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Both paper and author (D. R. McNeil, (1968)) will be referred to below as DRM. The said paper deals with the following situation: an intersection is controlled by a traffic light with a fixed cycle time, T; the possibility of other delays, e.g., due to turning vehicles, is ignored; arrivals at the light form a compound Poisson process; if vehicles arrive to find the light green and the queue empty they are not delayed, while in the contrary case they depart when they reach the head of the queue, providing the light is green, each vehicle taking a constant time to move off. The length of the effective red period is R. For further details and discussion, DRM may be consulted.

It has recently been found that my previous paper “On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem” Jap. Journ. Math. 19 (1949) — referred to as S. K. — contained in its Theorems 19 and 20 some errors. Nevertheless the writer has been able to correct them in suitable forms so that most parts of both theorems hold, even under a weaker assumption, and also subsequent theorems remain valid. These will be, together with some supplementary remarks, shown in the present note.

The proof of Theorem 1 in the paper “On Krull’s conjecture concerning valuation rings” (vol. 4 (1952) of this journal) is not correct. We want to give here a corrected proof of the theorem: From p. 30, l. 14 to p. 31, l. 7 should be changed as follows.

In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined byTheorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

We give another formulation of Theorem 1 of Ribenboim's paper, Bull. Austral. Math. Soc. 2 (1970), 81–88. Our condition on a module to be totally ordered such that this order extends a given strict order seems to be more useful for applications than condition (2) of Ribenboim. This is shown for example in Corollaries 3 and It of our paper. Corollary 4 is due to Ribenboim. Corollary 3 generalizes the corresponding well known result on abelian groups.

The purpose of this note is to point out some difficulties which arise in connection with the stochastic calculus used in the previous paper and to suggest some ways in which these difficulties may be overcome. It turns out that if one wishes to restrict the application of the stochastic calculus to stochastic processes whose sample functions are step-functions with probability one, then difficulties of the following sort arise. Suppose X(ω, t) is a stochastic process whose sample functions are, with probability one, right-continuous step-functions in t on the interval [a, b). Then for every ω outside a null set N and every t ɛ [a, b), there is a δ(ω, t) such that < 5(w, t) implies X(ω, t + h) = X(ω, t) which in turn implies (X(ω, t + h) — X(ω, t))/h → 0 with probability one as Since convergence with probability one implies convergence in probability, it follows that the right-hand derivative of X(ω, t) in probability is zero at every t ɛ [a, b). Consequently, if the left-hand t ɛ [a, b), then it is also zero, but in general left-hand derivatives in probability on [a, b) may not even exist as finite-valued random functions. It is also worthwhile to observe that if a process is not differentiable in probability, then it is not differentiable in quadratic mean. From these observations it follows that the claims in Section 6 of the previous paper regarding the existence of non-zero derivatives in quadratic mean are in error, for if a derivative in quadratic mean exists, then it is a derivative in probability and is therefore zero. Incidentally, it turns out that the Poisson process has a zero derivative in probability but is not differentiable in quadratic mean. The fact that the Poisson process has a derivative in probability seems to depend rather crucially on the fact that it is also a process with independent increments.

The proof of Proposition 2 and that of Corollary to Proposition 3 in my previous paper “On the structure of complete local rings” are not correct Here we want to correct them.

Recently Mr. Seizô Itô kindly called the author’s attention to the fact that the derivation of (4.15) in the above referred paper is insufficient since the paper does not contain the proof of the compactness condition:(1) uniformaly in a and b

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.

Let f(x) and F(x) be polynomials which are supposed to be decomposed into a series of n terms asFurther let θ be a distributive operation and φ(x) be a given function. Then the functional equationhas a solution of the formwhere yν (x) (ν = 1, 2, … n) is the solution of the equation

In this paper the laws of addition and subtraction of vectors were considered, and examples of their extreme usefulness in geometrical applications were given.

Tahara [1] has concretely determined the conjugate classes of finite subgroups of GL(3, Z). The group W5 in his list of groups of order 24 is in fact of order 12 and consists of the same matrices as the group W6 in his list of groups order 12. Hence, there are only 10 conjugate classes of subgroups of order 24 in GL(3, Z) and the total number of conjugate classes of finite subgroups is reduced to 73.

This paper provides simple proofs of two stardard .esults for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argum-nt is applicable to more general single server systems than the queue GI/G/1. STABLE GI/G/I1 QUEUE; STATIONARY VIRTUAL WAITING TIME; STATIONARY QUEUELENGTH; STATIONARY ACTUAL WAITING TIME The objective of this paper is to provide simple proofs of two well-known results for the stable GI/G/1 queue, which results give the distribution function of W* (stationary virtual waiting time) and the distribution of Q (stationary queue-length) in terms of the distribution function of W(stationary actual waiting time). Our approach does not require transform methods and is applicable to more general single server systems than the GI/G/1 queue. Let {V,, n 2 0} and {U,?1, n > 0} denote independent sequences of independent and identically distributed, real, non-negative random variables defined on the probability space (Q, Y, P). The variable VI, represents the service time of customer n, and U,+1 is the time between the arrivals of customers n and n + 1, n 2 0. Denote by F and B the distribution functions of Vo and U, , respectively, and by f and b the first moments of F and B. We assume B is non-lattice, that each off and b is strictly positive and finite, and that f W* and Q(t) Q, as n -+ oo and t oo, where ">" denotes weak convergence. If J is a proper distribution on the real line with J(O-) = 0 and finite first moment j > 0, denote by j the proper distribution with J(0-) = 0 and Received in revised form 4 December 1973. Research supported by N.S.F. Grant No. GK-39073. 849 This content downloaded from 157.55.39.244 on Fri, 17 Jun 2016 04:45:48 UTC All use subject to http://about.jstor.org/terms 850 AUSTIN J. LEMOINE J(x) = p-1f: [ -J(y)]dy. Then, if Y is a random variable having distribution J, E{Y A x} = pf(x) for x = 0, where u A v = min(u, v) for real numbers u and v. Let H denote the distribution function of W, K = H * P, where "*" denotes convolution, and p = f/b. Let Bo denote the distribution function assigning unit mass to the origin, and Bi the j-fold convolution of B with itself, j _ 1. With the above notation, there are the following basic results from queueing theory; see Takics (1963) or Cohen (1969). Theorem 1. P{W* 1, d = Vo 0+ + V,_I, t, = U1 + -+ U,, and S, = d,t,. Next, set n = inf{n: n > 1 and S, 0, so that the queueing system is aperiodic in the sense of (6) from [2]. Since U, has a non-lattice distribution so does the variable ?. And, if m = E{7t}, then E{jt} = mf and E{r} = mb on account of Wald's Lemma. Hence, by (9) and (22) from [2], for x > 0, (1) mH(x) = E I1wx) }X n=O and (2) mbP{W* < xj} = E 1(W(s)xdS We have S1(w(S)x)ds I 1 1 Fd,-xds o n=1 . , = I I, I_-t_ I _yldy = 1 lis,-i+v.,-sy)dylls-_,+v._,_dy n=1 0 (3) = [(x+ U -S_1V._)+ -_(x-S--1V,,-W)] n=l1 This content downloaded from 157.55.39.244 on Fri, 17 Jun 2016 04:45:48 UTC All use subject to http://about.jstor.org/terms On two stationary distributions for the stable GI/G/1 queue 851 n-I = [(-S,)+ (x-S,V,)+] +x-S,-(x-SoVo)+

It is known [9] that geometric or exponential decay of a Markov chain is preserved under derivation (as defined by Cohen [2]). In this paper we consider the inverse problem, i.e., does a derived Markov chain with a geometrical or exponential decay necessarily arise from a Markov chain having the same property. For a large class of Markov chains (including time reversible chains) a complete solution is found. The method used in this paper proved accurate for obtaining exact results on the value of the decay parameter. An example extending results of Miller [8] and Teugels [9] illustrates the procedure.

The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

We assume that the number of claims occur according to a renewal process and treat the number of claims that occur and are reported in a certain time interval as a renewal process with random displacements. We obtain a renewal equation for the mean value function and an integral equation for the Laplace transform of the distribution of the claims that are reported. We also give asymptotic expressions for the mean value function and calculate the generating function in the case where the renewal process is a Poisson process. This matter is a part of the IBNR-problem in insurance mathematics. INCURRED BUT NOT REPORTED CLAIMS; RENEWAL PROCESS WITH RANDOM DISPLACEMENTS ntroduction At a given point of time one has only partial knowledge of an insurance business. For instance, there is always a time lag between the occurrence of a road accident and the reporting of and full knowledge of the claim amount due to that accident. This lack of information generates many questions. One, which this paper touches, is: if we observe M(t) occurred and reported claims during the interval (0, t), how many claims occurred in that period? On the accounting date one has to answer this question and make an estimate of the number of claims which occurred during the last year. The starting point will be assumptions of the number of claims that occur in (0, t). Since the days of Filip Lundberg one has assumed that this is described by a Poisson process. Several authors have tried to generalize Lundberg's model. Without intending to make a complete list we mention Ove Lundberg, Sparre Andersen and Jan Grandell. We will pursue the course of Sparre Andersen who in 1957 presented a paper [1] where the epochs of claims form a renewal process. As to the distribution of the time lag between the occurrence of a claim and the report of that claim there are several possible ways. We will make the assumption that the lags are independent and equally distributed, which seems to be a reasonable assumption as a first approximation at least when the claim amounts are of a fairly equal size. Received in revised form 21 March 1973. 382 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 383 A model We assume that Feller [3], ch. XI is known. Assume that claims occur according to a delayed renewal process on [0, oo). Number the claims 0, 1, 2, -.. in the order in which they occur. So is a stochastic variable representing the time between the starting point zero and the occurrence of the first claim and So has the distribution function Fo. T, is a stochastic variable representing the time between the occurrence of claim number n 1 and claim number n, n = 1,2, ---. The variables TI, T2, -" have the same non-arithmetic distribution function F, with F(O) = 0. Put P, = E T, and aF = var T. and assume that ao 1, This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms 384 JAN-ERIK KARLSSON A fundamental property of the renewal process is that N(t) > n + 1 if and only if S, n if and only if S, + X, . We omit the index Fo, if P(So = 0) = 1. Thus we have MFo(t) = I e,(t) 0 and VFo(t) = P((t) P((t) = 1) = (G*Fo*F*)(t) 0 0 or putting Go = G,*Fo VF,(t) = (Go * U) (t). Thus VFo(t) is the solution of the renewal equation VFo(t) = Go(t) + VFr(tx)F {dx}. Conditioning by So, we see that VFo(t) = (V* Fo)(t). We can also get the renewal equation for VFo(t) by the usual renewal reasoning: EMFo(t) = f E(MFo(t) lSo + X0 = S, So + T1 = y)Pso+xo{ds} Pso+r,{dy} where (0 if s > t, y > t, E(MF(t) So + Xo = s, S + T1 = ) = 1 if s t, y> t, EM(t y) if s > t, y t, 1 +EM(t y) if s ?t, y t. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 385 This technique will be used later. According to the renewal theorem for a delayed renewal process, we have for h > 0 lim (VFo(t + h) VFo(t)) = h I/,. If we put z(t) = VFo(t) t /ip it can be shown that Z(t) satisfies Z(t) = z(t) + f (t x) F{dx} with z(t) = (1 F(y))dy (1 Go(t)). Since z(t) is directly integrable (we assume that Po = ESo All use subject to http://about.jstor.org/terms A stochastic model for time lag in reporting of claims 387 References [1] ANDERSEN, E. SPARRE (1957) On the collective theory risk in the case of contagion between claims. Transactions XVth International Congress of Actuaries, New York. Vol. II. [2] DALEY, D. J. AND VERE-JONES, D. (1972) A summary of the theory of point processes. Stochastic Point Processes: Statistical Analysis, Theory and Applications (P. A. W. Lewis, ed.), Wiley, New York. 299-383. [3] FELLER, W. (1970) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edition. Wiley, New York. [4] PRABHU, N. U. (1965) Queues and Inventories. Wiley, New York. [51 THEDtEN, T. (1967) On point .v.vtoms in R, under random motion. Doctor's thesis, Royal Institute of Technology, Stockholm. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form eO(m10gm) should hold, but I have not so far succeeded in proving such a result.

This paper continues the author's study of age-dependent branching processes allowing immigration. In this paper the multidimensional case is considered. A sufficient condition is obtained for the existence of a legitimate limiting distribution. Several corollaries are obtained, which generalize many of the results of the discrete theory and those of the one-dimensional continuous time model.

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived. MARKED BIRTH-AND-DEATH PROCESS; MULTIVARIATE LAGRANGE DIFFERENTIAL EQUATIONS; MULTITYPE BRANCHING PROCESS Introduction In a recent paper Williams (1969) has considered the following process. We have a population of elements developing as a linear birth-and-death process and each element bears a number of inanimate marks. When a marked element dies then it and its marks are removed from the population; when a marked element divides, its marks segregate independently and without loss, each with probability I into one of the two daughter elements. It is assumed that marking an element does not affect its birth and death rates (cf. Clifford and Sudbury (1972)). Williams' process was proposed as a model of an experiment conducted on bacteria (the elements) infected by phage (the inanimate marks), by Meynell (1959). (The bacteria are lysogenic with respect to the phage, which means that the phage do not interfere with the behaviour of the bacteria.) His object was to use the marking process in order to separate the birth and death rates of the bacteria; if we denote birth and death rates by A and p respectively, then the mean-value measures on the bacterial population (assumed to develop as a linear birth-and-death process) will estimate the difference (A p). As the marks are inanimate, however, they will only be affected by p and so mean-value measures on the mark population enable us to estimate p (see Williams (1969)). The papers of Meynell and Williams taken together provide a most interesting example of a virological experiment and the probability model of that experiment. Received in revised form 10 August 1973 Research sponsored by a Science Research Council grant and a World Health Organisation research traineeship. 423 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms 424 BYRON J. T. MORGAN Williams considered the mean-value solution of his model; this was adequate for Meynell's experiment in which population sizes were large. Here we shall give a fuller probabilistic statement of the model and make some contributions to its solution. Two differences from Williams' work lie in the adoption of an upper bound, s, to the number of marks any element can have, and the assumption (except where stated) of birth and death rates independent of time. We 'shall assume general initial conditions which, in Meynell's experiment, correspond to the distribution of phage over bacteria once these have been initially exposed to a phage population. Williams took this distribution to be Poisson; the distribution of phage over bacteria has, however, been the subject of much probabilistic study (e.g., see Gani (1965) and Morgan (1971)). Finally, let us note that the value s = I, on which we shall particularly concentrate, probably gives the most realistic description of Meynell's experiment. 2. The probability model The model is a continuous-time Markov process with discrete states. Let N,(t) be a random variable denoting the number of elements with i marks at time t, i = 0, 1,2, ..., s. Let n = (n0,n1, l , n.., ), Iu = (Uo, u1, *,us), N(t) = (No(t), N(t), --,N,(t)), and suppose N,(O) = ai, i = 0, 1,2, --, s. Let P(n; t) = P(N(t) = n), then corresponding to death we have the transitions, (No, ..., Ni, ---, N,) (No, --, Ni 1, , N,) at probability rate pNi, i = 0, 1,2, ,s, while corresponding to birth we have the following transitions (No, N,) ,(No + 1, .., N,) at probability rate, 2 =1 N 2-' +,ANo, (No, , Ni ,N2i,..., X (No, Ni + 2, , N2iI, at probability rate, This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms On the distribution of inanimate marks over a linear birth-and-death process 425 where we use [is] to denote the integral part of is, and (No, " , Ni, * * *, Nk i Nk, "', Ns) -*(No, ..., Nj + 1, ..., Nk-i + 1, ..., Nj 1, ..., Nl) at probability rate 2N 2-k' i = 1,2,...,C[(k 1)], k = 3,4,...,s (smaller k give transitions already considered above). It is readily verified that the sum of the transition rates is (A + yp) X = oN,, as required in the overall birth-and-death process. Introducing the operator Di, which increases the argument ni of a function of n by 1, we then deduce that the Kolmogorov forward equations (for all possible n) are, d S S (1) +(n; t) P(n; t)(A+ t)1 (n2 + D,P(n;t)(n +1) dt ==0 1=0

In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X.

The aim of this paper is to establish a conjecture of Shapiro (10) that an F-space (complete metric linear space) with the Hahn-Banach Extension Property is locally convex. This result was proved by Shapiro for F-spaces with Schauder bases; other similar results have been obtained by Ribe (8). The method used in this paper is to establish the existence of basic sequences in most F-spaces.

The proof of divergence of Fourier series which was constructed in the authors' earlier paper [Bull. Austral. Math. Soc. 8 (1973), 289–304] is not complete. The object of this paper is to show that the unestimated terms do not disturb the divergence.

A ring R is called a qc-ring if each cyclic R-module is quasi-injective. For various properties of these rings we refer to Ahsan (1) and Koehler (15). In this paper we shall obtain some additional results related to qc-rings. The scheme of the paper is as follows. Section 2 contains various preliminary definitions and results. In Section 3, we shall prove that every commutative hypercyclic ring is a qc-ring. In this section, we shall also show that a qc-ring which satisfies the ascending chain condition on its annihilators has nilpotent Jacobson-radical. Finally, in Section 4, we shall study rings all of whose proper factor rings are qc. Such rings will be called “ restricted qc ”.

In the previous paper [3] the author has shown a possibility to construct a series of sfields by taking sfields of quotients of split crossed products of a sfield. In this paper the same problem is treated, and, by considering general crossed products, a generalization of the previous result is given: Let K be a sfield and G be the join of a well-ordered ascending chain of groups Gα of outer automorphisms of K such that a) G1 is the identity automorphism group, b) Gα is a group extension of Gα-1 by a torsion-free abelian group for each non-limit ordinal α, and c) for each limit ordinal α. Then an arbitrary crossed product of K with G is an integral domain with a sfield of quotients Q and the commutor ring of K in Q coincides with the centre of K.

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.

In a recent paper [5] one of us has given a sufficient condition to be satisfied by a given property of radical classes within a universal classwin order that, for any subclassℳofw, there should be a smallest radical class having the given property and containingℳ. The sufficient condition is that the classof all radical classes with the given property can be characterised as the class of all radical classes fixed by an admissible function F (see Section 1 below). In this paper a necessary and sufficient condition is derived and the corresponding result for semi-simpleclasses is also presented. These results are given in Section 2.