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This gives the desired result. It might be pointed out that Theorem 4 shows that Ilfn-gn I-->O In ?4 it is tacitly assumed that the measurable set E can be partitioned into measurable sets E,,. This is always the case in separable spaces. To proceed without this assumption it will not be necessary to assume that J is metric. The class So(E) is defined as the class of functions finitely valued on E. Such a function is one for which there is a decomposition of E into a finite number of disjoint measurable subsets on each of which it is constant. This basis necessitates only a slight rewording in a few places. In Lemma 1 the set E should be taken as a set in A. In Theorem 2 the words "functions uniformly continuous" should be replaced by "functions finitely valued." In the proof of Theorem 11 the sentence "Fix . . . continuous on e" should be worded "Fix e with 18(E-e)

3. In the following places the exponent should be 1/2: (6), line 3 and 4 from bottom on p. 700, line 3 from bottom on p. 701. 4. In the last display on p. 701, the second half of the formula should precede the first half. 5. On p. 700, line 11-12, W(n, , n, yo , 1) should be W(n, , n, yl, 1). This necessitates a modification of the arguments leading from (6) to (7) on pp. 700-701. The correct argument is sketched as follows. We write

An algebraic explanation is offered for the graphical method of seriation proposed by Meighan. A suggested simplification of the graphing technique makes it possible to plot the seriation on ordinary graph paper rather than on three-pole graph paper. TN THE PRECEDING PAPER Meighan proposes a method for the field seriation of archaeological materials. This paper is an attempt to explain its algebraic foundations. In addition, a slight modification of the proposed method is suggested. The technique consists of selecting three artifact types, and using their sum in each lot as 100%, finding the percentages of each. The three percentages from each lot are used to plot one point on a three-pole graph. The points are then fitted by eye with a "best" line and a perpendicular is drawn from each point to the line. The order of the points of intersections along the line is considered to be the ordering of the lots in time. As each point is placed on the three-pole graph, it can be noted that any two of its three coordinates completely define the point. Since we take their sum to be 100%, we have a, b, and c=100-a-b, which give only two linearly independent quantities. For this reason, choosing two of the types and graphing their percentages on ordinary graph paper will give the same result. However, percentages should still be computed on the basis of at least three types. When alttempting to filt a line to the points, we suggest the use of a least-squares "best" fit. Although the use of least-squares curve fitting may seem needless since the data are not precise, there are three reasons it should be used: (1) the method of least-squares is designed as a data smoothing or correcting technique for observed data; (2) it is preferable to use a technique which, given the same data, leads to the same results regardless of the user; and (3) it seems wasteful to further compound errors that may exist by using a subjective "best" line. A seeming drawback to using this method of curve fitting is that it assumes that one set of observed values is being fitted to a set of more accurate values. In this case both sets of percentages being used are observed data. A solution to this difficulty is to fit two "best" lines, each minimizing the deviations along one coordinate. Although this might seem like undue effort, it may serve as an internal check indicating whether the graphical technique is vaild for the data being used. The method discussed in the remainder of this paper is that described by Meighan, but modified in the two respects noted. Our primary interest is to explain why the graphed points sometimes tend to be a line, and why this line reflects the temporal sequence of the points. Let us assume that we are dealing with samples from lots whose dates are known. If we graph the percentage of a particular type from each lot against the date of the lot, we would expect the graph to show a gradual rise until a peak is reached and then a gradual decline, provided that the dates of the lots cover the range of existence of the type. This type of curve is the basis of other seriation methods (Ford and Willey 1949; Robinson 1951). However, if only a short span of time is covered by the lots, we might observe only a rise or only a decline. Consider a hypothetical case in which the percentages of Types A and B from lots with known dates (Table 1) form lines when graphed against time. If we ignore the dates and apply the graphical technique, we arrive at the graph in Figure 1 a. It is found that both "best" lines are identical and that they are not approximating the points which have been plotted. Rather, all the points lie on !the line and are in their proper temporal order. If instead of using the graphical technique, we approach the data algebraically, we can see why we arrive at that line and the point ordering on the line. Each of the sets of percentages is a linear function of time. The percentage of A has Equation 1 and that of B, Equation 2 (Table 2). If we treat time as a parameter, we can eliminate it by solving Equations 1 and 2 TABLE 1. DATA FOR HYPOTHETICAL CASE. Percentage Percentage of A of B Date 10 30 900 20 32 920 30 34 940 40 36 960

Summary A table and a nomogram are given to find the growth rate of a regression line drawn on so-called semi-logarithmic graph paper. The growth rate is determined by: a) the slope of the regression line, b) the length ratio between the logarithmic and the metric unit, in other words, the number of years on the time axis (if a year is the time unit) corresponding in length to one cycle on the logarithmic axis (say, 1 to 10, or 100 to 1000). With an example, the use of the table and nomogram is explained.

Concept of n-width and extremal subspaces, first introduced by Kolmogorov, plays an important part in mathematical problems of approximation of classes of functions and in engineering problems of signal representation and reconstruction. In this short paper, explicit expressions for n-width and extremal subspaces are obtained for a class which is of some engineering importance.

This short paper is to provide additional informa· tion which came to light too late for the original publication of "The Second Spectrum of Ni (Ni II)." It is unfortunate but true that it is impossible to publish as complicated a paper as this one without some errors creeping in. Also, the very acts of preparing the paper for publication and the correcting of proof make evident some additions that should be made to the analysis. 1. One important line 1..3709.456 was inadverently omitted from Table I. It is included below in a list of changes. in that table. 2. A number of combinations between known levels were found. They are also listed below. 3. Two levels with which I was dissatisfied have been eliminated and new ones found to replace them. They are w 4 G~' / 2 and w 4G~ ' /2 and they are included below in a list of changes in Table III.

In previous papers I have broached the question of the two-dimensional partitions of numbers—or, say, the partitions in a plane—without, however, having succeeded in establishing certain conjectured formulas of enumeration. The parts of such partitions are placed at the nodes of a complete, or of an incomplete, lattice in two dimensions, in such wise that descending order of magnitude is in evidence in each horizontal row of nodes and in each vertical column. No decided advance was made in regard to the complete lattice, and the question of the incomplete lattice is considered for the first time in the present paper. I return to the subject because I am now able to throw a considerable amount of fresh light upon the problem, and have succeeded in overcoming most of the difficulties which surround it. In fact, I am now able to show how the generating functions may be constructed in respect of any lattice, complete or incomplete, in forms which are free from redundant terms. I have not succeeded, so far, in giving a general algebraic expression to the functions, but, in the case of the complete lattice, I have shown that an assumption as to form, consistent with all results that have been arrived at in particular cases, leads at once to the expression that has been for so long the conjectured result. For the complete lattice of two rows, and for the incomplete lattice of two rows, the results have been obtained without any assumption in regard to form, and must be regarded as rigidly established.

Two recent papers in these Proceedings have dealt with certain problems of probability which are of very great importance for psychology in particular, but in themselves are quite general. As both papers made reference to some of my own, the following considerations may not only be a further positive contribution to the topic, but incidentally serve to clear up some misunderstandings. Suppose that the product moment coefficients of correlation have been determined between any set of variables a, b, c, d , ... z , and have been set forth in a square Table thus:—

The object of this paper professes to be to ascertain whether the principle of Fermat’s theorem of the polygonal numbers may not be extended to all orders of series whose ultimate differences are con­stant. The polygonal numbers are all of the quadratic form, and they have (according to Fermat’s theorem) this property, that every number is the sum of not exceeding, 3 terms of the triangular num­bers, 4 of the square numbers, 5 of the pentagonal numbers, &c. It is stated in this paper that the series of the odd squares 1,9,25,49, &c. has a similar property, and that every number is the sum of not exceeding 10 odd squares. It is also stated, that a series con­sisting of the 1st and every succeeding 3rd term of the triangular series, viz. 1,10,28,35, &c., has a similar property; and that every number is the sum of not exceeding 11 terms of this last series, and that this may be easily proved [it was proved in a former paper by the same author]. The term “Notation-limit” is applied to the num­ber which denotes the largest number of terms of a series necessary to express any number; and the writer states that 5,7,9,13,21 are respectively the notation-limits of the tetrahedral numbers, the octa­hedral, the cubical, the eicosahedral and the dodecahedral numbers; that 19 is the notation-limit of the series of the 4th powers; that 11 is the notation-limit of the series of the triangular numbers squared, viz. 1,9,36,100, &c., and 31 the notation-limit of the series 1,28,153, &c. (the sum of the odd cubes), whose general expression is 2 n 4 — n 2 .

It should not be forgotten that Whitehead was a professional mathematician for most of his academic life and that he spent ten years or so collaborating with Bertrand Russell on Principia Mathematica. Logico-mathematical methods and procedures must have become thoroughly ingrained in his habits of thought by the time the later metaphysical works were written. The suggestion that Whitehead may have forgotten these methods or no longer trusted them must surely be in error. For him, it would seem, to think at all was to think mathematically. Even in Process and Reality 1 it is interesting to discern mathematical or quasi-mathematical notions, definitions, and statements creeping in almost unawares on every page. The great categoreal scheme of this book, in fact, it will be contended, may be viewed as a kind of logico-mathematical system in disguise. It is the purpose of this present paper, in any case, to take a few first, tentative steps towards substantiating such a claim.

In a beautiful paper which appeared in 1939 ([4]), F. Gantmacher made a thorough study of automorphisms of semi-simple Lie algebras over the field of complex numbers. Among other things, he defined the index n(G i ) of a connected component G i of the automorphism group G = G(8) as the minimum multiplicity of the characteristic root 1 for elements of G i . The main purpose of this note is the determination of these indices. It is somewhat surprising that this does not appear in Gantmacher’s paper since all the methods for deriving the formula for index G i are available in his paper. The secondary purpose of this note is to extend Gantmacher’s theory to the case of Lie algebras over arbitrary algebraically closed base fields of characteristic 0. This can be done by using algebraic group concepts and techiques which are by now well known. Nevertheless, it seems worthwhile to carry out the program in detail since Gantmacher’s results give a real insight into the action of an automorphism in a semi-simple Lie algebra. For example, as we indicate, they can be used to give a new derivation and sharpening of theorems on fixed points which are due to Borel and Mostow ([1]).

Introduction. In 1954 the mathematical entity called a (sequential) machine was found to be a valuable tool in designing sequential switching circuits [2; 8; 9]. Since then there has been considerable mathematical activity by mathematicians and nonmathematicians relating to the analysis and the synthesis of these machines. As was to be expected of a topic which arose because of an engineering need, most of these results have appeared in engineering and computing journals. Recently though, some of the papers have appeared in mathematical journals [3; 4; 5; 6; 10]. Also, much of the recent literature has dealt with questions almost exclusively of mathematical, as contrasted with engineering, interest [1; 3; 4; 5; 6; 10; 12]. The present paper is written in that spirit. The Moore-Mealy (complete, sequential) machine is defined [8; 9] as a nonempty set K (of "states"), a nonempty set D (of 'inputs"), a nonempty set F (of "outputs"), and two functions a (the "next state" function), and X (the "output" function), 5 mapping KXD into K and X mapping KXD into F. Then 5 and X are extended to sequences of inputs I1 ... Ik (written without commas) by

Čebyšev centers have been studied extensively. In this paper an alternate concept of center for infinite bounded point sets is introduced. Some of the results in this paper for this new type of center are similar to previous results for Čebyšev centers.

This paper is an outgrowth of my old battle with the open sentence problem for the theory of finite groups. The unsolvability of the word problem for groups (cf. [1] and [4]) entails the undecidability of the open sentence problem for the elementary theory of groups and thus strengthens the original undecidability result for this theory (cf. [7]). The fact that the elementary theory of finite groups is also undecidable (cf. [2] and [6]) therefore justifies my interest in the open sentence problem for that theory. This paper contains a construction of groups that might lead to a negative solution.

1. IntroductionThroughout this paper a "group" will mean a "finite soluble group".Carter, Fischer and Hawkes [1] call a group G critical for a class X of groupsif every proper subgroup of G, but no itselft G , belongs to X. We shall say a groupis £-criticcl if it is critical for £(/j) for some integer n (where Q(h) denotes theclass of groups of nilpotent length at most ri). Thus a group is fi-critical if itsnilpotent length is greater than that of each of its proper subgroups.The class of ii-critical groups is not closed under homomorphic images. Forexample, the group G = AB where A is cyclic of order 3, B is cyclic of order 4,and a generator of B takes a generator of A to its inverse is fi-critical, whereasG/A is not. Knowledge of when a homomorphic image of an 2-critical group isC-critical can be useful when proving results on fi-critical groups by inductionmethods. The main theorem of this paper implies that a homomorphic image ofan .C-critical group is fi-critical if it possesses a complemented minimal normalsubgroup. This result is then used in the last section of this paper, which is anattempt at giving as complete a description of the ^-critical X-groups as possible.G4-groups are groups whose Sylow subgroups are all abelian.) The author was,however, unable to give a complete classification up to isomorphism classes ofthese groups.This paper is an extract of the author's M.Sc. thesis, submitted to the Aus-tralian National University in February, 1972. The valuable assistance of Dr. JohnCossey, who suggested the problems and supervised the research, is gratefullyacknowledged.2. Main TheoremWe first give some notation and definitions and note down some results from[lj that will be needed to prove the main theorem.357

Let F be either a field of char 0 with all roots of unity or a field of char p > 0. Let G be a countable group. Then all simple NIG]-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The condition that all simple modules over a ring be injective first appeared in a theorem due to Kaplansky: a commutative ring satisfies the condition if and only if it is von Neumann regular. Several people have studied the property for noncommutative rings, a recent example being [3]. The authors of that paper suggest the problem of characterizing group algebras with this condition. In this paper we make substantial progress by offering Theorem 3. Let F be either a field of char 0 with all roots of unity or a field of char p > 0. Let G be a countable group. Then all simple FtG]-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The proof is divided into three parts. In ? 1, we show that F is injective as an 14G]-module if and only if G is locally finite with no elements of order char F. In the second and crucial section, we show that for a certain class of rings ("locally Wedderburn algebras") the condition that all simple modules are injective is equivalent to the property that all simple modules are finite dimensional over their commuting rings. In -?3, we prove the main theorem by showing that if all simple modules are finite dimensional over their commuting rings then G is abelianby-finite. We would like to thank D.S. Passman for suggestions that shortened and improved our work. 1. Results of Villamayor. Most of this section can be gleaned from several of Villamayor's papers. However, the statements and proofs that appear here are new. The field F is a right 14G]-module under the trivial action (i.e. if k E F, then k * a = k(a)e where E is the augmentation map). We characterize those G with F being 14G]-injective. By considering maps from ideals of FtH] to F, the reader can casily prove Received by the editors February 5, 1973. AMS (MOS) subject classifications (1970). Primary 16A26, 20C05; Secondary 16A52, 16A64.

The algebraic theory of inverse semigroups has received considerable attention in recent years with satisfactory structure theorems being obtained for extensive classes of these semigroups. As is the case with so much of semigroup theory, many of these structure theorems were obtained by, or had their origin in fundamental papers of, A. H. Clifford. For example, the structure of inverse semigroups with central idempotents (semilattices of groups) was obtained by Clifford in [1], that of Brandt semigroups was obtained in [2], while, in [3], Clifford described bisimple inverse semigroups in terms of their right unit subsemigroup. The third of these papers has been the basis for much of the research on structural aspects of inverse semigroups. On the one hand, its direct descendants include Reilly's papers on bi-simple inverse semigroups in terms of RP-systems [10], [11], and the author's description of O-bisimple inverse semigroups in terms of groups and semilattices [5]. On the other hand, it has led to Reilly's description [9] of w-bisimple inverse semigroups and Warne's characterisations of various classes of bisimple inverse semigroups with special semilattices of idempotents. These, in turn, have led to the determination of simple inverse semigroups with special semilattices of idempotents.

A p-ring (or generalized Boolean ring) P is a ring of fixed prime characteristic p in which ax=a for all a in P. In this paper P is partially ordered by a relation which is a generalization of the usual Boolean order. A subset S of P is then called quasiorthogonal if ab(a-b)=O for all a, b in S. It is shown that P is injective in the category of p-rings if and only if every quasiorthogonal subset has a supremum under this partial order. Sikorski [4] has shown that in the category 9 of Boolean rings the injective objects are the complete Boolean rings. The purpose of this paper is to present a generalization of this result to the category 9 of p-rings, where 9 is understood to be the category with objects rings P of fixed prime characteristic p in which aP=a for all a e P and with morphisms the usual ring homomorphisms. If P is a p-ring, then the set B(P) of idempotents of P is a Boolean ring under the multiplication of P and the new addition defined by aE3b=a+ b-2ab. Batbedat [2] has used B to establish an isomorphism between b? and R. (See also Stringall [6].) Hence, the injective objects in 9 are simply those P for which B(P) is a complete Boolean ring. In this paper the irnjective objects in b? will be characterized by a kind of completeness of a particular partial order that is an extension of the usual partial order on the Boolean ring of idempotents. DEFINITION ]. For a, b E P, a_b if and only if aP-1b=a. It is easily shown that _ is a partial order on P (see, e.g., Abian [1]), but in general (P, _) is not a lattice. It is, however, a lower semilattice with meet defined by aAb=a-a(a-b)P-1. A simple calculation shows that a(bAc)=abAac and aAO=O for a, b, c e P. Foster [3] has shown that every element a of P is uniquely expressible as a sum of elements {ej(a): jE Z.} of B(P). In particular, v-1 (1) a = Zjej(a) i=l Presented to the Society, January 26, 1973 under the title Injectivity in the category of p-rings; received by the editors February 27, 1973. AMS (MOS) subject classifications (1970). Primary 06A70; Secondary 06A40, 06A23.

The cutting number of a point of a connected graph is a measure of the extent to which the removal of that point cuts the graph. The cutting center of the graph is the set of points of maximal cutting number. In an earlier paper [1] all possible structures for the cutting center of a tree were determined and examples constructed which realize them. In this paper we extend those results to all graphs.

O. Introduction Let (X, Y) be a locally convex linear topological space with topology Y and denote by C its cone of continuous seminorms. Then the two main results of this paper are a bipolar theorem for subsets of C and a Grothendieck-type representation of the completion of C with respect to some topologies of uniform convergence on subsets of X. The completeness theorem is based on an approximation lemma for seminorms corresponding to the one for linear functionals; we prove: If M is a convex circled subset of X and h a seminorm on X whose restriction to M is continuous then for every e > 0 there exists a continuous seminorm h~ on X such that [(h-h~) (m)l < e for all m e M. These results will be established in Sections 1-3 by sublinear techniques in the somewhat more general framework of sublinear functionals. The completeness theorem will provide some new insight in properties of locally convex spaces from the point of view of seminorms and will, consequently, turn out to be a very useful tool in considering special types of those spaces, both in characterizing and handling them. Moreover it will yield a unified approach to several problems in locally convex theory. It is the primary object of Sections 4-6 to carry out some work in this direction. In Section 4 some types of locally convex spaces are characterized by completeness properties of their cones of continuous seminorms. Section 5 is devoted to the problem of characterizing the finest locally convex topology q on X which agrees with the initial topology ~of X on every member Ak of an increasing sequence (Ak)k~ of convex circled subsets of X whose union is absorbing. This problem has been investigated in detail in the papers of Garling [5] and Roelcke [14]. As for this problem the above mentioned Grothendieck-type representation theorem will show that the cone C, of t/-continuous seminorms on X is nothing else but the completion of C with respect to the topology of uniform convergence on the sets Ak, k e N. This result is entirely analogous to the corresponding one concerning the dual space of X with respect to t/(see [14]).

A common method of obtaining the classical modal logics, for example the Feys system T, the Lewis systems, the Brouwerian system etc., is to build on a basis for the propositional calculus by adjoining a new symbol L, specifying new axioms involving L and the symbols in the basis for PC, and imposing one or more additional transformation rules. In the jargon of algebraic logic, which is the point of view we shall adopt, the “necessity” symbol L may be interpreted as an operator on the Boolean algebra of propositions of PC. For example, the Lewis system S4 may be regarded as a Boolean algebra ℒ together with an operator L on ℒ having the properties: (1)Lp ≤ p for all p in ℒ, (2)L1 = 1, (3) L(p → q) ≤ Lp → Lq for all p, q in ℒ, and (4) Lp = L(Lp) for all p in ℒ. Here, of course, → denotes the material implication connective: p→q = p′ ∨ q. It is easy to verify that property (3) may be replaced by either (3′) L(p ∧ q) = Lp ∧ Lq for all p, q in ℒ, or by (3″) L(p → q) ∧ Lp ≤ Lq for all p, q in ℒ. In particular, it follows from (1) through (4) above that L is a decreasing, idempotent and isotone operator on ℒ. Such mappings are often called interior operators.In a previous paper [5], we considered the problem of introducing an implication connective into a quantum logic. This is greatly complicated by the fact that the quantal propositions band together to form an orthocomplemented lattice which is only “locally” distributive. Such lattices are called orthomodular. For definitions and further discussion, the reader is referred to that paper. In it, we argued that the Sasaki implication connective ⊃ defined by p ⊃ q = p′ ∨ (p ∧ q) is a natural generalization of material implication when the lattice of propositions is ortho-modular. Indeed, if unrestricted distributivity were permitted, p ⊃ q would reduce to the classical material implication p → q. For this reason, we choose ⊃ to play the role of material implication in an orthomodular lattice. Further properties of ⊃ are enumerated in Example 2.2(1) and Corollary 2.4 below.

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

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

Let C denote the category of compact Hausdorff spaces and H : C → H C H:C \to HC be the homotopy functor. Let S : C → S C S:C \to SC be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and H n ( X ) {H^n}(X) denote n-dimensional Čech cohomology with integer coefficients. Let A x = char H 1 ( X ) {A_x} = {\text {char}}\;{H^1}(X) be the character group of H 1 ( X ) {H^1}(X) considering H 1 ( X ) {H^1}(X) as a discrete group. In this paper it is shown that there is a shape morphism F ∈ Mor S C ( X , A X ) F \in {\text {Mor}_{SC}}(X,{A_X}) such that F ∗ : H 1 ( A X ) → H 1 ( X ) {F^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping f : X → A X f:X \to {A_X} such that S ( f ) = F S(f) = F and thus that f ∗ : H 1 ( A X ) → H 1 ( X ) {f^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. This result is applied to show that if X is locally connected, then H 1 ( X ) {H^1}(X) has property L. Examples are given to show that X may be locally connected and H n ( X ) {H^n}(X) not have property L for n > 1 n > 1 . The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, H n ( X ) / Tor H n ( X ) {H^n}(X)/{\operatorname {Tor}}\;{H^n}(X) has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that H n ( X ) {H^n}(X) itself may not have property L even if X is a finite-dimensional movable continuum.