Countable Case Research Articles

Infinite inequality systems and cardinal revelations

Many economics problems are maximization or minimization problems, and can be formalized as problems of solving “linear difference systems” of the form \(r_i-r_j \geqq c_{ij}\) and rk-rl > ckl, for r-unknowns, with given c-constants. They typically involve strict as well as weak inequalities, with infinitely many inequalities and unknowns. Since strict inequalities are not preserved under passage to the limit, infinite systems with strict inequalities are notoriously hard to solve. We introduce a unifying tool for solving them. Our main result (Theorem 1 for the countable case, Theorem [2] for the not-necessarily-countable case) introduces a uniform solvability criterion (the \(\omega\)-Axiom), and our proof yields a method for solving those that are solvable. The axiom’s economic intuition extends the traditional ordinal notion of revealed preference to a cardinal notion. We give applications in producer theory, consumer theory, implementation theory, and constrained maximization theory.

A Yosida–Hewitt decomposition for minitive set functions

We prove for minitive set functions defined on a σ -algebra, a similar decomposition theorem to the Yosida–Hewitt's one for classical measures, this way any minitive set function can be decomposed in a fuzzy minitive measure part and a purely minitive part. A particular attention is given for the countable case where the canonical description of any σ -continuous necessity is fully elicited and provide a simple way to compute the Choquet integral of any bounded sequence.

Functional graphs

A graph G is said to be a functional graph if there exist two mappings f and g from V ( G ) into a set F such that xy is an edge in G whenever f ( x ) = g ( y ) or g ( x ) = f ( y ) . Chvátal and Ebenegger proved that recognizing functional graphs is an NP-complete problem. Using the compactness theorem, we prove that if G is an infinite graph such that any finite subgraph of G is a functional graph, then G is a functional graph. We give an elementary proof of this fact in the infinite countable case. In the finite case, we prove that for n large enough, any graph of girth n containing at most 3 n − 7 vertices is a functional graph. It will be shown by an example that this bound is the best possible. To cite this article: A. El Sahili, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Decompositions of infinite graphs: I—bond-faithful decompositions

We show that a graph can always be decomposed into edge-disjoint subgraphs of countable cardinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinal. We also show that this result, with the assumption of the Generalized Continuum Hypothesis, can be generalized to any uncountable cardinal. As applications of such decompositions we prove some results about Seymour's double-cover conjecture for infinite graphs, and about the maximal number of edge-disjoint spanning trees in graphs having high edge-connectivity. However, the main motivation for introducing these decompositions can be found in the second part of this paper where, to achieve a complete solution of the circuit decomposition problem (i.e. the problem of characterizing the graphs that admit decompositions into 2-regular connected subgraphs), we use the results of this first part to carry out a reduction to the countable case.

The Löwenheim–Skolem–Mal'tsev Theorem for \mathbbHF-Structures

We deal with the problem asking whether hereditarily finite superstructures have elementary extensions of the form \(\mathbb{H}\mathbb{F}(\mathfrak{M})\). In so doing, we settle the question whether a theory for some hereditarily finite superstructure have \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) models of arbitrarily large cardinality. A Hanf number is shown to exist, and we provide an exact bound for the countable case.

Uncountable Homogeneous Partial Orders

A partially ordered set (P, ≤) is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of (P, ≤). Assuming the set-theoretic assumption ⋄(ϰ1), it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not (k + 1)-homogeneous. This is impossible in the countable case for k ≥ 4.

GENERAL ENTROPY OF GENERAL MEASURES

The concept of entropy is an important part of the theory of additive measures. In this paper, a definition of entropy is introduced for general (not necessarily additive) measures as the infinum of the Shannon entropies of "subordinate" additive measures. Several properties of the general entropy are discussed and proved. Some of the properties require that the measure belongs to the class of so-called "equientropic" general measures introduced and studied in this paper. The definition of general entropy is extended to the countable case for which a sufficient condition of convergence is proved. We introduce a method of "conditional combination" of general measures and prove that in that case the general entropy possesses the "subset independence" property.

On pseudocompactness of remainders of topological groups and some classes of mappings

We investigate, when a topological group G is pseudocompact at infinity, that is, when bG⧹G is pseudocompact, for each compactification bG of G . It is established that if G is a non-discrete topological group such that every compact subspace of G is scattered, then G is metrizable if and only if it is not pseudocompact at infinity. More generally, a topological group in which every compact subspace has a point of countable character is either metrizable or pseudocompact at infinity. In particular, each non-metrizable topological group of cardinality at most ω 1 is pseudocompact at infinity, while it need not be countably compact at infinity even in the countable case. We observe that every pseudocompact group is pseudocompact at infinity. None of these results generalizes to the class of all Tychonoff spaces. On the other hand, every P -space is shown to be countably compact at infinity. We also introduce a curious concept of a crowded mapping, unifying those of a condensation and of an open continuous mapping with a dense-in-itself image. From the general results obtained it follows that if a countable space Y is an image of a dense-in-itself metrizable space under a condensation, then Y is not countably compact at infinity. The importance of the notion of a crowded mapping for the theory of topological groups may lie in the following fact established in Section 3: every continuous homomorphism of one topological group into another with a non-open kernel is crowded.

Lattice-embedding scales ofL p spaces into orlicz spaces

We study the setPX of scalarsp such thatLp is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0 2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofLp-spaces for different values ofp. We also show that, in general, the associated setPX is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus, we construct a family of Orlicz spaces lF(I), with symmetric basis and indices fixed in advance, containing lp(Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T1]), we show that the set of scalarsp for which lp(Γ) is isomorphic to a subspace of a given Orlicz space lF(I) is not in general closed.

A test for expandability

A model \(M\) of countable similarity type and cardinality \(\kappa\) is expandable if every consistent extension \(T_{1}\) of its complete theory with \(|T_{1}|\leq \kappa\) is satisfiable in \(M\) and it is compactly expandable if every such extension which additionally is finitely satisfiable in \(M\) is satisfiable in \(M\). In the countable case and in the case of a model of cardinality \(\geq 2^{\omega}\) of a superstable theory without the finite cover property the notions of saturation, expandability and compactness for expandability agree. The question of the existence of compactly expandable models which are not expandable is open. Here we present a test which serves to prove that a compactly expandable model of cardinality \(\geq 2^{\omega}\) of a superstable theory is expandable. It is stated in terms of the existence of a certain elementary submodel whose corresponding theory of pairs of models satisfies a weak elimination of Ramsey quantifiers.

Partitions of Unity for Countable Covers

This paper should be considered as expository classroom notes for the instructor. Existence of partitions of unity for metric spaces is usually proved via some rather exacting set-theoretical and topological arguments using the (equivalent) concept of paracompactness. Although the standard proof of paracompactness of metric spaces is the one given by M. E. Rudin [Ru], in his 1965 PhD thesis, Michael Mather showed that it is easier, for metric spaces, to show directly the existence of locally finite partitions of unity for arbitrary covers [Ma]. This proof does not seem to be so well-known although it has been reproduced in [Bo] (see also the appendix of [Do]). It is astonishing that Mather's arguments do not appear in recent textbooks. We hope that the following exposition for countable covers will popularise it. An advantage of the method is that the same proof can be used in the smooth category. Although it is formally covered by the countable case, we will first explain the method in the case of finite covers. T/his will show how easy it is to obtain partitions of unity for compact metric spaces. For the sake of completeness, let us recall a few definitions. If X is a topological space, the support of a continuous function : X > R is the closed set supp(f ) = f xl ( x) + o} . A partition of unity on X is a family (<Pi)i E I of continu

On the Countable Recognition of Finitary Groups

A stage in the classification of the periodic simple finitary linear groups by J. I. Hall and others, is a partial reduction to the countable case; specifically a simple group is isomorphic to a finitary linear group if and only if its countable subgroups are isomorphic to finitary linear groups. It is natural to ask, how common is it for a group to have a faithful finitary linear (or possibly only skew linear) representation, whenever each of its countable subgroups has such a representation? In very recent work F. Leinen and O. Puglisi have shown that periodic primitive finitary linear groups are countably recognizable, see [4]. It is easy to see that every abelian group is isomorphic to a finitary linear group, for example over the subfield of the complexes C generated by all roots of unity. Thus the first non-obvious case, at the opposite extreme to the simple groups (and also in some sense to the primitive groups), is that of the nilpotent groups. We show the following, which, it should be noted, covers the case of periodic nilpotent groups.

The LP approach in average reward MDPs with multiple cost constraints: The countable state case

The objective of this paper is to study the linear programming for countable state average reward Markov decision processes with multiple cost constraints. A corresponding linear programming and its dual are formulated and their solvability and absence of a duality gap are proved under some reasonable conditions. Also, the ergodic assumption derives the existence of a constrained optimal stationary policy.

Complementation in the lattice of equivalence relations

The aim of this paper is to study those pairs of complementary equivalence relations on a fixed set which are maximal as families of mutually complementary equivalence relations. The existence of such pairs on uncountable sets was proved by Steprāns and Watson (1995). They conjectured that such pairs do not exist in the finite and in the countable case. Here we disprove this conjecture by proving that they exist in huge quantity in both cases. We study in detail the case when: (a) one of the equivalence relations in the pair has precisely two equivalence classes; (b) one of the equivalence relations has at most three equivalence classes; (c) in one of the equivalence relations all but one equivalence classes are singletons. In cases (a) and (c) we describe all pairs of complementary equivalence relations having this extremal property. In the case (a) the non-extremal pairs are related to Turan graphs.

Lattice generation of small equivalences of a countable set

Given a countable set A, let Equ(A) denote the lattice of equivalences of A. We prove the existence of a four-generated sublattice Q of Equ(A) such that Q contains all atoms of Equ(A). Moreover, Q can be generated by four equivalences such that two of them are comparable. Our result is a reasonable generalization of Strietz [5, 6] from the finite case to the countable one; and in spite of its essentially simpler proof it asserts more for the countable case than [2, 3].

Transboundary extremal length

We introduce two basic notions, ‘transboundary extremal length’ and ‘fat sets’, and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short proof is given to Koebe's conjecture in the countable case: every planar domain with countably many boundary components is conformally equivalent to a circle domain. This theorem is further generalized in two direction. We show that the same statement is true for a wide class of domains with uncountably many boundary components, in particular for domains bounded byK-quasicircles and points. Moreover, these domains admit more general uniformizations. For example, every circle domain is conformally equivalent to a domain whose complementary components are heart-shapes and points.

On the structure of nonarchimedean exponential fields I

Given an ordered fieldK, we compute the natural valuation and skeleton of the ordered multiplicative group (K >0, ·, 1, <) in terms of those of the ordered additive group (K,+,0,<). We use this computation to provide necessary and sufficient conditions on the value groupv(K) and residue field\(\bar K\), for theL ∞ε-equivalence of the above mentioned groups. We then apply the results to exponential fields, and describev(K) in that case. Finally, ifK is countable or a power series field, we derive necessary and sufficient conditions onv(K) and\(\bar K\) forK to be exponential. In the countable case, we get a structure theorem forv(K).

On the structure of nonarchimedean exponential fields I

Given an ordered fieldK, we compute the natural valuation and skeleton of the ordered multiplicative group (K>0, ·, 1, <) in terms of those of the ordered additive group (K,+,0,<). We use this computation to provide necessary and sufficient conditions on the value groupv(K) and residue field\(\bar K\), for theL∞e-equivalence of the above mentioned groups. We then apply the results to exponential fields, and describev(K) in that case. Finally, ifK is countable or a power series field, we derive necessary and sufficient conditions onv(K) and\(\bar K\) forK to be exponential. In the countable case, we get a structure theorem forv(K).

Mixture models for time series

In this paper we extend the class of zero-order threshold autoregressive models to a much richer class of mixture models. The new class has the important property of duality which, as we show, corresponds to time reversal. We are then able to obtain the time reversals of the zero-order threshold models and to characterise the time-reversible members of this subclass. These turn out to be quite trivial. The time-reversible models of the more general class do not suffer in this way. The complete stationary distributional structure is given, as are various moments, in particular the autocovariance function. This is shown to be of ARMA type. Finally we give two examples, the second of which extends from the finite to the countable mixture case. The general theory for this extension will be given elsewhere.

Mixture models for time series

In this paper we extend the class of zero-order threshold autoregressive models to a much richer class of mixture models. The new class has the important property of duality which, as we show, corresponds to time reversal. We are then able to obtain the time reversals of the zero-order threshold models and to characterise the time-reversible members of this subclass. These turn out to be quite trivial. The time-reversible models of the more general class do not suffer in this way. The complete stationary distributional structure is given, as are various moments, in particular the autocovariance function. This is shown to be of ARMA type. Finally we give two examples, the second of which extends from the finite to the countable mixture case. The general theory for this extension will be given elsewhere.