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LetA be aC*-algebra on the separable Hilbert space ℋ, and let ℛ be the von Neumann generated byA. LetG be a topological group anda→φ(a) a representation ofG into the group of *-automorphisms ofA. Suppose that each φ(a) extends to a *-automorphism of ℛ, and suppose thata→〈φ(a)(T)x, y〉 is continuous for eachT inA andx, y and ℋ. Then, for a large class of groupsG, one has automatically thata→〈φ(a)(T)x,y〉 is continuous for allT in ℛ andx, y in ℋ.

DR. WATSON has shown in his letter to NATURE of March 31 (p. 512) how the general methods of Maxwell and Boltzmann may be applied to the particular problem discussed by Prof. Burnside. He has also pointed out an error in Burnside's reasoning—namely, the non-introduction of the factor u − U + c?, whereby Burnside's conclusions at variance with the Maxwell-Boltzmann law of partition of energy are vitiated.

AN easy solution of the paper-band puzzle described by Prof. C. V. Boys in NATURE of June 9, p. 774, is obtained as follows: Hold the hand with thumb up and palm towards you; place the paper band over the index finger, letting the ends hang down. Observe which way the original four half-twists were applied. Treat the nearest of these to the index finger on the palm side of the hand as if it were that of an ordinary single half-twist band; which complete, by looping up one-half of the band over the finger (the other twists being pushed out of the way into the remaining half). Then apply the surfaces one upon another at the finger; and turn the other half of the band inside out so as to get rid of two of the twists. It will be found to fit exactly upon the first half, as required.

The right hand side of Equation (3.10) of this paper should read g(x, Uo)NAlb(x) Uo rather than g(x, Uo)+NAlb(X)Uo. This error in sign is sufficient to invalidate partially the results claimed. With the corrected sign the right hand side of (3.10) is always positive if (i) N = 0 or if (ii) g(x, u )>NAlb (x ) u for u > 0 for some positive integer N > 0. In the first case our analysis demonstrates existence of a positive solution under hypotheses H-1 to H-5 for all 2 in 0 NAt b(x) u for u > 0 for some integer N > 0 ; we then conclude that a positive solution exists for all 2 in the internal 0__ O. The author wishes to point out that for the problem (1.1), (1.2) of nonlinear reactor dynamics the existence of a solution for all 2 > 0 follows from the work of NORMAN LEVJNSON (Dirichlet Problem for A u=f(P, u), J. Math. Mech. 12, 567-575 (1963)). We have not yet been able to establish that a positive solution must exist for 2>A~. An unpublished theorem of H. B. KELLER shows that positive solutions are unique for 2 > 0. Thus, on the basis of the corrected results we know that there exists a unique positive solution for all 2 in 0 A~ still remains open, though we conjecture that it must be valid.

IN the recent article by Prof. Robert B. Grieves and Dibakar Bhattacharyya1, Fig. 2 shows a relation between two variables with the statement “… The residual DSS concentrations are related in Fig. 2 to the ratio of DSS to trivalent iron in the feed. As xi/zi was increased, xr remained relatively constant and then rose sharply above a feed ratio of approximately 1.0.”. In view of the close attention given to statistical aspects of data analysis in other parts of the article, the statement about the data of Fig. 2 appeared somewhat questionable. Accordingly, I read off the numbers from the graph as best I could with a piece of superimposed graph paper and replotted the data on log–log co-ordinates. The data suggested a relation of the form xr=√(2xi/zi), so I prepared a graph as shown in Fig. 1. Within the variability of the data, it does not appear to me that the data can be considered approximately constant below xi/zi of 1, but rather that a regular increase is observed over the full range of the values. Certainly the variability of the data does not warrant a definite conclusion on the relationship.

A NEW volume of Proceedings of the Bologna Mathematical Congress begins with a short paper by Max Bruckner, on the old problem of how many different polyhedra are possible of n sides—with the limiting condition that all the corners shall be trihedral. In my book on “Growth and Form” I dealt with the kindred problem of the possible number of arrangements of a plane assemblage of cells, their partition-walls all meeting three-by-three, as is actually the case in a system of soap-bubbles or of living cells. The problem of the polyhedra is just as interesting to the biologist; for any natural clump of cells, such as a totally segmented egg or ‘morula’, may be looked on as a polyhedron and may be studied accordingly.

It has been shown by T. Head that if G is an abelian and S = Je~A G is a union of abelian groups, then group there exists an isomorphism e:G@S ~UacA (G@G) defined by e(g@s) = g@s for all g (G, s ~ S. In this paper it is shown that if S = UaeA Ga and T = U 8~B H8 are both unions of abelian groups then the surjective homomorphism 4: S~T ÷ Ue~AUS~B (Ga@H~) defined by ~(s ~t) = s ~t is an isomorphism if and only if either S or T is a group. This theorem is then used to show that if S is an abelian semigroup and T = ~e~A Ge is a union of at least two disjoint groups, then there exists a natural surjective homomorphism ~: S~T~e, A S~G which is an isomorphism iff S is archimedean. In this paper all semigroups will be abelian. Z and 0 will be the additive group of integers and singleton semigroup respectively. Homomorphisms will be referred to as maps. The following development of the union of groups U and the surjective homomorphism ~ in Theorem 1 are a generalization of results by T. Head [2]. Given S = ~A Ge and T = US~B HS, unions of groups

Let G toe a 2-transitive permutation group of a set !! of npoints and let P be a Sylow p-subgroup of G where p is aprime dividing \G\ . If we restrict the lengths of the orbitsof P , can we correspondingly restrict the order of P ? In theprevious two papers of this series we were concerned with thecase in which all P-orbits have length at most p ; in thesecond paper we looked at Sylow p-subgroups of a two pointstabiliser. We showed that either P had order p , orG > A , G = PSL(2, 5) with p = 2 , or G = M of degree 12with p = 3 . In this paper we assume that P has a subgroup Qof index p and all orbits of Q have length at most p . We2conclude that either P has order at most p , or the groupsare known; namely PSL(3, p) S G5 PGL(3, p) ,ASL(2, p) £ G 5 AGL(2, p) , G = PrL(2, 8) with p = 3 ,G = M with p = 3 , G = PGL(2, 5) with p = 2 , or G > A2with 3p - n < 2p ; all in their natural representations.Let G be a doubly transitive permutation group on a set ft of npoints and let P be a Sylow p-subgroup of G where p is a primedividing \G\ . The previous two papers [9, JO] were concerned with thesituation in which P has no orbit of length greater than p . We showedessentially that either G contains the alternating group or P has orderp . The general problem is the following:Received 21 May 1975.

In this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are: (i) To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1...ξn), for such sequences. (ii) To obtain limiting laws of the form $$\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!} $$ where Mn(r)is the r-th largest of ξ1...ξn, and Prξ1>un∼Τ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results. (iii) As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d. (iv) To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.

Let Y be a topological space and X a subspace of Y. We assume that X is the union of an increasing sequence of subspaces KS such that every quasi-compact subset of X is contained in some KS and the singular homology groups of all KS are finitely generated. The object of this paper is to give a purely algebraic characterisation of the following subgroup of Hq (X,Z): it consists of all those elements in Hq (X,Z) whose image in each of the Hq (KS,Z) lies in the image of the induced homomorphism Hq (Y,Z)→Hq (KS,Z), These subgroups are encountered in Runge approximation theory. Partial results were obtained in an earlier common paper with K. Stein, [1].

In the paper [1] I have given an example showing that the relation of equivalence of automata defined by Dr. R. Nowakowski1 is not a congruence with respect to the operation of coupling with retardation, In this paper I shall prove that a certain re? lation ? which seems to me to be inessential contraction of the relation of equivalence of automata defined by Nowakowski ? is the congruence with respect to the opera? tion of connecting and coupling with retardation2. Nowakowski's definition of equi? valence of automata is based on the notion of homomorphism3 of automata. Here I shall use a certain contraction of this notion. The contraction consists in putting identities for the functions b and c appearing in this definition. The relation of equi? valence of automata will be contracted to these automata in which the sets of states, inputs and outputs are the sets of sequences consisting of O's and l's, and the output function is defined on the set of states only. It is the only difference between the no? tion of equivalence of automata I shall use in this paper and the notion of equivalence of automata defined by Nowakowski. I give some lemmata which are necessary in proofs of basic theorems of this paper. Lemma 1. // the automaton is homomorphic to the automaton , then the automaton F?{0, 1}"\ {0, 1}% {0, l}Pl, ?j, Aj? w homomorphic to the automaton F ?{0, l} 2, {0, l}% {0, l}\ S2, A2?. Proof. Let 8J, A^, S2, A2 be the transit functions of the automata F ?{0, 1}% {0, l}\ {0, 1}% o\, Ai? and F?{0, lp, {0, 1}"?, {0, l}\ S2, A2? respectively. In view of the definition of coupling with retardation it is easy to prove the equalities: F?{o,i}"s{o,i}vo,i}'?,$i,x;? = = 5 1. F ?{0, 1}"., {0, 1}-., {0, i}*, s25 x2? = ; = .

In a recent note [6] the first author has announced the discovery of a family of transformation groups (G,), > o which have the following property: G, acts on the n! elements of the symmetric group ~ , and the number of its orbits is equal to the n-th tangent or secant number, according as n is odd or even. The purpose of this paper is to give a complete description of these groups. Applications to enumeration problems will appear in a subsequent paper. The tangent (or Euler) number are defined by the series expansion of tan u

In a series of papers [20, 211 and [22], Schmid obtained several important results on the discrete series for semisimple Lie groups. The purpose of this paper is to prove Schmid's results by somewhat different methods and to relax as much as possible the restriction imposed on the regularity of the parameters of discrete classes. Though the basic line is similar to Schmid's, the main difference is that our methods do not rely upon complex analysis on the non-compact flag manifold G/T as developed in [20] and [21]. Rather, we just rely on some elementary differential calculus on the symmetric space G/K. This difference gives rise to less restrictive assumptions since the results on the vanishing of L2-cohomologies in the symmetric space situation seem to be sharper, so far. Our work also leads to some interesting results concerning the multiplicity formula of discrete classes in L 2 (/' \ G). In our development, we shall give an alternative proof of the key fact (Theorem 1, w 4) which is obtained in [20] using the complex analysis on G/T. Our proof, given in w 5, will be carried out directly in the symmetric space situation, and some standard theory of sheaf cohomology centering the Borel-WeilBott theorem on the compact flag manifold KIT will be used as in [20]. In this proof, our methods seem to be quite elementary. We now give a more precise description of the contents of the paper. Let G be a non-compact real semisimple Lie group with discrete series g2 4= qS. Assume, for simplicity throughout the paper, that G is a connected real form of a simply connected complex semisimple Lie group G e. Harish-Chandra 1-7] showed that there exists a compact Cartan subgroup T of G and that, if one denotes by T' the set of regular characters of T, there exists a distinguished surjection co: T'---, g2. We fix T and a maximal compact subgroup K containing T once and for all. Let ge, t e denote the complexifications of the Lie algebras g, t of G, T respectively. In considering the discrete class co (A) for a given regular character A e T', we always choose a positive root system P for the pair (go, t~) such that A is regular dominant with respect to 19, i.e., P = {~; (A, c0>0 }. Here as

This paper deals with the question of the existence of classical solutions for the equations $$\frac{{\partial ^{2} u}{\partial t^{2} }} + \sum_{\begin{subarray}{l} |\alpha| \leqslant m \\ | \beta | \leqslant m \end{subarray}} D^{\alpha} (A_{\alpha \beta } (x,t) D^{\beta} u) = f (t,x,u)$$ on [0,T] × G. G is a bounded or unbounded domain; the differential operator in the space variables is elliptic; the initial values of u are prescribed and Dαu (t,x) vanishes for (t,x) ∈ [0,T] × ∂G, |α|≤ m−1. First we develop a method for solving regularly linear wave equations. In contrast to the usual compatibility conditions, our method requires less differentiability in t but imposes some boundary conditions on f(t). It allows some applications to nonlinear problems which will be treated in the second part of this paper and which e.g. enable us to solve ∂2 u/∂t2−A(t)u+u3=f.

In the paper [2] there are given the following definitions: Definition 1. By an automaton (S,X, Y,8,X) we mean the system of three sets S, X, Y and two functions 8:SxX->S, \:SxX-+ Y. Arbitrary automata will be denoted by capital Latin letters. Definition 2. We say that an automaton is homomorphic to the automaton if there exist functions (not necessarily one-to-one) a and b and a one-to-one function c a:S->S;, b:X-+X', c:Y->Y' such that for every pair (s,x}eSxX have a [8 (s, x)] = l'\a (s), b (x)] and c [A (s, x)] = A'[a (s), b (x)] . If the functions a, b, c are one-to-one then the homomorphism of automata will be called isomorphism of automata. We shall write A horn B if the automata A and B are homomorphic and A iz B if the automata are isomorphic. It is easy to see that homomorphism of automata is a transitive relation. In this paper1 we shall give an example of equivalent automata in the sense esta? blished by R. Nowakowski in the paper [3]2; however, if we perform the operation of coupling with retardation on them then they cease being equivalent. It means that the relation of equivalence in the considered sense which is also an equivalence rela? tion in the algebraic sense is not a congruence with respect to the operation of coupl? ing with retardation. Let {0, l}k be the set of all sequences x = (xx,..., xk}, where xt e {0,1}, and k is a positive integer. By the operation of coupling with retardation3 we mean the ope? ration defined in the following way

Introduction Around 1900 Hilbert, Hecke ([7]), Blumenthal ([2]) and others started the study of certain 2-dimensional complex spaces, which are closely related to the classification of special types of 2-dimensional abelian varieties. These complex spaces can easily be described. In fact, let n be a natural number, n > 1, which is square free and let K = Q q/n). Then, if o K is the ring of algebraic integers of K, the group SLz(oK) operates in a natural way on ~ • ~ and on ~ • ~ where ~ is the upper and ~ the lower half plane of C. The quotients of See • ~ and Jg x J g by the action of SL2(ov,) are the 2-dimensional complex spaces mentioned above. If the field K has a unit of negative norm, then the two actions on ~ • ~ and ~ • J g are isomorphic. This is true if n is a prime congruent 1 rood 4, the only case we shall consider in this paper. Therefore, from now on we assume that K= Q (l,/p), where p is a prime congruent 1 mod 4. The complex space Jg x .)~/SLz(oK) can be compactified by means of a finite number of points, called the cusps, to a compact 2-dimensional complex space. After resolving the cusps and also the quotient singularities on ~ x ~ / S L 2 (OK) , both in a canonical, explicit way, a nonsingular compact complex surface Y(R) is obtained, which in fact is an algebraic surface. The field of meromorphic (i. e. rational) functions on Y(p) is isomorphic to the field of meromorphic functions of • ~ / S L 2 (OK). On the other hand, although no complete classification of algebraic surfaces is known, there exists a rough classification in several classes (for most of which the surfaces contained in that class can be classified completely, at least in principle). In big outline this classification was already known to the italian school, but its precise formulation (this time also covering the non-algebraic case) and many of the proofs involved are due to Kodaira. Now the question considered in this paper is the following: where are the surfaces Y(p) to be placed in the rough classification of algebraic surfaces?

This paper presents the mathematical derivation of a Bayes solution for the symmetric multiple comparisons problem presented earlier by Waller and Duncan [4]. Because the earlier paper details the model and assumptions on which the solution is based, only the functions and results needed to complete the development are stated here. Typically the data for a multiple comparisons problem consists of n independent normally distibuted t r ea tmen t means (~, z2, " " , ~ ) which est imate n t rue means (pl, p~, . . . , p~) and an independent estimate, say 8~E with f~ degrees of freedom, of the error variance a~. Fur ther , we suppose tha t each mean ~ is based on r replications. The problem is then to choose for each of the h = n ( n 1 ) ordered pairs of means ei ther decision d~: ~,>;~j or decision d o , j : /~ not ranked relative to/J j . However, Duncan [2] shows tha t under the assumptions of symmet ry and an additive loss model, the Bayes rule for the full multiple comparisons problem is the simultaneous application of the Bayes rule to select ei ther d~ or d[~ for the means (/~,/~2) to all h pairs of means. Thus our major concern is to obtain the Bayes rule for selecting one of the decisions

In attacking the problem of this paper (see Section 1), the authors were confronted with finding the distribution of a (k×k) matrix of random variablesR=P′VP, wherePP′=Σ -1, and where the (k×k) symmetric matrix Σ-1 has the Wishart distribution, matrix [(n−1)V]−1, and degrees of freedom (n−1), withV a (k×k) symmetric positive definite matrix of constants. This distribution (whenP is lower triangular with positive diagonal elements), and a related result, has recently been found by the authors and given in Tan and Guttman [7]. In this paper we use these results (stated here without proof in Theorems 1.1 and 1.2) to help us construct a β-expectation tolerance region, when sampling is from thek-variate normal,N(μ,Σ), where Σ is positive definite.

This paper provides a result necessary in the classification of simple finite groups with abelian S2-subgroups. In classifying these groups in another paper, we shall determine the structure of the centralizers of the involutions in such groups (cf. [24]). In particular, for those groups which have S2-subgroups of order 8, we shall verify the hypothesis of the principal theorem of this paper. Then from this theorem follows the conditions by which the recent papers of JANKO and THOMPSON [16], JANKO [15], and WARD [25] show that the simple group in question is either the one recently discovered by JANKO (cf. [15]) or has the character table given by Ward of the simple group of automorphisms of the simple Lie algebra of type (7 2 over a finite field of characteristic 3 discovered by REE [18]. The latter type of group is defined in Section 1 as an R-group. Theorem. Let (5 be a group with abelian S2-subgroups of order 8. Let = (J) be the subgroup generated by an involution J. Suppose that the centralizer .~=C~(J) of J in (5 contains a normal subgroup which is isomorphic to 1 PSL(2, q). Then one of the following holds.

[-5, 6] promises to be the subject of many investigations. We give here a short proof that this lattice is characterised by some of its simplest properties. Although we must quote two theorems to open and close the proof, the reader can take these on trust if he wishes, and he will find that the proof is otherwise completely self-contained. In particular the Leech lattice will itself be defined in the course of its characterisation, so that no acquaintance with it is presupposed, although for the convenience of certain readers we have adhered to the notation of [1, 2]. The methods of this paper immediately give the order of the group which is the theme of our two papers [1, 2], and can be used to give other information about the group and the lattice. But the main interest of this paper probably lies in the possibility that the argument can be extended to yield constructions for other interesting lattices. We conjecture, however, that any such extended argument must be considerably more complicated, since it is very doubtful that the extreme "tightness" which the reader will observe so often in our proof can ever occur again. We proceed at once to the proof. We shall show that Leech's is the only lattice in fewer than 32 dimensions which has one point per unit volume and in which the square of every non-zero distance is an even integer greater than 2. A lattice with one point per unit volume is called unimodular, and then also even if every squared distance is an even integer. In an even unimodular lattice of dimension d we use u, for the number of vectors of squared length n. Our theorem then asserts that Leech's is the only even unimodular lattice with d < 32 and u 2 = 0.

Let G be a topological group and H a closed normal subgroup of G. !f G contains a closed subgroup K isomorphic to the quotient group G/H such that Hc~K={1}, the identity of G and that the map (h, k ) ~ h k is a homeomorphism of H x K onto G, then G is said to be a semidirect product of H and K, and we say that G splits over H. In recent years, a great deal has been learned about the structure of connected locally compact groups on the one hand and of compact totally disconnected groups on the other. The natural synthesis of these two seems to be locally compact groups which are compact modulo component of the identity. Let [C] denote the class of all such groups. In this paper, we are primarily concerned with study of [C]-groups. In studying the structure of [C]-groups, the chief purpose we have in mind is to see how far these groups are away from being semidirect products of connected locally compact groups and totally disconnected compact groups. Even though one can hope for such a splitting in rare instances, we are able to find a totally disconnected compact 'supplement ' to the identity component of a [C]-group. The proof of existence of such a supplement given in this paper uses the method which HOFMANN and MOSTERT [6] have developed in their investigation of splittings of topological groups over vector subgroups. As is well known, a study of [C]-groups sometimes requires aconsiderable knowledge on Lie algebras, since these groups may be approximated by Lie groups. Thus we devote the first chapter to an investigation of automorphisms of Lie algebras. Although this chapter is not an integral part of our topological investigations, it is closely related to the following chapters. The following are outlines of results in the order in which the chapters containing them occur.

Let 7~: (X, X ) -~ (M, (9) be a holomorphic mapping of an analytic space (X, JC) onto a complex manifold (M, (9). For each t eM, the fibre X t : r c l ( t ) is an analytic set in X. Let m t c (gt be the ideal sheaf of germs of all holomorphic functions which vanish on the point t and (mt o re) the idealsubsheaf in generated by g o 7c, g ~ mr. Let us put ~ , . . = ~ / ( m t o ~z) l X . Then (X t , 2/f t) is an analytic space. Thus a holomorphic mapping lr: X --~ M gives a family of analytic spaces X~, t ~ M . A family is reduced if all Xt are reduced analytic spaces (or the analytic spaces which are defined by Cartan and Serre). A family ~: X ~ M is flat at x ~ X if TOrl~(2/f~, (9]mt)=O where l r ( x ) t . Grauert and Kerner give in [43 an equivalent condition to flatness. In this paper, we shall give an another equivalent condition to flatness, that is. rc is flat at x if and only if the sequence of homomorphisms t~, ..., ~,, of ~'~x into 24~x is a regular sequence where (t I . . . . , t,) are local coordinates in a neighborhood of ~z (x). We shall give that if zr is an open mapping, then rc is flat. It is showed in [6] that a reduced family zr: X ~ M is an open mapping if and only if ~ is flat. Let re: X --+ M be a flat reduced family and let Xo be the fibre at a point 0e M. Then there exists a natural homomorphism of the sheaf ~-* (M) into the sheaf ~Y~(Xo), where J-* (M) is the sheaf which the hotomorphic vector field on M is lifted to X and 3-;~(Xo) is the sheaf of the infinitesimal deformation of Xo. If the natural homomorphism ~ * ( M ) ~ ( X 0) is surjective. ~ is called complete at x s X . Kerner proved in [6] that ~z is complete at a regular point of X. In this paper, we shall give that taking X ' = ~z-~ ( M 3 instead of X o in the theorem proved by Kerner, a similar consequence holds, where M' is any submanifold of M.

The conditions on N can be replaced by the assumption tha t N is at least 6 and the intersection number/~ has the value ( q § 1) 2. The cases N----4 and 5 are unsettled even under the assumption tha t /~ has this value. We number the main theorems of this sequences of papers consecutively, Theorems I and I I having appeared in [4]. Theorem I I I can be regarded as a linear analogue of Theorem I I , which in turn corresponds to the case q---1 of Theorem I I I . The subgroups of P F L N (q) transitive on the set of 4-simpliees in PN-1 (q) can be regarded as linear analogues of 4-fold transitive permutat ion groups of degTee N. I t does not appear to be known whether PSLN (q) is always the minimal such group. Since this paper was submitted, D. PE~R~ has proved tha t a subgroup of PTLN (q), q 9 2, which is transitive on the set of k-dimensional linear subvarieties of PN-I(q) for some/c, 2 ~ / c _< [2//2] 1, contains PSLN(q). This implies in particular tha t a subgroup of PI'LN(q), N _~ 6, q :~ 2, which has rank 3 on the set of lines

This paper reviews properties of the negative multinomial distribution and some related distributions. On the negative binomial distribution (NBn) much has been wri t ten and the contributions were summarized in two recent survey reports ([3], [9]). In the course of researches on the NBn its multivariate extension has been tried. The notion of the negative multinomial distribution (NMn) was first introduced in the model of the inverse sampling in multiple Bernoulli trials and accordingly the parameter " k " was limited to integral values. Later, in the papers discussing statistical theory of accident, absenteeism and contagion, the NMn was introduced under the name " multivariate negative binomial d i s t r ibu t ion" ([1], [2], [4], etc.). Among them Bates and Neyman's paper [4] was the first which t reated the NMn systematically. Surveying the properties of the NMn we remark the relations among distributions, which make clear the probabilistic s t ructure of the individual distributions. We notice especially tha t the relation between the binomial distribution (Bn) and the NBn is quite similar to tha t between the multinomial distribution (Mn) and the NMn, so the name NMn is preferable to the multivariate NBn. On the way of discussions a multivariate extension of Fisher 's logarithmic series, the negative hypergeometric distribution (NHg) and its multivariate extension will be treated. Here, the name NHg is proposed, though this distribution has been discussed in literatures under different names.

The problem has been solved by Stoll for the case p ( n 1 ) [6] and for the case where n(r, A) is bounded [9], [10] and [11]. In the first part of this paper the case p = 0 is treated. The second part of the paper is devoted to the case 0 < p < (n 1) with the additional condition that the given analytic set A in Vis regular on an ( n 2 -p) -d imens iona l projective subplane of the infinite plane of the projective closure of V. The author wishes to thank Professor Wilhelm Stoll for his advice and encouragement.

In an earlier paper [12], the author studied the effect of departures from assumptions on the Ftes t for two associate PBIB designs with 2~=0 and 22=1. It was found tha t as in the case of Randomized Blocks, Latin Square and BIB designs, the Ftes t works reasonably well as an approximation to the randomization test. In this paper, the behavior of the Ftes t is studied for the same class of PBIB designs when the observations are assumed to have arisen from a t runcated normal population. Such samples arise fairly often in practical situations. Before proceeding fur ther , it must be pointed out tha t several research workers have investigated the effect of general non-normality" on the analysis of variance test . The important among them are Pearson [9], Bar t le t t [3], Geary [8], Welch [13], [14], and David and Johnson [6], [7]. These workers have found tha t the Ftes t for means is remarkably insensitive to general non-normality of the parent populations. In view of this, it may be reasonably concluded tha t a mild truncation will not effect the usual Ftes t to any appreciable extent . In fact, Agarwal and Gutman [1], [2] have found that when the truncation is symmetric, the change in the power of the one sided ttest for the mean of a normal population is very little even when all values beyond +(1.5)z (where G ~ is the variance of the complete population) are discarded. However, t runcation results in the test becoming more conservative in the sense that the rejection of the null hypothesis is more difficult. In spite of the assurances an experimenter has about the robustness of the Ftes t to truncation, a careful s tudy of the effect of t runcation seems to be very important for the following reasons

In [2] Knapp found necessary and sufficient conditions for two elliptic transformations to generate a discontinuous subgroup of L f(2, IR), the group of linear fractional transformations. In [6] Lyndon and Ullman give conditions for two hyperbolic transformations whose fixed points separate each other to generate a discrete free group of rank 2. In this paper we find necessary and sufficient conditions for the group generated by any pair A, B e L f ( 2 , IR) to be the discrete free product of the cyclic groups {A} and {B}. The results of this paper along with those of [2] resolve the question of when {A, B} is discrete in all cases except when the trace of the commutator a ( [A,B])= __2cos rrc, where r is rational. I would like to thank Professor Joseph Lehner for guiding me towards this problem. Also, the referee has kindly pointed out that Fricke and Klein discuss this problem as well as the case [A, Bl n-1 in [1].

This paper connects with Theorem 3 of the author’s paper [1], in which two criteria for type (B)d convergence ([3]) are shown to be incomparable to each other by presenting two examples. However, the statement of the theorem is not complete. In the present paper, we shall modify the statement of the theorem and give a proof by presenting a new example.

In this paper we consider some results on the orbits of groups of collineations, or, more generally, on the point and block classes of tactical decompositions, on symmetric balanced incomplete block designs (symmetric BIBD = (v, k, 2)system=finite 2-plane), and we consider generalizations to (not necessarily symmetric) BIBD and other combinatorial designs. The results are about the number of point and block classes (or orbits, i.e. sets of transitivity) and the numbers of elements in these classes. In Sections 2, 3 and 4 below we exhibit the key role of the rank of the incidence matrix of a design, while the remainder of the paper uses more specific properties of the incidence relations. Included in Section 2 is a simple new proof of the theorem of DEMBOWSKI [7] on the equality of the numbers of

The main result of this paper is the following theorem on the generalized fixed point index for locally condensing mappings: Let X be a subset of a Banach space such that there exists a locally finite covering {Ci|i∈I} of X by closed, convex subsets of X. If s=pt with p prime and t∈N, M an open subset of X and g: D[g]→X with D[g]⊂X and M⊂D[gS] such that g|M and gS|M are locally condensing and the fixed point set F of gS|M is compact and g(F)⊂M, then iX(gS,M)≡iX(g,M) mod p. This theorem can be applied in the theory of asymptotic fixed point theorems: An example may be found at the end of this paper.

JAFFARD ([3], Chapt. t , Sect. 3) has shown that a directed partially ordered group may be imbedded, by means of an ideal extension, in a complete lattice semigroup. In this paper we show how this procedure may be generalised to a wide class of partially ordered semigroups, with, in addition, preservation of existing least upper bounds. In particular, we g~ve a construction which is applicable to a class of semigroups which includeS that of all residuated semigroups. Further, we obtain necessary and sufficient conditions under which a partially ordered semigroup m a y be imbedded in a conditionally complete group. We introduce in Section l the terminology used in the paper, and in the next Section the notion, of an ideal extension,-in terms of which we find necessary and sufficient conditions that a partially ordered semigroup may be imbedded, with preservation of least upper bounds, in a (conditionally) complete lattice semigroup. I t is shown that not every partially ordered semigroup may be so imbedded. We describe the result mentioned above for residuated semigroups, and complete the Section by proving some general results on ideal extensions. In Section 3 relations between different ideal systems are considered, and in the fiflal Section we establish necessary and sufficient conditions under which a partially ordered semigroup may be imbedded in a conditionally complete group.

In a previous paper [1], it has been shown that for a partially balanced incomplete block design with two associate classes the nulldistribution of the F-statistic under the ' to ta l ' null-hypothesis (i.e., treatment-effects being all equal to zero) can be approximated by the familiar central F-distribution even under the Neyman model (i.e., an intra-block analysis model with both the unit errors and the technical errors), if it is randomized. As was announced in that paper, the approximate distributions of the F-statistic under the 'par t ia l ' null-hypothesis have been left to fur ther discussion. In the present article, the authors are concerned with this problem. They set forth the problem for a partially balanced incomplete block design with m associate classes and consider the null-distribution of the F-statistic for testing a 'par t ia l ' null-hypothesis, so that it includes the ' t o t a l ' null-hypothesis as a special case and they reached the conclusion that the null-distribution of the F-statistic can be approximated, after the randomization, by a certain central F-distribution with appropriate degrees of freedom, if certain uniformity conditions are imposed on the unit errors and the number b of the blocks is sufficiently large. In section 1 the spectral decomposition of the matrix NN', where N being the incidence matrix of the design under consideration, is given and this is useful for the later discussions. The null-distribution of the F-statistic for testing a partial nullhypothesis before the randomization under the Neyman model is presented in section 2, and this turns out to be a non-central F-distribution whose non-centrality parameter depends upon the quantities ~ and ~ both being the quadratic forms of the unit errors.

w 1. Introduction In this paper, "group" is to mean finite group. We shall consider certain properties of groups admitting fixed-point-free automorphisms (that is, automorphisms which leave only the identity element fixed). Such groups have been studied off and on for many years. BURNSIDE proved before 1897 that a group admitting a fixed-point-free automorphism of period two is abelian, and FRo~ENItJS proved in 1901 that a group admitting one of period three is nilpotent of class at most two (see [2]). The conjecture of FROBENIUS that a group admitting a fixed-point-free automorphism of prime period is nilpotent was proved for solvable groups by WITT about 1936 (unpublished); that the class of nilpotence is bounded by a function of the period was proved by G. I~IGMAN [9] in 1957. The proof of the FROBENIUS conjecture was completed by THOMVSON [I4] in 1959, when he showed that a group admitting a fixedpoint-free automorphism of prime period is solvable. In 1961, GOPCE~STEIN and HERSTEIN [6] proved that a group admitting a fixed-point-free automorphism of period four is solvable and has a nilpotent commutator subgroup. The question of the solvability of groups admitting fixed-point-free automorphisms of arbitrary order has not been settled, and will not be studied here, where we shall assume the groups to be solvable. (Because of the theorem of THOMVSON and FEIT [4], that all groups of odd order are solvable, this restriction is not very severe.) The main result of this paper is a proof that, except possibly in certain special cases, a solvable group admitting a fixed-point-free automorphism of period pn, p a prime, has nilpotent height at most n (see p. 265). We now introduce some of the properties studied in this paper. Let G be a solvable group. Let V(G) be the maximal nilpotent normal subgroup of G. Because the product of two nilpotent normal subgroups is also a nilpotent normal subgroup, V(G) is a weU-defined characteristic subgroup of G. We let V 1 = V(G), and inductively define Vk SO that Vk/Vk_ 1 = V(G/V~_ 1)" V(G) is called the Fitting subgroup of G. If V, = G, the Fitting length or nilpotent height of G is =n. We note that if G and H are groups, V(G x H)= V(G) x V(H). Thus we see easily that the Fitting length of the direct product of two groups is the maximum of the Fitting lengths of the factors. We further note that if 1 c Ni N z ~... ~N~ = G is a characteristic series for G with nilpotent factor groups

in which each term is normal in its successor and each factor satisfies either the maximum or the minimum condition for subgroups. A group G has the subnormal intersection property if the intersection of each collection of subnormal subgroups of G is itself a subnormal subgroup of G, that is, if G contains a subnormal closure of every subgroup of G. For this to happen it is sufficient, but not necessary, that the defects of the subnormal subgroups are bounded. In [6] Robinson shows that a finitely generated soluble group has the subnormal intersection property if and only if it is finite-by-nilpotent. Hence, in particular, soluble groups satisfying the maximum condition for subgroups and having the subnormal intersection property must be finite-by-nilpotent. On the other hand a soluble group satisfying the minimum condition for subgroups is an extension of a periodic radicable abelian group by a finite group, by a well-known result of Cernikov [2], and as such it has the subnormal intersection property (see, for instance, Lemma 2.2 of [7]). In this paper we show that a soluble minimax group with the subnormal intersection property is an extension of a radicable abelian group satisfying the minimum condition by a (torsion-free nilpotent)-by-finite group (Theorem A). However a soluble minimax group with this structure need not have the subnormal intersection property, as the infinite dihedral group shows. Thus Theorem A does not give a complete picture of the structure of soluble minimax groups having the subnormal intersection property. The structure of these groups is probably rather complex. However there is a subclass of the class of groups under consideration which is easy to characterise. A soluble minimax group whose every subgroup has the subnormal intersection property is an extension of a group satisfying the minimum condition by a nilpotent group, and conversely every subgroup of a group in the latter class has the subnormal intersection property (Theorem B). The material in this paper forms part of a Ph.D. thesis for the University of London. I wish to thank my supervisor Dr. D. J. S. Robinson for his valuable

Let S be a regular semigroup. The lattice of all idempotent-separating congruences on S and the lattice of all group congruences on S are both modular sublattices of the full lattice of congruences on S. It is evident that the set theoretical union of these two sublattices, (S), is also a sublattice of the full lattice of congruences on S. It is natural to ask: Under what conditions is the sublattice (S) modular? In this paper we obtain a necessary and sufficient condition for the sublattice (S) to be modular when S is what we call a θ-regular semigroup. Bisimple ω-semigroups and simple regular ω-semigroups are θ-regular semigroups and so this paper extends the work of Munn [5] and Baird [1].

This paper concerns the concept of upper semicontinuity of variable sets, precisely the variant of Kuratowski's definition of upper semicontinuity that Cesari has denoted as property (Q). This concept has been used by Cesari in most of his papers on existence theorems for optimal solutions, and later used by Olech, Lasota and Olech, Brunovsky, Baum, Suryanarayana, and Angell. First, criteria are given for property (Q) in addition to those which had been already given previously. Then, it is shown that a slight restriction in the concept can be expressed in a form which is similar to Tonelli's concept of seminormality for free problems of the calculus of variations. Thus, the property (Q) appears to be a generalization to Lagrange problems of control of the well-known concept of seminormality for free problems.

This paper deals with first-order matrix partial differential operators of the form $$L = - i \mathop \sum \limits_{j = 1}^n L_j (x)D_i + L_o (x)$$ (1) where x=(x1, ..., xn)∈Rn, Dj=∂/∂xj, and the Lj(x), j=0, 1, 2, ..., n, are m′×m matrix-valued functions of x. Let L2, m, be the Hilbert space of square integrable m-vector-valued functions on Rn. The operator(1) determines a closed linear operator L : L2, m→L2, m′. L is said to be coercive on a subspace V⊂L2, m if there is a constant μ>0 such that $$\mathop \sum \limits_{j = 1}^n \left\| {D_j u} \right\|^2 \leqslant \mu ^2 (\left\| {Lu} \right\|^2 + \left\| u \right\|^2 )$$ (2) for all u∈D(L)∩V, where D(L) denotes the domain of L and ∥·∥ denotes the norm in L2, m. L is said to have constant deficit k in Rn if the symbol\(L(p,x) = \mathop \sum \limits_{j = 1}^n L_j (x)p_j\) has constant rank m–k for all x∈Rn and p∈Rn−{0} (L is elliptic if and only if k=0). The paper gives criteria for nonelliptic operators L of constant deficit k to be coercive on subspaces. In particular, operators of the form\(\Lambda = - iE(x)^{ - 1} \mathop \sum \limits_{j = 1}^n A_j D_j\) are considered where E(x) and Aj are m×m Hermitian matrices and E(x) is positive definite.Λ defines a self-adjoint operator on the Hilbert space H with inner product\((u,v)E = \mathop \smallint \limits_{R^n } u^* Ev dx\). It is shown, under suitable hypotheses on E(x) and Aj, thatΛ is coercive on the subspace N(Λ)⊥, the orthogonal complement in H of N(Λ), the nullspace ofΛ.

In an earliner paper the authors proved the follwoing inequalities for the index (I(r,f) of an entire function f $$\max (0,\lambda - 1) \leqslant \mathop {\lim }\limits_{r \to \infty } \sup \frac{{l\mathop o\limits^ + g I(r,f)}}{{\log r}} \leqslant \lambda $$ , where λ is the order of f. It is known that the upper bound is sharp. In this paper the authors prove that the lower bound cannot be sharpened if λ≥1 is a rational number. In this direction it is shown that for certain solutions of linear differential equations $$w^{(n)} + a_{n - 1} w^{(n - 1)} + \ldots + a_o w = 0$$ with polynomial coefficients aj in the left side of the above inequality “≤” and “lim sup” are replaced by “=” and “lim”. Also it is proved that the differential equation has constant coefficients if and only if every solution is of bounded index.

In this paper, power comparisons are made for tests of each of the following two hypotheses based on individual characteristic roots of a matrix arising in each case: (i) independence between ap-set and aq-set of variates in a (p+q)-variate normal population withp≦q and (ii) equality ofp-dimensional mean vectors ofl p-variate normal populations having a common covariance matrix. At first, a few lemmas are given which help to reduce the central distributions of the largest, smallest, second largest, and the second smallest roots in terms of incomplete beta functions or functions of them. Since the central distribution of the largest root has been discussed by Pillai earlier in several papers ([6], [8], [9], [11], [12], [13]) cdf’s of the three others in the central case are given. Further, the non-central distributions of the individual roots forp-3 are considered for the two hypotheses and that of the smaller root forp=2; that of the largest root forp=2 has been obtained by Pillai earlier, (Pillai [11], Pillai and Jayachandran [14]).

In this paper we prove three theorems, each of which gives a set of absolutely independent group axioms. Theorem 2 is an extension of Morgado's theorem obtained by using a weaker associativity axiom. The axioms of Theorem 3 are the first absolutely independent group axioms we have encountered for ,,~ich there is no k such that the axioms are absolutely independent (rood k) [2, p. 758]. The systems of Jacobson-Yoco~n and Morgado and the systems which appear in Theorems 1 and 2 of this paper are absolutely independent (rood ,~0). We conclude the paper with two axioms which define a group and which are very [l] (and in this case absolutely) independent (rood n) for every integer n ~ 1. We will write "xy" instead of " x , y" in all proofs. The following lemma is quite useful.

This paper is a sequel to a paper by Boasson and Nivat "Sur diverses familles de langages fermees par transduction rationnelle" [1] and uses in a crucial way the main result of [1]. Greibach in two papers [7, 8] has proved many AFL's to be non principal using a number of similar lemmas the prototype of which appears in [7] and concerns the substitution of one semi AFL in another semi AFL. All the AFL's dealt with in this manner are closed under some binary operation. In these pages we study two unary operators on families of languages, translation and closure under bracket for which we establish lemmas similar to the lemmas of Greibach. We can thus prove some well known AFL's to be non principal.

A subgroup V of G is called an j in jector of G if Vc~ N is F-maximal in N (that is, Vn N is a maximal ~-subgroup of N) for each subnormal subgroup N of G. It is known that the ~-injectors of a group G form a conjugacy class of subgroups 1,2]. If ~-injectors are always normal then ~ is called a normal Fitting class. In I-1] and [-3] some work has been done to classify normal Fitting classes. It has been shown in these papers that to any normal Fitting class there corresponds an object called a Fitting pair (f, A) defined as follows: A is an Abelian group and to each finite soluble group G there exists a homomorphism fG: G~A such that for all N ~ Gf~ = f~lN, where f~lN is the restriction of fG to N. Each such pair defines a Fitting class ~ given by G ~ if and only if fG(G)= 1, 1,Satz 3.1, 1.]. We use this to construct a Fitting class which contains $3, the symmetric group on 3 letters but does not contain the dihedral group of order 18. For some time it had been undecided whether the Fitting class generated by S 3 was the class consisting of all 3-groups extended by 2-groups. For a general survey see the paper of Cossey read at the conference on group theory held in Canberra, Australia 1973. The construction will actually provide uncountably many normal Fitting classes for all of which, the injectors have index 2. We will define and prove the existence of these classes by using Fitting pairs. Let f2 be the set of powers of odd primes and let 0 E f2. Let A be the cyclic group of order 2 and define, for any finite soluble group G

MY note in NATURE of October 8 may be extended. I found that one of my pupils, Mr. Glasgow, was assuming that the expansion and compression parts of a gas engine diagram followed laws of the type pvn constant; in cases where there was a probability that the clearance had not been measured accurately, so that v being the measured volume and c the constant error, he assumed p(v+e)n to be constaznt, and he was enabled to find c from the curve. There is no reason to believe that these curves ought to have such a law, although, curiously enough, following this assumption, the clearance obtained from the compression curve is usually not very different from that obtained from the expansion curve. Mr. Glasgow's method of finding c is much the same as what I shall now describe. An empirical formula of the type y=a+bxn would be exceedingly useful in many parts of pure and applied science if, when given a table of values of y and x, we could readily find a, b and n. I have often sought for a method of working, but without success. If a is zero, we have only to plot log y and log x as the coordinates of points on squared paper. If a is not zero, there is a clumsy method of using logarithmic paper which may be adopted, but it is not satisfactory. We now have a method easy of application. Thus values of x and y being given, draw the curve AB shown in the figure. Set off any convenient angle DOX. Select the point P. Draw PD, XF, FEQ, EH, &c, the lines XF, EH, &c, being at 45°. Project horizontally from the points PQR, &c, to M, U or N, W or V, &c, letting lines at 45° from M, U, &c, meet the horizontals at N, V, X, &c. If the above law holds, N, V, X, Z lie in a straight line. If they lie only approximately in a straight line, draw the line N′O′ lying most evenly among them. Then OO′ is the value of a, and n is log (1+tan N′O′Y)/log (1+tan DOX), and b is readily found.

Dr. G.L.C. Bond of the University of Reading has pointed out a mistake in my above paper. In the proof of Lemma 3.2 I made the inadvertent assumption that all rings are commutative, and in fact the lemma fails for the ring of 2 x 2 upper triangular matrices over Zp, p prime. Fortunately the error can be confined to Lemma 3.2, as was also pointed out by Dr. Bond. The only place where 3.2 is used is in the proof of 3.3, and here its use can be avoided as follows: we have R / J ~ T 1 G " "• 7", where each T/~292, and we pick elements ejeR which map by the canonical homomorph i sm R ~ R / J to the identity of T i. We find that e , = x + a where x e J and aeAnn(J ) . At this point we wish to assert that e,ZsAnn(J) for some integer 2>0 . In the paper we invoked 3.2; instead we show that )~ = c 1 will do, where c is the index of nilpotency of J. For then c-1 =(x+a)C-1 en

For a class of repeated two-person zero-sum games with incomplete information it was proved byAumann andMaschler that limν n exists,ν n being the value of the game withn repetitions. If the players know at each stage the moves done by both players at all previous stages,Aumann andMaschler could prove that the error termδ n=¦ν n — limν n ¦ satisfiesδ n≤c/√n for somec>0. It was then shown byZamir that this bound is the lowest possible. In this paper it is shown that if previous moves are not always announced,δ n may be of higher order of magnitude e.g.δ n≥c/n 1/3 for somec>0. New upper bounds forδ n are given for two classes of games.

This paper gives a characterization of the syntactic monoid of a hypercode H over a finite alphabet X, a hupercode being a non empty set of non empty words over X, which are pairwise incomparable relatively to the embedding partial order of X.

This paper deals with the behaviour of power series with coefficients in a non-archimedean valued field Ω for arguments in a non-archimedean Banachalgebra E/Ω. Beyond a trivial augmentation, the domain of convergence is extended in relation to Ω by certain elements b ∈ E for which\(\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{||b^n ||}}{{||b||^n }}} \right) = 0\). These elements are called pseudo-nilpotent and characterized by\(\mathop {\lim }\limits_{n \to \infty } (||b^n ||^{\frac{1}{n}} )< ||b||\). The examples of exponential functions, logarithms and powers show the changes in relation to the methods and results in Ω caused by the extension of the domain of convergence and the absence of norm multiplicativity and invertibility in E. Finally the algebraic-topological structure of a part of the domain of convergence of the exponential function is presented.

The paper contains constructions of finite transformation monoids such that the number of transformations from one point into another is a constant and whose groups of units are not transitive.

It is well known that in a topological semigroup S with an identity 1 the maximal subgroup H (1) must be open if 1 has an euclidean neighborhood (Mostert-Shields [7]). If multiplication in S is only “separately” continuous, i.e. x↦yx and x↦xy is continuous for all y∈S, the statement remains true if the underlying space of S is a compact manifold (Berglund [2], Lawson [6]). In this paper the case of a compact semigroup with only one-sided continuity (i.e. x↦yx or x↦xy is continuous for all y∈S) which is defined on an interval or a circle is investigated. It is also shown that a group, defined on the line or on a circle, must be a topological group if it satisfies this very weak condition.