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This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement "that distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with G [p] the same subgroup of G" is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of G supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].

n/a

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

Craig (1962) described an application of order statistics from a binomial distribution and gave an approximate expression for the mean value of the minimum of two binomial variables having the same probability of success and the same number of trials. In this note we suggest another approximation for the mean and also for the standard deviation of the minimum, using normal order statistics tabulated by Teichroew (1956).

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = -" The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.

IN this paper Haldane points out (p. 142) a difference between his results for the mean and variance of x2 in a 2 x n-fold contingency table when the expectation p is fixed and the results obtained by me in my paper (Annals of Eugenics, Vol. VII, part III, p. 211). The difference is that I have (n 1) throughout where Haldane has n. Haldane writes, "my own results would appear to be slightly more accurate than Cochran's", which might, I think, give the impression that both Haldane's results and mine are only approximations. In fact, both results are mathematically exact, the difference between them being one of definition of x2. My paper is almost entirely concerned with the distribution of x2 when the expectation p is not known. In the results which I gave for the distribution of x2 when p is known, I retained the term S (x -x-)2 in the numerator of x2 instead of S (x np)2, to facilitate comparison between this and my other results. Thus my x2 has (n 1) degrees of freedom, whereas Haldane's x2 has n degrees of freedom. Unfortunately I did not emphasize this point in the passage concerned, and as it may have appeared misleading to others besides Haldane, I welcome this opportunity of drawing attention to it. Haldane's x2 is, of course, the one which is normally appropriate in testing the departure from independence when the expectation is known.

The incorrectness of Birkhoff's result, with which the paper [2] is concerned, was noted already by F. R. Gantmacher in 1954 in his book Theory of matrices (Russian); cf. the recent English translation of the second part of this, [l, pp. 175-176]. Example C in [2] is, in fact, the analogue of Gantmacher's counter-example when z= a> is taken instead of 2 = 0 as the singular point. Gantmacher does not, however, discuss the cases in which the result is correct; cf. [2, D]. Unfortunately, the writer became aware of Gantmacher's work only when it was too late to have paper [2] withdrawn or even to have a note added to it.

1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

F(k) ={xkj x F}; F,=F; Fk= (Fk_l,F). Let (u, v, 0)=u, (u, v, 1)=(u, v), (u, v, n)=((u, v, n-1), v). Then Sanov [3] proved that (1. 1) (u, v, apa _1) p-a E F(p#)Fapa+?, /3 a = 1, 2, where p is a prime. In this paper, (1.1) is proved for the cases a= 1, 2; 3 arbitrary. A slight generalization of these results is also proved. Sanov's proof involved an investigation of ideals in a Lie Ring. In this paper, Hall's Collection Process will be used. The method also yields other formulas, e.g. (1.2) (u, v, p2 1)P' (E F2p2_pF(p) (1.3) (u, v, pa+l 1)p~l C F2pa+l paF(pl), a = 1 2, and can be used to produce numerous formulas of a similar nature. The author hopes that some of these formulas and/or the method may be of use in solving other group-theoretic problems. The author was unable to use the method to prove (1.1) for a =3. Note that for a=f = 1, (1. 1) becomes (1.4) (U, v, p 1) E F(p)Fv+1. (1.4) plays an important role in the theory of the Restricted Burnside Problem.

The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows

In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions

In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together

It has been brought to my attention by Mr. N. E. Steenrod that the lemma of page 708 in the paper referred to is in error. The final assertion of the proof is false. It is therefore necessary to point out that the paper is not "disturbed" by this fault. For if one requires that the Pn, «= 1,2, • • •, of the lemma be arcs then the (altered) lemma does hold, since it is true that the connected sum of a perfect continuous curve and an arc is "perfect." One verifies that this restricted lemma is sufficient for the uses of the paper.

In this short paper, we show that there does not exist a noncompact Type-I [Formula: see text]-solution of the Ricci flow with positive curvature in dimension 3.

J. L. Palacios claimed [2] that the author’s paper [1] contained errors. This note refutes those claims by showing that Palacios misunderstood [1] and adopted assumptions different from those of [1].

If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q

There is a mistake in each of [3] and [4]. The purpose of this note is to correct the error in [3] and to salvage what can be saved in [4]. In each case the notation will be that of the paper under discussion. Professor Homer Bechtell has kindly informed me of an error in the proof of the Theorem of [3] which occurs in the case MG1c G. The theorem is true however by altering the proof at this point. Assume that Mc: MG1=M1lc:G. M1 is an invariant subgroup of G, and since G1c: Soc(G), G1 is a direct sum of minimal invariant subgroups of G. By Clifford's theorem (p. 70 of [1]), each minimal invariant subgroup of G is either M1-central or M1-hypereccentric, and hence

The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.

Walsh's Theorem II states that if the roots of f (z) are in the circular region C, and if z is exterior to C, then ai lies in C. This is precisely the result obtained by Laguerre as given on page 59 of volume I of his collected works, if expressed in non-homogeneous coordinates as on page 57. We shall use another form of Laguerre's Theorem (loc. cit., p. 57) to prove the underlying proposition on which Walsh bases his proof of his Theorem I, as follows: Every circle through any point z and its "derived point" a as defined by (1) either passes through all the roots off ( z ), or else has at least one root in the region interior to it, and at least one root in the exterior region. I have recently called attentiont to the fact that this is a corollary of a theorem of Bocher's on jacobians which has served as a starting point in Walsh's earlier papers. With a slight change in Walsh's notation, whereby we substitute z's for a's, the proposition from which Theorem I is deduced may be stated thus

It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.

This paper contains a number of disturbing misprints: Equation (2) p. 560 should read Gpf(x) = aoxPfm+ a1xPf11 + *** + amixpf + amx. Line 17 p. 561 read Ap(x) XBp(x) instead of Ap(x)Bp(x). The term perfect (volkommen) in Theorem 1 is used in the sense of Steinitz, Algebraische Theorie der Korper, edited by Hasse and Baer, pp. 50-51. Line 21 p. 562 should read Fp(x) = Qp(x) X (xP ax) + Ax. Equation (9) p. 562 should read A = aoa(Pm-l)/(P-1) + aia(P'1)/(P-1) + + am,2a0P+1 + am-,a + am. In the expression line 9 p. 563 the last term should be A(")x. Equation (17) p. 564 should read F'"(x) = Fnl(x)P -Fn-1(Wn)PFn-1(X). Line 8 from below p. 574 should read

The author regrets that there are certain errors in [1] and would like to give the following corrections.

Among other results, it is proved that if a sequence {t 1l) of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all a-compact sets or all open sets, then there exists a maximal open set U such that ft,1 ( U) O-* { f t,) being the associated submeasures. In [5], J. D. Stein, Jr. considers some versions of Phillip's lemma and the Vitali-Hahn-Saks theorem for sequences of regular scalar-valued Borel measures on Hausdorff spaces. The measures he considers are bounded, regular, and finitely additive Borel measures which are easily seen to be countably additive (to prove this, first note that I pi, the variation of such a measure /L, is regular, bounded, and finitely additive [5, Lemma 1]) and so for a sequence {Bi) of Borel sets, B1j0, and > 0, C a sequence of compact sets (Ki), Ki c Bi and vi(Bi \ K1) some no and so I til(BiWO), an observation which enables one to prove his results easily, in more general forms, and under weaker conditions. Theory of submeasures developed in [1] will be used. Let G be an Abelian Hausdorff topological group, X a Hausdorff topological space, and yt a countably additive, regular, G-valued measure on X (by regularity we mean that for any Borel set B in X and a 0-nbd U in G, there exists a compact set C c B such that tt(K) C U, VK c B \ C, K Borel). If G is normed [1, p. 270], we have an associated submeasure ft, AK(B) = sup{14(A)I:A c B,A Borel) = sup{ I(C)I: C c B, Ccompact), which is finite [1, p. 279, Corollary 4.1 1], exhaustive, a-subadditive, and order-continuous [1, II]. It is also regular in the sense that given E > 0 and a Borel set B, 3 a compact set K c B such that i(B \ K) < (proof by contradiction, using exhaustivity). For a collection of measures or subReceived by the editors August 25, 1976. AMS (MOS) subject classifications (1970). Primary 28A25; Secondary 46G10.