,. with each Er a finitely generated free i?-(bi)module. The dual E* ~ Hom^ (E, R) can also be regarded as a chain complex of length i. a*. E*:Ef<. δ*. E*i^<. δ*. ^—E* ,. where 3* = Horn (9, 1); each E* is also a finitely generated free JB(bi)module. (Note: our definition of the boundary operator in E* differs by a sign from that given in [2].) If A is an i?-module, it can be considered as a complex (in dimension zero) with trivial boundary operator. If E is a complex as in the previous paragraph, then by a mapμ:E—*A we mean a chain transformation of complexes, i.e. an jR-module homomorphism μ:E0—>A such that the composition. is zero. If E and F are two complexes as above then E (g) F and E*($Z)F* are chain complexes of length 2ΐ of finitely generated free jβ-bimodules (denote the boundary in these complexes by 9 and δ respectively). If A is an .ff-bimodule, then by a map μ:(E<g> F)i-^ A [or μ: (E* (g) JF*); —> A] is meant a bimodule homomorphism such that the composition (E(g) F)i+1 [or. -^(E^F^-^A. (E* (g) i ί 7 * ^ —. ( # * <g> F*), -^-> A]. is zero. If AιR, RA2R, , ^Aj"1, β A π are J?-modules, we shall define a certain group in terms of generators and relations, which (to avoid confusion in the long run) we call Multf'^A 1 , « ,AW). We shall eventually show that this is precisely the group defined in § 1. But until that time we shall use Mult; to refer to the group defined below and not to the group defined in § 1. 1 n We take as generators of Multf'^A , ••«, A ) all elements: <μ(l),. E\ μ(l, 2), E\ μ(2, 3), E\. , μ(n - 2, n - 1), E*~\ μ(n)>, where (for. r = 1, 2, •••,^ — 1) Er is a chain complex of length ί, with each £7^ finitely generated free ίϋ-module; t h e μ's are maps, μ(r, r + 1): (Er (g) E ^ 1 ) ; -> A r + 1. (2 ^ r ^ n - 1, r even).

(65) 1. n. A DESCRIPTION OF MULT* (A , •••, A ) GENERATORS AND RELATIONS. μ(r, r + 1): (£T <g) Er^% -> A^1 %. λ. w. μ(n): E ~ —> A. r odd). (w odd). %. μ(w): JE*" * -> A 1. ( B r ^ - 1 ,. 63. (n even) .. These generators are subject to the following relations. Suppose (for r — 1, , n — 1) Er and £ r are chain complexes of length i as above, λ r : Er —+ Er is a chain transformation, and there are maps μ(r, r + 1): (Er ® S r + 1 ), -> A r + 1 1. μ(r, r + 1): C£T ® JE^ *^ — A %. 71 1. μ(n): E ' —> A %. x. r+1. (r even) (r odd). (n odd). n. ^(^): S ~ * —> A. (n even) .. Then we require that the following relation hold. (1) <jS(l)λlf E\ μ(l, 2)(1* ® λ?), S 2 , jδ(2, 3) (1 <g) λ3), S 3 , //(3, 4) , μ(n - 2, n - 1)(1* ® λ^,), # n ~ \ ^ ) >. (1* <g> λf),. - <jδ(l), S S Ml, 2)(λf (8) 1*), J?2, jδ(2f 3)(λ2(8) 1), ^ 3 , ^(3, 4) (λ3* ® 1*),. - , μ(n - 2, n - l)(λ*_a ® 1*), E*-1, μ(n)\n^y. (^ is assumed odd here; the same relation, with the obvious changes in the last entry holds for even n). Thus two generators of Mult; are equal, provided one can be obtained from the other by a finite number of applications of the above relation. When no confusion can arise we shall often write generators of Mult^ as (μ,E1,μ,E2y •••>. Mult; ( A 1 , — , An) is made into an abelian group by definining addition as follows. If a: X ® Y—+D and β: X(g) Ϋ—> D are iϋ-module homomorphisms, we denote by a*β the map. which is the composition. ( I 0 Ϊ ) ® ( Γ 0 Ϋ) = (X(g) Y) 0 (X(g) Y) 0 (X® Γ) 0 (X(g) Γ) where π is the projection onto the two end summands and VD is the usual codiagonal map. This definition is extended in the obvious way to the situation where X, X, Y9 Ϋ are chain complexes of finite length ϊ Ϋ)4 — £>. Now define ζμ, E\ μ,E\to be the element. 1. , E"" , μ) + <fi, E\ μ,E\. *., E*~\ μ>.

(66) 64. THOMAS W. HUNGERFORD ι. <TAμ © μ), E 0 E\ μ*μ, E> 0 E\ μ*μ, • •-,μ*μ, E-1 e E»-\ FAμ 0 μ)>. It is easily verified that this addition respects the defining relation (1). For ^-modules X, Y let ω = ω(X, Γ ) : X 0 Γ ^ Γ 0 X be the map given by ω(x, y) = (y, x). Let Δx\ I - ^ I φ I and Fx: X®X->X be the usual diagonal and codiagonal maps. Then the following identities hold. (2) (3) (4). Fx = Fxω if α : X — X, β: F — Ϋ, then ω(α θ /3) = (/3 θ α)ω: I φ Γ - Γ 0 X if α:X(g> Γ— D,β:X® Ϋ->D, then (α /9)(ω (gι 1) = (/3*α)(l ® ω): (X 0 X) ® ( Γ 0 Ϋ) — β. ( 5). if α, /3 are as in (4), then a*β(ω ® ω) = /3*«: (X 0 X) (g) ( f © Γ) - D. (6) (7) (8) ( 9) (10). ω(X, Y)* = ω(X*, F*) if α,/S are as in (4) and γ : X ® Γ—D, then (α*/3)*7 = α*(^*τ): if « : X ® Γ— D, then a(VΣ ® 1) = α α(l ® J r ) : ( X 0 X) ® Y—1>; {Ax)* = Fx> and (F x )* = Δx.\ if β:X->D, then /3FX = F ^ 0 β): X 0 X — i ) .. Using (l)-(6) in a manner analogous to that in [2] one verifies that addition in Multί (A1, •• ,AW) is commutative. Associativity follows from (7) and the associativity of the diagonal and codiagonal maps. The zero element is <0, 0, •• ,0>, (where the zeros are either zero maps or zero complexes of length i). The inverse of <μ, E1, •>• is < - μ , E\ - - •> since VΛ,(μ 0 (-//)) = 0 . Using (1) and (8)-(10) one verifies that the generators (μ,E\μ, ...,E*~\μ> are additive in the μ's; i.e.. 1. = <^, .K ,. , ^(r, r + 1) + Jδ(r, r + 1),. 1. , S - , μ>. Finally if (for r = 1, 2, •••,») α(r): A r -^ Ar are i?-module homo1 morphisms, Mult, (A , , A") becomes a covariant functor of w varia-.

(67) A DESCRIPTION OF MULT* (A 1 , •••, An) GENERATORS AND RELATIONS. 65. bles to the category of abelian groups by defining α(r)*<ju, E\. , μ(r - 1, r), 1. , E*~\ μ>. a(r)μ(r - 1, r), E*~\ μ) .. = ζμ, E ,...,. 3* The main theorems* A1,. 3.1. If isomorphism:. THEOREM. natural. 1. 2. are R-modules, 1. n. A (g) A <g) Proof.. , An. then. there is. a. n. (g A = Mult^^A ,. ,A ) .. Define a map /: A1 <g). . (g) Aw — Multo (A1,. , Aw). by /(fli <g) <g) αw) = <^(αθ, -R, ^(α 2 ), , R, μ(an)y, where ^(α r ): R = r iί(8)jB[=i2*(g)jB*]->A is given by μ(α r )(l) = α r . / respects the defining relations on the generators of the tensor product and hence induces a well defined homomorphism. If a:A—>Ά and aeA, then aoμ(a) — μ(aa): R-* A it follows that / is natural in A r (r = 1, Next define a map έ o ί A. 1. ,. . . . , A^^A1®. , ri). ••• (g)A%. as follows. If ζμ, E\ , En~\ μ) is a generator of Mult0, with each r E finitely generated free, choose a basis {re(ir) \ ir e Ir} for each Er. Let r e*(ί r ) be the dual basis for Er*. Then define gζμ, E\ . , # — \ /i> to be the element Σ ^[^(ii)] ® J^Γβ*^) <g> 2β*(ί2>] (g) ^[βe(i2) (g) *e(i3)] (g) •. (g) ^ [ w ^ * ( ί . _ 2 ) (g) - 2 e*(i w -!)] (g jur^ίV-i)] ,. where ΐ r e I r and the sum is taken over Ix x x J Λ - 1 ; (n is assumed odd here; for n even the final terms should be changed in the obvious way). The proof that g is well defined is straight-forward (and analogous to the proof Theorem V. 7.3 of [2]). It is immediately verified that gf = 1 and hence / is an epimorphism. In order to show that / is in fact an isomorphism we need the following two lemmas. 3.2. If An is free, • • ,A ) is an isomorphism. LEMMA W. a. then f: A1 (g) . . (g An —> Mult0 (A1,. 3.3. If 0 —> A >B > C —> 0 is a short exact sequence of R-modules, then there is an exact sequence: LEMMA.

(68) 66. THOMAS W. HUNGERFORD. Multo (A1,. , A—1, A) - ^ U Multo (A1,. , A—1, B). -^U Multo (A1,. , A - 1 , C). The proofs of these lemmas will be given below. free iϋ-module such that (1). 0. > K-^. is exact (K = ker β). with exact rows. A1®. F-^. Let F be a. >0. A". Consider the following commutative diagram. (gA%-:L(g)iίΓ—>Ax(g). Multo(A\. >0. ® A*"" 1 ® 2^—-•A1®. k. k. 1. ® A*-^. k. , A - , 2SΓ) — Multo(A\ . . . , F) — Multo(A\. , A ) -> 0 ,. with horizontal maps induced by the sequence (1). Since F is free the middle map / is an isomorphism; since the other maps / are epimorphisms, it follows from the five-lemma that ® An -> Multo (A1,. /: A1 ®. , An). is an isomorphism. Except for the proofs of the lemmas this completes the proof of Theorem 3.1. Proof of Lemma 3.2. It suffices to assume that An generated and hence that An = R. Consider the diagram: A1 ®. A1 (g). ® A"-1 ® R -^-> MultJ(A\. k. ,. ί«. (g) A11-1. ^=4 MultJ-^A1,. is finitely. , A—1, 22). , A1-1) ,. where λ is the usual isomorphism and G is defined by (this makes sense since En~2 (g) i2 [or £>~2*(g) i2*] can be identified with i?w~2 [or £r%~2*]). It can easily be verified that G respects the defining relations in Multo"1 and hence induces a well defined homomorphism. Define a map H: MultJ(A\ by. n. ]. , A ~\ R) — MultΓ (A\. n. , A ~').

(69) 1. n. A DESCRIPTION OF MULT* (A , •••, A ). GENERATORS AND RELATIONS. 67. (this is for n odd; for n even, last entry is μ(l 0 v)). This makes n 2 n 2 sense if we consider /i(l*0ι>*) as a map on E ~ ® R* — E ~ * (similarly for n even). It can be verified that H induces a well defined homomorphism and that HG — 1 and GH — 1; hence G is an isomorphism. Finally one verified that the above diagram is commutative, i.e. fn — G/%_1λ. Since fn^ is known to be an isomorphism for n — 3 (cf. [2]) the conclusion of the lemma now follows by induction on n. Proof of Lemma 3O3. If ζμ,E\ , En~\ ιi) is a generator of Mu.lto(A\ « ,A%~1, C), then the fact that En~\ is free implies that n x there is a map 7: E ~ —> B such that βy = vo Hence. and β* is an epimorphism. The rest of the proof is analogous to the proof of Theorem V. 5Λ of [2] and is omitted here. n. r. 3O4. If F\ , F ~\ A are iϋ-modules and each F is finitely generated free, with basis {re(ir) | ir e Ir}, then every element of F10 F2 0 0 F71-1 0 A can be written uniquely in the form: PROPOSITION. where a(ίu •••, v J e A and the sum is taken over The proof follows from the fact that i2 0 • turally isomorphic to A under the map given by rι 0. 0 rn_i 0 α -> (rx 71. r^Oα .. 1. Suppose that F\ , F " are finitely generated free iϋ-modules, the basis of Fr being {re(ίr) \ ir e Ir}. Denote the dual basis of F? r r by { e*(ir)}. For r odd let F be the finitely generated free iϋ-module 1 r F 0 0 F ; it has a basis {^(iO 0 0 rβ(ίr)} which we shall β r denote by {^(ί^ ,ΐr)}. For r even, let F be the finitely generated free iϋ-module F1* 0 0 i^r*; denote its basis by {re*(ίi, , ir)}. Define maps: π(r): Fr~' (g) Fr-± Fr 1. r. (r odd, r ^ 3). π(r): F*- * 0 F * —> i^. r. (r even, r ^ 0). as follows. πir^-'eii,,. , i r - 1 ) 0 re{ju. , i r )] = Π δ(ik9 jk)re(jr). where δ(i, i) is the Kronecker delta, and. ik,jkelke.

(70) 68. THOMAS W. HUNGERFORD. 3.5. If F\ , Fn~\ A are j?-modules, with each n F finitely generated free, then every element of Mult0 (F\ , F ~\ A) can be written uniquely in the form: PROPOSITION. r. <1, F\ τr(2), F\. , π(n - 1), F*~\ v> ,. where v: Fn~H*] —> A.. Proof. Under the natural isomorphism of Theorem 3.1, the elen ment <1, F\ π(2), F\ , F ~\ y> is mapped onto where the sum is taken over Ix x x JΛ_1# Hence by Proposition 3.4 the values ^["""^(ii, •••, V-i)] are uniquely determined and therefore so is v. It is also clear from Proposition 3.4 and Theorem 3.1 that every element of Mult0 can be written in the required form. We are now in a position to prove the main result, that Multf'** (A1, •••, An) as defined by generators and relations is isomorphic to the ΐth left derived functor of the functor A1 (g) , (g) An. Recall that to define this functor it suffices to take free resolutions of only n — 1 of the n modules. 3.6. Let A\---,An be R-modules and K\ , Kn~1 1 1 free resolutions of A , , A*" . Then there is a natural isomorphism (for each i) THEOREM. F: Multf '"(A1,. , An) = H^K1 ®. ® K"-1 (g) An) ,. Proof. Let <μ, E\ , Έn~\ μ} be a generator of Mult^A1, , An). By the lifting theorem for chain complexes there exist chain transformations h over the respective identity maps as follows. E\. >. Ud, i) K\ (Er* (g) Er+1\ I \h(r + l,i) i KΫ1. > El -?-> A1. > E\ U(l, 1). >. > K\. U(l f 0) >K\-^. A1. >. (Er* (g) Er+1% - ^ Ar+1 I \h(r+1,0) i > Kr0+1 —?-> Ar+1. >. > (Er (g) Er+% -^-> A r + 1. >. for r odd, r ^ 1; (Er (g) Er+%. U(r+l,0). 4 7^r.

(71) A DESCRIPTION OF MULT*(A 1 , •••,An) GENERATORS AND RELATIONS. for r even, r ^ 2.. Note that. h(r + 1, p): (Er' <g> Er+ι%_P r. r. λ(r + 1, p): (E (g) E ^)i+P L. We define [in (K1 0. 69. -»K;+1 — if;. (r odd). +1. (r even) .. n. F(μ, E , -•-, E ~\ μ) t o be 0 ϋΓ"-1 (g) Aκ)i] of the element. the. homology. class. Λ—1. where the sum is taken over all (p19. , pw-i). s u c. h that ^ Pr = ί, and r= l. Pr = Pi + V2 + + Pr (r odd); Pr = ί - 2>i - 2>£ ~ - Pr (r even) the sign ( — 1)* is determined as follows. let. e(k) = X i .. Given (pu. For any positive integer fc,. , p Λ - 1 ) such that X p r = ΐ, let. l. ζ ( p r ) = ε(i — Pi — p 2 — •-- - Pr) + Pr+i. >r) = ε(Pi +. + Pr) + Pr+i. (r o d d , r ^ 3). (v even, r ^ 4) .. Then set (-1)* = Strictly speaking the maps Z^(r, p r ) in (2) are actually the restrictions of these maps to suitable sub-modules; for example, if r is odd h(r, pr) is defined on (E^1 0 Er)i+Pr and the map h(r, pr) in (2) is the restriction to E;;^1(g)E;r^:(Er~1®Er)i+Pr. Note that for each r, h(r,pr) is a map into KrPr; if w is even pn_1 — i and μ: Eΐ'1*—> An; if ^ is odd pΛ-i = 0 and ]M: ^ ί " 1 — Aw. Thus in every case F<μ, E\ , .S^-1, ^> is^an element of degree i of the group Mult0 (K\. , IT—\ Aw) = i ί 1 (8). - ® Z"^ 1 ® 4^ .. In order to show that i^7 is well defined we must verify that F is independent of the choice of the maps h(r, —) and that the image of .Fis in fact contained in the group of cycles of (K1® (QK^^A*)^ Let x = <μ, E1, , S—1, /i > ε Mult* (A1, , An). As an element of 1 71 1 % K ® <g) if " (g) A , Fαj has boundary,. (3) Σ ( - l [ r—1. where p 0 = 0, u(r) = Σ PΛ. an(. i Σ (—1)* *. s. a s.

(72) 70. THOMAS W. HUNGERFORD. Using the facts that the maps h{r, —) are chain maps (and thus commute with the various boundary operators), the additivity (in μ) of the <(• , μ, •)>, the defining relations in Mult0, and the fact that μ o (boundary) is zero in each case, it follows that (3) becomes (for n odd): Σ (-1)*<M1, Pi - 1)3, E\χ h{2, p2),. + Σ (-!)*[ Σ (-l^X. •> # w i. ^^Mr-l,^,. r even. E£.1+ι, Mr, pr -1), ^ , •> + ( - l ) «"+3v< • - , # £ , , h(r, pr_0, E;r+1, h(r + 1, pr+1)(3 Θ 1), ^ ^ •>]. + Σ (-!)*[ Σ (-iY*K ' 3^rg%2. r. ,E Cvh{r,pr-l)(d®l),Elr,. + Σ (-!)*<• , EnC3, h(n - 2, pn_2)(l (8) 9), tf£!ϊ+1, (A similar statement holds for n even.) After a suitable change of indices (in the terms with r even) and careful attention to signs, it follows that all the terms cancel and hence the boundary of Fx is zero. To show that F is independent of the choice of the maps h(r, —), it suffices to assume that for some t, g(t, —) is another such choice. (For convenience, assume t is odd; similar statements hold for even t.) Then there is a chain homotopy s: (E*-1 (8 E') -> K* specifically,. and g(t, p) = h(t, p) + ds(P + 1) + s(p)d.. (where 9 is the boundary in E1"1 (8) Eι). Thus it suffices to show that the element Σ (-l)*O(l, Pi), Elί9. - ,E^19 ds(p. is a boundary in i ί 1 (8) β β ® iί"" 1 (8) ^ % This fact follows from the repeated use of the defining relations for Mult0 and the fact that maps h(r — ) are chain maps. For convenience we shall now assume that K1, , K71"1 are finitely.

(73) 1. A DESCRIPTION OF MULT* (A , •••, A") GENERATORS AND RELATIONS. 71. generated; (more precisely, we use suitably chosen finitely generated subcomplexes, cf. the argument in Theorem V.8 β l of [2]). Denote by Kr the complex Kr " c u t off" beyond dimension i and let Kr be the complex K1 0 ® Kr (from dimension i through 0) for r odd and 1 r K * (g) 0 K * (from dimension 0 through i) for r even. Denote a free basis of Krp by {rkp(ur)} where ur runs over a finite index set; the dual basis of Kζ is denoted by {rk*(ur)}. If (r x , •• , r i ) is a ί-tuple of nonnegative integers such that Σ rj —r, we denote by {*fc(r)(w, , wt)} the free basis r2. (u 2 ) <g>. <g>. %t(ut)}. of if ^ ®. 0 £ί4 s ^j. (ί odd).. Similarly {*&(*(%!, •• ,u ί )} denotes the free basis of Kl\ (8). <8> ^. S ^U. (ί even) .. Strictly speaking this notation is somewhat ambiguous; but in context it will be clear. Define as follows chain transformations π: (K** (8) Kt+1*) — Kt+1 π: {Kι (8) Kt+1) -> Kt+1. (t odd) (t even) ,. where (K** ® Kt+1*) runs from dimension 0 to i and {K* (8) ^ ί + 1 ) from dimension 2i to i. For ί odd, let 2/ = *&£)(!*!,. , wt) (8) ί+1fc(ί_S)(^,. (where (r 1? , rt) = (r); (su t+1 tor of (K** (8) K \. Define. , vt+1) ,. , st) = (i — s); r + s — n), be a genera-. where ε(r) is as above and δ is the Kronecker delta. If πy Φ 0, then rj = Sy ( i ^ ί) and ί+l. t. hence st+1 — i — r — s = i — n and therefore. as desired (if πy = 0 there is no difficulty)..

(74) 72. THOMAS W. HUNGERFORD. For t even, let , ^ ) (g) t+%S)(vlf. y = '*(*<_,.,(%!, (where (r u , r f ) = (i — r), (s^ ί+1 tor of φ ® # ) ί + . Define. , v*) ,. , st) = s, r + s = i + w), be a genera-. πy = Note that if πy Φ 0, sj — r, (j ^ t) and ί+ l. = Σ 3=1. ί s. i = Σ. r. s. i + ί+i = i — r. j=l. hence st+1 = r + s — i = ί + n — i = n and therefore as desired. A laborious calculation shows that the maps π commute with the various boundary operators and thus are chain transformations. This calculation depends in part on the following facts (which will also be used below). Suppose E is a finitely generated free chain complex of finite length; denote the free basis of Er by {er(u)} and the dual basis of JK? by {β?(w)} Let G be a finitely generated free JS-module with basis {f{w)}\ define a map π: E by π(e*(u) (g) es(v) (g) f(w)) = δ(r, s) δ(u, v) f(w) , where δ is the Kronecker delta. Then π[d*e*(u) (8) er+ί(v) ® f(wy\ - π[er(u) ® 3er+1(v) ® jf(w)] . This is true since the map d: J57r+1 —> Er can be described by matrix (ruυ) such that d(er+1(v)) = J^ruver(u); hence 3*: J57r —> Er+ί is given by d*(e*(u)) - Σ ^ e ? f l ( ^ ) . V. To show that the map F is an epimorphism, let z be a cycle in (ίC1 (g) (g) i^"" 1 (g) An)iu Then « can be written uniquely in the form Σz(Pi> * >P»-i)> where the sum is over all (p 1? •• ,p»_i) such that 2 , ^ = i and ^(^, , p w - 1 ) e KlH® (g) JKΓ*~± (g) A%. Each sfo, ,p ^ ) can be written uniquely in the form: 1. \, π(2), frPvPi, π(Z), KlvW. τr(4),.

(75) 1. A DESCRIPTION OF MULT* (A ,. , A») GENERATORS AND RELATIONS. 73. [cf. Proposition 3.5; the subscripts on the Kr are necessary to distinguish the z(pu # ,2V-i)]. Hence. where the sum is as above. Let Kr and π be as in the previous paragraphs; we can consider the various KrPv...iPr as submodules (in various dimensions) of Kr; then the maps π(r) are just the restrictions of the maps π to these submodules (except perhaps for a sign). Consider the element. x = <ε, K\ επ, K\ επ, K\ eπ, where v is defined as follows.. , eπ, Kn~\ v) ,. If n is even, then. ffn—l*. ^Γ». τrn—l*. (where t h e sum is over all (pl9 •• ,j? Λ -i) such t h a t ΣιPj = i)9 and v: KΓ1*-> A is given on K*v...tPn^. and ζ(pj) as above).. by ( - l ) * M P i » •• , P » - I ) , where. Similarly, if n is odd, n—l V7» ffn-l 0 — ZΛ J^p1,. ',pn-1. (sum as above) and v:K™~ι—>An is given on Knv~^..)Vn_Ύ by. (sign as above). Assuming that x is a well defined element of Mult; (A1, •• ,A%) it follows [since π = ( - l ) 5 ^ ^ + 1)] that Fx is the homology class of the cycle z [Choose the identity for h(l, —) and π for h(r, —), r > 1.] Hence to show F is an epimorphism we need only show that x is in fact a well defined element of Mult; (A1, , An). For n odd this amounts to showing that vd — 0, where d is the boundary operator in K"-1. (Similarly for n even, we must show that v9* == 0.). The proof of this fact is tedious but straightforward and we omit most of details. One first computes dz and notes that an element of the form <1, K*Pί, τr(2),. , K;-]..,Pr_iy π(r), K'Pl,...,Pr,. can be written in the form ± <(1, K\v τr(2),. , Kl~*..,pr_iy π,. Kpv...,Pr^u. ---,K£LtPr-1,...tPn_1,v(-)d>, where (for n odd) 3 is t h e map 1 ® •• ® 1 ® 3 ® 1. ® 1 on.

(76) 74. THOMAS W. HUNGERFORD. KιVl <g) (g) KrPr-i Θ Θ ^ C - i T h i s i s a consequence of the definition of the map π9 repeated use of the defining relations in Mult0 and the fact that π = (-l)*®fiπ<j + l)(j = 1, , n - 2). Since dz = 0 the uniqueness statement of Proposition 3.5 implies that v(ply , p n - 1 )3 = 0. It then follows that vd — 0 as desired. Hence x is well defined and F is an epimorphism. In order to prove that F is a monomorphism we need the following lemma.. , An) can be written. 3.7. Every generator of Multi (A1, in the form: LEMMA. where the Kr. are formed. as above from. suitably. chosen. r. r. finitely. generated free subcomplexes of free resolutions K —^—> A of the. A\. The proof is given below; assume the lemma for the present. Suppose x = <ε, K\ επ, K2y , sπ, Kn~\ v) is a generator of Mult4 (A1, •••, An) and that Fx — 0, i.e. Fx is a boundary in TCι 62\ . . . (shi If™—1 fi?\ Δ. n. Then there is a chain , ^ x , π(2), X5 1>Pa ,. , Knp-l.,Pn_. (where the sum is taken over all (pu , p n - 1 ) such that ^ i Pi — ^ + 1) and du = x. The remarks above show that du can be written in the form:. where the sum is over all (pu , p n - 1 ) such that Σ i Pi — * a n ( i ^ ^ s n ι the boundary in K ~ (if w is odd: replace 3 by 3* for n even; recall that X Kn~x S ^ % - 1 ) . It also follows that i^x can be written in the form V ί —ΓlVl K(sum over all (pu Proposition 3O5 that. π K-. , pn_λ) such that Σ i P i v = ± ζ3 .. Hence,. 1. i/>. Ό. ^. π ••• K-' =. follows from.

(77) 1. A DESCRIPTION OP MULT4 (A , • • , A») GENERATORS AND RELATIONS. 75. x = <ε, K\ επ, K\ επ, • , Kn~\ v> , επ(l* <g> 9*), £ " - 1 , ζ>. ( i). = ± <β, K\. (ϋ). = ±<β,K\ ". (iii). = ± <ε3, K\ eπ,. *.*,K^\. (i) results from applying the defining relations in Mult^; (ii) follows since by the definition of the generators of Mult;, εττ(3* (g) 1*) ± εττ(l* (g) 3*) = 0 (where 3 is used to denote either the boundary in Kn~2 or Kn~ι). Repeated use of this finally gives (ii); by definition ε3 = 0 and hence x ~ 0. Thus F is a monomorphism, and therefore an isomorphism. Except for the lemma, this completes the proof of Theorem 3.6, and justifies the use of the notation Mult; (A1, , An) to denote unambiguously either the group defined by resolutions (§ 1) or the group defined by generators and relations (§ 2). Proof of Lemma 3.7.. Given a generator. of Mult; (A2, •••, An), there is a chain map &: .E1—> iΓ1 lifting the identity map on A1 (K1 is a free resolution of A1 as above). Let K1 be a finitely generated free subcomplex of length i of K1 which includes the image of h. Then. x = <μ, E\. , E«-\ v"). Note that K1 can be taken as K\ We now proceed by induction and assume that x can be written in the form. <ε, K\ eπ, K\ . . . , eπ, K\ μ, E^\ . . . , E«~\ ^> . For convenience, assume t is even. We wish to define a chain map φ:Et+1—>Kt+1 such that μ — eπ(l (g) φ). If we can do this, then x - <ε, K\ . - -, K\ eπ, K^\ μ(φ* (g) 1*), . . . , v> and the induction is completed. In order to define φ, note that there is a chain map h:.

(78) 76. THOMAS W. HUNGERPORD. I' (where we take (Kι (g) Et+1) from dimensions 2ί to i; note that ι t+1 t+1 (K (g) E ) is finitely generated). Let K be a finitely generated t+1 free subcomplex of length i of K which includes the image of h. Denote the generators of E\ Et+\ K\ Kt+1 as above and define +1. φ: Ei. +1. —> K\. on a generator. t+1. e8(v). by. where the second sum runs over all (r) = (ru , r j such that X TV = r and r takes all values from 0 to s; for each (rl9 , rt) the first sum runs over all (uί9 m fut), where the generators of Krrj are indexed by the uj. Note that *kW)(uu •• ,ut) is a generator of &\tr and λ [ ^ U (g) El+1] s ίΓ'ίJ; hence. and therefore <px e J^ί*1"1. A tedious calculation shows that φ is in fact a chain map and that eπ(l ®φ) — μ\Kt® Et+1 —> At+1. The procedure for t odd is similar. This completes the proof of the lemma. REFERENCES 1. T. Hungerford, Multiple Runneth formulas for Abelian groups, Trans. Amer. Math. Soc. 118 (1965), 257-276. 2. S. MacLane, Homology, Springer, Berlin, Gottingen, and Heidelberg, 1963. UNIVERSITY OF WASHINGTON.

(79) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. THE DISTRIBUTION OF CUBIC AND QUINTIC NON-RESIDUES JAMES H. JORDAN. For a prime p = 1 (mod 3), the reduced residue system S3, modulo p, has a proper multiplicative subgroup, C°, called the cubic residues modulo p. The other two cosets formed with respect to G°, say C1 and C2, are called classes of cubic nonresidues. Similarly for a prime J J Ξ I (mod5) the reduced residue system S5, modulo p, has a proper multiplicative subgroup, Q°, called the quintic residues modulo p. The other four cosets formed with respect to Q°, say Q1, Q2, Q3 and Q4 are called classes of quintic non-residues. Two functions, fs(p) and /5(p), are sought so that (i) if p = 1 (mod 3) then there are positive integers di^C1, i = 1, 2, such that aι < /3(p), and (ii) if p Ξ l (mod5) then there are positive integers α;eQ% i = 1, 2, 3, 4 such that α; < /5(p). The results established in this paper are that for p sufficiently large, (i) fs(p) = pa+s, where a is approximately .191, and (ii) /5(p) = pβ+*, where .27 < β < .2725. Davenport and Erdos [3] raised the general question about the size of the smallest element in any given class of kth power nonresidues. The special cases k = 3 and k = 5 are of primary concern in this paper. They proved a quite general theorem of which two special cases are: A. For sufficiently large primes p = 1 (mod 3) and e > 0 each class of cubic non-residues possess a positive integer smaller than p™ι™+\ THEOREM. B. For sufficiently large primes p = 1 (mod 5) and ε > 0 each class of quintic non-residues possess a positive integer smaller than pmi396+\ THEOREM. In the same paper Davenport and Erdos used a result of de Bruijn [2] to improve the constant of Theorem A to approximately .383. Recently D. A. Burgess [1] succeeded in improving Polya's inequality concerning character sums. Burgess' result is THEOREM C. If p is a prime and if χ is a nonprincipal character, modulo p, and if H and r are arbitrary positive integers then Received April 8, 1963 and in revised form May 4, 1964. 77.

(80) 78. JAMES H. JORDAN Σ. I. X(™) < H1-ίlr+1p1'ir. In p ,. any integer n, where A < B is Vinogradov's notation for some constant c, and in this Theorem c is absolute.. for for. \A\<cB. The application of Theorem C to the arguments of Davenport and Erdos cuts each of the exponents of p in half. The achievement of this paper is to obtain the same result about cubic non-residues by an argument which is independent of the de Bruijn result, reduce the exponent on the result for quintic nonresidues by a similar argument, and indicate a method of obtaining results for any primeth power non-residues. 2* Cubic Non-Residues* A well known result about sums of inverses of primes is: Π : ΣiVQ = lnlnx + K+ O(l/ln x) , where if is a positive constant. LEMMA. 1. 1. // 0 ^ v < 1/2 then Σl/g=Γ χl — v. l/ydy + O(l/lnx),. Jl—v. where the error term is independent of v. The proof of Lemma 1 follows directly from II. LEMMA. 2. If 0 ^v < 1/2 then ;. Λί~. »/?i. Σ. Σ l/ϊift =. 3(1-10/2. r i/2. ri—y. (yzy'dzdy + O(l/ln x) , J(l-<ιO/2 Jy. qx. where qx and q2 run only over primes and the error term is independent of v. Proof. Σ (l-v)/2. x. Σ VQ1Q2 = Σ qλ. χ(l-v)f2. l/ffi (In In (x/q,) - In In q1 + O(l/ln x)). — Σ VQi (ln ln (X/QI) ~ ln I71 Qi) + O(l/ln x) . Now by a well known summation Theorem,2 1. See for example LeVeque, Topics in Number Theory, Vol. 1, Th. 6-20, p. 108. See for example LeVeque, Ibid, Th. 6-15, p. 103, with λn the nth prime, cn = l/λn, and f(y) = lnln(x/y) — Inlny + 0(l/lnx). 2.

(81) THE DISTRIBUTION OF CUBIC AND QUINTIC NON-RESIDUES 7. 79. (Inlnt + K+ 0(l/ln x))ln x dt — v)/2) + Inlnx + K % x)){ln ((1 + v)/(l - v)) + O(l/ln x) . s. If the change of variable t — x is made then Σ Σ. Σ Σ 1/?!?. = 1 2 T?. d-»)/2 s ( l. — δ). Jd-tθ/2. - (In((1 - «)/2) + lnlnx. s ( l — δ). +K x). t>)/2 S ( l — S). + Oilβn x). S. l/2. fl-i/. (yzy'dzdy + O(l/Zw x) . (l-«)/2 Jv For any positive integer r and primes p = 1 (mod 3) let a? = 4(^2py.+y3]# Let <^(αθ = Cj Π {m | 0 < m ^ a?}, i = 0, 1, 2, and j let N(C (x)) be the cardinality of Cj(x). The following is a special case of a general theorem of Vinogradov [4], [5]. LEMMA. 3. N(C\x)) = xβ + O(x/ln x), j = 0,1, 2.. Proof. Formula I with H — x, and n = 0 reads as Σ χ(m) < x 1 - 1 / r + y / 4 r In p ^ (pwir)H(in*pγ+iγir+ipinr 1 +ll4r. =. p" {1+1. = p. 2. ιn. p. r. (ln*p) -lnp. r+1. ^ι\ln p) /ln. p .. In other notation. J^ χ(m) = 0(x/ln x) . Let χ3>:p be the cubic residue character for primes p = 1 (mod 3). By the above there is an absolute constant, Ku such that III:. ^-1 TO = 1. A3,ί. Kxx\ln x ..

(82) 80. JAMES H. JORDAN. Set N(C'(x)) = x/S + TJf j = 0,1, 2. Notice that To = - 2\ - 2Γ,. Let w = cos 2π/3 + i s m 2π/3. It now follows that. Σ j=0. T2)3/2 + ΐ(2\ - Γ2)τ/"3/2. Now by III: | Γx + Γ21 < 2iΓ^/3 Zw £ and | Tλ - T2 \ < 2K&/VT In χm / These inequalities imply that | Tx | and | T2 \ are less than 2K1x/V Ύ lnx» Hence | To \ < AK^jVΎ In x completing the proof of the lemma. THEOREM. 1. Let d be the solution of 1. !. fl/2. Γl-y. Vydy+\ 1-v. (yzY'dzdy. J(l-v)/2 Jy. For all sufficiently large primes p = 1 (mod 3) there is in each class of cubic non-residues modulo p9 a positive integer smaller than p<i-Λ>/4+.β ^ satisfies the inequality .234 < d < .235). Proof. Given ε > 0 let r = [1/ε] + 1. Define x in terms of p as. above and notice that as long as ε < d then. \(i_. for sufficiently large primes p. It therefore suffices to prove that for sufficiently large primes, each class of cubic non-residues contains a positive integer smaller than x1-**-'. Assume that Theorem 1 is false. Then, for some fixed ε > 0, there are infinitely many primes p = 1 (mod 3) such that one of their classes of non-residues fails to contain a positive integer less than x1-*^. Let pλ be one such prime. Notice that x, C°, C1 and C2 are defined in terms of pλ and will therefore be fixed in this argument. Without loss of generality C2 can represent that class of nonresidues modulo p1 that has no positive integers less than xx~a+\ Since C2 has this property it follows that C1 has no positive integer less than n-d+ε)i2 b e c a u s e α i n c1 implies a2 in C 2 . x Since C° is closed under multiplication, modulo pu an integer w in C2(x) must have prime factors not in C°. If w has exactly one.

(83) THE DISTRIBUTION OF CUBIC AND QUINTIC NON-RESIDUES. 81. prime factor, q, not in C° then q must be in C\ If w has exactly two prime factors, q1 and #2, not in C° then both q1 and q2 must be in C°. Further w cannot have more than two prime factors not in C° since the product of any three or more prime factors not in C° exceeds x. A consequence of the above conditions on positive integers in C2(x) is the following upper bound for N(C2(x)):. +. iv:. s+9^^ v^- ? wq^. where the q1 and q2 are taken only over primes β But Vld +. + e)/(l - d)) +. <x(-ln((l-d + \. \. (yxy'dzdy + KJln x. J(l-d)/2 Jy. - d)) + 1/3) + K2x/ln x , where K2 is a constant independent of x. But this inequality can hold only for finitely many primes to be compatible with Lemma 3. 3* Quintic Non-Residues* notations: Let 7X = I. It is helpful to adopt the following. 1/ydy. J (l-o) ri/2. ri-y. h =\. \. r(i-»)/2. {yz)-ιdzdy. ri-ί/. {yz)~ιdzdy. Λ=\ J(l-v)/4 Jl-t -2/. S. (l—1))/3 Γ(l—ΊJ—ϊ/)/2 Π - ί / - 2. \. /5 = \. (yzu)-ιdudzdy. I. (1 — V)/4 Jί/. Jl—V —7/—2!. \. \. {yzu)~ιdudzdy. J ( l - z>)/4 J ( l — » - » ) / 2 J z. I7 = I. \ \. \ \. (yzu)~ιdudzdy \ (yzutyxdtdudzdy .. In the following summation the g^ run only over primes.. Σ = ΣΣ ( l ) / 2. ff.

(84) 82. JAMES H. JORDAN (l-v)/2. x/q1. x. S3= Σ. Σ. (l-v)/4. VQΆ. χl-vjQl. x. (l-v)/3. </χ(l-v)ιQl. x. s*= Σ. xlQlq2. Σ. Σ. ( l ) / 4. l. a.(l-t;)/3. V^T^. a;/?!^. Σ. Σ Σ. y. i. i2. S6= Σ Σ Σ a;(l-t>)/3 ?i 4 _ 3.. S7=. 92 .. Σ Σ Σ a.U-tO/4. ?!. g2. g3. We can now restate LEMMA. 1. 7/ 0 <; v < 1/2 ίfce^ Si = Λ + O(l/ϊn a?), and. LEMMA. 2. If 0 S v < 1/2 £/^w S 2 = 72 + O(l/ln x), and similarly. LEMMA j + 1. If 0 ^v < 1/2 ίfee^ SΛ = /,- + O(l/Zw a?) /or i = 3, 4, 5, 6, 7.. The proofs of Lemmas 4, 5, 6, 7, and 8 are straight forward generalizations of Lemmas 1 and 2 and are much too lengthy to be exhibited here. For any positive integer r and primes p == 1 (mod 5) let x = 5[p (1+1/r)/4 (ί^ 2 p) r+1 /5]. Let Qj(x) = <y n {m 10 < m ^ x}, j = 0,1, 2, 3, 4, and let N(Qj(x)) be the cardinality of Qj(x). The following is another special case of the general Theorem of Vinogradov [4], [5]: LEMMA. 9. N(Q3(x)) - x/5 + 0(x/ln x), j = 0,1, 2, 3, 4.. Proof. Let χ6>p be the quintic residue character for primes p = 1 (mod 5). By the argument in the proof of Lemma 3 there is an absolute constant Kz such that V:. m=l. <KBx/lnx.. Set N(Qj(x)) = x/5 + Tj9 j = 0,1,2,3,4 β Notice that To = - Σ i = Λ Let p — cos 2ττ/5 + i sin 2τr/5. It now follows that.

(85) THE DISTRIBUTION OF CUBIC AND QUINTIC NON-RESIDUES. ΣXsM = Σ(Φ. 83. + T,)pi. j—0. m =l. = Σ Tip*. = Σ Tj(cos 2πj/5 - 1) + i Σ Tό sin 2πj/5 . Now from V it follows that (i) : I (Γ x + T4)(cos 2τr/5 - 1) + (Γ 2 + T3)(cos 4ττ/5 - 1) | < K3x/ln x and (ii) : I (T, - T4) sin 2ττ/5 + (Γ 2 - Γ3) sin 4ττ/5 | < JSΓ8α?/Zw a?. Notice that χ ^ is also a character and by the argument in the proof of Lemma 3 of § 2 there is an absolute constant K4 such that < K^xjln x .. VI: But on the other hand. t. .7=0. =Σ j=0. = Σ. Γ. i (cos. 47r. 3=1. 3=1. Now by VI it follows that (iii) : I (2\ + Γ4)(cos 4τr/5 - 1) (iv): I (Γ x + T4) sin 4τr/5 + (Γ 3 With a little manipulation of I Tj I < Kδxβn x, ^ = 0,1,2,3,4, independent of x, proving Lemma THEOREM sufficiently. 2.. large. i / 5 - 1) + * Σ Γy sin 4τri/5 .. + (Γ 2 + Γ3)(cos 2ττ/5 - 1) | - Γ2) sin 2π/5 | < iΓ4x/ϊ^ x. (i), (ii), (iii), and (iv) one can obtain where K5 is an absolute constant 9.. Let d denote the solution of 1/5 = Σ*=i ^ primes. j ) = l. (mod 5) ίfeere. is. w. quintic non-residues, modulo p, a positive integer p(i+d)/4+e (^ satisfies the inequality .08 < d < .09),. eαc/i c ί α s s. smaller. o/. than. Proof. Given ε > 0 let r — [1/ε] + 1. Define x in terms of p as above and notice as long as ε < d then α;1~d+ε < pt1-*w+β for sufficiently large values of p. It will suffice to prove that for sufficiently large primes p Ξ= 1 (mod 5) that each class of quintic non-residues modulo p contains a positive integer less than ^1~<z+e. Assume that Theorem 2 is false. Then, for some fixed ε > 0, there are infinitely many primes p = 1 (mod 5) such that one of their classes of non-residues fails to contain a positive integer less than x1'^*..

(86) 84. JAMES H. JORDAN. Let px be one such prime. Notice that x, Q°, Q\ Q\ Qs and Q4 are defined in terms of pλ and will therefore be fixed in this argument. Without loss of generality Q4 can represent that class of non1 d+e residues modulo Pi that has no positive integers less than a? ~ . Since 4 1 2 3 Q has this property it follows that Q , Q and Q have no positive (1 d+e)/4 1 3 4 4 integers less than x ~ because a in Q or Q implies α in Q and 2 2 a in Q implies α in Q\ Since Q° is closed under multiplication, modulo plf an integer w 4 in Q (x) must have prime factors not in Q°. One of the following conditions holds depending on the exact number of primes, qiy not in Q° that divide w. ι a+t ( i ) There exists a prime q± such that q1 \ w and x ~ <; gx ^ x, since QΊ is in Q\x). (ii) There exist primes q1 and g2 such that qxq2 \ w and (iii) There exist primes qu q2 and g3 such that qλq2q3 \ w and ι-a+ε is in Q4(x). χ < qiq2q3 </ x > s i n c e qiq2q3 (iv) There exists primes qu q2, g3, and g4 such that q^q^q^ \ w and x1-^' < q^q^A < %, since q&iq&t is in Q4(α?).. It should be noticed that w cannot have more than four prime divisors which are not in Q° since the product of any five or more primes not in Q° would exceed x. The number of w's that could possibly satisfy (i) is less than or equal to Σ. [*/?il. a i-ci+ε. The number of w's that could possibly satisfy (ii) is less than x(S2 + S3). The number of w's that could possibly satisfy (iii) is less than x(S4 + S5 + S6). The number of w's that could possibly satisfy (iv) is less than xS7. Combining the above we have. N(Q\x))< ff. [/ΪJ. l-d+ε. Σ i=2 7. χ. Σ l-d. [aj/</J + x Σ St il. x. - ε/(l — d)) + 1/5) + Kδx/ln x , where K6 is a constant independent of x. But this inequality can hold only for finitely many primes to be compatible with Lemma 9. 4* Remarks* The techniques of the previous sections generalize for Kth. power non-residues when if is a prime. In these cases the definition of d involves (K2 — 3K + 4)/2 integrals ranging from multi-.

(87) THE DISTRIBUTION OF CUBIC AND QUINTIC NON-RESIDUES. 85. plicity 1 through K—l. There are K — 1 possible divisibility conditions κ k imposed on the elements of A ~\x). The upper bound for N{A ~\x)) 2 involves (K — 2K + 4)/2 summations ranging from multiplicity 1 through K—l. The contradiction is reached in the same manner. The details are lengthy but straightforward. For example for seventh power residues the results of Davenport and Erdos imply an exponent of p equal to 959/3840. While using methods exhibited in § 3 one obtains an exponent smaller than 25/104. When K is composite the job is more difficult since the subgroup of ifth power residues and the K—l cosets form a cyclic group of composite order. These cyclic groups have proper subgroups. The "without loss of generality" comment is no longer valid and some arguments concerning the number of prime factors of K must be called upon. The author intends to present these techniques at a future date. REFERENCES 1. D. A. Burgess, On character sums and primitive roots, London Math. J. 12, No. 45, (1962), 179-192. 2. N. G. de Bruijn, On the number of positive integers < x and free of prime factors < y, Indagationes Mathematicae, 13, No. 1, (1951), 50-60. 3. H. Davenport, and P. Erdos, The distribution of quadratic and higher residues, Publications Mathematicae, 2: (1952), 252-265. 4. I. M. Vinogradov, Sur la distribution des residues et des non-residus des puissances, Journal Physico-Mathematical Society University Perm, No. 1 (1918), 94-96. 5. , On a general theorem concerning the distribution of the residues and non-residues of powers, Trans. Amer. Math. Soc. 29 (1927), 209-217..

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(89) PACIFIC JOURNAL O F MATHEMATICS Vol. 16, No. 1, 1966. CONVEXITY WITH RESPECT TO EULER-LAGRANGE DIFFERENTIAL OPERATORS J.. COLBY. KBGLEY. This paper is concerned with the problem of characterizing sub-(L) functions, where L is the Euler-Lagrange operator for the functional ICd[y] — \. 3. 2. Σpj(.D 'y). , with n a positive integer,. [c, d] a subinterval of a fixed interval [a, b], and p0, pί9 - 9pn continuous real-valued functions on [α, 6] with pn(x) > 0 on this interval. Under certain hypotheses on the operator L, it is shown that if / is a function in the domain of L on a subinterval [c9 d] of [a, b]9 then the statement that / is sub-(L) on [c9 d] is equivalent to each of the following conditions: (i) (-l)nLf(x) ^ 0 on [c, d] (ii) Icd[y] ^ Icd[f] whenever y is a function having continuous derivatives of the first n — 1 orders with Dn~λy having a piecewise continuous derivative on [c, d] such that Dj~xy and Dj~xf have the same value at c and at d for j in {1, •••,%}, and y(x) — f(x) ^ 0 on [c, d]. Section 2 is devoted to the definition and equivalent formulizations. of Euler-Lagrange operators and to a discussion of adjoint operators. In § 3 it is shown that, under a hypothesis which is equivalent to the operator L being nonoscillatory on [α, 6], L admits a factorization of the form ( — l)nL$L0, where Loy = Yΰ^TjD'y for suitable r0, r19 , rn. Under the further hypothesis that the operator LQ possesses the "property W" of Polya [3], it is established that L can be written as a composition of first order real linear operators. In § 4, the analogue of Polya's mean-value theorem in [3] is obtained for L under the above hypotheses on L and Lo. This theorem is used in §§ 5 and 6 to give characterizations, which are generalizations of results of Bonsall [1] and Reid [5] on convexity with respect to second order operators, of sub-(L) functions in terms of the operator L and the functional Ied, as well as a theorem on the constancy of sign of the Green's function for a certain incompatible boundary-value problem. Finally, in § 7, it is proved under the above assumptions on L and Lo that the null-space of L is a 2^-parameter family in the sense of Tornheim [7], although the relationship between sub-(L) functions and Received July 24, 1964. The author wishes to express his sincere appreciation to Professor W. T. Reid for his valuable suggestions and assistance throughout the preparation of this Ph.D. dissertation. (State University of Iowa, August, 1964). The research for this paper was performed while the author was a Research Assistant, supported by the Air Force Office of Scientific Research under grant AF-AFOSR438-63. 87.

(90) 88. J. COLBY KEGLEY. functions which are convex with respect to this family remains undecided. Matrix notation will be used throughout; in particular, a vector is a matrix having one column. If M is a matrix, then M* denotes its transpose. If M is a symmetric matrix, (he., M = M*), then ikf is nonnegative (M^ 0) if and only if ψMη is a nonnegative real number for all admissible vectors η. The symbol Eh is used to denote the k x k identity matrix, 0 is used to denote the zero matrix of arbitrary dimensions, and, for j in {1, , n), ej denotes the vector [δ<i]*=1. If M is a function matrix, (i.e», a matrix of real functions), then M is said to be continuous, differentiate, etc., whenever each of its elements has the corresponding property. If M is a differentiate function matrix, then DM denotes the matrix of derivatives of the elements of M. All functions appearing are real-valued. In particular, if A; is a nonnegative integer and [c, d] is a subinterval of [α, 6], then Ck[cf d] denotes the class of all real-valued functions which have continuous derivatives of the first k orders on [c, d]. The symbol zΓ[c, d] will stand for the class of all functions w in Cn~\c, d] for which On~xw has a piece wise continuous derivative on [c, d], and Δ$[c, d] is the class of all those functions w in An[c, d] such that D'^wic) — 0 = D5~ιw{d) for j in {1, •••,%}. Also, Rk denotes the class of all fc-tuples of real numbers. Finally, if / is an integrable function and c is a point in its domain, then \ / denotes the function whose value at x is \ / . Jc. Jc. 2. Properties of differential operators* Let [α, b] be a nondegenerate compact interval and, for each a and each β in {0,1, , n}, let faβ be a continuous real-valued function on [α, &]„ The first problem of this section is to examine the form of the Euler-Lagrange operator L which corresponds to the functional Ied, where [c, d] is a subinterval of [α, b], defined on An[c, d] by. (2.1). Uy] = ΓΓ Σ faβDavD*yλ . JcLα>,/3=0. J. By definition, a function y belongs to the domain of L on a subinterval [c, d] if and only if y e Cn[c, d] and there exists a function φ[y] in C°[c, d] such that for every w in Δl\c, d], the relation (2.2). ί T Σ faβD"yD*w] = \*φ[y]w. holds. In this case, φ[y] is uniquely determined, and Ly is defined to n be ( — l) φ[y]. The following result gives an explicit form for the operator L..

(91) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 89. 2.1. If L is the Euler-Lagrange operator for the functional defined by (2.1), then y belongs to the domain of L on a subn interval [c, d] of [α, b] if and only if ye C [c, d] and there exist functions μ^y],m fμn[y] in C\c, d] such that THEOREM. (2.3) μi-i[y] = Σ/»i-i°*ϊ - Dμly] ,. , n} .. i in {2,. In this case, Ly = {-lY^ίϋμM. - Σ fc,oDay) ,. that is,. Ly = D(D(. D(D(± fanD«y) - Σ Ln-^l n. \. n. First, if y is in the domain of L on [c, d], then y satisfies (2.2) with φ[y] = { — l)nLy. Let ply], p^y], - , pn[y] be the functions defined recursively by po[y] = Σ f*J)ay. <2.4). - φ\v\. r Ply] = Σ faiD'y cύ=0. f. i in {1,. pi-M ,. , n) .. Jc. Then, for every w in Δl\c, d] and each Jk in {1,. , ri\,. ( T i t Σ faβDayD?w) + lO^Ji/lί?*-1^] = 0 . In particular, \ί h — n then \ = 0,. S. d. c. pn[y]Dnw. — 0.. Since w is. an arbitrary member of Δl\c, d], the fundamental lemma of the calculus of variations implies there is a polynomial function Qn^ of degree at most n — 1 such that pn[y] = Qw_i. If Qn-i-j denotes the j t h derivative of Qn^ for y in {1, , n — 1}, then, for i in {1, , n} let μ{[y] be.

(92) 90. J. COLBY KEGLEY. Then a. μάv] = pΛy] + ί P«ΛV\ = Σ fanD y , Jo. and, for i in {2,. o>=0. , %}, .D^JIJ/] exists, is continuous, and. Thus, the relations (2.3) hold, and, since Qo is a constant function, n cύ=0. SO. (. \. n. Df*ι[y] — Yxfa*Day) Conversely, suppose y e C \cy d] and there exist functions μ^y], μn[y] in Cλ[c, d] satisfying (2.3). If n. ,. φ[y] = then, for any w in J?[c, d], Σ ΛβD β i/Z)^l = \Λ\μJίv]D w + Σ Φ^+1[2/] + μA.v\)B*n + Φμλy] + <plv\)w\ JcL. β=i. J. Σ Hence, y is in the domain of L with +1. α. Ly = (-l)VM = ( - i r ( ^ i M - Σ /»o-D2/) . Since the coefficients faβ are only assumed to be continuous, L is in general not a 2wth order differential operator in the classical sense but is an example of what has come to be known as a "quasidifferential operator". However, if the "leading coefficient" fnn vanishes at no point of [a, b], then the equation Ly — φ is equivalent to a first order 2^-dimensional vector system..

(93) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 2.2. Suppose fnn(x) n x n matrices THEOREM. Φ 0 on [a, δ], A and B are the. 0. 0 •. 0 f. 0 ___ Jon/Jnn. Jln/Jnn *. 91. Jn—lnlJnn __. 0 0. 0 l/Λ.. respectively, F is the n x n matrix Λ . . . 0 f If v. 1. J nOlJ nn. JnllJnn n—i. _. #. J nn—llJ nn. and C is the n x n matrix whose element in the ith row and jth column is f^U-i — (/«<-i/j-m)//»* Then Ly = φ if and only if u = [-Di"1l/]?=i, v = [μi[y]]t=i is a solution of (2.5) Du = Au + Bv , Dv = Cu + Fv + {Moreover, if fnn{x) > 0 on [a, b], then the matrix B(x) ^ 0 on [α, δ], and if the matrix [faβ\Z=o β=o is symmetric then so is the matrix C, and (2.5) becomes (2.6). Du = Au + Bv ,. Dv = Cu - A*v + ( -. The first part of the theorem follows immediately from Theorem 2.1, particularly the fact that the functions μ^y], •• •,/*«[#] are determined uniquely by (2.3) for a given y in the domain of L. The last statement of the theorem is obvious from the definitions of the matrices involved. We shall be concerned in particular with the homogeneous vector systems (2.5'). Du == Au + Bv ,. Dv = Cu + Fv ,. (2.6'). Du^ Au + Bv ,. Dv — C u — A*v .. and the related matrix systems (2.5"). DU = AU+ BV ,. (2.6"). DU = AU+ BV ,. For convenience, if each of U and V is an n x r function matrix, then (U; V) will stand for the 2n x r function matrix whose ith column consists of the functions U13 , , Un3, V13, , Vn3..

(94) 92. J. COLBY KEGLEY. The following property of the system (2.5') will be especially important for discussing the oscillation properties of L in the case that fn%(χ) > 0 and the matrix [/αβ] is symmetric. 2.3. The system (2.5') is identically normal on [a, δ], that is, if (u; v) is a function vector which satisfies (2.5') and there is a nondegenerate subinterval I of [a, b] on which u vanishes identically, then both u and v vanish identically on [a, δ]. THEOREM. If (u; v) satisfies (2.5') with u(x) = 0 on a nondegenerate subinterval I of [a, b], then the relations vn = fnnDun. +. 71-1. Σ. Vi-l = Σ fai-lUa + 1 + fni-lDUn — D ^ ,. ί in. {2,. cύ=0. imply that v(sc) Ξ= 0 on / and, therefore, both u and v must vanish identically on all of [α, 6]. Indeed, if (u; v) is a solution of (2.5') with u±(x) = 0 on a nondegenerate subinterval I of [α, δ], then the first n — 1 component equations of (2.5') imply that u(x) = 0 on /, so u and v vanish identically on I. Thus, in view of the results of Theorems 2.2 and 2.3, together with the elementary existence and uniqueness theorems for first-order vector differential equations, it follows that if fnn(x) Φ 0 on [α, δ] then the null-space of L has a basis of 2n linearly independent functions, so that L deserves to be called a "2wth order operator". We conclude this section with the well-known formulization of the adjoint Lo* of an operator Lo which is defined by (2.7). Loy =. Σ 3=0. where the coefficients r0, rlf , rn are continuous real-valued functions on [α, δ]. By definition, a function z belongs to the domain of L* on a subinterval [c, d] of [α, δ] if and only if z e C°[c, d] and there exists a function φ[z] in C°[c, d] such that, for every w in Δt[c, d],. S. d. c. Cd. ZLOW — \ φ[z]w . Je. In this case, φ[z] is unique, and Ltz is defined to be φ[z\. Using much the same integration-by-parts technique, and subsequent application of the fundamental lemma of the calculus of variations as in the proof of Theorem 2.1, we find that z belongs to the domain of Lo* on [c, d] if and only if z e C°[c, d] and there exist functions vx[z\y , vn[z] in C\c, d] such that.

(95) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. (2.8). i in {2,. 93. ,n} ,. in which case Lo*£ = roz — It is easily verified that if rn(x) Φ 0 on [α, b] and G is the n x n function matrix 0 E. G =. 0 _ -ro/rn. -rn_Jrn „. -rjrn. then Loτ/ = / if and only if there exists a function vector u = [t6<]?=1 such that Z)u = Gu + (f/rn)en and 2/ = ^ x , and Lo*^ = 0 if and only if there exists a function vector v = [/yί]Γ=i such that Dv = — G*v — ge1 with 2 = <vn/rn. 3* Factorization of Euler-Lagrange operators* In this section we shall consider a particular functional of the form (2.1) which is given by. IJy] =. (3.1). Σ c Lj=o. where p0, plf , p % are continuous real-valued functions on [α, 6] with P*(#) > 0 on this interval, and [c, cZ] is a subinterval of [α, 6]. We then have the following special case of results of § 2. 3.1. If L is the Euler-Lagrange operator for the functional Icd given by (3.1), then a function y belongs to the domain of L on a subinterval [c, d] of [α, b] if and only if ye Cn[c, d] and there exist functions μ^y], •••, μn[y] in Cx[c, d] such that THEOREM. n. (3.2). μlv] = PnD v, μi-iiv] = Pi-ιΌ{-ιy - Dμly] ,. In this case Ly = ( — l)n+1(DμXy] Ly = D(D(.. D(D(pnDny). i in {2,. — pQy), that is, - p^D^y). .). Moreover, the equation Ly = φ is equivalent mation ut = D'-'. , n) .. i in {1, i in {1,. under the transfor-. , n} , , n).

(96) J. COLBY KEGLEY. 94. to the vector system (3.3). Du = An + Bv ,. Dv = Cu - A*v +. where 0. Po. 0 0 0 ••• 0. 0. Pn-l. In particular, transformation. the equation Ly = 0 is equivalent under the above to the identically normal system. (3.3'). Du = Au + Bv ,. Dv = Cu — A*v .. As was indicated in § 2, we shall also make use of the related matrix equation (3.3"). DU = AU + BV ,. ΰ 7 - CU - A*V .. In particular, consider the following condition: (Hi). There exist solutions y19 of Ly — 0 such that if 9yn U= [D^yjll^ t, and V= [μlyj]]^^ then U*(x)V(x) - V*{x)U(x) = 0 on [α, 6] and U(x) is nonsingular on [α, &]. Since the matrix (U; V) based on the matrices U and V of (H:) is a solution of (3.3"), U*V — F*Z7 is a constant function matrix, and the particular condition that this constant matrix be the zero matrix is what has been termed the condition that (U; V) be a "select solution" of (3.3"), or that the columns of (U V) be "mutually conjugate" or "conjoined" solutions of (3.3'), (see, e.g., Reid [4]). Hypothesis (Hx) has an important bearing on conditions of oscillation involving L and on the variational behavior of the functional Icd. At the present, however, we are concerned with the following property of L. THEOREM 3.2. // (H^ holds, then there exist continuous realvalued functions r0, ru , rn on [α, b] with rn(x) > 0 on this interval such that if Lo is the nth order differential operator defined by. (3.4). LoV = Σ. and L is the Euler-Lagrange operator for the functional (3.1), then.

(97) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. Moreover, the functions the null-space of Lo.. y19 * ',yn. specified. in (H^ form. 95. a basis. for. It is useful for the proof of this theorem to introduce the following notation. Let R be the n x n matrix 0. let P—C, and let ω be the function defined on [α, b] x Rn x Rn by the formula 2ω(x, σ, τ) = τ*R(x)τ + σ*P(x)σ. Then L is also the EulerLagrange operator for the functional 2ω(x, Ύ](x), Dη(x))dx subject to the restraints DVi = Vi+i f. i w. {1,. , n — 1} .. Now, if U and V are as in (HO, and, for a subinterval [c, d] of [α, 6], y e Cn[c, d] and w e zlw[c, d], then with. we have 1. # }. This identity is essentially formula (5.3) in Reid [6]. Since ϋD[I7-γ] = UiDU-1)^ + Dη* and the matrix U is independent of both y and w, as is also the matrix R, it follows that there exist continuous functions r 0 , n , , rn independent of y and w such that if Lo is defined by (3.4), then hf]) = (LQy)(Low) .. (3.6) 2. In particular, r w = pj/ , so rn(a?) > 0 on [α, δ]. If w also belongs to Δl{c, d], then (3.5) and (3.6) imply that (3.7).

(98) 96. J. COLBY KEGLEY. Theorem 2.1, with faβ — rarβ, and the remarks at the end of §2 concerning the adjoint Lo* of an operator LQ of the form (3.4), show that ( — ϊ)nL*LQ is the Euler-Lagrange operator for the functional given by Γd. \ (L0y)2 on An[c,d] and that y belongs to the domain of ( — l)nLίL0. if. Jc. and only if y e Cn[c, d] and \(LQy)(LQw)] = c. \d[Lί(L,y)w] Jc. whenever w e AH[c,d]o On the other hand, the left-hand member of (3.7). S. d. c. Σj=QPjD3yDjWo. These remarks together with the definition. of L show that a function y in Cn[c, d] belongs to the domain of L if and only if it belongs to the domain of ( — l)nL$L0 and, in this, case, Ly = ( — ΐ)nLfLoy. Finally, if y is one of the functions y19' ,yn specified in (H^) and rf = [J5<"1i/]Γ=i, then £/~γ i s constant and (3.6) implies that LQy = 0. The linear independence of {yly , yn) follows from the assumption that U(x) is nonsingular on [α, 6]. In [3], Poly a showed that, under a certain hypothesis which he called "property W", the operator Lo can be written as a composition of first order operators. We shall show that, under this same hypothesis, the operator Lo* can also be written in this form, and, therefore, so can L if the additional hypothesis (Hx) holds. The "property W" of Polya shall be referred to in this paper as: (H 2 ). There Wk denotes the. exist solutions Wronskian. y19. * ,yn. of. Loy = 0 such. that. if. W(yίf ••-,»*) = det [D<-1yJ]ii1 A ,. (3.8). then Wk(x) > 0 on [α, b] for each k in {1,. , n}.. It should be noticed that if hypotheses (Hx) and (H2) were always to be applied simultaneously, then one could assume without loss of generality that the functions yu , yn specified in (Hx) also satisfied the condition on the corresponding Wronskians which is stated in (H 2 ). This follows directly from the last statement of Theorem 3.2 and the identical normality of (3.3'). However, we shall be interested in certain statements which are true under (H2) alone. The following known property of Wronskians is stated here for convenience. If each of fl9 ,fkff belongs to Ck[a, 6], and W(f19 vanishes at no point of [α, 6], then LEMMA.. ,fk).

(99) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. D[W(fu •••,/*-!,/)/TΓ(Λ, •••,/*-!,/*)] = W(f19 -, Λ_0 TΓ(Λ, - , Λ_χ, Λ, /)/[ TΓ(Λ,. 97. , Λ_χ, Λ ) ] 2 .. This equality is most easily seen by noting that each side is the value at / of a kth order linear differential operator whose null-space has {/i, •••,/*.} as a basis. Hence, the two expressions must be proportional, and examination of the leading coefficients shows that the expressions are, in fact, identical. THEOREM 3.3. If (H2) holds, then there exist positive functions τr0, π19 , πn with π3 in Cn~'[a, b] for j in {0, 1, , n} such that if Γj and Jj are the operators defined recursively by:. Γoz = πnz , (3.9). Aoy = (l/πo)y ,. Γj-z = π^jDΓj^z. ,. j in {19 - ,n — 1} , A3y = π3DA3_λy ,. L o = An and Lo* = JΓW. It is to be emphasized that a real-valued function / belongs to the domain of Γ3 (respectively, A3 ) on a subinterval [c, d] of [α, b] if and only if / is continuous on [c, d] and if j e { l , • β , / ^}, then Γj^f (respectively, A^J) is in C\c, d]. By a theorem of Polya [3], if Wo = 1, ^ is as specified in (H2) for k in {1, . . . , n}, π0 = Wu πά = W!/(W^Wi+1) for j in {1, , n - 1}, and 7ϋn = rnWJWn^.l9 then L0 = An. Furthermore, since each yk appearing in (H 2 ) is necessarily in Cn[a, δ], it follows t h a t each π3 is in Cn~j[a, δ], and there exist continuous functions pi3ii in {0,1, β ,w}, y in {0,1, , n}, such that (3.10). A3y = ± p^Dhj ,. for j in {0, 1,. , n} .. Moreover, pj3 = W3 /W3+1 for j in {0,1, , n — 1} and pnn = r Λ , so that Pu(x) > 0 on [α, δ] for i in {0,l, ,w}. This implies that a function tt; is in JJ[c, cί] for a subinterval [c, d] of [α, δ] if and only n if w e A [c, d] and A3w vanishes at c and at d for i in {0,1, , n — 1}. As to the factorization (3.9) of Lo*, notice that if z is in the domain of Γn on a subinterval [c, d] of [α, δ] and weJJ[^,(ί], then repeated use of integration by parts and the fact that A3 is of the. S. d. C. Γd. Γd. zLQw — 1 zAnw — \ wΓnz. J C. Hence, by definition of. J C. Lo*, z belongs to the domain of L* on [c, d], and Γnz = L£z. In particular, since Γn is clearly a linear operator, the null-space of Γn has dimension at most n. On the other hand, suppose that, for k in {1, β ,^},.

(100) 98. J. COLBY KEGLEY. (3.11). zk = W(y». , yk-» yk+ly. , yn)/(rn Wn) .. Then {zu * ,zn} is a basis for the null-space of Lo*. For the discussion of adjoints in § 2 shows that LQy = 0 and L*z = 0 are, respectively, equivalent to vector systems (3.12) (3.13). Dv = - G * v .. But if F is the function matrix [D^y^Zijlu then Y is a fundamental matrix for (3.12), by choice of yu - *,yn. Hence, the matrix Y*" 1 is a fundamental matrix for (3.13). It follows that the elements in the last row of Y*"1J each multiplied by l/r Λ , form a basis for the nullspace of Lo*. But these elements, after a proper choice of sign, are just the functions zk. Now, zn = ^ / ( r . W J - l/πn, so D ( τ τ A ) = 0 and Γnzn = 0. For k in {1, , n — 1} and j in {0,1, , n — (k + 1)}, it will be shown by induction on j that Γάzk is defined, and Γβh. = W(l/i,. , Vlc-l, yic + 1,. ' , 1/—i)/ Wn-i-l. For the case j = 0, Γozk = πw2Λ = W ^ , , ?/&_!, 2/ft+1, , yn)/ Wn_u since TΓ^ = rnWJWn-!. Assume the result holds for some index j in {0,1, , n - k - 2}. Then , #*_!, 2/Λ+1,. = W(yu. , 3/n_i)/ W(ylf. , T/^!, ?/&+1,. , yn^-u yk). Since both Wronskians appearing have at least one derivative, so does ΓjZk, and by the above lemma, u Vk-l,. ,y k _» yk+1,. Vjc+1, -•',. ,yn^ί9yk)]*. Vn-j-l). yk, Vn-i). Therefore, DΓ^fc = W(yl9 and then Γj+1zk =. π^^DΓjZj,. which completes the induction. In particular, Γ%_k_xzk = Wd/!, so. --,2/^1, yk+i)/Wk ,.

(101) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 99. Thus Γn_kzk = πkΌΓn_k_λzk = 1, so DΓn_kzk = 0 and Γnzk = 0. This shows that {zu , zn} is not only a subset of the domain of Γn, but, by the previous remarks, it is also a basis for the null-space of Γn. Now, suppose z is in the domain of Z/o* on [c, d\. The form of the operator Γn clearly implies the existence of a function zQ such that Γnz0 — L$z. But then z0 is in the domain of L£ on [c, d] as well, and Γnz0 = L*z0, so L*(z — z0) = 0. Therefore there exist constants cu * ,c w such that z — 20 = Σ*=i C Λ> so 2 is a linear combination of elements in the domain of Γn and must therefore be in the domain of Γn. Moreover, Γnz = Γ%20 + Σϊ=i GkΓnzk = Lo*2. Thus, the operators Lo* and / ^ are identical. Throughout the remainder of this discussion, Icd will denote the functional given by (3.1) for which L will he the Euler-Lagrange operator with the corresponding operators Lo and L* as in Theorem 3.2 and Γά and Aά defined by (3.9). 4* A mean-value theorem* In this section, theorems analogous to Polya's Theorem I, II, III of [3] are obtained under hypotheses (HO and (H2) for the operator L. For these theorems and certain preliminary results, we shall adopt the following terminology: if X is a finite set of real numbers, then a number x is said to be intermediate with respect to X if and only if x lies in the interior of the smallest compact interval containing X, unless X i s a one-point set {$}, in which case the only intermediate point is defined to be the point x. The first result is an analogue of Polya's Theorem I for the operator Lo*. THEOREM 4.1. Under hypothesis (H2), if z is in the domain of I/o* on a subinterval I of [α, b] and one of the following conditions holds: (i) z vanishes at n + 1 points tx<t2< < tn+1 of I, (ii) z vanishes at n points tλ < t2 < < tn of I and there is a j in {1, , n} for which D(rnz)(tj) = 0, then there is a point t intermediate with respect to the set {ίj such that Lfz(t) = 0.. Notice that no additional condition of differentiability of the function z has been asserted in (ii), since rnz has a continuous derivative whenever z is in the domain of Lf, (see 2.8). In case (i), it will be shown by induction that for every j in {0,1, , n} there exist n — j + 1 points t{ < t{ < < t{_j+1 in [tu tn+1] Bit which ΓjZ(t§ = 0. The assertion for j = 0 is just the condition (i). If the statement is true for some j in {0,1, ,n — 1}, then, by Rolle's theorem, for each i in {1, •• , ^ — j} there is a point +1 1 +1 t{ in (tl ti+1) such that DΓfltyi* ) = 0. Hence Γj+1z(t{ ) = 0 for i.

(102) 100. J. COLBY KEGLEY. in {1, . . . , n _ j}y a n d t i < < ίi < ^ ί T h u g t h e i n duction is complete, and, in particular, there is a point t which lies in (tu ί Λ+t ) at which £Q*2(ί) = Γ β «(ί) = 0. In case (ii), Rolle's theorem implies that for each i in {1, -,j - 1} there exists a point t\ in (ί i f t < + 1 ), and for each i in {i, . . . , n - 1} there exists a point t}+ 1 in (tif t < + 1 ), such that />(*}) = 0, i in {1, , j -. 1}, and Λs(tί + 1 ) = 0, i in {i, . . . , n - 1}. But. and WJWn^. has a derivative, so. since z also vanishes at ί,, . Hence,, if t)) = <y,, then then the the n n points points y ίϊ < ίί ί f ίϊ < ίί < < ίί of (ί1? ίn) satisfy Γ^(^) = 0 for i in {1, . , n}. The same inductive process used in the proof of (i) then gives the existence of a number t intermediate with respect to {tlf ••-,£} such that Lfz(t) = 0. Theorem 4.1, together with results of § 3, result in the following analogue oϊ Polya's Theorem 1 of [3] for the operator L. THEOREM 4.2β // (HJ and (H2) hold, y is in the domain of L on a snbinterval I of [a, 6], x, and x2 are points of I with x1 < χ2, and there is a point x0 of I different from x1 and x2 such that y{%0) = 0, white v sαtis^es tfie. (4.1). Dt-'yfa). = 0 = D*-ιy{x2) ,. is a point which Ly{t) = 0.. t intermediate. with. respect to {xQ, χu χ2} at. An induction argument will show that for each k in {1, « , n — 1} there exist points s* < s* < . . . < skk+1 which are all different from x, and x2 and lie in (xl9 x2), (χl9 χ0), or (xOi χ2), depending as x1 < xQ < a?2, α;2 < α?0, or α?0 < xί9 such that ^^(s^) = 0 for i in {1, , k + 1}, and Av&i) = 0 = Λj/(α2). First, the statement that J^(xO = 0 = Aky(x2) for fc in {0,1, ,w-l} follows from the fact that A # is of the form (3.10) and the hypothesis that y satisfies (4.1). Since Ay(x0) = (l/πo)y(xo) = 0, and a?0 is different from xt and a?2, an application of Rolle's theorem gives the assertion when k = 1. If the statement is true for some k in {1, , n — 2}, +1 fe+1 then points s? < s2 < . . . < stχ\ are chosen as follows. If xx < x0 < a?2> then the points sf, s2*, •• ,8Ϊ+i are in ( ^ a?2) and, by Rolle's theorem, +1 +1 choose sί in (xu βf), s{+i in (βj +1 , α;2), and s? in (sU, α?f), for i in {2, . . . , & + 1}, such that DJky(sf^) = 0 for * in { 1 , . . . , * + 2}. If, on the.

(103) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 101. other hand, x2 < Xo, then there is an index q such that Sq < x2 < sj + 1 while xλ < a* for i in {1, ,fc+ 1}. Therefore, choose sί + 1 in (xly sf), +1 k βj in (sti, sj) for ί in {2, , q}, s qχ\ in (sj, a?2), sJίJ in (&2, sj+1), and k+1 s in (st 2 , sj_i) for i in to + 3, , Λ + 2} such that DAky(sl+1) = 0, ΐ in {1, , k + 2}. A similar method of choice gives the values s^+1 in +1. case x1 lies between xQ and α?2. But then Ak+1y(sϊ ). k+1. ~ (πk+1DJky)(s ). =0. for i in {1, , k + 2}, and the induction is complete. In particular, there are points s?" 1 < sf-1 < < s^"1 different from xx and ίc2 at which An_1y vanishes, and, as the above construction shows, these points are also intermediate with respect to {x0, xu x2}. But An_λy also vanishes at xx and x2 so, applying Rollers theorem once more, there exist points tλ < t2 < < tn+ί different from xx and x2 at which Loy(U) = J ^ ( ^ ) = {π^A^y)^) = 0. By Theorem 4.1 there is a point t intermediate with respect to {tu β ,ί %+1 }, (hence, with respect to {x0, xu x2}), at which Ly(t) = ( — l)nL£(Loy)(t) = 0. Before continuing with the development of this section, we introduce an important property of the operator L. Since the equation Ly — 0 is equivalent to the identically normal system (3.3') in which the matrix B(x) is nonnegative on [α, 6], it follows, (see Theorem 5.2 of Reid [6]), that a necessary and sufficient condition for hypothesis (H^ to hold is that L be nonoscillatory on [α, 5], that is, if a g x1 < x2 fg &, then the boundary-value problem (4. '2). D'-Wx,) = 0 - D^y(x2) ,. j in {1, . . . , n} ,. is incompatible, i.e., has the function which vanishes identically on [^!,x2] as its only solution. The equivalence of Ly = 0 to (3.3') then implies that (HO is also equivalent to the statement that if {xuy\,yl, -- ,2/Γ) and (x2, y\, y\, , yΐ) are points of Rn+1 with α ^ xx < x2 ^ b and ^ G C 0 ^ , ^ ] , then there exists a unique solution of the nonhomogeneous boundary-value problem LV. (4.2'). D^vix,). =. Ψ. ' = y{,. i in {1, 2}, j. in{l,-..,n}.. This enables us to formulate the following extension of Polya's meanvalue theorem, the proof of which is identical to that of Polya. THEOREM 4.3. Suppose (Hj) and (H2) hold, f is a function in the domain of L on a subinterval I of [a, b], xL and x2 are points of I with x1 < xz, and y12 denotes the solution of. (4.3). LV. D'-WJ. =° ' D i W ). i i n { 1 , 2}, j i n {1, • - • , % } ..

(104) 102. J. COLBY KEGLEY. If hl2 denotes the solution of (4.4). *» =. X. '. D'-'yfc) = 0 ,. i m {1, 2}, j m {1, . . , n) ,. then for each point x in I, there is a point tx in I such that (M) f(x) = y12(x) + K(x)Lf(tx) . If x = x1 or x = cc2, then (M) holds for any choice of tx. If x e I and x is different from a^ and x2, then Λ12(a;) ^ 0 by Theorem 4.2, so there is a (unique) number cβ such that f(x) — y12{x) + h12(x)cx. Let ό1^ denote the function / — y12 — cxhl2. Then Θx is in the domain of L,Ds-1θΛ(xi) = 0 for i in {1,2}, j in {1, « ,^}, and 0β(α) = 0. By Theorem 4.2, there is a point ί s intermediate with respect to {x,x19 x2} at which Lθx(tx) = 0. But L0 β = I// — Lτ/12 — c β Lλ la = L/ — cxΛ , so Ca. = Lf(tx) and (M) follows. It was noted that the solution h12 of (4.4) does not vanish in [α, b] except at xx and at x2. We now determine exactly what the sign of h12 is on (xu x2) and on the union of [<z, x±) and (x2, b\. THEOREM 4.4. Under hypotheses (H^ and (H2), if h12 is the solution of (4.4), then. (-l)nh12(x). > 0,. A12(α?) > 0 ,. if x1 < x < x2 , if a S x < xλ or x2 < x ^ b .. Fix x1 and x2 in [α, b] with si^ < α?a, and suppose z — L0h12. As in the proof of Theorem 4.2, one obtains by use of Rollers theorem a set of n points # < £ ? < • • • < £ ? in (xu x2) such that z(tϊ) = 0 for Λ in {1, •• ,w}. Applying Rollers theorem as in the proof of Theorem 4.1, for each j in {1, , n} there exist n—j + 1 points t)<t)< < t]~j+1 +1 such that t) < ίj+i < ί i for i in {1, , n - 1}, k in {1, , n - j}, and Γj^zit)) — 0 for j in {1, , n}, k in {1, , n — j + 1}. If, for example, = η~j+\ then x1 < sn < §,,_! < < sx < ίc2 and Γ^zisj) = 0 for j " Sj in {1, •••,%}. But Lfc12 - (-l) w Lo*^, so L*z = ( - l J Therefore,. In particular, suppose sx< x ^ x2. Then z(x) > 0, because each of the functions πj is positive on [α, δ], and at the jth. stage of the indicated iterative procedure used to calculate z(x) the integral function.

(105) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 103. πol .]l n. J. JJ. is necessarily restricted to an interval with left end-point sn_j+1 and is therefore positive. However, Theorem 4.1 and the fact that L*z = ( — l)n imply that z cannot vanish at any point other than t\,tl, , if, and that z must change sign at each of these points. Since z(x) > 0 on (*Γ, «a] = (su x2], it follows that (-l)nz(x) > 0 on [xx, t\). But sfo) = LJbJpt) = [ΣiUr^h^ix,) = K D ΛJί^) for i in {1, 2}, and r.fo) > 0, n n n so ( — l) D h12(x1) > 0,D hl2(x2) > 0, and the conclusion follows. 5* Sub-(L) functions. We are now prepared to define the notion of a sub-(L) function and to examine some of the properties of functions of this type. Throughout this section, it is assumed that hypotheses (Hi) and (H2) hold. A function / which has derivatives of the first n — 1 orders on a subinterval I of [α, 6] is said to be sub-(L) on I if and only if for every pair of points xx < x2 in I, if y12 is the solution of the boundaryvalue problem (4.3), then f(x) ^ y12(x) on [ccj., x2], and a sub-(L) function / is strictly sub-(L) on I if and only if for every pair of points x1 < x2 in Ijf{x)<y12{x) on (xlfx2). We have the following characterization of sub-(L) functions. THEOREM 5.1. If f is a function in the domain of L on a subinterval I of [α, δ], then f is sub-(L) on I if and only if ( — l)nLf(t) <£ 0 on I. Moreover, if ( — ϊ)nLf(t) < 0 on I, then f is strictly sub-(L) on I.. Suppose ( — l)nLf(t) ^ 0 on 7. Let x1 and x2 be points of / with Xi < x2, let y12 be the solution of (4.3), and let h12 be the solution of (4.4). By Theorem 4.3, if x e I then there is a point tx in / such that (M). fix) - yl2(x) + hn{x)Lf(tx). .. But {-lfLf(tx) g 0 and, by Theorem 4.4, (-l)nh12(x) > 0 on (xu x2), so that if x1 < x < x2 then f(x) ^ y12(x) It is also seen that, since h12(x) > 0 outside the interval [x19 x2], n. (-l) f(x). n. ^ (-l) y12(x). if xel. and x $ [xu x2] .. Conversely, if / is sub-(L) on J, but there is a point t0 of I such n that ( — l) Lf(tQ) > 0, then there is a nondegenerate subinterval [xux2] of / on which (~l)nLf(t) > 0. Applying the mean-value formula (M) on [xu x2], one has f(x) > y12(x) on (xu x2), a contradiction. The last statement of the theorem clearly follows from formula (M). In view of the equivalence of Ly — φ to the nonhomogeneous first.

(106) 104. J. COLBY KEGLEY. order linear system (3.3) and the classical properties of the Green's matrix for the corresponding incompatible first order system which is equivalent to (4.2), it follows that the solution / of Ly = φ which satisfies the boundary conditions (4.1) is given by 2. f(x) = [ g(x, t)φ{t)dt , where the Green's function g is real-valued on [xu x2] x [xu x2] and has the following properties: ( i ) g and the first n partial derivatives with respect to its first argument are continuous. (ii) If i e {2, , n}, then, in the notation of (3.2), the mapping Ti:(x,t)—> f*i[g(t)](x) is continuous on [xu x2] x [a^, cc2] (iii) On each of {(x, t)\ x1 ^ x < t ^ x2} and {(x, t): x1 ^ t < x ^ x2), the mapping Ty\(x,t)^μ&g(t)\{x) is continuous, and if xι<t<x2, + then Ti(ί-, t) - T,{t , t) = (-1)\ (iv) If Xi<t < x2, then on each of the half-open intervals [xu t) and (t, x2] the function μ^git)] has a continuous derivative, and Lg(t) — 0 on each of these intervals; moreover, g(t) satisfies the boundary conditions (4.1). (v) g(x,1) ΞΞ g(t, x) on [xl9 x2] x [xl9 x2]. The following theorem on the Green's function gives a strengthening of the second assertion of Theorem 5.1. THEOREM 5.2. If a rg xx < x2 ^ b and g is the Green's function for the incompatible problem (4.2), then ( — l)ng(x, t) ^ 0 on [xu x2] x yjuu. JU2\.. If not, then, since g is continuous, there is a point (α?0,ί0) in (xί9 x2) x (xί9 x2) such that ( — l)ng(x0910) < 0. Using the fact that g(x, t) — 0 on the boundary of [xu x2] x [xί9 x2], let ίx denote IΛJB{t: xγ^t <t0, g(x0, t) — 0}, and let t2 denote GLB{£: tQ < t ^ x2, g(x0, t) = 0}. Then t1<t0< t2 and the continuity of # implies that ( — l)ng(x0, t) < 0 on (ίj., t2) andflf(α?0,ίi) = 0 = 0(&o, t 2 ). Suppose ψ is the function whose value at t is g(xOy t) for £ in [tu t2] and is zero otherwise. Then φ is continuous, and if / is defined on [xlf x2] by f(x) = \ V(^, t)φ(t)dt , s. then Lf—φ and, since ( —l)*<p(ί) ^ 0 on [asi, cc2], / i sub-(L). But 5 ι D^ffa) = 0 = D ~ f(x2) for j in {1, •••, w}, so, by definition of sub(L) functions, f(x) ^ 0 on [xl9 x2]. On the other hand,.

(107) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 105. which is positive, a contradiction. THEOREM 5.3. If f is in the domain of L on a subinterval I of [a, b] and f is sub-(L) on I, then a necessary and sufficient condition that f fail to be strictly sub-(L) on I is that there be a nondegenerate subinterval of I on which Lf(x) = 0.. If / fails to be strictly sub-(L) on /, then there are points xx < x2 in /such that if y12 is the solution of (4.3), then f(x) S yJix) on [^x,^2] and there is a point x0 in (xu x2) at which f(x0) = y12(x0). If φ = Lf, then φ — L(f — y12), and if g is the Green's function for (4.2), then. S on [xu x2].. x2. g(x, t)φ{t)dt. But then. S. x2. g(x0, t)φ{t)dt. = f(xQ). - yi2(x0). = 0 ,. and, since g(xo,t) and φ do not change sign on [#icc2], it follows that g(x0, t)φ(t) = 0 on.[xux2]. Now, the restriction of g(xo,t) to \xu x0], using the appropriate one-sided limits at xOf is a solution of Ly = 0. Hence, if g(xQ, t) vanishes on some subinterval of [xu x0], then g(xQ, t) vanishes identically on [xly xQ], Since at least the first n — 1 derivatives of the function g(x0, t) are continuous at x0, is follows that on [x0, x2], the function g(xOf t) is a solution of Ly = 0 D'-Wxo) = 0 = D^y(x2). ,. j in {1,. , n} ,. so g(xo,t) = 0 on [xo,#2] ^s well. But then g{x^t) = 0 on [cci,cc2], which violates the discontinuity condition which the function /*i[#(£0)] must satisfy at x0. Correspondingly, the assumption that g(x0, t) vanishes on some subinterval of [x0, x2] leads to a contradiction, so that any subinterval of [x19 x2] contains a point t at which g(xO11) Φ 0, which implies that φ vanishes at this point. Hence, φ is a continuous function whose set of zeroes is dense in [xu x2], so φ(x) = 0 on [xl9 x2]. This, in turn, implies that f(x) = y12(x) on [xu x2] and Lf(x) = 0 on \Xί9 Xϊl. The sufficiency of the condition is obvious. It is to be remarked that the result of Theorem 5.3 is weaker than the result that might be expected for sub-(L) functions. In the classical case where L = D2, any convex function which fails to be.

(108) 106. J. COLBY KEGLEY 2. strictly convex must be a solution of D y — 0 on some interval of its domain, and in [5] Reid generalized this statement exactly for a secondorder Euler-Lagrange operator. However, for higher-order operators a generalization stronger than the above theorem is not immediately apparent. 6* Variational properties of sub-(L) function* In addition to the classes Δn[c,d], Δ%[c,d], we shall be concerned with the class Δ%[c,d] consisting of those functions w in Δ%[c,d] for which w(x) ^ 0 on [c,d]. If M is any real linear functional on any of these three classes, then M is positive definite if and only if M[w] ^ 0 whenever w belongs to the given class with equality holding only if w — 0. The next two preliminary results are analogues of those in Reid [5], and the proofs are nearly identical to his. THEOREM 6.1. The statement that L is nonoscillatory is equivalent to each of the following conditions: ( i ) (fly holds; (ii) For each subinterval [c, d] of [α, b], the functional Icd is positive definite on 4J[c, d].. Since the system (3.3') is identically normal and the matrix B(x) ^ 0 on [a,b], it follows from Theorem 5β2 of Reid [6] that the nonoscillation of L is equivalent to each of the conditions (i) and (ii). THEOREM 6.2. The condition (H^ implies that if [c, d] is a subinterval of [a,b] and fedn[c,d], then the following conditions are equivalent: ( i ) Iedbf] ^ Iedlf] whenever y - / e Jf[c, d]. (ii) // Jcd is the bilinear functional defined on Δn[c, d] x Δn\c, d] by. then Jcd[(f, w)] Ξ> 0 whenever w e Δl\c, d].. lΐwe Δ%[c,d], then Icd[w] ^ 0 by Theorem 6.1. Also, if we lf[c,d] and t is any positive number, then tw e J?[c,d]. The result then follows from the identity W. + *w] = Icd[f] + 2tJea[(f, w)] + tflcd[w] .. We now obtain a characterization of sub-(L) functions which are in the domain of L in terms of unilateral variational property..

(109) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 107. 6.3. If (H^ and (H2) hold and f is a function in the domain of L on a subinterval [c, d] of [α, 6], then a necessary and sufficient condition for f to be sub-(L) on [c, d] is that THEOREM. (6.1). Iedlv] & IcJίf] whenever y -fe. J?[c, d].. If weΛH[c, d], then. which, by definition, means that. Therefore, Jea[(f, w)] Ξ> 0 for every w in Jf[c, d] if and only if ( — l)nLf(x) ^ 0 on [c, d]. The conclusion then follows from Theorems 5.1 and 6.2. It would be desirable to remove the condition that / belong to the domain of L from the hypothesis of this theorem. One possibility which might be examined is the simple case where L = D2n, for if / e An[c,d], w e J?[c, d], po[f] = pQf, and. for i in {1,. , n}, then J.a[(f, V>)] =. which is of the form \dDnwDnφ. ,. Jβ. exactly that which arises in considering the case L = D2n. It is to be noted, however, that the "sufficiency" part of Theorem 6.3 does not require / to be in the domain of L. 6.4. If (HJ and (H2) hold and feΛn[c,d], sub-(L) on [c, d] in case (6.1) holds. THEOREM. Suppose an arbitrary at x is zero the Green's that. then f. is. c ^ x± < x2 ^ d and y12 is the solution of (4.3). Let t be point in (xu x2), and let wt be the function whose value nJrl outside [xu x2] and is ( — l) g{x, t) on (xux2)y where g is function for (4.2). Then wt e J?[c, d] by Theorem 5.2, so.

(110) 108. J. COLBY KEGLEY. 0 :£ Jea[f, Wt] = (-l)«+f Σ D^i+1[g(t)]T. -fit). .. But, for arbitrary w in Δl\c, d],. [. n-l. "Id. ΐ=o. Λc. In particular, 0 = Jcd[Vi2, Wt] = ( - i r and, in view of the^boundary conditions of (4.3), 0 £ y12(t) - f(t) . Hence, / is sub-(L) on [c, d]. 7. Strong nonoscillation of L. Under hypotheses (Hx) and (H2) we are able to conclude that the null-space of the operator L is a 2nparameter family on [a, b], i.eβ, that there is exactly one solution of Ly = 0 which assumes 2n given values at 2n given (distinct) points of [a, 6]. We first establish the following result, the proof of which is modeled after a proof of Polya [3]. THEOREM 7.1. Suppose (H2) holds, {zu , zn} is the basis for the null-space of Lo* given by (3.11) and, for each k in {1, • *,n},Zk is the set of all linear combinations of {zk, , zn}. If ze Zk, then either z(x) Ξ 0 or else z has at most n — k zeroes on [a, b]. In particular, if a Ss tt < t2 < < tn ^ b, then the n-point boundary-value problem. (7.1). Uz = 0 ,. z(U) = 0 ,. i in {1, . .-, n} ,. is incompatible. If z — cnzn then, since zjx) > 0 on [α, 6], either z(x) = 0 or else z vanishes nowhere on [a, b]. Assume that k + 1 is an index for which the assertion is true and suppose ze Zk1 say z = J^=k CjZj. If ck = 0 then z e Zk+1, so either z(x) Ξ O or else z has less than n — k zeroes on [α, 6] If ck Φ 0, then z(x) ί 0 and we may write zk = (lfck)z — zQ, where z0 = Σj=k+i (Cj/ck)zj. If it were possible that there exist n — k + 1 points tu t2, , tn_k+1 at which z vanishes, then zk + z0 would also vanish at these points and, as in the proof of Theorem 4.1, there would exist a point t intermediate with respect to {t1912, , tn_k+1} at which Γn__k[zk + zo](t) = 0. But the proof of Theorem 3.3 shows that Γn_kz3- = 0 if j ^ k + 1 and Γn_kzk = 1. In particular, Γn_k[zk + zQ](t) = 1, a contradiction..

(111) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 109. THEOREM 7.2. Under (HO and (H2), if a^x,<x2< < xm ^ b and, for each i in {1, , m}, λ { e {1, 2, , n + 1} such that ΣΓ=i \ = fte m-point boundary-value problem. (7.2). ^ =° '. D'-WXi) = 0 , is incompatible.. ϋw{l,. ,m},iw{l,. , λj ,. It will be shown that if y is any function in the domain of the operator L o which satisfies the boundary conditions of (7.2), then there exist n points of [a, b] at which LQy vanishes. Let v — max {Xu λ2, , λ m }. If v = 1, then m — 2n, and repeated application of Rolle's theorem using the decomposition (3.9) gives the result. If v > 1, then for each k in {2, , v) let ak denote the number of points Xi at which Xi — k, and for j in {1, 2}, let s3-tk denote the set of integers r with j ^ r ^ v and r Φk. Now, if £ is an index such that Xi = k then, by (3.10), Λόy{x%) = 0 for i in {0,1, • , k - 1}, so Rolle's theorem implies that Aλy vanishes at β1 — m — 1 + Σfc=a αfc points of [a, b]. It will be shown that for each j in {1, •••, v — 1}, Ajy vanishes at βi = m - i + fi Φ - l)α 4 + i Σ Λ=2. k=3+l. αfc. points of [α, δ]. Since the assertion is known for j = 1, assume that it holds for some j in {1, ••-,!; — 2}. Applying Rolle's theorem, ^y+ii/ must vanish first of all at βs — 1 points, none of which will be an x{ with X{ ^ j + 2. But Λi+17/ also vanishes at exactly these points x{ as well, and it follows that βj+1 - m - j - 1 + Σ (k - l)α* + j Σ k. k. j+2. l j. -1 + Σι(k-. u. +2. V. 3+1. = m-j. a. «* + Σ. l)ak + (j + 1) X α* .. In particular, Av_λy vanishes at /3v-i = m - (y - 1) + Σ (fc -. l)ak. points of [α, δ]. But α* = g [Π. 1 > 4 (λ, - r)/(fc - r)] , so (ft - l)ak = Σ [ Π . . (λ4 - l)(λ 4 - r)/(fc - r)] ..

(112) 110. J. COLBY KEGLEY. Hence,. &_! = m - (v - 1) + g | Σ [IL2,fc (λ, - r)/(fc - r)]J(λ« - 1) . The expression in braces is a polynomial in λ< of degree at most v — 2 which has the value 1 for each of the v — 1 values λ4 = 2, 3, •• , v. Hence, this expression is identically 1 in Xi9 and βM. = m - (v - 1) + Σ (λ< ~ 1) = 2n - (i; - 1) ,. i.e., Λ-i2/ vanishes at 2n — (v — 1) distinct points of [α, 6], The same use of Rolle's theorem as that for the case v — 1 now gives the conclusion that Loy must vanish at n distinct points of [α, 6]. If y satisfies (7.2), then z = Loy satisfies (7.1) for some set {tu , tn} of points in [α, 6], so z = 0, i.e., Loy — 0, and, by Theorem II of Polya [3] for the operator Lo, it follows that y must also vanish identically. In particular, the problem (7.2) with m = 2n is incompatible, and the elementary solvability theorems for vector differential systems imply that the null-space of L is indeed a 2n-parameter family. Hence, it is possible to examine L-convexity in the sense of Tornheim [7] and Hartman [2], whereby a function / defined on an open subinterval (c,d) of [α, b] is L-convex if and only if for every set of 2n points x^x2< <x2n of (c,d), if y is the unique function satisfying. then (~iYy(x) ^ (~lYf(x). on. (xifxi+1).. However, the exact relationship between the two types of convexity remains undecided. It is also natural to ask about the properties of the operator L — n (— l) LQL*. It is easily seen that Ly — 0 is equivalent to an identically normal system of the type (2.6') and that if U and V are as specified in (HO then (ί/*" 1 ; 0) satisfies an analogous condition (Hi) for L. Moreover, in the notation of (2.8), if {zu -- , 2 j is contained in the domain of Lo* then for each k in {1, , n} we define the "generalized Wronskian" W*(zn, zn_u , zn_k+1) to be the determinant of the k x k matrix n [y*-i+i[3n-i+i]]iίi A I particular, if {zu •••,««} is the basis for the m null-space of Lo* defined by (3.11) and W* = W*(zn,zn_ί9 ,zn_k+1), then W* is equal to the lower right principal minor of order k in the matrix C/"*"1. Hence, if 11 is the adjoint matrix of Z7* and Uk is the lower right principal minor of order k in the matrix U, then a well-.

(113) CONVEXITY WITH RESPECT TO EULER-LAGRANGE. 111. known formula (see, eog., Hohn's Elementary Matrix Algebra, p. 61) gives Wϊ = Ufc/(det Uy = (det Ur. =. which, by hypothesis (H2), is positive. Thus, we have an analogue (H2) of (H2). However, it is not evident that properties of convexity, etc., with respect to L shed any light at all on the questions already raised concerning L. BIBLIOGRAPHY 1. F. F. Bonsall, The characterization of generalized convex functions, Quart. J. Math. Oxford Second Series, 1 (1950), 100-111. 2. P. Hartman, Unrestricted n-parameter families, Rendiconti del Circolo Matematίco di Palermo (2), 7 (1958), 1-20. 3. G. Polya, On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Soc. 2 4 (1922), 312-324. 4. W.T.Reid, Oscillation criteria for linear differential systems with complex coefficients, Pacific J. Math. 6 (1956), 733-751. 5. , Variational aspects of generalized convex functions, Pacific J. Math. 9 (1959), 571-581. 6. , Riccati matrix differential equations and nonoscillation criteria for associated linear differential systems, Pacific J. Math. 13 (1963), 665-685. 7. L. Tornheim, On n-parameter families of functions and associated convex functions, Trans. Amer. Math. Soc. 6 9 (1950), 457-467. STATE UNIVERSITY OF IOWA.

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(115) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. ON THE DETERMINATION OF CONFORMAL IMBEDDING TlLLA KLOTZ Two imbedding fundamental forms determine (up to motions) s the smooth imbedding of an oriented surface in E . The situation is, however, substantially different for the sufficiently smooth conformal imbedding of a Riemann surface R in Ez. Conventionally such an imbedding is achieved by a conformal correspondence between R and the Riemann surface ϋ?i determined on a smoothly imbedded oriented surface S in Ez by its first fundamental from I. We show that except where z H K = 0 on S, such an Rx conformal imbedding of R in E is determined (up to motions) by the second fundamental form IT on S, expressed as a form on R. In particular, / is determined by II on Rίf where H-K Φ 0 on S9 Similar remarks are valid for two less standard methods of conformal imbedding. If an oriented surface S is smoothly z imbedded in E so that H > 0 and K > 0, then II defines a Riemann surface R2 on S. And, if S is imbedded so that K < 0, then II' given by f. HΊI. =. KI-. with H' = -)/H*-. K. defines a Riemann surface R% on S. Thus a conformal correspondence between R and R2 (or R'2) is called an R2 (or Rζ) 3 conformal imbedding of R in E . We show that I on S, expressed as a form on R, determines the R2 or (wherever H Φ 0 and sign H is know) the Rί imbedding of R in E* (up to motions). In particular, I determines II on R2, and (where HΦO, and sign H is known) on R'2 as well. Finally, we give restatements of the fundamental theorem of surface theory in forms appropriate to Rlf R2 and Ri conformal imbeddings in Es. The two fundamental forms which determine (up to motions) the 3 smooth imbedding of an oriented surface in E are, of course, related by various equations. But neither form determines the other, except in very special cases. Thus, for instance, isometric imbeddings of a surface in Es may differ essentially unless (to cite a famous example) the surface is compact, and the common metric imposed by imbedding has positive Gaussian curvature. 2*. Consider an oriented surface S which is C3 imbedded in. E\. Received January 10, 1964. This research was supported under NSF grant GP1184. 113.

(116) 114. TILLA KLOTZ 3. We may introduce C isothermal coordinates x, y locally on S, so that 2. 2. I = X(dx + dy ) ,. with x + iy a conformal parameter on Ru and λ > 0 a C2 function of x and y. The Codazzi-Mainardi equations involving λ and the C1 coefficients L, M and N of I I become L, - Mx = -£*- (L + iSΓ) , ( 1 ). λ. The theorem egregium formula for TΓ. LN. -. M. 2. 5? may be written in the form. ( 2). 2. LN - M = -=^ j ( ^ ) + f ^ 1. 7. 2. luλ. \λλJ. Moreover, since λ > 0 while 2λ. the equations (1) may be solved for Xx/X and λy/λ, provided that H Φ Q. Substitution in (2) of the expressions so obtained yields (LN -. L + N '». \ L+N. If we add the assumption that K Φ 0, making LN — M2 Φ 0, then (3). LiV- M2. λ —. \\ L. N )*. \ L +N. Thus, we have established our original claim that II on Rx determines I wherever H'KΦ 0β It will be convenient to refer to the expression on the right side of (3) as λ(L, ikf, N). Of course, when Lyy, Nxx and M"x2/ exist, λ = λ(L, Af, iV). =. In any case, substitution of λ = λ(L, M", iV) in (1) yields conditions.

(117) ON THE DETERMINATION OF CONFORMAL IMBEDDINGS. K. 2\(L,M,N). Nx-My=. 115. }. L. ^ > M, N)}. 2λ(L, M, N). {L K. N). }. on L, M and iV wherever H Kφ 0, or, equivalently, wherever 2. (L + N)(LN - M ) Φ 0 . Suppose now that a C1 quadratic form Ω = Lcte2 + 2Mdxdy + is given on a Riemann surface i2. (Here x + iy is a conformal parameter on jβ.) Suppose also that (L + N)(LN - M2) Φ 0. Then the previous discussion establishes λ(L, M, N)(dx2 + # 2 ) as the only possible / for a C 3 i ^ conformal imbedding of R in ΐ/ 3 with Ω — II. Thus, if such an imbedding exists, λ(L, M, N) must be positive and C2, while (4) must be valid. On the other hand, if λ(L, M, N) is a positive C2 function, and if (4) does hold, then both (1) and (2) are valid with λ = λ(L, M, N). Thus the fundamental theorem of surface theory (see p, 124 of [3]) immediately implies the following result. 1. If Ω — Ldx2 + 2Mdxdy + Ndy2 is a C1 quadratic form on R with (L + N)(LN — M2) Φ 0, then necessary and sufficient conditions for the existence (locally) and uniqueness (up to motions) of a C 3 ^ conformal imbedding of R in E3 with Ω — II are that λ(L, M, N) be positive and C2, and that (4) be valid, THEOREM. 3* Consider an oriented surface S which is C4 imbedded in E3 so that H > 0 and K > 0. We may introduce C3 bisothermal coordinates x, y locally on S, so that 2. 2. II = μ(dx + dy ) 1. with x + iy a conformal parameter on R2, and μ > 0 a C function of x and y. Here the Codazzi-Mainardi equations involving μ, and the Christoίfel symbols for the coefficients E, F and G of I become. And the theorem egregium yields a complicated expression for.

(118) 116. TILLA KLOTZ. as a function of E, F, G and their first and second partial derivatives, which we refer to for convenience as K(E, F, G). Thus 2. (6). μ = VK(E, F, G)(EG - F ) ,. and we have established our original claim that / on R2 determines IL We will refer to the expression on the right side of (6) as μ(E, F, G). Here, substitution of μ = μ(E, F, G) in (5) yields conditions (. '. {μ(E, F, G)}y = μ(E, F, G)(Γ\2 - /*) ,. on E, F and G. Suppose now that a C2 quadratic form Ω = Edx2 + 2Fdxdy + Gdy* is given on a Riemann surface R. Suppose also that K(E, F,G) > 0. Then the previous discussion establishes 2. μ(E, F, G)(dx + df) as the only possible II for a C3 R2 conformal imbedding of R in E3 with Ω — I. Thus, if such an imbedding exists, μ{E, F, G) must be positive and C\ while (7) must be valid. On the other hand, if μ(E, F, G) is 1 a positive C function, and if (7) does hold, then both (5) and K — K(E, F, G) are valid, with μ = μ(E, F, G). Thus the fundamental theorem of surface theory immediately implies the following result. 2. 2. 2. If Ω = Edx* + 2Fdxdy + Gdy is a C quadratic form on R, then necessary and sufficient conditions for the existence {locally) and uniqueness (up to motions) of a C3 R2 conf ormal imbedding of R in E* with Ω = I are that K(E, F, G) be positive, that μ(E, F, G) be positive and C1, and that (7) be valid. THEOREM. A* Finally, consider an oriented surface S which is C4 imbedded in I£3 so that K < 0. We may introduce C3 disothermal coordinates x, y locally on S, so that IΓ = μ'(dx2 + dy2) with x + iy a conf ormal parameter on R2', and μr > 0 a C1 function of x and y. Since HΊΓ = KI - HII,.

(119) ON THE DETERMINATION OF CONFORMAL IMBEDDINGS. 117. HL + H'μ' = KE , ( 8). HN + H'μ' = KG , HM = KF.. But we show in [2] & that on iϋ2', (9). L=-N.. Thus κ. -{U + M>) EG-F*. =. must be given by the theorem egregium expression K(E, F, G). Using (8), we obtain HL = K(E - G) ,. (10). HM = KF,. so that H\U + ikP) = KX(E - GY + F2y . Division of this last equation by (L2 + M2) = - K(E, F, G) (EG - Fz) Φ 0 yields (11). H=. Thus H vanishes if and only if (E — G) + iF = 0. Of course, where H Φ 0, the orientation of S determines the sign of H. On the other hand, where H Φ 0, we may set. ME FG)-. J-K(E,F,G)(EG-F>). F. and N(E,F,G)=. -L(E,F,G) .. Using (10), we conclude that so long as H Φ 0, L = ±L{E, F, G) , (12). M= ±M(E, F, G) , N=. ±N(E,F,G) ,. with plus or minus signs consistently chosen in accordance with the.

(120) 118. TILLA KLOTZ. sign of H. Thus we have established our original claim that / on R[ determines // (if H Φ 0, and sign H is known). Note, however, that the Codazzi-Mainardi equations on Ri which read Ly - M. = L{Γ{2 + Π) + M(Γ\2 (. }. I\). Lx + My = L(Γ\2 + Γ{2) + M(Γ\2 - Γ\2). are not affected by the sign of H. (In particular, if L, M and N solve (13), so will —L,—M and —N.) Thus, whichever the choice of signs in (12), the Codazzi-Mainardi equations yield the following conditions {L(E, F, G)}y - {M{E, F, G)}x = L(E, F, G){Γ\2 + I\) + M(E, F, G)(I\ - I\) , (. }. {L(E, F, G)}x - {M(E, F, G)}y - L{E, F, G){Γ\2 + Γ\2) + M{E, F, G)(Γ222 - Γ\2) ,. on E, F and G, wherever (E - G) + iF Φ 0. Suppose now that a C2 quadratic form Ω - Edx2 + 2Fdxdy + Gdy2 is given on a Riemann surface R. Suppose also that K(E, F, G) < 0 while (E — G) + iF Φ 0. Then the previous discussion establishes (15). L(E, F, G)(dx2 - dy2) + 2M(E, F, G)dxdy. and (16). -L(E, F, G)(dx2 - dy2) - 2M(E, F, G)dxdy. as the only possible forms which could serve as II for a C3 R[ conformal imbedding of R in 2?3 with Ω — I. Thus, if such an imbedding exists, L(E, F, G) and M(E, F, G) must be C1 functions, while (14) must be valid. Finally, should such an imbedding exist with one choice (15) or (16) for II, composition with a reflection of S in a plane will leave I invariant while yielding the remaining choice for II. On the 1 other hand, if L(E, F, G) and M(E, F, G) are C functions, and if (14) does hold, then, given either choice (15) or (16) for II, both (13) and K = K{E, F9 G) are valid. Thus the fundamental theorem of surface theory immediately implies the following result. Theorem 3. If Ω = Edx2 + 2Fdxdy + Gdy2 is a C2 quadratic form on R with (E — G) + iF Φ 0, then necessary and sufficient conditions for the existence (locally) and uniqueness (up to motions 3 and reflections in planes) of a C R2 conformal imbedding of R in.

(121) ON THE DETERMINATION OF CONFORMAL IMBEDDINGS. 119. Es with Ω — I are that K(E, F, G) be negative, that L(E, F, G) and 1 M(E,F,G) be C functions, and that (14) be valid. 5* We close by noting a pair of statements of the type one gets by slight rewording of the results described above. Isometric oriented 4 B surfaces imbedded C in E so that H > 0 and K > 0 are congruent if and only if the isometry between them is conformal between their R2 structures. Similarly, such surfaces on which H Φ 0 and K < 0 are congruent if and only if the isometry between them is conformal between their Rr2 structures and preserves the sign of H. The weakness of these results amply illustrates the sense in which //, while inessential on R2 or i?2', is of fundamental importance in determining the imbedding of a surface, as distinct from the R2 or R[ conformal imbedding of a Riemann surface. None-the-less, more significant applications of Theorems 1, 2 and 3 should be possible. REFERENCES 1. Tilla Klotz, Some uses of the second conformal structure on strictly convex surfaces, Proc. of the Amer. Math. Soc. 14 (1963), 793-799. 2. , Another conformal structure on immersed surfaces of negative curvature, Pacific J. of Math. 13 (1963), 1281-1288. 3. Dirk J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley, Cambridge, 1950. UNIVERSITY OF CALIFORNIA, LOS ANGELES.

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(123) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. ON THE SPECTRAL ANALYSIS OF BOUNDED FUNCTIONS PAUL KOOSIS. The spectral analysis of functions /eLoo(— oo, oo), and the different possible ways of looking at it, are discussed. A method is given for constructing functions /eLoo(—oo, oo) having the property that 0 is in the spectrum of /, but 1 is not the weak limit of any sequence of linear combinations of translates of /. Preliminary discussion*. Let feL00(. — oo9 co).. As is customary,. we say that a real number λ belongs to the spectrum of /, and write Xespf, when the function eίλx is in the weakly closed (he., closed in the weak topology over Lx) subspace of L^ generated by the translates of /. (A translate of / is any function fh of the form fh(%)~f(%—h) for — oo < h < oo.) As an immediate consequence of Wiener's Tauberian theorem, one has, by duality (see, for instance [5], p. 128, [9], p. 185, [4], p. 106 or [7], p. 181) the Theorem on Spectral Analysis. everywhere, sp f is nonempty.. Unless / e !/«, is zero almost. This result is often referred to as Beurling's theorem. However, Beurling proved much more about a smaller class of functions /. If the ψn(n = 1,2, •) and ψ are bounded continuous functions on (— oo, co), let us agree to say that ψn —•*ψ narrowly as n—> °° iί \\ψn\\oo—>\\ψ\\QO and ψnW^Ψitt) uniformly on finite intervals for n—> oo. Then: Beurling's Theorem ([1]). Let f be bounded and uniformly continuous on ( — oo, oo). Then there is a real number X and a sequence {ψn} of linear combinations of translates of f, such that ψn(x) —• e. iλx. narrowly. as n —• oo .. Beurling used complex variable theory in his proof of this theorem. However, it is not hard to see that the main idea behind his reasoning is independent of his use of analytic functions, and a simple demonstration, based on this idea, can be given if one is willing to employ the above theorem on spectral analysis together with a well known Received September 22, 1964. The preparation of this paper was sponsored by the Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the United States Government. 121.

(124) 122. PAUL KOOSIS. elementary result on synthesis (the same argument thus applies to arbitrary locally compact abelian topological groups). For the reader's convenience, we sketch such a proof now. Let / be uniformly continuous, and let Ke L1(—oo9 oo). If, as in [1], one approximates the integral involved in the convolution K*f by finite sums, one sees easily that there is a sequence of linear combinations of translates of / which converges narrowly to K*f. (As usual, we write K*f for the function defined by. (K*f)(x) = Γ K(t)f(x - t)dt J-oo. It is therefore enough to show that if λ e sp /, there is a sequence of functions Kne Lλ such that (Kn*f)(x) —+ eiλx narrowly, n—> oo. For h > 0, let Gh denote the function in Lλ whose Fourier transform is zero outside (λ — h, λ + h), one at λ, and linear on each of the intervals [λ — h, λ], [λ, λ + h]o Since Xesp f, we cannot have Gh*f=0. Let, therefore (and this is Beurling's main idea) Xhe (— oo, oo) be such that I (Gh*f)(Xh) I ^ (1 - h) || G A * / | U , and put Kh(t) = Gh(t Xh)/(Gh*f)(Xk). Then Kh e L19 (ϋΓΛ*/)(O) - 1, and \\Kh*f\U^(lh)~ι for h < 1. If we denote by K the function in hx whose Fourier transform is one on [λ — 1, λ + 1], zero outside (λ — 2, λ + 2) and linear on each of the remaining two intervals, we see that K*Kh — Kh for h < 1, so that K*{Kh*f) — Kh*f, h < 1, which, with the previous inequality, implies that the functions (Kh*f)(x) are uniformly equicontinuous for h < 1/2. There is therefore, by Ascoli's theorem, a sequence hn —> 0 such that, if we write Kn = Khn, the functions Kn*f converge uniformly on finite intervals to some continuous function g as n —• oo. Clearly \\g\\o* ^ 1, and since (Kn*f)(Q) = 1, g(0) — 1, and the convergence of the Kn*f to g is in fact narrow. Now g(x) — eiλx. Indeed, g(0) = 1 and g is continuous, so sp g is not empty by the theorem on spectral analysis. On the other hand, sp g must reduce to the single point λ. For if μ is real and ^ λ , let ε = 1/2 I μ — λ |, and let G be the function in Lx whose Fourier transform is 1 at μ, zero outside (μ — ε, μ — ε), and linear on [μ — ε, μ]9 [μ, μ + ε]. Then G*Kh — 0 for all sufficiently small h which, together with the narrow convergence of the Kn*f to g, implies G*g — 0, proving μ£sp g, since the Fourier transform of G does not vanish at μ. The conclusion ίλx g(x) — e now follows by an elementary theorem on spectral synthesis (see, for instance, [4], p. 106 or [7], pp. 151 and 181). Before going any further we should say that it is clear, from Appendix 1 to [5], that Godement had the above proof in mind when he wrote his paper. He was, however, unable to carry it through because the theorem appealed to at the end of our argument was not then available, except as a corollary of Beurling's reasoning. A general.

(125) ON THE SPECTRAL ANALYSIS OF BOUNDED FUNCTIONS. 123. demonstration for locally compact abelian topological groups has indeed been given by Domar (see Ch. IV of [3]), but because his exposition is very long and complicated, we have thought it useful to include the above proof. Now the point we wish to make is that the topology involved in the statement of Beurling's theorem is much stronger than that of the one on spectral analysis. Whereas the second merely asserts that there are functions of the form eiλx in the weak closure of the set of linear combinations of translates of /, the first shows that for uniformly continuous /, such functions eiλx can be obtained as weak limits of sequences of such linear combinations which are uniformly bounded in norm. Since, for many of the limiting processes used in analysis, the weak closure property is useless without the boundedness condition, it is not entirely correct to say, as some have done (for example, the authors of [4] on p. 106 thereof), that the theorem on spectral analysis contains the gist of Beurling's result. Although the uniform continuity of / is used in an essential way at the beginning of the proof of Beurling's theorem, it is legitimate to ask whether or not one might, by another method, be able to establish some weakened form of it, valid for general fe L^ which would still affirm the important boundedness property. One would be satisfied with the following: Let / e L M . If Xesp f, the function eiλx belongs to the weak closure of some norm-bounded set of linear combinations of translates of /. PROPOSITION.. Godement recognized the value of such a result, for he states it explicitly in his article (remark on p. 131 of [5]), and uses it to prove Theorem D of that paper. It seems not to have been observed in the literature (neither in Segal's review of Godement's article, written for Mathematical Reviews), that Godement bases the above proposition on an incorrect statement. If V is an arbitrary subset of !/«>, closed with respect to the algebraic operation of taking finite linear combinations, let us denote by [V] the union of the weak closures of all normbounded subsets of V. Then Godement establishes the proposition by identifying [V] with the weak closure of V, where V is generated by the translates of /. However [V] is not in general equal to the weak closure of F. (Godement, by the way, cites [2] in support of this claimed equality, but the latter article contains no such affirmation.) It is well to observe here that Godement's proof of Theorem D on p. 131 of [5] is in fact wrong on two counts: the use of the above incorrectly established proposition, and, besides this, an additional appeal to the false relation [[V]] = [V], which may fail to hold even when V consists entirely of continuous functions..

(126) 124. PAUL KOOSIS. Various subsequent authors (with the exception of Loomis in [7]) have taken one of two positions on the matters we have been discussing: either they assimilate Beurling's result with the theorem on spectral analysis (as in [4], p. 106), or they go to the other extreme and treat the above proposition as an established part of the theory (the author of [8], on p. 8 thereof, seems to claim that the proposition was proved by Beurling in one of his lectures!) In view of this state of affairs, it is of interest to have a judgement as to the truth of the proposition. It turns out that it is false. The rest of this paper consists of a description of a method for the construction of counterexamples thereto, followed by an indication of our original application of this method. This construction, which is based on an interesting result of Rudin, is merely sketched, because it involves a considerable amount of technical detail, and especially because J.-P. Kahane has found a much simpler one, depending only on the most elementary principles of analysis. Kahane's work is presented in the paper immediately following this one, and the reader may, if he wishes, turn to it as soon as he has read § 1 below. It should be noted that both Kahane's construction and our original one yield counterexamples to Godement's Theorem D, as well as to the proposition discussed above. 1Φ If fe Loo(— oo, oo), we denote by Vf the set of finite linear combinations of translates of / . The method spoken of at the end of the above discussion in contained in THEOREM 1. Suppose a function fe !/«,(— oo, oo) has the following properties: 1° Oespf. 2° If X19 *,XM are distinct real numbers and Alf ,AM are complex, then. \ = ess sup. k. inf{ sup ± T>0 l-oo< α <oo ϊ. 7. Then 1 is in the weak closure of Vf, but not in the weak closure of any norm-bounded subset of Vf. Proof. Since 0 e sp /, 1 is in the weak closure of Vf. Let, on the other hand, E be any set consisting of functions ψ e Vf satisfying || ψ!!«, ^ K < oo. Then, if ψeE, it is of the form.

(127) ON THE SPECTRAL ANALYSIS OF BOUNDED FUNCTIONS. 125. f(x) = Σ Atf(x + Xk) k. with the Xk all different, and property 2 implies that ^k \ Ak | ^ K. If T > 0 is such that X)dx. <! 2K. for all real X, we will then have < — for any 1. feE.. ΓT. Since —\ l dx = 1, 1 cannot be in the weak closure of E. T Jo COROLLARY 1. If fe L^ possesses the properties of the theorem, then there is no sequence of elements of Vf which tends weakly to 1. For if φne Vf,n = l,2,39 •• , and t h e sequence {ψn} is weakly convergent in !/«,, t h e norms | | ^ w | | o o are bounded by a well-known theorem of Banach. COROLLARY 2. If feL^ is continuous and has the properties of the theorem, it provides a counterexample to Theorem D of [5].. Theorem D of [5] says that if feL^ is continuous and Oesp f, there is a sequence {ψn} of elements of Vf satisfying \\irn\\oo^K, such that ψn(x) —* 1 uniformly on finite intervals as n —* oo# Such a sequence would certainly converge weakly to 1 which is, however, impossible by the theorem. 2. Let us show a way of constructing functions / having the properties required in Theorem 1. By modifying a construction of Salem, Rudin [10] has shown that there exist perfect sets P c ( 0 , 1) of Lebesgue measure zero, having the following properties: I.. The elements of P are linearly independent over the rationale*. II. There is a nondecreasing continuous function F, defined on [0, 1] and constant only on the contiguous intervals of P, such that [ eίλζdF(ξ)->0 as λ->±oo. JP.

(128) 126. PAUL KOOSIS. A detailed exposition of Rudin's construction and the results of Salem on which it is based can be found in Ch. VIII of [6]. By using, if necessary, a set kP instead of P, with k suitably chosen e (0,1) we can ensure, besides properties I and II: endpoint of one of the finite intervals contiguous to P is a rational multiple of π. It is also easy to see (cf. [6], p. 104) that II in fact implies jjδis i eiλξdF(ξ)—>0 uniformly for all intervals J as λ—>±°o. Let us denote δy E the set obtained from P by removing from it the right endpoint of each of its finite contiguous intervals. F is strictly increasing on E and may, after suitable normalization, be taken to map it in one-to-one fashion onto [0, 1]. Denote by φ the inverse of F restricted to E; φ is strictly increasing on [0,1] and maps that interval onto E. We now define a certain function fe !/«>. First of all, for n ^ 2, we define intervals II, /*, •••,/£ as follows: II = [2* + 1, 2n + 4). II = [2n + 2r-\ 2* + 2r) ,. 3^ r ^ w.. Denote by lrn the left endpoint of /;, and let {Nr} be a rapidly increasing sequence of positive integers. Then, for x ^ 4, put /(α?) = exp Γ(ΛΓr + n ) ^ ^ 7, | ; ) ] if x e Iζ . For x < 4, let fix) = Σ 2-» exp ( - | ^ α ) . It is clear that / e L M and | | / | U = l THEOREM 2. 1/ ί/^e sequence {Nr} increases rapidly enough, fix) has properties 1, 2, and 3 of Theorem 1.. Sfceέcfr o/ Proof. In the first place, fix - 2*) -> ΣΓ 2~m exp i2πij2m)x m weakly in L^ as iV—> oo. Therefore each of the numbers 2τr/2 , m = 1, 2, 3, belongs to sp f (one way of seeing this is to apply the method use to prove Beurling's theorem in the preliminary discussion of this paper). Since sp f must be closed, we get 0 e sp f, establishing.

(129) ON THE SPECTRAL ANALYSIS OF BOUNDED FUNCTIONS. 127. property 1. Property 2 follows from the definition of f(x) (for x ^ 4) by Kronecker's theorem (see Appendix V of [6]), since φ maps [0,1] in one-to-one fashion onto E which forms, together with π, a set linearly independent over the rationals, according to I and Ihi\ his To establish property 3, observe first of all that II can be restated thus: I eiλφ{x)dx —> 0 uniformly for all intervals I ϋ [0, 1] as λ—>±oo. With this in mind one sees straightforwardly how, looking at the definition of f(x), to assign to {Nr} a rate of increase sufficiently rapid so that f(x + a)dx shall tend to zero uniformly in a as M—> oo. 0. COROLLARY. If {Nr} increases sufficiently rapidly, GodemenVs proposition fails for the function f{x). REMARK. The function f(x) which we have just constructed is not continuous. One can, however, modify the procedure used above so as to obtain a continuous (but not uniformly continuous!) bounded function g(x) which still has properties 1, 2, and 3 of Theorem 1. Essentially, this modification consists in using, on the successive intervals I*, better and better continuous pointwise approximations to the increasing function φ(x), instead of φ(x) itself. This construction is quite straightforward, but somewhat technical in its details, and we shall not discuss it further. In any event, it leads to a counterexample for Godement's Theorem D. In conclusion, I would like to thank Professors Arens and Straus for some helpful discussions.. BIBLIOGRAPHY 1. A. Beurling, Un theoreme sur les functions bornees et unifor moment continues sur I'axe reel, Acta Mathematics, 7 7 (1945), 127-136. 2. J. Dieudonne, La dualite dana les espaces vectoriels topologiques, Annales de ΓEcole Normale Superieure, 59 (1942), 107-139. 3. Y. Domar, Harmonic analysis based on certain commutative Banach algebra, Acta Mathematica, 96 (1956), 1-66. 4. I. Gelfand, D. Raikov, and G. Shilov, Kommutativnye normiroυannye koΓtsa, State Publishing House of Physico-Mathematical Literature, Moscow, 1960. 5. S. Godement, Theorems tauberiens et theorie spectrale, Annales de ΓEcole Normale Superieure, 64 (1947), 119-138. 6. J.-P. Kahane, and R. Salem, Ensembles parfaits et series trigonometriques, Fermann, Paris, 1963. 7. L. Loomis, Abstract Harmonic Analysis, Van Nostrand, New York, 1953. 8. B. Nyman, On the One-dimensional Translation Group and Semi-group in Certain.

(130) 128. PAUL KOOSIS. Function Spaces, Thesis, Uppsala, 1950. 9. W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962. 10. , Fourier Stieltjes transforms of measures on independent sets, Bull Amer. Math. Soc. 6 6 (1960), 199-202. UNIVERSITY OF CALIFORNIA, LOS ANGELES.

(131) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. ON THE CONSTRUCTION OF CERTAIN BOUNDED CONTINUOUS FUNCTIONS J.-P.. KAHANE. We give an elementary method for constructing continuous functions fulfilling the hypothesis of Theorem 1 of the preceding paper. Such functions thus constitute counterexamples to the proposition and theorem discussed therein. THEOREM.. suppose (i) (ii) (iϋ). Let φ(x) be continuously differentiate. 9>(0) = 0 φ'{x) is nonnegative, and strictly φ'(x)lφ(x). on [0, oo), and. increasing to °o on [0, <*>). —> oo , x —• CXD .. Put (1). /(«) = Σ 2 - » e x p ( ^ a ? ) ,. (2). f(x) = eiφ{x) ,. x<0 x ^ 0.. Then the bounded continuous function f(x) 3 o/ Theorem 1 m ίfte previous paper.. has properties. 1, 2, and. Proof. That 0 e sp / follows from (1) as in § 2 of the previous paper. To establish property 3, let us show that. ψ\*/(x + Φx - 0 uniformly in a as T —> co. If / is any interval of length T, denote by A the part of I lying to the left of 0, and by B that part lying to the right. We have, by (1),. The quantity in brackets is always in absolute value ^ 1 , and tends to zero independently of the position of A as | A \ —• co (this fact belongs to the rudiments of the theory of almost periodic functions, and can here be verified by direct calculation). Since | A | ^ T, we have (3). —I f(x)dx—>0. independently of the position of I as T —> °o .. JL J -4. Received December 22, 1964. 129.

(132) 130. J.-P. KAHANE. The integral I f(x)dx. is bounded for all intervals B of the form. [0, b]. Indeed, if b> 1, ri. b. S. o. rb. f(x)dx = \ f(x)dx + \ f(x)dx . Jo. Ji. Since φ'{x) ^ 0 we can, by (2), make the substitution <p(x) = ξ in the second integral on the right, getting for it the value. S. b. i. Λf. Cφ{b). eίφ{x)dx. =. J<P(i). eίζ-^-. φ'{x). .. In view of (ii), this last is in absolute value ^ 4/V(l) by the second r. mean value theorem. It follows that I f(x)dx is bounded for all interJB. vals B lying to the right of the origin, whence ( 4 ) —I f(x)dx—>0 independently of the position of / as T—+ oo . T JB. From (3) and (4) we see that 1/Γl f(x)dx. is small in absolute. value for all intervals / of length T, if only T is sufficiently which is property 3. It remains to verify property 2. We show that if 0 < Xx < and the Ak are complex numbers (5). large, < XM. sup %>0. So as not to lose the reader in details, we do this for the case M — 2; it will be clear how to extend the reasoning to any value of M. Let ε be given, 0 < ε < π/2, and, choosing a positive determination of the argument, put, for k = 1, 2, 3,. ( 6). ak = φ-'Uπk + a r g ^ - ε) - X,. φ^ylπk +. (7). -It.. k-. Clearly CJ,k < bk < ak+1, a/. —> o°. (8). bk — ak —-o,. Λ. as. and by (ii. Also,. (9). ^{Axeiφ^x. 1') ^ (1. -. ε2). for. ak S. I claim t h a t φ(bk + X2) — φ(ak + X 2 ) —> oo as fc —• oo# by (ii):. If c > 0,.

(133) ON THE CONSTRUCTION OF CERTAIN BOUNDED CONTINUOUS FUNCTIONS 131. φ\x + c) ^ - — —. ^. φ'(x + c) •. c—. •> φ'(x + c) — —. ^. •. C—. φyx + c) — φ\X). φ (X). ,. φ\X + C). whence /"I. (+s \ Άs "I. f\\. (10). ^. v. ^. v. I. ^ —•> co. y. X ^ > oo ^. in view of (iii). Since X2 > Xi, there is, by (8), a c > 0 such that, for all sufficiently large k, c + bk + X1 ^ αfc + X2. We thus have, from (6), (7), (ii), and (10): φ(bk + X2) -. 2ε V'^ + ^i + as k —> cχ3 f since 6^ —> co? k —> oo. This is the desired result which implies, in particular, the existence, for all sufficiently large Jfc, of an xk G [ak, bk] such that <£>(α^ + X2) = arg —. (mod 2π) .. For such xk we have A 2 β^ (Xfc+X2) = | A2 \ which, together with (9), yields (5) for the case M — 2, since ε > 0 is arbitrary. Suppose φ{x) is even, and fulfills condition (i), (ii), and (iii) of the theorem. Besides this, let it be twice continuously differentiable, and be such that φ"(x) ^ C > 0 (example: φ{x) = eχ2). Then, if f(χ) — ei(ρ(χ\ e*λ* is not, for α^τ/ Xesp /, in the weak closure of any bounded subset of Vf (notation as in the preceding paper). (This observation is due to P. Koosis.) Indeed, the function f(x) clearly has property 2, according to the above work. A glance at the proof of Theorem 1 in the preceding paper now shows that the desired result will certainly follow if we establish, for all real λ, that REMARK.. — Γ / ( a + X)e-iλxdx-*0. uniformly in l a s. Γ->oo,. But by a. lemma of Van der Corput ([1], vol I, p. 197), IfT I JO. ϊ -1/2. (. e. τ[φ{x+X). λx]. ~ dx. S 1 2 - ^ inf. φ"(x. + X)\. lo^a^Γ. for all T, which implies the desired statement.. ^ 12C~ J. 1/2.

(134) 132. J . - P . KAHANE REFERENCES. 1. P. Koosis, On the spectral analysis of bounded functions, Pacific J. Math. 1 5 (1965), 2. A. Zygmund, Trigonometric Series, Second Edition, Cambridge, 1959. FACULTY OF SCIENCES, ORSAY, UNIVERSITY OF PARIS.

(135) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. A THEOREM ON PARTITIONS OF MASS-DISTRIBUTION V.. V.. MENON. A 'bisector' of a continuous mass-distribution M in a bounded region on the plane is defined as a straight line such that the two half-planes determined by this line contain half the mass of M each. It is known that there exists at least one point (in the plane) through which pass three bisectors of M. THEOREM. Let, for a continuous mass distribution M, the point P through which three bisectors pass be unique. Then all bisectors of M pass through p.. The following corollary also is established: For a convex figure K (i.e., compact convex set with nonempty interior) to be centrally symmetric, it is necessary and sufficient that the point through which three bisectors of area pass be unique.. In what follows, M stands for any continuous mass-distribution in a|eompact domain in the plane. A line I is called a bisector of M if the two half-planes determined by I contain equal masses of M. The following results are well-known regarding bisectors of M. (see, for example, [4], Problem 3-1, 3-2, and [1]). (1) Let I be any line in the plane. There is a bisector of M parallel to I. (2) There exists a point P in the plane and two perpendicular lines through P such that the portions of M contained in each of the four 'wedges' determined by these lines have the same mass, namely, a quarter of that of M. (3) There exists a point in the plane through which three distinct bisectors of M pass. Further, let lQ be a bisector of M and 0 a fixed point on lQ. Let l(a) be a bisector of M, inclined to l0 at an angle a and intersecting l0 in Pa. It is easy to verify that we can choose the bisector l{a) such that the distance 0Pa is a continuous function of a. We shall make use of this observation in the following. In this paper we shall investigate the nature of the points through which three distinct bisectors of M pass. Specifically, let P be a point Received January 25, 1964, and in revised form July 30, 1964. 133.

(136) 134. V. V. MENON. in the plane such that three distinct bisectors llf l2, l3 of M pass through P; and let l(a) be a bisector of M not passing through P. We shall prove the existence of a point PXΦ P such that three distinct bisectors pass through P1 too. First, let l{a) be parallel to one of l1912, l3; say, to llm Since lλ and l(a) are both bisectors, it follows that the portion of M contained between these lines lλ and l(a) has zero mass (see Fig. 1).. Fig. 1. Since M is enclosed in a bounded domain D, we can choose a point Pi midway between l1 and l{a), and three distinct lines through P1 such that each of these three lines intersects (if at all) lx and ϊ ( α ) outside Zλ In other words, these three lines are bisectors of M. Secondly, let l{a) intersect the lines l19 l2, h (see Fig. 2), and let X be the point of intersection of l{a) with i 2 . (We number the lines llf l2, ϊ3, such that X lies between the points of intersection of l{a) with k and l3). With reference to a fixed direction, let θx and θ2 be the directions of lλ and l3 respectively, and let a be that of lia).. Fig. 2.

(137) A THEOREM ON PARTITION OF MASS-DISTRIBUTIONS. 135. When 0 varies from θλ to θ2 we can choose the bisectors 1(0) such that 1(0^ = llf l(θ2) = ϊ3, and PX = x(θ) is a continuous function of θ. (The equality 1(01) = lx means that the lines 1(0 J and lλ coincide). Since x(θ1) = 0 = x(θ2), and for the given bisector λ(α), a (α) =£ 0, if follows that there are two distinct values ax and <z> for which αK^i) = x(a2) Φ 0 . Let P1 be the position of X corresponding to x(a1). Thus three distinct bisectors l2, l(aλ) and l(a2) pass through P19 and P x =£ P. This proves the required assertion, that is, if a bisector r of M does not pass through P and three distinct bisectors pass through P then there is a point Pt distinct from P through which also pass three distinct bisectors. Hence we have the Let, for a continuous mass distribution M, the point P through which three distinct bisectors of M pass be unique. Then all bisectors of M pass through P. (In particular, every line through P bisects M). THEOREM.. Something more can be asserted about the mass distribution M in the following special case. Consider a compact convex figure K (i.e., a compact convex set with nonempty interior) and interpret mass as the area. Since the bisector in any direction is unique, it follows from the above theorem that every line through P is bisector of K where P is the unique point through which three bisectors pass. Consider two such bisectors inclined at a small angle 0, as in Figure 3.. Fig. 3. Let l1 intersect the boundary of if in A and B, and let PA = rlf PB = r 2 Denote by Aλ and A2f respectively, the areas of the portions of K in the two wedges (shaded in the figure) between l19 l2. We have, for small θ, the approximate equalities.

(138) 136. V. V. MENON. L^i-rJ*. Ax = A2 since ^, ί2 are both bisectors, and hence rx = r 2 by making θ approach zero. As this is true for any position of l19 it follows that the figure K is centrally symmetric and P is its centre. Of course, the converse also is true because any line through the centre of any centrally symmetric figure (convex or not) is a bisector of it. Thus we have the following corollary. COROLLARY. Let K be a compact convex figure. The following four statement are equivalent: (a) the point P through which three bisectors of K pass is unique, (b) all bisectors of K are concurrent in P; (c) there exists a point P such that any line through it is a bisector of K; (d) K is a centrally symmetric figure with P as its centre. REMARKS. 1. K. Zarankiewicz appears to have proved a similar theorem for convex figures (see [3], page 264, note 10). Our result is in a more general setting, and is, surprisingly, quite strong. The author believes that his proof is different from that given by Zarankiewicz.. 2. A stronger statement of the theorem is not possible, in the sense that out of the four statements (a), (b), (c), (d) mentioned in the corollary, it is not true in general that (c) implies (b), (since a bisector in a direction need not be unique). Also mass-distributions can be constructed easily for which (a) is true but (d) is not. (I am grateful to the referee for bringing to my notice an example where (a), (b), (c) are true but (d) is not). 3. Consider the set of points through which three bisectors pass. Very little is known about this set (see, however, [2]). Acknowledgement. I am thankful to the referee for his helpful suggestions, and for having pointed out the reference [2]. REFERENCES 1. R.C. Buck and E.F. Buck, Equipartίtίon. of convex sets, Math. Mag. 22 (1948/49),.

(139) A THEOREM ON PARTITION OF MASS-DISTRIBUTIONS. 137. 195-198. 2. M. Goldberg, On area-bisectors of plane convex sets, Amer. Math. Monthly 7 0 (1963), 529-531. 3. B. Grϋnbaum, Measures of symmetry for convex sets, Proceedings of symposia in Pure Maths. Convexity, 7 (1963), 233-270. 4. I.M. Yaglom and V.G. Boltyanskii, Convex Figures, Holt, Rinehart and Winston, New York, 1961. INDIAN STATISTICAL INSTITUTE.

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(141) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. THE ENUMERATION OF HAMILTONIAN POLYGONS IN TRIANGULAR MAPS R. C. MULLIN. A finite nonseparable topological graph G in E2 is said to be a triangular map if all its finite faces are triangular. Edges and vertices of G are external if they are incident with the infinite face, otherwise they are internal. The maps considered are rooted by distinguishing an external vertex and incident external edge. A polygon in the graph G of such a map is Ήamiltonian if it includes all vertices of G. In this paper, the average number of Hamiltonian polygons in a member of the class of nonisomorphic rooted triangular maps with n internal and m + 3 external vertices is determined. Asymptotic estimates are included for the results obtained. An unexplained coincidence is shown between the number of Ήamiltonian polygons in rooted triangular maps and in their duals, rooted nonseparable trivalent maps. Triangular maps and Hamiltonian polygons* Let R be a simply connected closed region in the Euclidean plane E2 whose boundary is a simple closed curve C. A triangular map is a representation of R as the union of a finite number of disjoint point sets called cells, where the cells are of three kinds, vertices, edges and faces (said to be of dimension 0, 1 and 2, respectively) where each vertex is a single point, each edge is an open arc whose ends are distinct vertices and each face is a simply connected open region whose boundary consists of the closure of the union of three edges. Two cells of dimension are incident if one is contained in the boundary of the other. Two cells of the same dimension are incident if their closures are not disjoint. Vertices and edges are external if they are contained in the boundary of R. Otherwise they are internal. A rooted triangular map is a triangular map in which one external vertex is distinguished as the root vertex and an external edge incident with the root vertex is distinguished as the root edge. Two triangular rooted maps T and T * are isomorphic if there exists a bi-unique mapping / of the cells of T onto the cells of T* which preserves dimension and rooting, and both / and f~x preserve incidence. Isomorphism is clearly an equivalence relation and, as usual, we enumerate only the number of isomorphism classes of such maps. A map is said to be of type [n, m] if it contains precisely m + 3 external vertices and n internal vertices. Received April 14, 1964 and in revised form June 4, 1964.. 139.

(142) 140. R. C. MULLIN. With every triangular map there is an associated linear graph whose edges and vertices are the edges and vertices of the map and whose incidence relations are those induced by the map. A polygon in a linear graph is a connected di-valent subgraph. If the polygon contains all vertices of the graph, it is called Hamiltonian. In the following we refer to a polygon in the graph of a map as a polygon in the map. A map will be called an iϊ-map if it contains at least one Hamiltonian polygon. A Hamiltonian-rooted map is a rooted iί-map in which a Hamiltonian polygon is distinguished as root polygon. A Hamiltonian polygon H in a map T is said to be internal if no external edge of T is an edge of H. 2. Enumeration of Hamiltonian polygons. We enumerate Hamiltonian-rooted maps in terms of the parameter k = m + 3. We begin with an important special case. A map of type [0, m] will be called a sliced polygon. Every sliced polygon contains exactly one Hamiltonian polygon, namely that consisting of external edges and vertices. It is well-known that there are {l/(fe - 1)} (Ψ ~f) sliced rooted polygons of type [0, m] where k = m + 3. To determine the number of Hamiltonian-rooted maps and the number of Hamiltonian maps in which the root polygon is internal, we introduce the concept of a Λ-perm. We define a k-perm as an ordered set of k objects, these objects being either directed edge graphs (that is, a graph consisting of a single edge one of whose ends is distinguished as the root) or sliced rooted polygons. We shall show a one-to-one correspondence between. Figure 1.

(143) HAMILTONIAN POLYGONS IN TRIANGULAR MAPS. 141. Hamiltonian-rooted maps and Λ -perm, sliced rooted polygon pairs. Let H be the root polygon of a Hamiltonian-rooted map of type [n, m] m > 0. Then H contains n + k vertices. Beginning at the root vertex one can enumerate the vertices of the polygon in one of two orders Sλ and S2. In one of these, say S19 the other end of the root edge will be the second external vertex to appear in the sequence, since the external vertices must appear in the Hamiltonian polygon in the same relative orders as they appear in the external polygon. Their order of appearance, of course, depends on the direction in which the polygon H is traversed. (See Figure 1.) To construct the corresponding &-perm, one may proceed as follows. Consider the sequence of vertices S l β Label the root vertex elf the other end of the root edge e29 and the remaining external vertices as ^3, β\ > **» βfc in accordance with their order of appearance. This defines k arcs of H, A1 = e^, A2 — e2e3, , Ak = efcele Then in position j in the &-perm, (j = 1, 2, •••, k) one constructs an arc Bj homeomorphic to Aj, with ends labelled bj, bj+1 being the correspondents of ej9 ej+ί respectively, the labelling being done modulo k. If B3 consists of more than one edge, we join bj and bj+1 to complete the polygon which is now labelled B. Two vertices are joined by a line segment across the interior of the polygonal region bounded by Bj if and only if they are joined by an edge interior to the region defined by the correspondent of Bj in the original Hamiltonian rooted map. Clearly this produces a sliced polygon in position j of the λ -perm, this may be rooted by taking the vertex bj and edge bj bj+1 as root vertex and edge respectively. This produces a ά-perm with k ^ 2. To obtain the associated sliced polygon, one selects a polygon P homeomorphic to the Hamiltonian polygon H. The vertex corresponding to the root vertex of H is taken as the root vertex of P and the edge defined by the first pair of elements in S± is the root edge of P. Vertices are joined by a line segment across the interior of the polygonal region determined by P if and only if their correspondents are joined across the interior of region bounded by H. This produces a rooted sliced polygon. The total number of vertices in the /c-perm is 2k + n. Up to isomorphism, the /b-perm and polygon pair produced in unique. Evidently if one is given a &-perm with 2n + k vertices and sliced rooted polygon with n + k external vertices, one can construct a corresponding Hamiltonian rooted triangular map. We further note that the root polygon of such a map is internal if and only if the corresponding λ -perm has no elements consisting of a single edge only. Let anΛ denote the number of Λ-perms which correspond to a Hamiltonian map with n internal vertices. Using the formula for sliced rooted polygons,.

(144) 142. (2.1). R. C. MULLIN. an,k =. This is evident if one considers the fc-perms in which the ith component contains the correspondents of a{ internal vertices. Noting that 2x (which we shall denote by A(x)) we see that antk is the coefficient of xn in the expansion of Ak(x). Noting that A(x) is defined by the equation (2.2). A(x) = 1 + xA\x) ,. we employ Lagrange's power series expansion theorem [10, p. 132] to obtain iϊo. n\(n + k)\. (There is no ambiguity involved here since (2.2) has only one solution which is analytic at the origin). Hence ntk. _ k(2n + fe - 1)1 n\ (n + k)\. If VnΛ denotes the number of Hamiltonian-rooted maps of type [n, m] then since there are (2n + 2k — A)\/(n + k — 1)! (n + k ~ 2)1 sliced rooted n + k — polygons to accompany the A -perms (2 3. ). v. k{2n + 2k- 4)! (2n + k - 1)! % (n + k - 1)! (n + k - 2)1 n\ (n + k)\. If we wish to enumerate internally rooted iϊ-maps of type [n, m], we must determine the quantity (2.4). bntk =. k. Π. v. 2-χ+oi. =n. 2. — ( " < ) where all a, > 0 .. «=i (Xi + 1 V MJ. Hence we must find the coefficient of xn in the expansion of Γ ( l - 2 a ) - τ / l -4a? T L 2x J This author sees no immediately apparent way of obtaining a closed form expression for bntk by use of the Lagrange formula. However we note that the differential equation for.

(145) HAMILTONIAN POLYGONS IN TRIANGULAR MAPS. - 2% - Vl - Ax]k. im = (. (2.5,. 143. is x[Ax2 - x]N"(x) + x[Qx - l]N'(x) + k2N(x) = 0 . This gives the recursion formula (2 6). h. 2 ( r. -. ~ 1 ) ( 2 r ~ *> b. with bktk = 1. Hence (2Ί). {. '. b. }. *-k(n ~~λ\. n k. '. 2. (2n ~ 1 ) ( 2 ^ ~ 3 ) " ' ( 2 f c - 3 ). U - &/ (fc +W)(fc +n - 1) ... (2k + 2). If qnίk represents the number of internally rooted iϊ-maps of type [n, m], then , *. ( 2 8 V ;. qn k. '. =. k(2n + 2k- 4)! (2n)l n(n + k- 1)! (n + k - 2)! (k + n)\ (n - k)\ '. and if rntk denotes the ratio qn>k/pn,k (2.9). rnth =. (2n)\ (n - 1)! - fc)! (2n + k- 1)1#. It is shown in [4] that the number of rooted triangular maps of type [n, m] is 2*+1(2m + 3)! (Zn + 2m + 2)! 2 (m + I)! nl (2n + 2m + 4)! Hence the average number of Hamiltonian polygons in a rooted map is (2 10). m. + 3 (m + ί)! 2 (2n + m + 2)! (2n + 2m + 2)! (2w + 2m + 4)!. Asymptotic formulae. In the following asymptotic approximations, we keep the number of external edges fixed, and observe the behaviour of the formulae as n —> oo. We find that (2.11) and. pn,k ~ —2 i n + * k - 5 n~* , π.

(146) 144. R. G. MULLIN. Therefore (2.12). rn,. Also the average number of Hamiltonian polygons in a rooted map is asymptotically 2. 8(m + 3)(m + I)! / 32 y + γ 32 y ^. /OK*. m. which increases without bound as n tends to infinity. 3* A coincidence? A rooted trivalent map is the dual of a rooted triangular map in which the external boundary contains three edges . The trivalent map is rooted by calling the edge corresponding the root edge of the triangular map the first major edge, and the edge corresponding to the external edge not incident with the root vertex the second major edge, and the correspondent of the remaining external edge the third major edge. Tutte [6] has shown that if un denotes the number of rooted trivalent maps in which a Hamiltonian polygon passing through the first and second major edges is distinguished as root polygon, then _ 1 (2n)\ (2n + 2)! 2 n\ (n + I)! 2 (n + 2)! ' We note that in this enumeration a restriction is placed on the root Hamiltonian polygon, namely it must pass through the first and second major edges. If we reduce the restriction to merely insist that the Hamiltonian polygon pass through the first major edge, the number of Hamiltonian rooted trivalent maps is »_. (2n)l (2n + 2)! 2 n\ (n + I)! (n + 2)! '. We compare this with the number of Hamiltonian rooted triangular maps of type [n, —1] as given by equation (2,3). =. 2. (2n)l (2n + 1)! nl2 (n + 1)! (n + 2)!. =. (2n)\ (2n + 2)! n\ (n + I)! 2 (n + 2)!. =. #. However u* is the number of Hamiltonian polygons in the class of maps obtained by dualizing the class of rooted triangular maps of type [n, —1]; we call these duals "almost trivalent maps" and observe.

(147) HAMILTONIAN POLYGONS IN TRIANGULAR MAPS. 145. that they have 2^ + 1 vertices. Under dualization we understand that the root face of the original map is the root vertex of the dual, etc. Indeed rooted "almost trivalent" map with 2n + 1 vertices is equivalent to a rooted trivalent map of 2n vertices obtained by considering the divalent root vertex and its incident edges as a single edge, which we take as the first major edge. The remaining end of the root edge is incident with two more edges which are taken as second and third major edge according to a suitable convention, such as the second major edge is adjacent to the root face, whereas the third is not. Clearly every Hamiltonian polygon through the first major edge of the trivalent map is a Hamiltonian polygon in the "almost trivalent" map. Furthermore the above construction is reversible. Hence we conclude that the number of Hamiltonian rooted maps of type [n, — 1] is equal to the number of Hamiltonian rooted maps in its dual class, "almost trivalent" maps with 2n + 1 vertices. But the duals of some of the maps with Hamiltonian polygons contain no Hamiltonian polygons, so that the coincidence is not explained by duality. The author wishes to thank the referee for his useful suggestions in the revision of this paper. REFERENCES 1. W. G. Brown, Enumeration of non-separable planar maps Canad. J. Math. 15 (1963), 526-545. 2. , Enumeration of triangulations of the disc, Proc. London Math. Soc. 14 (1964), 746-68. 3. F. Harary, Unsolved problems in the enumeration of graphs, Publications Math. Inst. Hungar. Acad. Sci. 5 (1960), 63-95. 4. R. C. Mullin, The enumeration of rooted triangular maps, Amer. Math. Monthly, 7 1 (1964), 1007-10. 5. W. T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38. 6. , A census of HamAltonian polygons, Canad. J. Math. 14 (1962), 402-417. 7. , A census of slicings, Canad. J. Math. 14 (1962), 708-722. 8. , A census of planar maps Canad. J. Math. 15 (1963), 249-271. 9. , A new branch of enumerative graph theory, Bull. Amer. Math. Soc. 65 (1962), 500-504. 10. E. T. Whittaker and G. N. Watson, A course of modern analysis (Cambridge, 1940). UNIVERSITY OF WATERLOO.

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(149) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. FAMILIES OF PARALLELS ASSOCIATED WITH SETS E. E. ROBKIN and F. A.. VALENTINE. There exist sets S in Euclidean space En which have an interesting association with a family & of parallel lines. For instance S and & may be so related that each point of S lies on a member of & which intersects S in either a line segment or a point. There exist compact sets S c E2 such that every finite collection of points in S is contained in some collection of parallel lines each of which intersects S in a single point, and yet no infinite family & of parallel lines exists having the same property and covering S. This paper contains a theorem which enables one to determine the existence of a family of parallel lines each of which intersects S in a line segment or point and which as a family covers S. Secondly we show that the points, the closed line segments, the closed convex triangular regions, and the closed convex sets bounded by parallelograms are the only compact convex sets B in E2 which have the following property. If A is a closed connected set disjoint from B and if every 3 or fewer points of A lie on parallel lines intersecting J5, then A is covered by a family of parallel lines each of which intersects B. Finally, we obtain a theorem of Krasnoselskii type. Intuitively, this may be stated as follows. Suppose S is a compact set in En and suppose there exists a plane H such that every n points of S can see H via S along parallel lines. Then all the points of S can see H via S along parallel lines. The above results appear in Theorems 3, 2, 1 in that order. The appendix at the end contains the theorems of Helly, KrasnoseΓskii and other results used. Furthermore, the reader is recommended to consult the compendium "Helly's theorem and its relatives" by Danzer, Klee and Grϋnbaum [1]. In order to proceed logically we adopt the following notations. NOTATION. If S is a set in ^-dimensional Euclidean space En, then closure of S = cl S, interior of S = int S, boundary of S — bd S, convex hull of S — conv S. If x e En, y e EnJ xΦ y, then L{x, y) = line containing x and y, xy — closed segment joining x and y, intv xy = relative interior of the segment xy, R(x, y) = ray having x as endpoint and containing y. The empty set is indicated by 0 and the origin of En by 0 . Set union, intersection and difference are denoted by U, ΓΊ and ~ respectively. Received July 6, 1964. The preparation of this paper was sponsored (in part) by the National Science Foundation Grants GP 1368 and 1988. 147.

(150) 148. E. E. ROBKIN and F. A. VALENTINE. Parallels. THEOREM 1. Let S he a closed set in n-dίmensional Euclidean space En. (a) Suppose there exists a hyperplane H such that S Π H is compact. ( b ) Suppose for each integer s such that 1 ^ s ^ n and for each set of distinct points xlf x2, , xs in S ~ H there exist points Vi> y%i " y Vs in H such that x^i c S (i = l, ,s) and such that 9 χ aτ #il/i> %2VΪ> ** y sVs & parallel. (The y19 * 9ys need not be distinct.) Then there exists a family & of parallel lines such that for each point xeS~ H there exists a point yeH such that xycS and such that the line L(x, y) belongs to &m. Proof. To each xe S ~ H, let C(x) denote the union of all lines L(x, y) where yeH such that xy c S. Choose a point 0 as origin in En with 0 ί H, and let D{x) be that translate of C(x) so that x goes to 0 . Define M(x) as follows, when xe S ~ H, M(x) = conv (H n D(x)) . Since HΠ D(x) Φ 0, we must have M(x) Φ 0 for each xeS ~ H. Hypothesis (b) implies that every n or fewer members of the collection {M(x), xe S ~ H) have a point in common. Since the dimension of H is n — 1, and since the members of {M(x)9 xe S ~ H} are compact convex sets in H, Helly's theorem [2] (see Appendix) for (n — 1)dimensional space implies (1). Γl. M(x)Φ0.. Since 0 £ H, condition (1) implies there exists a line L through 0 such that L Π M(x) Φ 0 for each xe S ~ H. We let & denote the family of all lines parallel to L and intersecting S. We will prove that & has the desired property stated in the theorem. To do this choose a point xe S ~ H, and let L(x) be the line through x parallel to L. Let L(x) Π H = y and suppose xy φ S. Since S is closed then there exists a point u e intv xy and a closed solid sphere U(u, r) with center u and radius r such that S Π U(u, r) = 0. Let U(v, t) be a closed solid sphere with center vexu and radius t with t < r. Define K{v) as follows, K(v) = conv [U(u, r) U U(v, t)],vexu. .. Since S is closed, and since K(u) ΓlS = 0, K(x) Π S Φ 0, there exists a point w e xu nearest to u such that Sf)bd K(w) Φ 0, Sf] int K(w) — 0. Choose a point z such that.

(151) FAMILIES OF PARALLELS ASSOCIATED WITH SETS. 149. z e S Π bd K(w) φ 0 . Clearly D(z) contains no line L(z, y^ parallel to the line L(x) with y1eM{z). This contradicts the statement following (1). Hence xyaS, and Theorem 1 has been proved. In order to express the next theorem easily the following concepts are used. DEFINITION 1. A set A has the parallel property P(m) relative to a set B if every m or fewer points of A all lie on a family of parallel lines each of which intersects B. A set A has the parallel property P(A) relative to B if all the points of A lie on a family & of parallel lines each of which intersects B. THEOREM 2. Let B be a compact convex set in the Euclidean plane E2. Suppose that each closed connected set A in E2 which is disjoint from B and which has property P(3) relative to B also has the property P(A) relative to B. Then B is either a point, a closed line segment, a closed set bounded by a triangle or a closed set bounded by a parallelogram.. Proof. Since this theorem characterizes the sets B, the proof consists of two parts. Let sf = {A} denote the collection of all those closed connected sets A which are disjoint from B and which have property P(3) relative to B. First, suppose B is either a point, a closed line segment, a closed triangular region or a closed region bounded by a parallelogram. Case 1. Suppose bdB is a parallelogram with consecutive vertices alf a2, a3, α4. The four lines determined by the four edges of B divide.

(152) 150. E. E. ROBKIN and F. A. VALENTINE. the plane E2 into nine parts. Let V(ai) (i = 1, 2, 3, 4) denote the unbounded open V-shaped region abutting B at α* (i = 1, 2, 3, 4), and let V(a19 a2) be the closed unbounded region abutting B along axa29 etc. Since A e s^f is connected and disjoint from B9 if A Π V{a,i) Φ 0, A Π V(ai+1) Φ 0, (i = 1, 2, 3, 4; αδ = αx), the set A would not have the property P(2) relative to B. Hence, it would not have property P(3) relative to B. Therefore, we may relabel the vertices of B so that (2 ). A c V(a19 a2) U V(a2) U V(a29 α3) .. For xe A9 let C(#) denote the union of all rays emanating from x which intersect B, and let D(x) be that translate of C(x) which sends x to the origin 0 of E2. Since B is compact and convex, and since x 0 B9 the set C(x), and hence D(x)9 is a closed convex cone which is not all of E2. Let C be the unit circle with center at 0 so that C=[x:\\x\\ = ί\. Define M(x) as follows, C Π D(x) = M(x), x e A . Consider the collection of sets ^t ΞΞ {M(x)f x e A} . Property P(3) implies that there exists a semicircular arc CΊ of C such that every two members of ^ have a non-empty connected intersection with Cx. To see this observe that if in (2) we have A Π V(a2) Φ 0, then for each point xe AΠ V(a2) we have a connected intersection M(x) Π Mix,) Π M{x2) Φ 0 for every pair of points x19 x2 in A. If A Π V(a2) — 0 then either A c V(a19 a2) or A c V(a2, α3), and the above italicized statement is also still true. (It is instructive to observe that condition P(2) does not suffice to imply the above italicized sentence.) We may now apply Helly's theorem [2] (see Appendix) to the set ^/ί to yield the existence of a paint ue M(x) for all Me ^£. Let L be the line determined by 0 and u. For each xe A, let L(x) denote the line through x parallel to L. The above facts imply that L(x) Π B Φ 0, x e A9 by virtue of the definition of C(x). Hence A has property P(A) relative to B. Case 2. Suppose B is a closed set bounded by a triangle with vertices a19 a29 α3. As argued in case 1, we may relabel the vertices so that.

(153) FAMILIES OF PARALLELS ASSOCIATED WITH SETS. 151. if A e sf, where V{a^) is the open V-shaped region abutting B BX aly and where V(alf a2) similarly abuts B along α ^ . The rest of the proof is exactly the same as Case 1. Case 3. Suppose B is a closed segment α ^ . If A e Sf, then A either lies on L(a19 α2) or in one of the open half-spaces bounded by L(au α 2 ). The proof is either trivial or exactly the same as Case 1, or, for that matter, as Theorem 1. Case 4. If £ is a point, then A must lie on a line through B, and the conclusion is trivial. This completes the first part of the proof. (I) Secondly, to complete the characterization, suppose B is a compact convex set which is neither a point, a line segment, a triangular region or a set bounded by a parallelogram. We will prove that for such a set B there exists a closed connected set A, disjoint from B, which has property P(3) relative to B, and which does not have property P(A) relative to B. In order to construct A we use the familiar concept of "exposed point." DEFINITION 2. A point x in the boundary of a convex set S c E2 is an exposed point of S if there exists a line L of support to S at x such that S Π L = x.. To construct A, let xxx2 be a diameter of the set B described in the italicized statement (I). The points x19 x2 are exposed points of B since the line L { through xt (i = 1, 2) and perpendicular to xγx2 is such that Li Π B = Xi (i = 1, 2). Let H be one of the open half-planes bounded by L(xlf x2) such that H Π B Φ 0, since B (£ L(x19 x2). It is a well-known elementary fact that B contains at least one exposed point in H. If B contains one and only one exposed point in each of the open half-planes H and E2 — cl H, then B is a quadrilateral. (We have excluded the case of a parallelogram, here.) On the other hand, if B contains only one exposed point in H and none in E2 ~ cl H then B is bounded by a triangle, which is also excluded here. For the moment, suppose A contains at least two exposed points xB and x± in H. Without loss of generality, suppose xu x2, x4, x3 occur on bd B in that order as illustrated in Figure 2. (In Fig. 2, the dotted lines and curves, except for the points x19 x2, x3, x± miss B.) Let L(Xi) (i = 3, 4) be two lines such that BΓ\L(Xi) = xt. Observe that conv (Xi U cc8 U Xi U x2) c B . There exist points xtj (i, j = 1, 2, 3, 4, i < j) such that.

(154) E. E. ROBKIN and F. A. VALENTINE. 152. Xι. Fig. 2.. xd = L(xt) Π L(xs) (i < j , i, j = 1, 2, 3, 4, (i, j) Φ (1, 2)) . To construct the set A, illustrated in Figure 2, extend the segment xuxί3 to x34y13 and xMx2i to #34?/24 so that , Vu) ΓΊ L(x19 L(x2,. xs)Γ\Hφ Xt)Γ\Hφ. Recall that xΆ e £Γ, x4 e £Γ. Furthermore, replace a segment α6 of with midpoint x3 by a semicircular are C3 with endpoints a and 6 and with C3 Π J5 — 0. Introduce a corresponding arc C4 at cc4 (see Fig. 2). The set A defined as follows A = {ylza) U C 3 U φxu). U (x3ic) u C 4 U (dy2i). is illustrated in Figure 2. We may choose the arcs C3 and C4 sufficiently small so that A clearly has property P(3). To see this observe first that A ~ C4 has property P(2). Hence, to see that A has property P(3) one merely has to demonstrate that if xeC±,ye A, ze A, the triple {x, y, z) has property P(3) relative to B. However, clearly the set A does not have property P(A) relative to B. This completes the proof when B in statement (I) has two exposed points in H. The only case remaining in this part of the proof is that in which B is a quadrilateral which is not a parallelogram. So to complete the proof suppose B is such a quadrilateral. In this case there must exist some two vertices of B, say x± and x2, such that the other two vertices x3 and.

(155) FAMILIES OF PARALLELS ASSOCIATED WITH SETS. 153. x4 of B are interior to a strip bounded by two parallel line L(xλ) and L(x2) at xx and x2 respectively such that xs and x± lie on the same side of L(x19 x2). Hence, we have a situation which is essentially the same as in Figure 2 (perpendicularity was not essential), and the same construction can be carried out to yield a set A having property P(3) relative to B but not P(A). This completes the proof. There exist further results related to property P{m) and these will be presented in a subsequent paper. It should be mentioned that if in the hypothesis of Theorem 2 we replace P(3) by P(2) then B must he either a point or a line segment. Also it is easy to prove that if B is a compact strictly convex body in E2 and if m is a prescribed integer, there exists a compact connected set A which is disjoint from B, which has property P(m) relative to B but which does not have property P(A) relative to B. THEOREM 3. Let S be a closed connected set in the Euclidean plane E2. Suppose there exist two points a and b in S such that the following holds. If x1 and x2 are points in S then there exist some two parallel lines, denoted by Lγ and L2, such that L* ΓΊ S = a^ and such that Z^Π ab Φ 0 (i = 1, 2). Then there exists a family &> of parallel lines in E2 such that each point of S is contained in a member of £P which intersects S in either a line segment or a point.. Proof. If x e S, by hypothesis there exists a line L(x) through x such that (4). Sf] L(x) = x,abf) L{x) Φ 0 .. For x e S, let C(x) denote the union of all possible lines L(x) satisfying (4). We will prove first that C(x) is a two-napped cone, each nappe of which is convex, although it need not be closed. To prove this, we consider two cases. First, suppose xgab. Suppose L^x), L2(x) are two lines in C(x) through X. Choose an arbitrary line L(x) through x such that L(x) intersects ab between ab Π Lx{x) and ab Π L2(x). We will show that L(x) Π S = x, so that L(x) c C(x). The proof is indirect. Suppose a point y exists such that y e S PΪ L(X), y Φ X. By hypothesis, there exists a line L(y) through y such that S Π L(y) — y, ab Π L(y) Φ 0. (See Fig. 3. In this figure, the dotted lines, except for the points x and y, miss the set S.) Since L(y) (Ί intv ab Φ 0, L(x) Π intv ab Φ 0, it is a simple matter to verify that the deletion from E2~ S of an appropriate ray of L(y) together with an appropriate ray from Lt{x) or L2{x) separates the plane into two disjoint open parts, one of which.

(156) 1 5 4. E. E. ROBKIN and F. A. VALENTINE. >„. xeab Fig. 3.. contains a and the other of which contains 6. However, this violates the fact that S is connected. Hence, we have a contradiction so that bf]L(x) = χ. Hence, when x$ab, the set C(x) is a two-napped cone each nappe of which is convex. Secondly, suppose xe S n ab. The proof follows the same pattern as in the case x$ab, (see Fig. 3). The obvious details are made selfevident there. To complete the proof, choose an origin 0 in E2 so that 0 e L(a b) Let D(x) be that translate of C(x) so that * goes to 0 . Consider the collection ^f, defined as follows ^ T = {M(x) = L(a, b) n cl D(x), x e S}. We have shown that every two members of ^ have at least one point m common. Furthermore, if x$L(a,b), then M{x) is a compact interval. If xeab the set M(x) may be unbounded (although closed and convex), however, this will cause no difficulty because if Saab then n.e*M(x)Φ0 follows trivially, and if S £ ab Helly's theorem L2J can be used to yield the existence of a point u such that w e Π M(x) . xes. Let L be the line determined by the two points u and 0 , since 0 ^ w Let ^ denote the set of all lines in E2 which are parallel to L and which mtersect S. We will prove that & is a family as described in iheorem 3, and the proof is indirect. To do this, for xeS let L(x) denote that line through x which belongs to & so that L(x) and L.

(157) FAMILIES OF PARALLELS ASSOCIATED WITH SETS. 155. are parallel. Suppose a point xe S exists such that S Π L(x) is not connected. Since S is closed if S Π L(x) is not connected there exist points c and d in L(x) such that ce S, de S, c^d, S Γ) intv cd = 0. The segment cd is usually called a cross-cut of the complement of S. Since S is connected, there exists a component i£" of the complement of S such that the removal of intv cd from K yields two disjoint parts of K9 at least one of which is bounded (see Fig. 4) which we denote by. Kλ. L{x). There exist points yebdKx with y g L(c, d), sufficiently close to L(x) such that the L(y) e & parallel to L(x) intersects Kx and also intersects bdK in a nonconnected set. Since cl C(y) is the closure of a nonempty two-napped cone, each nappe of which is convex, there exists a line Lt{y) c C(y) through y9 sufficiently close to L(y) (in terms of angles), such that Ljiy) ί l S ^ t / , a contradiction (see Fig. 4). Hence, we have proved that & is a desired family, and the proof is complete. The hypotheses of Theorem 3 do not imply that a family & necessarily exists such that each x e S is contained in a member of & which intersects S in just the point x. In fact the following is true. There exists a compact set S(zE2 such that every finite collection of points x19 a?2, , x% in S is contained in some collection of parallel lines L19 L2, , Ln such that 1^ (Ί S = xi (i = 1, , n), and yet no family gP of parallel lines exist such that each point of S is contained in a member of ^ which intersects S in just one point. We exhibit such a set S as follows. Let (xi9 y^ denote rectangular coordinates of a point We define the sequence of points {pi9 i = 1, 2, •} in E2 as. EXAMPLE.. Pi in E2. follows,.

(158) 156. E. E. ROBKIN and F. A. VALENTINE. (5). so that (xlt yt) = (1, 0), (x2, y2) = (1,1), (x3, #,) = (1/3, 0), etc. Define S as follows S = cl U PiPi+i i. so that S is the increasing limit of a sequence of zig-zag polygonal paths. Beginning at (1, 0), the odd segments are vertical, and the even segments have finite positive slope. Observe that 2n-l *-~. X2n-2. — %2n-l. % >. - °°. =. ^. 2. so that the even segments have slopes approaching oo. It is a simple matter to verify this set S has the property described in the above italicized statement because of (5) and because x2n — y2n (n = 1,2, •). For the concluding result we need the following concepts used by Horn and Valentine [4]. DEFINITION 3. The set B is a set of visibility for a set S in En if for each point xe S there exists some point yeB such that xy c S. DEFINITION 4. A set SaEn is said to be an L2 set if each pair of points in S can be joined by a polygonal arc consisting of at most two line segments.. Horn and Valentine [4] proved that a simply connected compact L2 set in E2 is expressible as the union of convex sets every two of which have a point in common. No simple characterization of nonsimply connected compact L2 sets has ever been given. The following theorem is a step in that direction. 4. Let S be a compact L2 set in En (see Definition 3), ( a ) then each hyperplane in En has a translate which intersects S in a set of visibility for S, ( b ) also each (n — 2)-dimensional flat is contained in a hyperplane which intersects S in a set of visibility for S. THEOREM. Proof. For each point xe S, let S(x) denote the set of all points y such that xy c S. Also define C(x) to be C(x) — conv S(x) ..

(159) FAMILIES OF PARALLELS ASSOCIATED WITH SETS. 157. Since S is compact, the set C(x) is compact. Since every two members of the collection {C(x), x e S} have a point in common, a theorem of Klee [6] implies that each hyperplane H' has a translate H which intersects every C(x), x e S. Since S(x) is the union of rays having x in common, the fact H Π C(x) Φ 0 implies H Π S(x) Φ 0. Hence, for each point xe S, there exists a point yeHΠS such that xy c S. This establishes (a). In the same manner a theorem of Horn [3] implies that each (n — 2)-dimensional flat is contained in a hyperplane which intersects every C(x), xeS, and the remainder of the proof of (b) is identical to that given for (a). Appendix THEOREM (Helly [2]). Let ^ he a family of compact convex sets in En containing at least n + 1 members. If every n + 1 members of ^~ have a point in common, then all of the members of j ^ * have a point in common. THEOREM (KrasnosePskii [5]) Let S be a compact connected set in En. Suppose that for every n + 1 points xi e S (i — 1, , n + 1) there exists at least one point yeS such that x^a S (i = 1, , n). Then there exists a point pe S such that xpa S for each point xe S. THEOREM. Let j ^ ~ be a family of bounded closed convex sets in a Euclidean space E. Suppose Jf contains at least n members. Suppose every n members of J^~ have a point in common.. (Klee) Then every flat of deficiency n — 1 has a translate intersects every member of ^Z. which. (Horn) Every flat of deficiency n is contained in a flat of deficiency n — 1 which intersects every member of ^ . BIBLIOGRAPHY 1. L. Danzer, B. Grunbaum and V. Klee, Helly's theorem and its relatives, Symposium on Convexity, Proceedings of the Symposia in Pure Math. 17 Amer. Math. Soc. (1963). 2. E. Helly, Uber Mengen konvexer Korper mit gemeinschάftlichen Punkten, Jber. Deutch Math. Verein 32 (1923), 175-176. 3. Alfred Horn, Some generalizations of Helly's theorem on convex sets, Bull. Amer. Math. Soc. 5 5 (1949), 923-929. 4. Alfred Horn and F. A. Valentine, Some properties of L sets in the plane, Duke Math. J. 16 (1949), 131-140. 5. M. A. Krasnoselskii, Sur un critere pour qu'on domain soit etoile, Math. Sb. (61) 19 (1946), 309-310. 6. Victor Klee, On certain intersection properties of convex sets, Canad. J. Math. 3 (1951), 271-275. 7. F. A. Valentine, Convex sets, McGraw-Hill Book Co. (1964). UNIVERSITY OF CALIFORNIA, LOS ANGELES.

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(161) Pacific Journal of Mathematics Vol. 16, No. 1, 1966. COMMUTATIVE. F-ALGEBRAS. MELVIN ROSENFELD. We extend several theorems for commutative Banach algebras to topological algebras with a sequence of semi-norms (F-algebras). The question of what functions "operate" on an F-algebra is considered. It is proven that analytic functions in several complex variables operate by applying a theorem due to Waelbroeck. If all continuous functions operate on an F-algebra, then it is an algebra of continuous functions. However, unlike the situation for Banach algebras [6], it is not true that if V operates the algebra is C(Δ). This will be shown by an example. A theorem due to Curtis [4], concerning continuity of derivations when the algebra is regular is extended to F-algebras. The result is applied to an algebra of Lipschitz functions to show that it has only a trivial derivation. Preliminaries* Throughout this paper the letter A will stand for a commutative i^-algebra. An F-algebra is a topological algebra with topology determined by a sequence of algebraic semi-norms. The nth semi-norm of an element x in A will be written || x \\n. We may + and shall always assume that for all x in A, \\x\\ntί ||$IL+i. ^ will denote the topological space of all continuous multiplicative linear + functionals on A with the weak* topology. A will denote A minus the zero functional with the relativized topology. For x in A, x will + + be the function in C(/l ) (the continuous functions on A with the compact-open topology) defined by x{φ) = φ{x). A will be called regular if given φ0 in A and V a neighborhood of φ0, there is an element x in A such that φo(x) — 1 and φ{x) — 0 for φ $ V. A will be called semi-simple if x — 0 implies x = 0. A basic device in the study of F-algebras is to represent A as the inverse limit of a sequence of Banach algebras {An} where An is the completion of A\In with norm ||flc + i»|| = ||aj|L and In is the ideal of all x in A such that ||a?|| Λ = 0. The homomorphism πmtn\ An —> Am for m ^ n is defined as the completion of the mapping x + In —• x + Im. This representation enables one to construct an element in A by constructing a sequence {xn} such that for each n, Received September 8, 1964. This paper is based on a portion of the author's Ph. D. dissertation written at the University of California, Los Angeles under the supervision of Professor P.C. Curtis Jr. The research was supported in part by the United States Air Force, Office of Scientific Research under contract No. AFAFOSR 62-140 and in part under a contract from the National Science Foundation No. NSFG 18999. 159.

(162) 160. COMMUTATIVE F-ALGEBRAS. xne An and τrm>nxn = xm. The homomorphism πn: A —* An is defined as x—>x + In. Then TΓ*: (multiplicative linear functionals in An)—+A+ is continuous and one-to-one and so its range, which we shall denote by A+ is a compact subset of A+. If K is an arbitrary compact subset + of A , there is an integer n such that K £ At [9]. The following theorem, due to Silov, is also valid for F-algebras. + If C is a closed and open subset of A and the zero homomorphism is not in C, then there is an idempotent e in A such that + C = {φe A : φ(e) = 1}. The extension to F-algebras is proven via the device of the previous paragraph. With the aid of Silov's theorem the proof that if A is regular, then A is normal is essentially the same as for Banach algebras. Since so many of the theorems true for Banach algebras are also true for F-algebras with almost the same proofs, it is perhaps appropriate to remark that the difficulties introduced by the sequence of semi-norms are sometimes quite subtle. For example such a seemingly innocuous question as whether a multiplicative linear functional is necessarily continuous is still unanswered. Functions that operate on a commutative semi-simple Falgebra* A function /: D g C —> C is said to "operate" on an F-algebra A if foχe A whenever xe A and the range ΐ g ΰ . It is not difficult to adapt Katznelson's proof in [5] to show that if every continuous function operates on A, then A — C{Δ). However another theorem due to Katznelson which states: If A is a self-adjoint Banach algebra and V operates on the positive functions in A, then A — C(A) is no longer true for F-algebras; as the following example shows. Let H be the subalgebra of l°° consisting of those sequences {αj for which there is a number, a such that | an — a \lln —»0. Let Hr be the subalgebra of H consisting of those sequences for which a = 0. Let τ be the linear transformation from H' to the entire functions defined by r({αft})(λ) = Σ?=o <^λΛ For each integer N and for {an} e Ht defined || {an} \\N = sup [| r({αn})(λ) | : | λ | ^ N]o || — ||^ is evidently a vector space norm. It is also algebraic; for suppose {αj and {bn} e H\ f = τ{{an}), g - τ{{bn}) and F = τ({anbn}).. F(X) = (l/2πi) (. Then. f(ιv)g(X/w)dwlw .. J\w\=M. Hf is a complete .F-algebra under the sequence of norms defined above and H is the F-algebra obtained by adjoining a unit to H\ For n = 0,1, 2, , define zn as the sequence which is 1 in the nth coordinate and 0 in all the other coordinates. These elements f generate H (since the polynomials are dense in the entire functions).

(163) COMMUTATIVE J^-ALGEBRAS. 161. and together with the unit of H generate H. A(H) is homeomorphic to the one-point compactiίication of the integers, the point corresponding to the integer n being the functional sending zn into l β It is evident that H is a self-ad joint subalgebra of C(A(H)), and that H is semi-simple and regular. Yet, although 1/ operates on the nonnegative elements of H, H Φ C(A(H)). n For U an open subset of C let H(U) be the i^-algebra of all holomorphic functions on U with the compact-open topology. For σ n an arbitrary subset of C , let H(σ) be the direct limit of the i^-algebras H(U) for U ranging over open sets containing σ directed as follows: H(U)^H(V) if UQV. Let a19 , an be elements of a commutative F-algebra, say A, with n unit. For ψeA = A(A)9 let σ(φ) be the point in C {φ{a^, - - , φ{an)) and let σ — {σ(φ) : φ e A}. THEOREM. There is a continuous homomorphism τ from H(σ) to A such that φ(τf) — f(σ(φ)) for every φ in A and every f in H{σ) and z(z{) — ai9 i = 1, , n. {Evidently fe H(σ) defines a function on σ.). Proof. Waelbroeck, in [11], proved that such a continuous homomorphism exists for even more general topological algebras providing the elements, a19 "-,an are regular, i.e. have compact spectrum. An element of an _F-algebra needn't be regular, but an element of a Banach algebra is of course regular. We will apply Waelbroeck's theorem to each of the Banach algebras As where A is the inverse limit of {A,}. For every integer k let σk be defined as above for πka19 * ,πkan, let τk be the continuous homomorphism from H(σk) to Ak. V&: σk £Ξ σ and there is a continuous homomorphism vk\ H{σ) —+ H(σk). The essence of the proof is that the sequence {fk} where fkeAk is defined as ?k°Vk{f} satisfies πSitft = fs for s ^ t. For then the sequence {fk} defines an element τf in A. If each Ak were semi-simple, then it would follow that π8ftft = fs for s ^ t. For Waelbroeck's theorem implies that (π ^t/ί) 7 ^ = /,. However, even if A is semi-simple, it does not follow that each Ak is semi-simple. Let s and t be two fixed integers with s g ί . We shall examine the construction of f8. Let b{ = πsa{ for i = 1, , n. fe H(σ) may be considered as a function holomorphic in a neighborhood, say W, of σ and, therefore, of σ s . The following assertions are proven in [11]. (1) σs is convex in the following sense. There is a finite set of polynomials in n variables, say pl9 , pr and neighborhoods Dl9 , Dn of the spectrum of bu , bn respectively and neighborhoods Dn+l1 , Dn+r.

(164) 162. MELVIN ROSENFELD. of the spectrum of bn+1 = px{bu , bn), , bn+r = pr(blf ••-,&„) respectively such that the following two facts are true: (a). σsS-DQ W where D = {\eD1x x Dn : p<(λ) 6 D Λ + i for i = 1, r}. (b). If # = A x x Dn x . . . x D Λ + r and X = {(λ, ^(λ), , p r (λ)): λ e D}, then the restriction mapping, p, from £7 to X is a continuous open homomorphism of H(E) onto H(X) with kernel the ideal generated by {zn+k - pk(zu , zn): k = 1, , r}. By (a), / is a holomorphic function on D and determines a function FeH(X) where F(λ, j)(λ)) is defined to be /(λ) (i.e. F depends only on the first n coordinates). By (b), F = ρ(G) where GeH(E). (2) Define a: H(E) — As by +. a(H) = (l/2πi)* ' \ (X+r —ftn+ r ) " 1 ^ !. * d\n+r. where Γi is a rectifiable curve in D{ including in its interior the spectrum of bι for ΐ = 1, , ?ι + r. α is a continuous homomorphism and a(Zi) = 64 for i = 1, , n + r. Thus, by (b), if ^(Gy = |θ(G) = F, then α(G0 = α(G). f. is defined as α(G). (3) If the system of polynomials p19 , pr and the neighborhoods Du , Dn+r are replaced by another system which meets the condition os S D S ^ then the same element /, 6 As arises. Let {plf . . , p r , A , * ,i5Λ+r} be a system used to to define ft. Suppose Ci = πtα< for i = 1, - , n and cn+k = p^Cj, , c j for A; = 1, , r. Then. j GίλXλ, - cO"1. *..*/. = π.ft(l/2πi)»+' j .. J. (λ, + r - c . ^ ) - 1 ^ .. dXn+r - (l/2πί) % + r. J G(\)(\ - 6,)-1. (κ+r -. K+r^. - ° ° dXn+r — fs . For the system {pl9 •• ,ί?Λ, A , φ ',Dn+r} may be used to define /,: spφi) gΞ sp(^) g A for i = 1, , w + r and ffsg(7(gl)£ TΓ. Thus τ/ is well defined. If φ G z/, then φ& Δk for some integer &, say Vkfo=* for all & τkovkfa-+τk*vkf0 (i.e. for all A?.

(165) COMMUTATIVE F-ALGEBRAS. 163. This theorem, except for continuity of the operational calculus, is also proven in [1] via the Arens-Calderon theorem [2]. Continuity of derivations* A derivation on an algebra A is a linear operator D satisfying D(xy) ~ xDy + (Dx)y for every x and y in A. If A is a commutative jP-algebra, a linear transformation D: A—*C(A) satisfying D(xy) — xDy + (Dx)y will be called a derivation into C{A). It is conjectured that a derivation on a Banach algebra must be continuous. Curtis [4] proved that if a Banach algebra is regular, then any derivation is continuous, in fact any derivation from the algebra to C(A) is continuous. This theorem will be extended to allow the algebra to be an i^-algebra. It will then be applied to some .F-algebras to determine all derivations in these algebras. The following lemma is a modification of one in [3] and its proof is essentially the same. Let t be an algebraic homomorphism from a commutative F-algebra A to a semi-normed algebra B. Let {gk} and {hk} be two sequences of elements in A such that for all n: gnhn = gn and if m Φ n, then hnhm — 0. Then it is not possible that for all n HίflUI >n\\gn\\n\\K\\n. LEMMA.. COROLLARY. If D is a derivation from a regular semi-simple F-algebra A to C(Δ), then D is continuous.. commutative. Proof. Let {Ak} and {Ak} be defined as in the preliminaries. Since every compact subset of A is contained in some ΔN, it suffices to prove that if xn —> 0, then Dxn —> 0 uniformly on each ΔN. The procedure will be to show: (1) for all N there is an at most finite set FN s AN such that Dxn —> 0 uniformly on the closure of [AN\FN]; (2) if φ is isolated in Δ, then Dx(φ) = 0 for every x in A; and (3) if φ e AN is isolated in Δm for every m ^ N, then φ is isolated in A. (1), (2), and (3) imply that Dxn(φ) —* 0 for every φ and this together with (1) implies that Dxn —»0 uniformly on AN. This is basically the some proof as in [4]. The third step is the only novel point in the proof. It does not follow from the fact that every compact set is contained in some ΔN. The example of Arens' ([7] problem 2Έ) shows this. (3) may be proven as follows: Suppose φeAN is isolated in Δm for all m ^ N. By Silov's theorem, for each m ^ N, there is an idempotent em e Am such that φ(em) = 1 and φ'(em) = 0 if φf e Am and φ' Φ φ (identifying Δm with A(Am)). Then, because each /v em is an idempotent and (τr r , β e β ) = er for N ^ r ^ s, πr,se8 = er for N ^ r ^ s (two idempotents in Ar equal modulo the radical are iden-.

(166) 164. MELVIN ROSENFELD. tical). Thus {em} defines an idempotent e in A such that φ(e) — 1 and φ'(e) = 0 f or φ' Φ <p and <p' e Δ. Steps (1) and (2) will be sketched. Proof of (1): Let B be the semi-normed algebra which as an algebra is A, but with semi-norm || a; || = II x 11^ + II Dx \\N. Let F = {φ e ΔN : x —• Dx{φ) is not a continuous linear functional}. Since A is an .F-space, the principle of uniform boundedness applies. Since for each x in A {Dx(φ): φ e ΔN\F} is bounded (by || Z>α? H^-), Dxn—>0 uniformly on ΔN\F. F is a finite set. If not, then there is an infinite sequence {φn} gΞ F with mutually disjoint neighborhoods. Since the algebra is by hypothesis regular, there are sequences {yn}, {zz} such that yn(φn) = 1, ynzn = yn and s«s» = 0 if mΦ n. Then since φn e F, there is an xn in A such that Dxn(φn). \> n\\xn\\n. \\yn\\n-\\zn. \\n.. Thus letting gn = xnyn. and hn =. zn, we have \\gn\\ ^ || Dgn \\N > n \\ gn || n || few || n and this contradicts the previous lemma. Thus we may let F be FN. Proof of (2): Let φe Δ be isolated. Choose, by Silov's theorem an idempotent e such that φ(e) = 1 and ^/(e) = 0 for φf — φ. Then De(φ) = 0 and, by semisimplicity, ex = <p(x)e for any α? in A. Hence 0 = D(ex)(φ) = x(φ)De{φ) + Dx{φ) = Zte(9>) for any a; in i . By the closed graph theorem and the previous corollary, if D is a derivation on a regular commutative semi-simple F-algebra, then D is continuous. Let C°°(R) be the algebra of infinitely differentiable functions on the real line. For / in C°°(R), let 11/II. = Σ2=o sup [|/<*>(t) I : - n ^ t S n]/k ! . C°°(R) is a regular semi-simple F-algebra. If D is a derivation on C°°(R) and a? is the function mapping t into t, then for any polynomial p in x, Dp(x) = p\x)Dx. Since the polynomials in cc are dense in C°°{R) and since Z> is continuous, Df = /'Zto for any / in C°°(i?). As a second application of the previous corollary, we show that the following algebra of Lipschitz functions has no nontrivial derivations. Let a ^ 1. Let La be the subalgebra of C(R) consisting of functions of period 1 with finite norm || — || α where | | / | | α is defined to be sup[|/(t) \:teR]. a. + sup[|/(s + h) - /(β)|/| h \. :seR,hΦ0].. Let 1« = {/e La; ΪEm[| f(s + h) - /(s) |/| fc | α — 0 : A — 0] for s e R}. For α < 1, La is a Banach space, l α a closed subspace, and La is isomorphic to 1** [8]. Let an = 1 — 1/τι and L be Πl/«Λ with the sequence of algebraic norms {|| — | | α J . L may also be defined as the inverse limit of {LaJ. LUn+l £Ξ l t t j i g L α% and so L is also the inverse limit.

(167) COMMUTATIVE F-ALGEBRAS. 165. of {laj. This implies that L — L**, however even more is true: A bounded subset of L must have compact closure, i.e., L is a Montel space. For let S be a bounded set in L S l«w. 1«Λ is isometrically isomorphic as a Banach space with a subspace of C(W*) where W* is a compact set obtained as follows: Let U — {t e R: 0 g t g 1}, F = {(r, s): 0 ^ r ^ 1, 0 < r - s g 1/2} and W = U U F, then W is a locally compact space and W* is its one-point compactification. The isomorphism / - > / is defined by /(oo) = 0, f(t) = /(*), and. To see that S is precompact in L it suffices to show that S is precompact in each lttn or, equivalently, that S is equicontinuous. This follows from the fact that there is a number K such that feS=>\\f\\an+i<LK. The representation of la% as C(W*) is due to DeLeeuw [8]. A derivation D on L must map every element into 0. For L is a regular, commutative, semi-simple F-algebra and so it suffices to show that if fe L, then φ{Df) = 0 for any φ e zί(L). D(f - φ(f)) = X)/ and / — <£>(/) is in the kernel, M, of φ. So it suffices to show that D[M] s M". Since Λf is an ideal, D[M2] S ΛΓ. M2 ^ Λf, but Λf2 is dense in ikf and so, since D must be continuous, D[M] £ M. (Any maximal ideal M must be the set of all functions in L vanishing at some t0 where 0 ^ ί0 < 1. The function sin ([t — tQ]/2π) is in M but not in ikP. Sherbert [10] proved that M2 is dense in M for the Banach algebra l α , in fact for algebras of Lipschitz functions on more general spaces than the unit interval. His proof works as well for L.) The author would like to thank Professors P. C. Curtis, Jr. and R. Arens for their kind help during the preparation of this paper. REFERENCES 1. R. Arens, The analytic-functional calculus in commutative topological algebras, Pacific J. Math. 11 (1961), 405-429. 2. R. Arens and A. P. Calderon, Analytic functions of several Banach algebra elements, Ann. Math. 62 (1955), 204-216. 3. W. Bade and P. Curtis, Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608. 4. P. Curtis, Jr., Derivations of commutative Banach algebras, Bull. Amer. Math. Soc. 67 (1961), 271-273. 5. Y. Katznelson, Algebres caracterisees par les founctions qui operent sur elles, C.R. Acad. Sci., Paris 2 4 7 (1958), 903-905. 6. , Sur les algebres dont les elements non-negatifs admettent des racines carres, Ann. Ecole Norm. (3), 7 7 (1960), 167-174. 7. J. L. Kelley, General Topology, D. Van Nostrand, 1955..

(168) 166. MELVIN ROSENFELD. 8. K. De Leeuw, Banach spaces of Lίpschitx functions, Studia Math. 2 1 (1961), 55-66. 9. E. Michael, Locally multiplicatively-convex topolotical algebras, Memoirs, Amer. Math. Soc. 1 1 (1952). 10. D. R. Sherbert, Banach algebras of Lipschitz Functions, Dissertation at Stanford University (1962). 11. L. Waelbroeck, Le calcule symbolique dans les algebres commutatives, J. Math. Pures Appl. 3 3 (1954), 147-186. UNIVERSITY OF CALIFORNIA, LOS ANGELES.

(169) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. DERIVATIONS AND INTEGRAL CLOSURE A. SEIDENBERG Let d? be an integral domain containing the rational num1 bers, Σ its quotient field, D a derivation of Σ, and & the ring of elements in Σ quasi-integral over &. It is shown that if £? then Dέ?f c &'.. According to a lemma of Posner [4], which is also used by him in a subsequent paper [5], if 6? is a finite integral domain over a ground field F of characteristic 0 and D is a derivation over F sending έ? into itself, then D also sends the integral closure of έ? into itself. The proof of this in [4] is wrong, but the statement itself is correct and a proof is here supplied. More generally it is proved that if & is any integral domain containing the rational numbers and D is a derivation such that Z λ ^ c ^ , then Dέf'aέ?', where &' is the ring of elements in the quotient field Σ of έ? that are quasi-integral over <^. The theorem is not true for characteristic p Φ 0, but if one uses the Hasse-Schmidt differentiations instead of derivations, one gets the corresponding theorem for a completely arbitrary integral domain £?. Let & be an arbitrary integral domain containing the rational numbers, and let & be the integral closure of d?. The question whether D& c & implies Dέ?c έ? is related to the question whether the ring of formal power series <^[[t]] is integrally closed. Thus consider the statements: A. For every έ?, D^a έ? implies Dέ7<z. d?, and B. For every έ?, έ?[[t]] is integrally closed. We show that A and B are equivalent statements. (We also show: C. If έ?[[t\] is integrally closed, then D^aέ? implies D^aέ?.) Now according to the last exercise in Nagata's book Local Rings, [3; p. 202, Ex. 5], B is a true statement, but we give a counter-example, which also leads to a counter-example for A. 2* Criticism of Posner's proof* Posner purports to prove that if P is a place of the quotient field Σ of & that has F as residue field and is finite on d? and if g e Σ is finite at P, then Dg is finite at P. This is not so, as the following example shows. Let & — F[X, Y] be polynomial ring in two indeterminates over F. Let D = d/dX. Let Pλ be the place of F(X, Y) over F(Y/X) obtained by mapping X into 0, let P 2 be the place of F(Y/X)/F obtained by Received February 13, 1964 and in revised form June 25, 1964. This work was supported in part by the Air Force Office of Scientific Research. 167.

(170) 168. A. SEIDENBERG. mapping Y/X into any element of F, and let P be the composite place. Then X, F, Y/X are finite at P, but d(Y/X)/dX = -Y/X2 is not.1 One reason that Posner's proof fails is that there are no parameters such as those of which he speaks, except in the case that the degree of transcendency of d7/F is 1. In that case, Posner's argument yields a proof. 3* A generalization* Let έ?he an arbitrary domain, with quotient field Σ. An element ae Σ is said to be quasi-integral over έ? if all powers of a are contained in a finite ^-module contained in Σ, or, what comes to the same, if there is a d e ^ , d Φ 0, such that dap e &, p — 0,1, , (see [2]). If έ? is a Noetherian domain, then the concepts of integral dependence and quasi-integral dependence (for elements in Σ) become the same; but it is the concept of quasi-integral dependence, rather than that of integral dependence, which is adapted to our considerations. The elements in Σ that are quasi-integral over έ? form a ring &\ which in the case ^ is Noetherian is the integral closure & of &. The base field F plays little role, and it will be sufficient to assume that έ? contains the rational numbers. THEOREM. Let & be an arbitrary integral domain containing the rational numbers, let &f be the ring of elements in the quotient field Σ of έ? quasi-integral over &, and let D be a derivation of Σ. Then: if D ^ c ^ , then Ώ&1 at?'. Proof. Let Σ[[t]] be the ring of formal power series in a letter t over Σ and let Σ((t)) be its quotient field. The mapping left—> Σ(Dci)ti1 i Ξ> 0, CiβΣ, is a derivation of Σ[[t]] into itself and extends D; it has a unique extension to Σ((t))> which will also be denoted D. Let E be the expression 1 + tD + (f/2l)D2 + ••• ( = etD). Then a + tDa + (tf/2l)D2a + •••, to be denoted Ea, has a meaning for every e e ^ ^ [ [ £ ] L i M the partial sums converge in the topology defined by powers of (t); and the mapping a—>Ea is an isomorphism of Σ[[t]\ into itself, as one easily verifies.2 Its unique extension to Σ((t)) will 1. Far from all, or even infinitely many, valuation rings S3 centered at (X, Y) being sent into themselves by D — dldX, there is one and only one. In fact, restricting oneself to valuation rings % centered at (X, Y), if DS c 93, then XI Yφ S3, since D{XjY) = 1/Yφ S3. Hence YjXe®, and therefore D(Y/X), D*(Y/X), etc. are n also in S3. Since D^-^Y/X) = cnY/X (cneK), v(Y)^n v{X) for % = 1, 2, •••, where v is the valuation corresponding to S3. Thus S3 could not be other than the ring of the valuation in which v(X) is infinitely small with respect to v(Y); and for that ring one checks that DS3 c S3. 2 We only use that α -> Ea is a monomorphism, but it is actually onto Σ[[t]] as one sees from the identily etD(e~tDa) = a..

(171) DERIVATIONS AND INTEGRAL CLOSURE. 169. also be denoted E. Since D^a^, one has D^[[t]] c £?[[£]], and since έ? contains the rationale, E^[[t]] c ^[[t]\. Let α be quasi-integral over ^, and let d e ^ be such that p p <ta G ^ , p = 0,1, . . Then E(da ) = Ed(Eaf e έ?[[t]], p = 0,1, . p Hence dEd(Ea — α) e ^[[t]], p = 0,1, •; here we use that eZ and p p 2£<Z are in £?[[£]]. The coefficient of t in dEd(Ea - α) , i.e., the leading coefficient, is d2(Da)p; and this coefficient, as well as all the others, are in &. Hence Da is quasi-integral over έ?. Ifdeέ?and then d\Day e έ?, i = 0,1, COROLLARY.. aeΣ are such that da1 eέ?,i = 0,1, , p.. ,ρ,. Let K = {c I c e £?, c ^ ' c ^'}; then S is an ideal, which in the case £?' is the integral closure ^ of έ? is called the conductor of &. If D^a^, then D K c K . In other words, & is a differential ideal for any derivation (or any family of derivations) sending & into itself. COROLLARY.. Proof. If CGK and α e ^ ; , then (Dc)a = D(ca) - cDaeέ?, that also (Dc)έ?' c &'.. so. The last corollary can sometimes be used to prove that a given integral domain έ? is integrally closed (see [4]). We first restrict ourselves to a class of integral domains & such that & — &*', for example, the class of Noetherian domains. Then we restrict ourselves further to a class ^ of domains έ? such that έ? has a conductor < ^ : ^ ^ ( 0 ) , or equivalently, such that ^ i s contained in a finite έ?module (contained in Σ), for example, the class of finite integral domains (see [7; p. 267]), or quotient rings thereof, or the class of complete local domains (see [3; p. 114]). (For examples of Noetherian domains not having this property, see [3; p. 205 ff]; for an example in characteristic 0, see [l])β Then we can state: COROLLARY. Let & he an integral domain belonging to a class & defined just above, let έ? contain the rational numbers, and let {D} be a (finite or infinite) family of derivations of & into itself. Then, if & is differentiably simple under {D} (i.e., has no differential ideal other than (0) or (1)), then & is integrally closed.. 4* Extension of D to ^ The above is a simplification of our original proof for a finite integral domain. The idea was that since E sends έ?[[t]] into itself, it also sends the integral closure of.

(172) 170. A. SEIDENBERG. into itself. It was then sufficient to prove that έ?[[t]] is integrallyclosed; in fact, we have the following theorem for any integral domain έ? containing the rational numbers. C. If έ?[[t]] is integrally closed and D^aέ?, ?. (Here έ? is the integral closure of #.). THEOREM. then. Proof. If aeΣ,a = c/d, c,de^, then Ea = Ec/Ed, so Ea is in the quotient field of ^[[t]]. If a is integral over #>, then Ea = a + tDa + is integral over £?[[*]], hence in έ?[[t]], whence Da e έ?. Our proof that ^[[t]] was integrally closed for έ? a finite integral domain depended on the following observation, which holds for an arbitrary domain £?. THEOREM. If & is completely integrally closed (i.e., if' 0" = &), then so is έ?[[t]]. More generally, for any έ?, (£?[[*]])'c £. Proof. Let a(t) be quasi-integral over <£?[[*]]. Then there is p a d e έ?[[t]], d = d(t) Φ 0, such that da e έ?[[t\], p = 0,1, . Since ord d + p ord a ^ 0, p = 0,1, , one first observes that a e Σ[[t]]. Let d = d8ts + d8+1t8+1 + , d8 Φ 0, and let a = α r ί r + <*r+1Γ+1 + . Since the leading coefficient of ctap is in &, we have ώ8α? e ^ , whence α r is quasi-integral over £?. Now α — α r ί r is quasi-integral over ^[[t]], whence ar+1 is quasi-integral over &\ and in this way one sees that all the coefficients of a are quasi-integral over &. If & is Noetherian, then so is <^[[t]]. COROLLARY.. Hence:. // <?7 is an integrally closed Noetherian domain,. then so is This is Nagata's (47.6) in [3; p. 200]. Finally, if & is a finite integral domain, then so is ^ whence in this case ^[[t\] is integrally closed. Recalling that έ? is a finite <^module (see [7; p. 267]), one sees that ^[[t]] is even the integral closure of ^[[t]] in accordance with the following: Let έ? he an integral domain whose integral closure is Noetherian and is a finite ^-module. Then the integral closure of έ?[[t]] is <?[[t]]. THEOREM.. Proof. Let & = ^. +. + ^w,.. Then.

(173) DERIVATIONS AND INTEGRAL CLOSURE. 171. whence έ?[[t]] is a finite ^[[ί]]-module and thus integral over Let d be a common denominator of the w4 when written as quotients of elements in £?. Then ete?[[ί]] c ^[[t]], whence ^[[ί]] and £?[[*]] have the same quotient field. As we have already seen that έ?[[t\] is integrally closed, the proof is complete. Although not necessary for our considerations, we mention the following: THEOREM. If & is a Noetherian closed, where t abbreviates a set tu. domain, then £?[[<]] is integrally , tn of n distinct letters.. Proof. & is a Krull ring (see [3; p. 118]), hence from the definition [3; p. 115], &v is a Noetherian valuation ring for every minimal prime ideal p of έ?. Moreover ^ = Π ^ , where the intersection is taken over the minimal prime ideals of έ? (see [3; p. 116]). Since έ?P[[t]] is integrally closed, also έ?[[t]] = Π &p[[i\] is integrally closed. Now consider the statements A and B mentioned at the beginning. We say that A and B are equivalent. Recall that we are assuming that & contains the rational numbers. B=> A. This follows at once from C, the first theorem of this section. A => B. Let a be in the quotient field of &[[t]] and integral over έ?[[t]\. Then ae Σ[[t\], a — aQ + att + . From an equation of integral dependence for a on έ?[[t]], by placing t — 0, one sees that aQ e έ?. Now apply A to the ring έ?[[t]] and the derivation D — d/dt. Then daldt,d2a/dtf, ••• are integral over ^[[t]]f whence all the coefficients of a are in έ?. Now according to the last exercise in Nagata's Local Rings, B is a true statement; however, we will show that this is incorrect. THEOREM. If ^ is an (integrally closed) integral domain containing a field and there is a nonunit be & such that Π φp) Φ (0), then ^[[t]] is not integrally closed.. Proof. Let p be the characteristic and n > 1, an integer such n n 2 1{n that n Ξ£ 0(p). Then b + b ~ t has an nth root a = b[l + (t/tf)] = 2 2 4 6[1 + φ/b ) + c2(t /b ) + •] in ^[[ί]], where cu c2, are in the prime p field of Σ and cxφQ. If ae Π (b ) and a Φ 0, then aaeέ?[[t]], so.

(174) 172. A. SEIDENBERG. that a is in the quotient field of ^[[t]]. Now a is integral over but is not in ^[[t]]. Hence ^[[t\\ is not integrally closed. THEOREM. Let 95 be a (proper) valuation ring containing a field. Then 95[[ί]] is integrally closed if and only if 95 is of rank 1, i.e., if and only if there is no chain 0 < px < p0 < 95 of prime ideals.. Proof. If 95 is of rank 1, then it is well-known and can be checked at once, that 95 is completely integrally closed. Hence 95[[ί]] is completely integrally closed, hence integrally closed. On the other hand, if 95 is of rank > 1 and 0 < px < p0 < 95 is a chain of prime ideals in 95 and bep0 — pu then ^ c Π (&p), whence 95[[£]] is not integrally closed. To get a counter-example to Nagata's last exercise, one has but to take έ? to be a valuation ring of rank > 1 that contains a field.3 To get an example of a ring έ? and derivation D such that D& c & but Dέ?<£ έ7, let 93 be a valuation ring of rank 2 containing the rational numbers, let έ? — 95[[£]] and D — d/dt. Let b be a nonunit in 95 such that Π (bp) Φ (0), and let a = (62 + ί) 1 ' 2 where c19c2, ••• are rational numbers. Then a is integral over έ? = S3[[ί]] but Z>α is not. Concerning the proof spoken of at the beginning of this section, the author is obliged to Professor Mumford for the remark in context that if D is a derivation, then eD, formally at any rate, is an isomorphism. The introduction of the parameter t on the one hand prevents the computations from collapsing into meaninglessness, and on the other. 5. T h e case of characteristic p Φ ()• For p Φ 0, the theorem of § 3 is not true, even for curves. Thus consider the curve given by Yp - Xp - Xp+1 = 0. One checks that Yp - Xp - Xp+1 is irreducible (over the ground field F). Let (x, y) be a generic point of the curve over F. Let I) be a derivation of F(y)/F with Dy — 1; since x is separable over F(y), D can be extended uniquely to a derip vation, still to be denoted D, of F(y, x). One finds — (p + l)x Dx = 0, hence Dx = 0. Let £ ? = F[x, y]. Then D^a έ?. Now y/x is integral 3. In reference to the exercise, Nagata [3; p. 221] cites Sugaku, Vol. 9, No. 1 (1957), p. 61, which we have not been able to locate; and while he notes that the proof there is not complete, he remarks that "a supplement is expected to appear soon'7..

(175) DERIVATIONS AND INTEGRAL CLOSURE. 173. over ^ , since (y/x)p — 1 + x, but D(y/x) — 1/x is not, as otherwise it would be integral over F[x]. However, if one uses the Hasse-Schmidt differentiations [6] instead of derivations, one gets the corresponding theorem. 4 Recall that a differentiation D of a field Σ into itself is a sequence D — (δ0, δu δ2, •) of mappings of Σ into itself with δQ — 1 and satisfying the properties: δi(x + y) = δiX + δty. By D^d. g? we now mean δ ^ c ^ for every i. Then. still yields an isomorphism and can be used instead of our previous E to get the conclusion Dέ?' c έ?'. (After obtaining δ^έ?' c £?' as before, we argue that d*Ed{Ea - a - ^ x a ) p e έ?[[t\], (0 = 0,1, , whence 4 p ώ (δ2α) e ^ , (O = 0, 1, , and δ2a is quasi-integral over <^, etc.) In the case of characteristic 0, the same argument shows one can drop the assumption that 6? contains the rationale (Le., if one uses differentiations instead of derivations). The corollaries of the theorem of § 3 also have easily stated generalizations, with similar proofs. Since (1 + (1 + 4£)1/2)/2 e Z[[t\], the last two theorems of §4 hold without the field condition. REMARK.. REFERENCES 1. Y. Akizuki, Einige Bemerkungen uber prίmάre Integritάtsbereich mit Teϊlerkettensatz, Proc. Phys.-Math. Soc. Japan, 3rd ser., 17 (1935), 327-336. 2. W. Krull, Beίtrάge zur Arίthmetik kommutativer Integritάtsbereiche, II, Math. Zeitschrift, 4 1 (1936), 665-679. 3. M. Nagata, Local Rings. New York, 1962. 4. E. C. Posner, Integral closure of differential rings, Pacific J. Math. 10 (1960), 1393-1396. 5. Integral closure of rings of solutions of linear differential equations, f Pacific J. Math. 12 (1962), 1417-1422. 6. F. K. Schmidt, Noch eine Begrundung der Theorie der hoheren Differ entialquotienten in einer algebraischen Funktionenkorper einer Unbestimmten. Zusatz bei der Korrektur, J. fur die reine u. angewandte Math. 177 (1937), 223-237. 7. 0. Zariski and P. Samuel, Commutative Algebra, Vol. 1. New York, 1958. HARVARD UNIVERSITY AND UNIVERSITY OF CALIFORNIA, BERKELEY 4. For some useful information on differentiations, see K. Okugawa, "Basic properties of differential fields of an arbitrary characteristic and the Picard-Vessiot theory", J. of Math, of Kyoto Univ., Vol. 2 (1963), pp. 295-322..

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(177) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. ON THE STABILITY OF THE SET OF EXPONENTS OF A CAUCHY EXPONENTIAL SERIES S. VERBLUNSKY. It fe L(— D, D) and Q(z) is a meromorphic function whose poles, all simple, forms a sub-set of the set {λv} (v = 0, ± 1, ± 2, •), then the C.E.S. (Cauchy exponential series) of / with respect to Q(z) is Σcveλ^x9 where -D. Suppose we are given a class A of functions / each of which can be 'represented' in (— D9 D) by its C.E.S. with respect to Q(z). We define a set of neighbourhoods U of {Λ}. Then {λv} is stable if there is a U such that to each {tcv} eU there corresponds a meromorphic function q(z) whose poles, all simple, form a sub-set of {tcy} and which is such that each fe A can be represented in (— D, D) by its C.E.S. with respect to q(z); and {λv} is unstable if there is no such neighbourhood. The case in which λv — iv, A is BV[— D, D], 'representation of / in (- D, DY means ' Σ M ^ V ^ V * -» 1/2 (f(x +) + f(x -)) boundedly within (D, D)' is considered. It is shown, in particular, that with reasonable conditions on the set of neighbourhoods U9 {iv} is unstable if D > 1/2 π, and stable if D = 1/2 π.. Let D > 0 and feL(—D,D). Let Q(z) be a meromorphic function whose poles, all simple, form a sub-set of the set {λj(v = 0, ± 1, •••)• Here, and in what follows, the use of the symbol {λv} implies that λ v Φ λ>/ if v Φ v'. The C. E. Sβ (Cauchy exponential series) of / with respect to Q is ^cveλvX where. Suppose that the set {λj is such that, for a class A of functions /, the C.E.S. of / 'represents' / in (—D,D)O Then we may consider the question of the stability of the set {λv}. We define, in some way, a set of neighbourhoods U of {λv}. Then {λj is stable if there is a neighbourhood U such that to each {Λ:,} e U, there corresponds a meromorphic function q(z) whose poles, all simple, form a sub-set of {Λ J , and which is such that each fe A can be represented in (— D, D) by its C.E.S. with respect to q(z); and {λv} is unstable if there is no such neighbourhood. The stability of {λj depends on the value of D, the class A, the, particular meaning we give to the 'representation' of /, Received July 27, 1964. 175.

(178) 176. S. VERBLUNSKY. and finally on the definition of the set of neighbourhoods U. In this note, we confine our attention to the simplest case: j\, v = iv, A is the class of functions / which are BV[ — D, D] and satisfy 2f(x) = n / ( # + ) + / ( # — ) i (— D, D), 'representation' of / means 'bounded convergence to f(x) within (— D, D)', i.e., for each δ satisfying λ x 0 < δ < D, Σivi<^ cve * —>f(x) boundedly in the segment | x \ ^ D — δ. We recall that if D = π, then each feA can be represented by its C.E.Sc with respect to Q0(z) — 1/2 coth πz, since, in this case, the C.E.S. is the Fourier series of /. Let us suppose that to each neighbourhood U there corresponds an ε > 0 such that {μu} e U if Σ I ^^ — λ J < ε; and to each δ > 0 there corresponds a neighbourhood UB such that if {μu} e U8 then sup | μu — λ„ | < δ. What we prove, implies that {iv} is unstable if D > π/2, and stable if Ό — π/2. We shall, however, prove more than this, viz. THEOREM 1. Let {£J be a real set not containing every integer, such that l» is an integer for \ v \ ^ N. If D > π/2, then there is no meromorphic function q(z) whose poles, all simple, form a sub-set of {Hv} and which is such that each feA can be represented by its C.E.S. with respect to q. THEOREM. which satisfy. 2. Let lu = v + ocv + iβu where aV9 βv are real numbers Πrn I au \ < — ,. IvHoo. g. ϊίrn | ft, | < oo . |v|-*oo. If D — π/2, there exists a meromorphic function q(z) whose poles, all simple, form a sub-set of {ilv} and which is such that each feA can be represented by its C.E.S. with respect to q. THEOREM. 3. The conclusion of Theorem 2 holds if the condition on a» is replaced by sup | ex J < 1/4.. The relation between Theorem 2 and the work of Korous [1] is explained in § 6. The relation between Theorem 3 and the work of Levinson [2] is explained in § 7. 2.. Let 0 < D ^ π, and let A have the meaning specified in § 1.. LEMMA 1. If Hn(t) e L(- 2Ό, 2Ό) for n ^ n0, then, in order that for each feA,. f(t)Hn(t-x)dt~->f(x).

(179) ON THE STABILITY OF THE SET OF EXPONENTS. 177. boundedly within (— D, D), it is necessary and sufficient that. S. 1. t. o. Hn(u)du -* — sgn t 2. boundedly within (— 2D, 2D). Proof. Let sin in + - /.(«)=. x. v. 2 π. 2. Bin|«. Then for each feA, Γ f(ί)Jn(t-x)dt-+f(x). J-D. boundedly within (— D, D), and. S. i. 1. Jn{u)du o. —> — i 2. boundedly within ( - 2D, 2D). Let Kn(u) = Hn{u) - Jn(u). to prove: in order that for each fe A, Γ. It suffices. f(t)Kn(t-x)dt->0. boundedly within (— D, D), it is necessary and sufficient that kn{t) =. boundedly within ( - 2D, 2D). Sufficiency. (1). We have. Γ f{t)Kn{t - x)dt = f(D)kn(D - x) - / ( - D)kn{-. D-x). J-D. - Γ kn(t - £B)d[/X*) J-D. and the second member tends to zero boundedly within (— D, D). Necessity. In the first place, it is necessary that for each τe(-2D,2D),kn(τ)-*0 as n-+co. For let a, βe ( - D, D) and let x = a. Let f(t) — 1 in the open interval, and let f(t) = 0 outside the closed interval, whose end points are a, β. Then.

(180) 178. S. VERBLUNSKY. kn(β - a) = Γ Kn(t - a)dt -» 0 . Since a, β can be chosen so that β — a has any assigned value in (— 2D, 2D), this proves our assertion. By (1), for each xe(— D, D), the functions kn(t — x) of t, for n Ξ> n0, form a sequence of elements of C[ — D, D] such that Ό. is convergent for each fe A. it follows that. kn(t - x)df(t) By the principle of uniform boundedness,. sup. I kn(t - x) I < co .. teί-D, z>]. Choose x = D — δ. Then kn(t) is uniformly bounded in [— 2Ώ + δf δ]. Choose x — — D + δ. Then &„(£) is uniformly bounded in [—<5, 2Ό — <?]. Hence &„(£) is uniformly bounded within (— 2D, 2D) as required. 3. Proof of Theorem 1. We may suppose that D ^ π. Let ω be chosen to satisfy π < ω < 2D. We choose the notation so that if 0 e {lv} then 0 = ϊo If a meromorphic function ^(2J), with the properties mentioned in the enunciation, exists, let Cn denote a contour which contains in its interior precisely those ilv for which | v \ g n, and which does not pass through any of the ϊlv. Let \ <•. If Σ c»eihx is the C.E.S. ofv / with respect to q(z),{ then (3) Σ Cβ" = Σ res?(«)Γ f(t)e? -"dt. =Γ J. We have (4). l. 1. eZX. f flr.(tt)dw = - L ( q(z) ~ Jo 2π%)on z. dz. where r v is the residue of g(^) at iϊ v and where, if l0 = 0, we use the convention -I. (5). λ. -ilQt. p. -f. I. ^,-iZί 1. =lim ~. e. = t ..

(181) ON THE STABILITY OF THE SET OF EXPONENTS. 179. By Lemma 1, it is necessary that (6). Σ^d-^-. boundedly within (— 2D, 2D), and hence in [— ω, ω]. Let xe (— ω, ω — 2ττ). Then for | v | Ξ> iV, the terms on the left are unaltered on replacing x by x + 2τr. By subtraction, it follows that. for such x, and hence for all x. We note that if k — 0, the term with v — 0 is — ro2π. At this point, we distinguish to cases, (a) l0 Φ 0, (b) l0 = 0. In case (a), we integrate (7) over (— X, X), divide by 2X, and let X—>oo# We obtain a contradiction. In case (b), we take mean values as in case (a), and deduce that the term with v — 0 is — 1. Then (7) implies that ^ ye. y j ——e 0<|v|<iV Hu. ^. — l.). :=z. u. ίor all x. If we multiply this by its conjugate, and take mean values, we deduce that Iγ. (8). 2. 12. '— Γ. O<|V|<Λ. si*1* π^ — 0. \\. By (6),. boundedly within (— 2D, 2D). (9). Σ. Considering odd parts, its follows that. ^ sin kx -> - ί sgn x -. o<ιvi^τz lv. 2. ^ 2ττ. boundedly within (— 2J9, 2D). By hypothesis, there is an integer μ say, which is not one of the lv; and μ Φ 0 since Zo = 0. By (8), r v = 0 if £„ is not an integer. Hence, on multiplying both sides of (9) by μ sin μx and integrating over (— π, π), we obtain 0 = 1, a contradiction. 4. Proof of Theorem 2. For all sufficiently large n, the circle Γn:\z\ ~ n Λ- 1/2, contains in its interior the points ilv for | v \ 5g n, and every point on Γn is at a distance greater than 3/8 from all the points ilu. Let q(z) be a meromorphic function whose poles, all simple,.

(182) 180. S. VERBLUNSKY. form a sub-set of {ilv), and define Hn(u) by (2) with Cn replaced by Γn. Using the notation of §§ 1, 2, we have. and therefore, as in § 2, it suffices to prove that we can choose q(z) so that X. \ Kn(u)du = - L ί Jo. e. (q(z) - QQ(z)) * ~ ~°* dz -> 0. 2mirn. z. boundedly within (—π, π). Write. In § 5, we shall prove LEMMA 2. As \z\-+<*>, P(z) = o(\z\iί2eπM). o(n e~πlrezl) as n~+ oo.. On. Γn9. \P{z)\~ι. =. ll2. The meromorphic function QQ(z)P(z) is regular, except possibly at the points iv, which are at most simple poles of residue P{iv)j2π. By Lemma 2, P(iv) — o{\ v | 1 / 2 ). Hence we can define the meromorphic function. +Σ W 2π L %. +4 \z — %v. which has the same principal parts as Q0(z)P(z).. %. Thus. Q0(z)P(z) = R{z) + S(z) where S(z) is an integral function. We can write q{z)P(z) — F(z)r where F(z) is an integral function. Then. In § 5, we shall prove LEMMA 3.. On Γn, R(z) = o{niμ) as n—• oo.. We choose F(z) so that the numerator in (10) will not be of a greater order of magnitude than R(z). This means, since F and S are integral functions, that F = S + c where c is a constant. Theorem 2 will follow if we show that.

(183) ON THE STABILITY OF THE SET OF EXPONENTS. tends to zero boundedly within (—π, π). Lemmas 2 and 3, R. ^. P(z). Write z — (n + Ij2)eiθ.. = o(ne-nπlcosΘ]). 181. By-. .. If then I x \ ^ π — δ, δ > 0, we have In(x) = o 5* write. In order to prove Lemmas 2 and 3, it will be convenient to P(iz) = ip(z)9. so that. and (11). Λ(w) = r(z). We need the following result, which is a special case (a — 0) of [3] Theorem 1 (with a change of notation). 4. Let L, M be positive numbers. Let s v = v + σv + iτu, where σv1 τv are real numbers which satisfy \σu\ ^ L9 \ τv \ ^ M for all v. Suppose that there is a δ > 0 such that \su\*zδ for all v. Let LEMMA. Then there is a positive constant C (depending only on L, M, δ) such that, (i) for all z, \ f{z) \<C(l+\z |) 4 V liW21 (ii) if I z - sy I ^ δ for all v, then \ f{z) \~ι <C(l + \z |)«e-|C>*wιr| . Proof of Lemma 2. We can find a positive number L < 1/8 such that \au\ :g L for | v \ > N say; and a positive number M such that I βv I ^ M for all y. In Lemma 4, choose s v = lv for | v | > N; —v for.

(184) 182. S. VERBLUNSKY. 0 . By Lemma 4 (with <? = 3/8), there is a positive constant -D such that (i) I p{z) \<D\z | 4 V K m z | if \z\ is sufficiently large; (ii) if z is on Γn and τ& is sufficiently large then | p(z) I""1 < Dn4Le~πlίmzl (the condition | z — su | Ξ> 3/ 8 for all v being satisfied). Since P(z) = ip{ — iz), and 4L < 1/2, the lemma follows. Proof of Lemma 3. By (i) above, p(v) = O(| i; | 4 Z ). will suffice to prove that if z is on Γn, then V. By (11), it. (z - v). The left hand side is. o. 2-. ^y4Z. + -. v. n. ~ v\ v. 1. n. - -1). +. ^. ?zr 4 z - 2 ". The first and second sums are O(w4Zlogw). The third sum is This proves the lemma. In Lemma 4, we could replace 4L by 2L, if the σy satisfy the further condition. Σ -^T ii^. = θ(i). 1. 2. This follows from [3] Theorem 2. Hence, as the preceding proof shows, we can replace 1/8 by 1/4 in Theorem 2 if we add the condition. Σ - ^ V = 0(1). 2 6*. Let. The function g(z) of §4 is given by. <zo(z) = iq(iz) = — cot πz + 2. If Σ c»eil*x is the C.E.S. of / with respect to q(z), then, for all sufficiently large n,.

(185) ON THE STABILITY OF THE SET OF EXPONENTS. (12). Σ ^. q()[πl2. M. I. 183. J-jr/2. 2π%. J-π/2. n. Suppose now that βv — 0 for all v, and that c is real. Then qo(z) is real for real z, so that qo(z) = qjz). If ry = res g(«) = res go(s), then r v is real. Let / be real.. Write. S. π/2. f(t)e-il^dt .. -JΓ/2. Equating real parts in (12), we get (13). 2. α. >c o s ^ + 6^ s i n ^^ = — : I qo(z)dz \. f(t) cos z(x — t)dt. We thus obtain the class of trigonometric series investigated by Korous [1]. Theorem 2 shows, in this special case, not only that (13) converges boundedly to f(x) within (— π/2, π/2), but also that 2. av sin lux — b» cos lvx. converges boundedly to zero. 7. We now turn to the proof of Theorem 3. We again suppose that the notation has been chosen so that if 0 e {lv}, then 0 = lQ. It will suffice to prove LEMMA 5. Under the conditions of Theorem 3, there are complex numbers wv such that. — sgn x 2. boundedly within (— π, π). For then, by the classical theorem of Mittag-Leffler, there is a meromorphic function q(z) whose poles form a sub-set of {ilv}, the principal part at ilv being ilvwj(z — ilv) if lv Φ 0. If l0 = 0, we allow the origin to be a regular point. Defining Hn(u) by (2), we have 1. q(z)-. o. — zx. —dz.

(186) 184. S. VERBLUNSKY —. /. Λ. β. Wv\l.. By Lemma 5, • — o ,. -2-1. ^v6. -γsgno?. boundedly within (— π, π). Thus, Theorem 3 will follow from Lemma 1. One way of proving Lemma 5 is to generalize the following theorem of Levinson [2, 48]: if the real numbers λ v satisfy | λ v | ^ P < 1/4, then there are numbers wv such that 2π. J-:. converges uniformly to zero within (— π, π) if / e L 2 ( — π, π). The generalization consists in showing that we can replace the real λ v by v + av + iβU9 where \au\ ^ P and lim | βv \ <.°°.. However, we only. |v|-»oo. need the result for the function f(t) — 1/2 sgn t. It seems worthwhile to prove this special case, for which the argument of Levinson can be given a rather simple form. This is done in § 9. 8*. We need the following deduction from Lemma 4.. LEMMA 6. Let Sv = v + σv + iτv, where σv, τv are real numbers which satisfy \σv\ ^ P, \ τv \ ^ Q for all v, where 0 0. Let. Then there is a constant K {depending only on P and Q) such that. I Ψ(z) I < K(l + I z \)ipeπlίmz]. (14). .. and there is a constant Kε (depending only on P, Q and έ) such that I Ψ(g) I"1 < Kε(l + I z |«y* 1 < I M |. (15) if. I z — S» I ^ ε for. not. all. v.. Proof. In the following proof, and in § 9, the symbols K, Ke do necessarily denote the same constants at each occurrence. In. Lemma 4, choose s0 = — P , s v = S v for v φ 0. I sv I > — . 4 (16). For | v \ ^ 1, we have. By Lemma 4 (with δ = min (1/2P, 3/4)), \ψ(z)\.

(187) ON THE STABILITY OF THE SET OF EXPONENTS. 185. Now. y(s) = - £ ( ^ ^. (17). 2 \ z — s0 and I (z - S0)/(z - s0) | < K for | s - s01 ^ 1/4. For such 2, (14) follows from (16). Finally, | Ψ(z) \ S K inside \z — so\ ^ 1 / 4 since this is true on the boundary. This proves (14). Let I z — Sy I ^ ε for all v. If | z - s01 ^ ε then \ψ(z)\-ί<K.(l + \z\ype-*»»*. (18). by Lemma 4, and | (2 — so)/(z — So) | < JfiΓβ so that (15) follows from (17) and (18). If, however, | z — s0 \ < ε, then for small ε the disc Δ : I z — s01 < ε is outside each disc |« — S y | < ε (v = ± 1, ± 2, •)• If it is outside the disc / : | z — So \ < ε, then (^(z))- 1 is regular in Δ and so | Ψ(z) I"1 ^ iΓe in Δ since this is true on the boundary. If Δ meets Δf we apply this argument to the portion of Δ which is outside Δf. 9* Proof of Lemma 5. By the hypothesis (of Theorem 3), there are positive numbers P, Q such that \au\ ^ P < 1/4, | β„ | ^ Q, for all y. Let C n denote the rectangular contour whose vertices are ± (n + 1/2) ± ni. Let. We define If". G(u)φ(u). where φ(u) =. 1 — COS. u. Then ^». Wvβ. fX. «^ = - ^ Γ G(U) φ(u)du\ 2. 4τr J-~. dζ. Jc7w G(ζ)(w — ζ). 4ττ2 J - ~. jon u — ζ. The last term is I. fw + l/2. LiJZ v—(?l + l/2). — sgn x. I. — sin uxdu /. f/(/.

(188) 186. S. VERBLUNSKY. boundedly within (— π, π). Hence it suffices to prove that In(x)—>0 boundedly within (— π, π), where fX. In(χ) = ί~ G(u)φ(u)du\. dζ .. Since G(z) is a function Ψ(z), we have by (15), | G(ζ) I"1 < Kn*pe~πn on the horizontal sides of Cn. Further, ι. ι. | β * " | ^ β '"i, | w - ζ | - < ί Γ ( l + | u | ) - \. | ^(w) | < JBΓ(1 + | u\)~ .. Since \G(u)\<K(l+ \u\)iρ by (14), the contribution to In of a horizontal side of Cn does not exceed in absolute value. and tends to zero uniformly within (— π, π). It remains to consider the contribution to In of a vertical side of Cn, say the right side. This contribution is G(«)^)dW. —. J-G(Λ (19). i +. |. +. -dζ. C)(—— i c ). = β*"^1'2' Γ G(U + n + ± ) ΨU + n + 1 ))d^ e. x f". —. dζ .. For all v, we define l[ = — w + i^ +w .. G(g) _. (z — ^o) -fi(w — lQ). =. _. i. Then. (g — £v)(a —Li;) (w — lv)(w. — Lv). g - n - Z; J J (g - w - l'v_n)(z -n - V^_n) w — n — l'o i (w — n — l[-n)(w — n — i_ v _J Gn{z-n) Gn(w - n). where. and i; = v + < + i/S;, a[ = α v + % , ^ = &+.. Then | < | ^ P, | /3; | ^ Q. Hence Gn(z) is a function Ψ(z) (of Lemma 6) and satisfies the inequalities (14), (15) with constants K, Kε independent of n. In (19), we use the equation.

(189) ON THE STABILITY OF THE SET OF EXPONENTS. 187. It follows that. ±)\φ[u. + n + ±)\J\du. where P.ixi. J. \y. Gn(ζ + !.)(« - ζ). and 7 denotes the path from — in to in modified by replacing the segment (— iβ, iβ) by the right half or the left half of the circle I ζ I = 1/8, according as u < 0 or u > 0. On 7, re(ζ + 1/2) is between 3/8 and 5/8, and therefore ζ + 1/2 is at a distance greater than 1/8 from all the zeros of Gn(z). By Lemma 6, | Gκ(ζ + 1/2) I"1 <Ke~*M(l + | η |), where η — imζ. Further u ζ I""1 < ϋΓ(l + | u \)~\ and so K. w. •s:. (l + \u\)(π-\x\y Since |GJu + 1/2) \<K(l where. + \u])4P, it remains to prove that Hn->0. ί.=j". du. u +n +— 2. and d = 1 - 4P > 0. If m is a positive integer, then. and the first integral tends to zero as n 1 1 and let g- + p- = 1. Then. c>o. Choose p so that pd > 1.

(190) 188. S. VERBLUNSKY. so that lim Hn = 0, as required. Added in proof. A result similar to Theorem 2 was proved in a Ph. D thesis by J. A. Anderson. REFERENCES 1. J. Korous, On a generalization of Fourier series, Casopis Pest. Mat. 7 1 (1946), 1-15. 2. N. Levinson, Gap and density theorems, (New York 1940). 3. S. Verblunsky, On a class of infinite products, Proc. Cambridge Phil. Soc. 6O (1964), 847-854..

(191) PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966. SOME AVERAGES OF CHARACTER SUMS H. WALUM Let χ and φ be nonprincipal characters mod p. Let / be a polynomial mod p and let aίf , ap be complex constants. We will assume αy = ak for j = k(p), and thus have an defined for all n. Define (1). S = γι. and (2). J.(c) = Σ,φ(r)χ(r. - c) ,. r. where the variables of summation run through a complete system of residues mod p. The averages in question are. and 2. (4). A 2= Σ \ S \ ,. where the sum in (4) is over the coefficients moάp of certain fixed powers of the variables in /. Exact formulae for Ai will be obtained in all cases, and for A2 in an extensive class of cases. Specifically, the following theorems are true. mi. THEOREM I.. 2. Let f(r) = yr + xr™ + g(r) and assume (m2 — mu p — 1) = 1. Let the sum in (4) be over all x and y mod p. If g has a nonzero constant term and neither mγ nor m2 is zero, then (5). A2. = P(P. 2 r=l. + p21 a012. Otherwise, p-1. (6). A = p(p -i)Σ. \ar. 2. r-l. THEOREM I I . Let d = (n,p— 1), ψ(t) = e. 2πi{rind. {t)ls). , where, naturally, s I (p — 1), (r, s) — 1 and g' = t(p) for g a primitive root moάp. If ds)((p — 1), then A1 = 0. If ds \ (p — 1) and ψχn is nonprincipal, then A1 = p(p — l)d. If ds\(p — 1) and ψχn is principal, then Ax ~ p(p - l)(d - 1) - (p - 1). mά{t). The following is an immediate consequence of the first theorem. Received November 21, 1963 and in revised form June 16, 1964. Research done under the auspices of the National Science Foundation. 189.

(192) 190. H. WALUM. III. Let f be as in Theorem I, and assume | ar \ — 1, r = 1, , p. Then there exist x09 y0, x1 and yx depending on χ, such that the S, as in (1), for x0 and y0 satisfies \S\ < Vp and the S, for x1 and ylf satisfies λ/{p — 2) < | S | . THEOREM. Proof of Theorem II. Our principal device is the fact that a function which is periodic modp has a unique expansion by means of the characters modp[2]. That is if h(r) — h(s) for r = s(p), then for n ξέ 0(p). (7). h(n) =. where θ runs through the characters mod p. (8 ). bθ is given by. (p - l)bθ =. Regarding Jn(c) as a periodic function modp of c, and expanding Jn{c) in the form (7), we obtain, by standard methods, ( 9). Jn(c) =. Σ. π. Φy X)P(C). where π(a, β) is a Jacobi sum [1] (10). π(a, iδ) = Σ ^ ) / S ( l - r) . r. The sum in (9) is over all characters p which satisfy the indicated condition. The expansion (7) has a Parseval identity /Λ i \. "SΓ~l I Z , / J Λ ί=l. 12. ί/γ\. 1 \ ^V"1. si. 2. θ. Thus we can evaluate A1 by means of (11) and (9) when we know the value of | π(a, β) | 2 . Now [1] | π(a, β) |2 = p when a Φ ε, β Φ ε and aβ Φ ε, where ε is the principal character. If a — ε or β — ε, then π(a, β) |2 = 1. If aβ = ε with α ^ ε or /S ^ ε, then | π(α, β) |2 = p. By hypothesis, % is nonprincipal. Thus | π(p, y) | 2 is p unless p = ε or PX — ε. It p = ε, then p = ε and ^%% is principal. If /?χ = ε, then p — X and |0% = α/rχΛ implies ψ = ε which is excluded by hypothesis. n n n Let N be the number of solutions of ρ = ψχ . If ψχ is nonprincipal 2 then I π(ρ, χ) | = p for all iSΓ of the p and Ax = p(p — 1)N. If ψχn is principal, then | π(ρ, χ) |2 = p for N— 1 of the p and |π(/δ>, χ) |2 = 1 2 for p = ε. Thus, in this case, Aj. = (p — l)(p(iV — 1) + 1) = Np(p — I) . n n N, the number of solutions of p = ψχ , is the number of solutions n of σ = ψ. It is a standard lemma from the theory of cyclic groups of order k that an — b has (w, &) or 0 solutions according to whether.

(193) SOME AVERAGES OF CHARACTER SUMS. 191. or not order b \ k/(n, k) . Also, N is the number of solutions of xn — ψ(g), for x, in (p — 1) — st roots of unity. From either description of N, it follows that N = d or N = 0 according as ds\(p — 1) or ds)({p — 1), and the theorem follows. Proof of Theorem I.. Referring to the hypotheses of Theorem I,. IS I2 = Σ araaχ(yrmί. + £r m * + g(r))χ(ysmί. + xsm* + g(s)). r,s. and thus, (12). A2 = Σaβ.Σχiyr^. + xr~* + #(r))χ(τ/s^ + xs™* + g(s)) = Γ,. Γi is the sum of the terms in (12) such that r =έ 0 and s ξέ 0. T2 is the sum of the terms in (12) such that r Ξ= 0 or s = 0. ϊ 7 ! can be witten (13). Γi = Σ α r α g χ m i (r r=έθ,s. where. Now, A(α, 6; c, d) = Σ Σ x( V Except when (α — c)a? + (6 — d) = 0(p), ^ J V + (α - Φ + (6 - cZ) \ 2/^0. V. 7/. =. _. 1. ^. /. Also, (α — e)x + (6 — d) = 0(p) when a? = ((6 — d)/(α — c))(p) or when α Ξ c and 6 = d. Thus, if α ^ c or δ ^ d, then A(α, 6; c, d) = - ( p - l ) + p - l = 0 . If α = c and b = d, then A(α, 6; c, d) = p(p - 1) . In view of this (13) r Ξ£ 0 Ξ£ S and r W 2 " W l p — 1) = 1, we have r s such that r Φ 0 ^ s. becomes the sum over all r and s such that = sm2-^i? r~mιg{r) — s~mig(s). Since (m2 — mu Ξ= S. Thus the sum in (13) is over those r and and r = s. Thus. T, = p(p - 1) Σ r=l.

(194) 192. H. WALUM. Now (14). T2 = Σ Wo Σ X(yrmί + xr™* + g(r))χ(g(O)) r=έθ. x,y. g(s)) x,y. [2 Σ χ(ff(θ))χ(ff(o» = p21 «„ | 2 except when m1 — 0 or m 2 = 0. Thus, if 0(0) = 0, Λ = P(P - l) Σ I «r I2 and if gf(O) ^ 0, then A2 = p(p - 1) Σ I ar Γ + P21 O. when mx = 0 or m 2 = 0, then χ(#(0)) in (14) must be changed to X(V + 0(0)) or χ(x + g(0)), and A2 is given by (6). REFERENCES. 1. H. Davenport, and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fallen, Journal fur Math. (Crelle) 172 (1934), S. 151-182. 2. M. J. Delsarte, Essai sur Vapplication de la theorie des fonctions periodiques a Vrithmetique, Annaies Scientifiques LΈcole Normale Superieure, series 3, 62 (1945), 185204. OHIO STATE UNIVERSITY.

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(196) Pacific Journal of Mathematics Vol. 16, No. 1. November, 1966. Larry Armijo, Minimization of functions having Lipschitz continuous first partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Edward Martin Bolger and William Leonard Harkness, Some characterizations of exponential-type distributions . . . . . . . . . . . . . . . . . . . James Russell Brown, Approximation theorems for Markov operators . . . . . . Doyle Otis Cutler, Quasi-isomorphism for infinite Abelian p-groups . . . . . . . Charles M. Glennie, Some identities valid in special Jordan algebras but not valid in all Jordan algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas William Hungerford, A description of Multi (A1 , · · · , An ) by generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James Henry Jordan, The distribution of cubic and quintic non-residues . . . . Junius Colby Kegley, Convexity with respect to Euler-Lagrange differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tilla Weinstein, On the determination of conformal imbedding . . . . . . . . . . . . Paul Jacob Koosis, On the spectral analysis of bounded functions . . . . . . . . . . Jean-Pierre Kahane, On the construction of certain bounded continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. V. Menon, A theorem on partitions of mass-distribution . . . . . . . . . . . . . . . . Ronald C. Mullin, The enumeration of Hamiltonian polygons in triangular maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eugene Elliot Robkin and F. A. Valentine, Families of parallels associated with sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Melvin Rosenfeld, Commutative F-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Seidenberg, Derivations and integral closure . . . . . . . . . . . . . . . . . . . . . . . . . S. Verblunsky, On the stability of the set of exponents of a Cauchy exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Herbert Walum, Some averages of character sums . . . . . . . . . . . . . . . . . . . . . . .. 1 5 13 25 47 61 77 87 113 121 129 133 139 147 159 167 175 189.

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