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It is the purpose to examine Krafts arguments and proofs on the basis of the evidence which gives reality. His derivation of principles with unlimited validity is not succesful, judgements concerning “basic valuations of human culture“ being inhomogeneous and historically bound. The way to introduce norms into scientific statements by “teleological” foundation is only valid for “normal” principles, basic must be excluded. The fruitfulness of the first step is appreciated and illustrated by economic examples. Jointly in connection with this is the third question. In accordance with Kraft, the “ranking” of value judgements of different kinds must be solved without privileges. Scientific examination and foundation of value systems has its legitimacy, but relevant decisions of basic principles can only happen in the pre-scientific sphere. In order to avoid other value judgements in scientific contexts, one ought to dispose of really operational criterions. Today, there is an almost complete lack in the practical field. This is partly due to theoretical and dogmatic interpretations of science procedures. Critical science (“critical realism”) has to accept the provisional character of all scientific research and must, therefore, integrate those value judgements, which result from not fully testable hypotheses. Their “internalization” into the scientific progress of knowledge allows together to delete a great deal of this crucial point.

A random discrete-time system { x n }, n = 0, 1, 2, … is called stochastically stable if for every ϵ > 0 there exists a λ > 0 such that the probability P [(sup n ∥ x n ∥) > ϵ ] ϵ whenever P [∥ x 0 ∥ > λ ] λ . A system is shown stochastically stable if some local Lyapunov function V (·) satisfies the supermartingale definition on { V ( x n )} in a neighborhood of the origin; earlier proofs of stochastic stability require additional restrictions. A criterion for x n → 0 almost surely is developed. It consists of a global inequality on { U ( x n )} stronger than the supermartingale defining inequality, but applied to a U (·) that need not be a Lyapunov function. The existence of such a U (·) is exhibited for a stochastically unstable nontrivial stochastic system. This indicates that our criterion for x n → 0 is “tight,” and that the two stability concepts studied are substantially distinct.

This paper refers to the brand-preference parameters of a previously accepted empirical model, and axiomatically generates several basic propositions and their proofs, which define the retail store's most profitable short-run blend of product brands.

In attempting to organize the mass of material we felt deserved coverage, we found it necessary to impose a classificatory structure on it. The structure we imposed is in terms of the nature of the dependent variables and the number of populations of measurements (e.g. treatments) being explored. There is a considerable degree of arbitrariness in our classification scheme, and the classes turn out not to be fully mutually exclusive. We feel, however, the reader will find the system useful and convenient. We have attempted to write this chapter at a level which will be accessible to the sophisticated researcher in psychology and to the relatively ad· vanced graduate student. Some papers cited, however, require a considerable degree of mathematical sophistication on the part of the reader for full understanding. We wish to encourage readers to realize that it is not necessary to be able to follow mathematical proofs in detail in order to apply the results. Our classification scheme's first order concerns whether single or multiple depen­ dent variables are under study; that is, the traditional dichotomy between multivari­ ate and univariate analyses. The second order of classification considers whether a single or multiple populations (treatments, experimental conditions, etc) are in­ volved. With single populations, the questions considered are ordinarily the testing of some hypothesis or fitting of some structural model; multiple populations ordi­ narily involve testing hypotheses about population differences. So far we have a fourfold order of classification. The remaining order of c1assifica· tion is in terms of the nature of the measurements involved and has three levels: (a) continuous interval scale data; (b) ordinal data, and (c) categorical data. These

T HE facsimile of the extant manuscript of Winterreise, published by Birenreiter in 1966, shows that of the twelve songs in Part I ten are in rough copy and only two (No. 1, Gute Nacht, and No. 8, Riickblick) in fair copy. All the twelve songs of Part II are in fair copy. It is not surprising, therefore, to find that the textual discrepancies between the manuscript and the first edition are far more numerous in Part I than in Part II. In Part II, indeed, they are so few and unimportant that for the purposes of this article they may be ignored. Schubert, as will be seen, evidently had a good deal of trouble vwith Haslinger, the publisher, over various details, important and slight, in Part I (perhaps Haslinger also had trouble with Schubert); and Schubert's meticulous fair copies of the Part II songs, composed some months later, may well have been prepared as a result of these experiences; they certainly enabled him to carry out the task of correcting the proofs on his deathbed, which he could hardly have done had they been in the confused state of the Part I proofs. Typographically minor errors which may, however, be musically important are almost bound to occur in transcribing from rough copies of manuscript like most of Schubert's in Part I. Such errors I define as discrepancies of individual notes in regard to written pitch or note value, disagreement on accidentals and rests, including their omission or insertion, and the more sig

It is apparently “well known” that the image of the closed unit interval under an open map is homeomorphic to the closed unit interval (see [13], [11], and [15]). In this paper, we generalize this result to chainable continua. In particular, the fact that the open continuous image of a chainable continuum is also chainable is proved, answering a question of A. Lelek (see [10]). This fact, as well as its proof, implies that the open continuous image of the pseudo-arc is also a pseudo-arc. An additional corollary (of the proof) is that a local homeomorphism of a chainable continuum is actually a homeomorphism. The proofs are all very elementary.

Lloyd Metzler's classic article on "Wealth, Saving, and the Rate of Interest"-which appeared in this Journal over 20 years ago-is noted for its clear and searching analysis of the implications of the Pigou or real-balance effect, and for the crucial distinction it made in this context (from the viewpoint of optimum-portfolio-composition, and hence neutrality) between a monetary change generated by a current deficit and one generated by an open-market operation. But in addition to this valuable substantive contribution, Metzler's article also made an important contribution to technique by providing a graphical analysis of the stability of the system in terms of what is now termed a "phase diagram." And though such a diagram is essentially to be found much earlier in the literature,' Metzler's article is, to the best of my knowledge, the first instance of its being applied to the Keynesian system.2 The purpose of this note is first to make a minor refinement of Metzler's diagrammatic analysis, and then to point out that its geometric proof of stability in the small is actually inadequate-as is also the case for the corresponding proofs that accompany other phase diagrams in the literature. For convenience, let me carry out the discussion in terms of the version of the Metzlerian diagram that I have used elsewhere (Patinkin 1965, pp. 232-33 et passim). The curve CC in figure 1 represents the locus of combinations of the rate of interest, r, and price level, p, which satisfy the equilibrium condition in the commodity market

In the theory of optimization, in connection with ordinary and partial differential equations, a number of closure and lower closure theorems have been obtained in different contexts and under a variety of conditions and modes of convergence. In particular "seminormali ty" conditions (property (Q) and its variants) have played different roles. In this paper we first prove closure and lower closure theorems for orientor fields in a rather abstract context, all based on weak convergence and MAZUR'S theorem (w167 and 5). In the context of orientor fields, these theorems can be given the most satisfactory formulation and simplest proofs (see, e.g., th. (4.i), (5.i)). Furthermore, in the present new approach, the interplay of "seminormali ty" conditions and modes of convergence can be easily seen: the stronger the mode of convergence, the weaker are the "seminormali ty" conditions that are needed. From these theorems we then derive, as corollaries, closure and lower closure statements for Mayer and Lagrange problems (~ 6 and 7) and lower semicontinuity statements for free problems (w Under suitable hypotheses, no seminormality condition (or property (Q)) is needed. Further theorems without seminormality conditions, as well as other details, are discussed in [3]. In particular, we show that seminormality conditions can be removed, not only under standard Lipschitz requirements, as expected, but also under much more satisfactory simple growth conditions, as proposed some time ago by E. H. ROTHE for free problems. Applications to multidimensional Lagrange problems are discussed in [8].

This paper shows that the final solutions to the executive and divisional programs in a linear fractional decomposition furnish the complete primal and dual solutions; it extends the proofs already presented for the linear decomposition problem.

The intuitionistic propositional logic I has the following (disjunction) property.We are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property. Systems with the disjunction property are known, for example the Kreisel-Putnam system [1] which is I + (∼ϕ → (ψ ∨ α))→ ((∼ϕ→ψ) ∨ (∼ϕ→α)) and Scott's system I + ((∼ ∼ϕ→ϕ)→(ϕ ∨ ∼ϕ))→ (∼∼ϕ ∨ ∼ϕ). It was shown in [3c] that the first system has the finite-model property.In this note we shall construct a sequence of intermediate logics Dn with the following properties:These systems are presented both semantically and syntactically, using the remarkable correspondence between properties of partially ordered sets and axiom schemata of intuitionistic logic. This correspondence, apart from being interesting in itself (for giving geometric meaning to intuitionistic axioms), is also useful in giving independence proofs and obtaining proof theoretic results for intuitionistic systems (see for example, C. Smorynski, Thesis, University of Illinois, 1972, for independence and proof theoretic results in Heyting arithmetic).

Recently Biedenharn and Gamba (1972) described a simple group-theoretical method to calculate the quantitative splitting of a degenerate energy level under the action of a symmetry-breaking hamiltonian. The authors provide the general proofs for the rules of the method and clarify the group-theoretical background. Furthermore they discuss the various kinds of multiplicities which can arise and the difficulties which they entail.

In this paper we provide proofs, in a precisely formulated model, of the intuitive notion that any continuous consumer can be approximated arbitrarily close by a smooth and regular one (ail these terms will be carefully defined later on). The proper formal framework for such a question is a topological space of consumers. In this context the problem has been posed by Debreu [7, p. 301 as a natural generalization of an earher result of Grodal [14]. His specific conjecture receives an answer below (Theorem 2). Recently (see Delbaen [9], Dierker and Dierker [IO], and Hildenbrand [17]) a good deal of attention has been devoted in general equilibrium analysis to problems (on the continuity of the equihbrium correspondencep for example) which do essentially necessitate smoothness hypotheses. One can look at the theorems here as giving an assessment of their restrictiveness. If, as it is the case, economic data are not exact, then our results imply that differentiability hypotheses cannot be seriously in conflict with observation. Indeed, the density of the set of elements satisfying an economic assumption could be suggested as a first criterion for deciding if this assumption is in the nature of a regularity or a sub

Ethical naturalists have claimed, and their opponents denied, conceptual connections between what both have been happy to call facts and moral values. Neither camp has in the past made it clear whether what is at issue is the existence of a connection between linguistic performances or between propositions expressed in such performances. This omission, encouraged by the terms 'moral evaluation' and 'description', has been responsible for much confusion on both sides about what a speaker can and cannot say without thereby morally evaluating or else in some obscure sense committing himself to a moral evaluation. To give just one instance of confusion: a good deal of what both Naturalists and Prescriptivists have said makes sense only on the assumption that if a moral conclusion is entailed by given descriptive statements, then one who makes those statements must do so with a view to drawing and stating the moral conclusion which they entail. So availability is confused with inescapability. If in a given case an entailment holds, and if I propose to draw and state a moral conclusion rather than some other conclusion, or no conclusion at all, then there is indeed one and only one moral conclusion which I must draw and state; but that is all. It is tempting to see beneath this confused assumption about the consequences of Naturalism the curious view that acknowledgement of and allusion to a certain sort of facts can have one point only: the derivation of moral conclusions. Statements of such facts are thus cast in the restricted and restrictive role of premises for moral conclusions, the implicit model being proofs in mathematics. But it cannot be a logical truth about the nature of moral evaluation, and it is unlikely to be a moral rule, that whenever a moral conclusion is available it must be stated. There are, indeed, exercises, the rules of which demand this: 'Describe and morally appraise the life of Saint Teresa' specifies one. But this is quite a different matter. Are these strictures fair? Well, let us, in accord with recent

It is sometimes objected that if we require that knowers always be able to ground what they know, provided it is not self-evident, then what ordinarily passes for knowledge would be little of the sort. I happen to know, for example, that Alamogordo is north of El Paso, but I doubt that I could establish it froin memory: I cannot cite any authoritative inap or encyclopaedia where I saw or read it, etc. In spite of this, however, it seems to me that even the traditional account cali be defended here. For even though I may not remember wvhat specific evidence I had in coming to know the relative position of the two towns, I can still remember that I once had such evidence, and perhaps what type of evidence it was, and that I have not since then found any contrary evidence. And this, I think, is enough to justify my continuing to believe. Indeed, even just the fact that one seems to remember may be enough for those of us with a good inemory. Much of our mathematical knowledge has a similar basis. Few of us can produce mathematical proofs at will. Even if we cannot liow produce a proof, nevertheless, we may still remlember well that we once did, or at least that we once saw it don-e, and this somletimles suffices to justify present belief. Other doubts about the traditional account cannot be remo(jved so easily. Suppose, for exaimiple, that I hallucinate a bear in the

Introduction. Recently, Sussmann has proved that the accessible sets of a system of vectorfields on a C manifold M are immersed submanifolds of M [1], [2]. In this paper we state a general theorem on accessible sets of collections of 'arrows' and indicate how it implies (a) the above result; (b) the fact that the orbits of an arbitrary 'isotopically connected' subgroup of Diflf(M) form a foliation with singularities; and (c) a similar result for groupoids of germs of local diffeomorphisms. The complete proofs will be published elsewhere. The results of this paper were obtained independently of Sussmann's work.

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? = ; = .

The arithmetic error detecting and correcting capabilities of product (AN) codes in residue number systems (RNS) are described. The redundancy necessary and sufficient to allow single residue digit error detection or correction is determined, under the hypothesis that the error affects either an arbitrary legitimate number or a number in overflow. It is shown that single-bit errors are also correctable, provided that the residue digits are conveniently encoded. Two different approaches to this problem are discussed. Simple procedures for error detection and correction are presented, and it is shown that the additive overflow detection is a by-product of such procedures. Proofs and examples are given.

Stability and error estimates in the maximum norm are derived for Friedrichs’ operator applied to a strongly hyperbolic system in two space variables. The proofs are based on the theory of Fourier multipliers.

The statement that appears in [3] under Lemma (2,3) is incorrect and part (c) of the statement of Theorem (2.4) has to be deleted t. [3] is erroneous since it can happen that one is confronted with ),{0}=1 (,:, as on p. 147 of [3]). Therefore condition (c) is only sufficient. The theorem that is given in this note replaces (2.4). (The remainder of the paper is not affected. Here are some errata: p. 153 read Tq instead of T, pp. 154, 155 in (3.2) and (3.3) and their proofs read R instead of N, p. 160, in (33) read +s(i) instead of -s(i) , and in (36) read the exponent } as -1 . ) The terminology and notation is that of [3].

ℓ-matrices are defined and shown to be the incidence matrices of certain matroids (lines). The well-known characterizations of binary matroids of W.T. Tutte and A. Lehman are proved along with a new characterization. The proofs use ℓ-matrices.

We examine new low rate error upper bounds for M equally likely code words used over discrete input channels. When optimized over the code ensemble probability distribution, these bounds coincide with the optimized expurgated bounds and the error exponents satisfy rate distortion equations for natural Bhattacharyya distances. Proofs for these error bounds do not require expurgation of code words, and for certain “modular” channels including all binary input memoryless channels, the bounds extend to convolutional codes.

A semilinear parabolic initial-boundary-value problem of order 2m in a possibly unbounded domain Ωx(O,T), Ω⊂Rn, is considered within the framework of the Lp-and Cα-theory. In the first case a proof is given of the existence of a “strict” solution of the corresponding evolution equation. In the second case one can guarantee a classical solution, provided the homogeneous linear parabolic equation has a unique classical solution. Only local solvability is considered. The nonlinearity is a Holder-continuous function of the derivatives up to the order 2m-1 of the unknown solution. The principal tool is the semigroup-theory in Lp(Ω) as well as in Cα(\(\bar \Omega \)). In the latter case the semigroup is not strongly continuous, but it has sufficiently good properties to use it for existence proofs of classical solutions.

In the present paper, a parallel presentation of the theories of abstract families of languages (AFL) and abstract families of deterministic languages (AFDL) is given. This is done by introducing two families of languages. One of them is the one-way nondeterministic family of languages (1NFL). A 1NFL is a family of languages closed under special marked substitution and inverse nondeterministic a-gsm mapping. The deterministic counterpart of 1NFL is 1DFL. It is shown that 1NFL and 1DFL are equivalent to AFL and AFDL, respectively. These families of languages are then used to characterize, side by side and with alternate proofs, the families of languages accepted by AFA and AFDA. Moreover, it is also shown that 1NFL and 1DFL can be used to characterize the families of languages accepted by a closed class of 1NBA and 1DBA, respectively.

An attempt is made to apply information-theoretic computational complexity to meta-mathematics. The paper studies the number of bits of instructions that must be given to a computer for it to perform finite and infinite tasks, and also the time it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms.

An analytic proof is given for the author’s product formula for Jacobi polynomials and a new integral representation is obtained for the product $J_\alpha (x)J_\beta (y)$ of two Bessel functions. Similarly, a product formula for Jacobi polynomials due to Dijksma and the author is derived in an analytic way. The proofs are based on Bateman’s work on special solutions of the biaxially symmetric potential equation. The paper concludes with new proofs for Gasper’s evaluation of the convolution kernel for Jacobi series and for Watson’s evaluation of the integral\[\int_0^\infty {J_\alpha (\lambda x)J_\beta (\lambda y)J_\beta (\lambda z)\lambda ^{1 - \alpha } d\lambda .} \]

Elementary proofs of the first and second Griffiths-Kelly-Sherman (GKS) inequalities are given for higher-spin Ising systems with a Hamiltonian containing only a quadratic form in the spin variables and integer powers of single spin variables. These proofs are obtained using Gaussian random variables. A slight generalization of previous results has been obtained in that the coefficients of the even powers of the spin variables are allowed to be negative.

THE BEHAVIOR OF THE CORE of large economies has received considerable attention during the past decade during which two distinct lines of research have been followed. The first originates with Debreu and Scarf [7] who supplied a rigorous generalization and proof for a conjecture made already by Edgeworth [9], which since has entered economic textbooks as the statement that the core ".shrinks" to the set of competitive equilibria if the number of economic agents becomes infinitely large. The other line of research was started by Aumann [2] who used, as a representation of a large economy, an atomless measure space. Both approaches represent an attempt to describe the intuitive phenomenon of competition, that the power of any individual agent to influence the outcome of trading diminishes if there are sufficiently many other agents who also participate in the market. Recently, Hildenbrand [12] indicated that an atomless economy may be considered as the limit of a sequence of certain increasing but finite economies which provides a link between the two approaches. Most of the available results, however, deal only with pure exchange economies, except for Arrow and Hahn [1], Debreu and Scarf [7], Champsaur [6], and Hildenbrand [10] where production is treated in a very special way. Recently, the general case of the core with production has been formalized and existence proofs were given for the finite case in [3] and for the approach in a measure space by Sondermann in [14]. In [4] and [14] similar market equilibrium concepts were proposed which allow a comparison of the set of equilibrium allocations with the core. Since then a conjecture has been formulated which asserts that the same results may be obtained as in the case of pure exchange. This paper intends to show that this is, in principal, the case for the result of Debreu and Scarf with very general assumptions about the technology distribution. For the case of an atomless economy, however, the example in [5] indicates that the identity of the core and the set of equilibrium allocations may not be expected if the production correspondence is not strictly additive.

Anyone who has worked at proving theorems of intuitionistic logic in a natural deduction system must have been struck by the way in which many logical theorems “prove themselves.” That is, proofs of many formulas can be read off from the syntactical structure of the formulas themselves. This observation suggests that perhaps a strong structural identity may underly this relation between formulas and their proofs. A formula can be considered as a tree structure composed of its subformulas (Frege 1879) and by the normal form theorem (Gentzen 1934) every formula has a normalized proof consisting of its subformulas. Might we not identify an intuitionistic theorem with (one of) its proof(s) in normal form?

Consider N points labelled cyclically forming the vertex set of a planar convex polygon, pairs of which may be connected by bonds. Define a ladder graph to be a graph with no crossing bonds. New simple proofs are given for two results of Kirkman relating to the total number of ladder graphs with A bonds. Relations are derived for calculating the number of connected ladder graphs with A bonds, and asymptotic expressions are obtained.

This paper refines some results of Barwise [1] as well as answering the open question posed at the end of [1] about the Hanf number of positively. We conclude by showing that the existence of a Hanf bound for cannot be proved in the natural formally intuitionistic set theories with bounded predicates decidable of [3], [4] and [5].All notation not explained below is taken from [1]. In the Appendix, we give the axioms of ZF0, ZF1, and T in full. We remark that an important point about the axiom of foundation was not emphasized in [1]. This axiom was intended to be the axiom scheme (∀x)((∀y ∈ x)(A(y)) → A(x)) → (∀x)(A(x)), where y does not occur in A = A(x), instead of the more customary (∀x)(∀y)(y ∈ x → (∃z ∈ x)(∀w ∈ z) (w ∉ x)). This is of no consequence in the presence of full separation, but is vital when considering ZF0 and the T below, for with the customary form of foundation, these cannot even prove the existence of Rω+ω.In [1], a proof of the following is sketched.

Galileo's steps in the discovery of the law of free fall and its application to inclined planes are retraced from one of his letters and some manuscript notes. Proofs of two preliminary theorems are reconstructed, and his methods of calculation are analyzed. Eudoxian proportion theory, and not mean-speed analysis, was the foundation of Galileo's work on motion.

Lévy’s N N -parameter Brownian motion in d d -dimensional space is denoted by W ( N , d ) {W^{(N,d)}} . Using uniform partitions and a Vitali-type variation, Berman recently extended to W ( N , 1 ) {W^{(N,1)}} a classical result of Lévy concerning the relation between W ( 1 , 1 ) {W^{(1,1)}} and 2 2 -variation. With this variation W ( N , d ) {W^{(N,d)}} has variation dimension 2 N 2N with probability one. An appropriate definition of weak variation is given using powers of the diameters of the images of sets which satisfy a parameter of regularity. A previous result concerning the Hausdorff dimensions of the graph and image is used to show the weak variation dimension of W ( N , d ) {W^{(N,d)}} is 2 N 2N with probability one, extending the result for W ( 1 , 1 ) {W^{(1,1)}} of Goffman and Loughlin. If unrestricted partitions of the domain are used, the weak variation dimension of a function turns out to be the same as the Hausdorff dimension of the image.

A useful method for investigating the L 2 instability of the feedback interconnection of a linear unstable subsystem G and a nonlinear stable subsystem N is to resolve L 2 into two orthogonal spaces on the basis of the input-output behavior of G . This yields insight and permits simple proofs of various instability counterparts to stability theorems.

The application of priority arguments to study the structure of the upper semilattice of α-r.e. α-degrees for all admissible ordinals α was first done successfully by Sacks and Simpson [5] who proved that there exist incomparable α-r.e. α-degrees. Lerman and Sacks [3] studied the existence of minimal pairs of α-r.e. α-degrees, and proved their existence for all admissible ordinals α which are not refractory. We continue the study of the α-r.e. α-degrees, and prove that no minimal pair of α-r.e. α-degrees can have as least upper bound the complete α-r.e. α-degree.The above-mentioned theorem was first proven for α = ω by Lachlan [1]. Our proof for α = ω differs from Lachlan's in that we eliminate the use of the recursion theorem. The proofs are similar, however, and a knowledge of Lachlan's proof will be of considerable aid in reading this paper.We assume that the reader is familiar with the basic notions or α-recursion theory, which can be found in [2] or [5].Throughout the paper a will be an arbitrary admissible ordinal. We identify a set A ⊆ α with its characteristic function, A(x) = 1 if x ∈ A, and A(x) = 0 if x ∉ A.If A ⊆ α and B ⊆ α, then A ⊕ B will denote the set defined byA ⊕ B(x) = A(y) if x = λ + 2z, λ is a limit ordinal, z < ω and y = λ + z,= B(y) if x = λ + 2z + 1, λ is a limit ordinal, z < ω, and y = λ + z.

In this paper we extend some aspects of the theory of 'supersolvable lattices' [3] to a more general class of finite lattices which includes the upper-semimodular lattices. In particular, all conjectures made in [33 concerning upper-semimodular lattices will be proved. For instance, we will prove that if L is finite upper-semimodular and if L' denotes L with any set of 'levels' removed, then the M6bius function of L' alternates in sign. Familiarity with [3] will be helpful but not essential for the understanding of the results of this paper. However, many of the proofs are identical to the proofs in [3-I (once the machinery has been suitably generalized) and will be omitted.

This note gives elementary proofs for the characterizations of the Poisson distribution given by Rao and Rubin (1964) and Talwalker (1970) and disproves a conjecture of R. C. and A. B. L. Srivastava (1970). CHARACTERIZATION, POISSON DISTRIBUTION

This paper is an attempt to describe in general terms the pattern of development of women's movements. One of the main problems concerns the mobilization of women to participate in such a movement. In solving this problem I find it necessary to define women first as members of a society and then as participants in a movement. I shall discuss women as a social category, as a collectivity, and as a group. Moreover I shall identify the women's movement as a collective actor and discuss two aspects of its mobilization potential: the mobilization of women in relation to each other, i.e. by their membership of a group, and in relation to the women's movement as a reference individual. The empirical data cited in the discussion should be regarded as illustrations and not, of course, as proofs. The model presented is the result of a theoretical exercise, and should be assessed on the basis of its potential value for a future empirical study of women's movements in Sweden.