Fred Richman

CALCULATING MAXIMUM-ENTROPY PROBABILITY DENSITIES FOR BELIEF FUNCTIONS

A common procedure for selecting a particular density from a given class of densities is to choose one with maximum entropy. The problem addressed here is this. Let S be a finite set and let B be a belief function on 2S. Then B induces a density on 2S, which in turn induces a host of densities on S. Provide an algorithm for choosing from this host of densities one with maximum entropy.

Generalized real numbers in constructive mathematics

Abstract Two extensions of the real number system, one given by uppercuts the other by lowercuts, are developed within a constructive framework. The first includes distances to arbitrary subsets, the second includes norms of arbitrary bounded linear operators. The intuitive meaning of comparing such quantities to ordinary real numbers is preserved. Difficulties with encompassing both kinds of numbers in a single system are considered.

Existence theorems for Warfield groups

Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.

A Class of Rank — 2 Torsion Free Groups

This section has little to do with the main theme. I once thought that the notion to be discussed here was key to classifying finite rank torsion free groups; I still feel that it is the fundamental invariant for such groups. It seems to be in the back of the mind of averyone who studies these groups and it surfaces, more or less explicitly, in many discussions. I include it with the lope of giving it some formal standing.

Real numbers and other completions

A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete Archimedean Heyting field, a terminal object in the category of Archimedean Heyting fields. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Constructive Mathematics without Choice

What becomes of constructive mathematics without the axiom of (countable) choice? Using illustrations from a variety of areas, it is argued that it becomes better.

An extension of the theory of completely decomposable torsion-free abelian groups

We construct a class of strongly indecomposable finite rank torsion-free groups that includes the rank-one groups, and develop a complete set of invariants for them and their direct sums. 0. Introduction. Perhaps the only class of countable torsion-free abelian groups, for which a satisfactory complete set of invariants has been developed, is the class of countable completely decomposable torsion-free groups [FCHS, §85], The theory of completely decomposable groups begins by classifying rank-one torsion-free groups up to isomorphism by their type, then shows that direct sums of rank-one groups are isomorphic only if they are composed of the same number of groups of each type. As an added bonus it turns out that summands of completely decomposable groups are completely decomposable. We can attempt to generalize this elegant theory by starting with a larger class than the rank-one groups. A natural place to look is among the Butler groups: those groups that are isomorphic to pure subgroups of a finite direct sum of rank-one torsion-free groups [BTLR, ARNL]. In this paper we describe a class C of Bulter groups, that contains the rank-one groups and also contains strongly indecomposable groups of every finite rank, and exhibit a complete set of invariants for distinguishing among groups in C. Applying the global Azumaya theorems of [AHR] we also obtain a complete set of invariants for the class of direct sums of groups in C. Let « 2» 2 be an integer. We shall be concerned with «-tuples A = iAx,... ,A„) of subgroups A¡ of the rational numbers 0. For such an A we define the rank-(« — 1) group G i A) by G(A) = \(ax,...,an) EAX® ■■■ ®An: 2«, = oj. This construction is slightly more general than it might appear at first glance. Any pure subgroup K of Ax@ ••• ©An of corank one is the kernel of a nonzero Received by the editors August 24, 1982. 1980 Mathematics Subject Classification. Primary 20K15, 20K20; Secondary 20K25, 20K40. 1 Research supported by NSF grant MCS-8003060. ©1983 American Mathematical Society O002-9947/82/0OO0-2365/

Subgroups of p 5-bounded groups

03.75 175 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Church's thesis without tears

Each valuated module B with B(5) = 0 is a direct sum of simply presented valuated modules and copies of two valuated modules which come from (finite) hung trees. There are infinite-rank indecomposable valuated modules B with B(6) = 0.

Intuitionistic notions of boundedness in ℕ

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is. The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular if P is some property, then the statement “for each m there is n such that P(m, n) ” means that we can construct a (computable) function θ such that P(m, θ(m)) for all m . Churchs thesis has a different flavor in an environment like this where the notion of a computable function is primitive. One point of such a treatment of Churchs thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Churchs thesis, or any consequence of Churchs thesis. Hence counterexamples such as Speckers bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.