Countable Case Research Articles

The Minimax Property in Infinite Two-Person Win-Lose Games

We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we prove that a game satisfies the minimax property along with all its subgames if and only if none of its subgames is isomorphic to the “larger number game.” This generalizes a recent theorem of Hanneke, Livni, and Moran. We also propose several applications of our results outside of game theory.

On the packing/covering conjecture of infinite matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family (Mi:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}:i \\in \\Theta)$$\\end{document} of matroids on the common edge set E is a system (Si:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({S_i}:i \\in \\Theta)$$\\end{document} of pairwise disjoint subsets of E where Si is panning in Mi. Similarly, a covering is a system (Ii: i ∈ Θ) with ∪i∈ΘIi=E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\cup _{i \\in \\Theta}}{I_i} = E$$\\end{document} where Ii is independent in Mi. The conjecture states that for every matroid family on E there is a partition E=Ep⊔Ec\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E = {E_p} \\sqcup {E_c}$$\\end{document} such that (Mi↾Ep:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}\\upharpoonright{E_p}:i \\in \\Theta)$$\\end{document} admits a packing and (Mi.Ec:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}.{E_c}:i \\in \\Theta)$$\\end{document} admits a covering. We prove the case where E is countable and each Mi is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.

Maximal elements with minimal logic

There is a remarkable argument for Krull's maximal ideal theorem in the countable case—which is constructive, the essence of which traces back to Krivine, with no choice involved, and requiring minimal logic only. A simple extension to abstract domains is possible: principles of intuitionistic set theory on top of minimal logic suffice to ensure that every countably based bounded complete dcpo has a maximal element. The aforesaid theorem is a ready instance.

On a new class of trigonometric thin sets extending Arbault sets

In this article we introduce a new class of trigonometric thin sets, namely statistical Arbault sets properly containing the class of classical Arbault sets [1] as well as a large subfamily of N-sets (also called “sets of absolute convergence”) [17]. In particular this class happens to contain the types of N-sets (that includes again strictly the so called N0-sets) which have been extensively used in the literature (see [1,23,24]), but at the same time being distinct from the class of N-sets. We observe that this is a new class strictly lying between the class of Arbault sets and the class of Weak Dirichlet sets.The motivation for thinking of such a possible class comes from the well known observation that the characterized subgroups of the circle group forms a basis of Arbault sets and the recent introduction of the notion of statistically characterized subgroups [14] extending the notion of characterized subgroups. In Section 2 of the article it is shown that there are statistically characterized subgroups which can't be characterized by any sequence of integers establishing the “novelty” of the notion. Other significance of this result is that it presents an example of a polishable subgroup of the circle group that can not be characterized by a suitable sequence of integers unlike the countable case whose answer is positive [5]. This naturally paves the way for a new class of sets generated by the class of statistically characterized subgroups as basis which we name statistical Arbault sets.

Inf-convolution and optimal risk sharing with countable sets of risk measures

The inf-convolution of risk measures is directly related to risk sharing and general equilibrium, and it has attracted considerable attention in mathematical finance and insurance problems. However, the theory is restricted to finite sets of risk measures. This study extends the inf-convolution of risk measures in its convex-combination form to a countable (not necessarily finite) set of alternatives. The intuitive meaning of this approach is to represent a generalization of the current finite convex weights to the countable case. Subsequently, we extensively generalize known properties and results to this framework. Specifically, we investigate the preservation of properties, dual representations, optimal allocations, and self-convolution.

Structural results on lifting, orthogonality and finiteness of idempotents

In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized in terms of certain connected sets related to that ideal. Then as an application, we obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that lifting idempotents preserves the orthogonality in countable cases. The lifting property of an arbitrary morphism of rings is characterized. As another major result, it is proved that the number of idempotents of a ring $R$ is finite if and only if it is of the form $2^{\kappa}$ where $\kappa$ is the cardinal of the connected components of Spec$(R)$. Finally, it is proved that the primitive idempotents of a zero dimensional ring are in 1-1 correspondence with the isolated points of its prime spectrum. These results either generalize or improve several important results in the literature.

A zero-one law for Markov chains

We prove an analogue of the classical zero-one law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event A from the tail σ-algebra of MC , for large n, with probability near one, the trajectories of the MC are in states i, where is either near 0 or near 1. A similar statement holds for the entrance σ-algebra when n tends to . To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by or in both the finite and countable cases. This extends a well-known result due to Kolmogorov. Further, in our discussion, we note an interesting dichotomy between two commonly used definitions of MCs.

Unit representation of semiorders II: The general case

Necessary and sufficient conditions under which semiorders on uncountable sets can be represented by a real-valued function and a constant threshold are known. We show that the proof strategy that we used for constructing representations in the case of denumerable semiorders can be adapted to the uncountable case. We use it to give an alternative proof of the existence of strict unit representations. In contrast to the countable case, semiorders on uncountable sets that admit a strict unit representation do not necessarily admit a nonstrict unit representation, and conversely. By adapting the proof strategy used for strict unit representations, we establish a characterization of the semiorders that admit a nonstrict representation. Conditions for the existence of other special unit representations are also obtained.

ON GRAEV’S THEOREM FOR FREE PRODUCTS OF HAUSDORFF TOPOLOGICAL GROUPS

AbstractThe paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of $k_\omega $ -groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the $X_0$ -topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.

Proof of Nash-Williams' intersection conjecture for countable matroids

We prove that if M and N are finitary matroids on a common countable edge set E then they admit a common independent set I such that there is a bipartition E=EM∪EN for which I∩EM spans EM in M and I∩EN spans EN in N. It answers positively the Matroid Intersection Conjecture of Nash-Williams in the countable case.

Noncountable Tuberculosis Case Reporting, National Tuberculosis Surveillance System, United States, 2010-2014.

Supplemental federal funding is allocated to state and local tuberculosis (TB) programs using a formula that considers only countable cases reported to the National Tuberculosis Surveillance System (NTSS). Health departments submit reports of cases, which are countable unless another (US or international) jurisdiction has already counted the case or the case represents a recurrence of TB diagnosed ≤12 months after completion of treatment for a previous TB episode. Noncountable cases are a resource burden, so in 2009, NTSS began accepting noncountable case reports as an indicator of program burden. We sought to describe the volume and completeness of noncountable case reports. We analyzed 2010-2014 NTSS data to determine the number and distribution of noncountable cases reported. We also surveyed jurisdictions to determine the completeness of noncountable case reporting and obtain information on jurisdictions' experience in reporting noncountable cases. In addition, we prepared a hypothetical recalculation of the funding formula to evaluate the effect of including noncountable cases on funding allocations. Of 54 067 TB case reports analyzed, 1720 (3.2%) were noncountable; 47 of 60 (78.3%) jurisdictions reported ≥1 noncountable case. Of 60 programs surveyed, 34 (56.7%) responded. Of the 34 programs that responded, 24 (70.6%) had not reported all their noncountable cases to NTSS, and 11 (32.4%) stated that reporting noncountable cases was overly burdensome, considering the cases were not funded. Complete data on noncountable TB cases help support estimates of programmatic burden. Ongoing training and a streamlined reporting system to NTSS can facilitate noncountable case reporting.

On the support of extremal martingale measures with given marginals: the countable case

AbstractWe investigate the supports of extremal martingale measures with prespecified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure Q and the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a financial context as the set of all semistatic trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (WEP) and provide two combinatorial sufficient conditions, called the ‘2-link property’ and ‘full erasability’, on how the points in the supports are linked to each other for granting extremality. When the support of the first marginal is a finite set, we give a necessary and sufficient condition for the WEP to hold in terms of the new concepts of 2-net and deadlock. Finally, we study the relation between cycles and extremality.

On Weak and Strong Ergodicity

We establish some classical results on ergodicity of inhomogeneous Markov chains in a more general context. Madsen (Ann Math Stat 42(1):405–408, 1971) extended the characterization of weak ergodicity presented by Paz (Ann Math Stat 41(2):539–550, 1970) from the countable case to the continuous case, admitting, however, that the transition probability functions have associated density functions. Here, we show that Madsen’s extension remains valid if the transition probability functions are not associated with density functions. In addition, we extend the criteria for strong ergodicity of Dorea and Rojas Cruz (Sankhya Indian J Stat (2003–2007) 66(2):243–252, 2004) which is a refinement of a classical result given by Isaacson and Madsen (Markov chains, theory and applications, Wiley, New York, 1976).

Some transfinite natural sums

AbstractWe study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages. We show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a mixed sum (in an order‐theoretical sense) of the ordinals in the sequence; in fact, it is the largest mixed sum which satisfies a finiteness condition. We introduce other infinite natural sums which are invariant under permutations and show that all the sums under consideration coincide in the countable case.

Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

Let K→N be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of K→N in which every monochromatic path has density 0.However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every (r+1)-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length ℓi for any colour i≤r can be covered by ∏i∈[r]ℓi pairwise disjoint monochromatic complete symmetric digraphs in colour r+1.Furthermore, we present a stability version for the countable case of the latter result: We prove that the edge-colouring is uniquely determined on a large subgraph, as soon as the upper density of monochromatic paths in colour r+1 is bounded by ∏i∈[r]1ℓi.

On the Modal Logic of Jeffrey Conditionalization

We continue the investigations initiated in the recent papers (Brown et al. in The modal logic of Bayesian belief revision, 2017; Gyenis in Standard Bayes logic is not finitely axiomatizable, 2018) where Bayes logics have been introduced to study the general laws of Bayesian belief revision. In Bayesian belief revision a Bayesian agent revises (updates) his prior belief by conditionalizing the prior on some evidence using the Bayes rule. In this paper we take the more general Jeffrey formula as a conditioning device and study the corresponding modal logics that we call Jeffrey logics, focusing mainly on the countable case. The containment relations among these modal logics are determined and it is shown that the logic of Bayes and Jeffrey updating are very close. It is shown that the modal logic of belief revision determined by probabilities on a finite or countably infinite set of elementary propositions is not finitely axiomatizable. The significance of this result is that it clearly indicates that axiomatic approaches to belief revision might be severely limited.

Automated Deduction in Gödel Logic

This article addresses the deduction problem of a formula from a countable theory in the first-order Gödel logic. We generalise the well-known hyperresolution principle for deduction in Gödel logic. Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. We introduce a notion of quantified atom: a formula a is a quantified atom if a = Qx , p ( t 0 , …, t n ), where Q is a quantifier (∀, ∃), p ( t 0 , …, t n ) is an atom, and x is a variable occurring in p ( t 0 , …, t n ); for all i ≤ n , either t i = x or x does not occur in t i . Then an order clause is a finite set of order literals of the form ϵ 1 ⋄ ϵ 2 , where ϵ i is either an atom, or a truth constant ( 0 , 1 ), or a quantified atom, and ⋄ is either a connective ≖, equality, or ≺ strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [ 0 , 1 ], respectively. On the basis of the hyperresolution principle, a calculus operating over order clausal theories is devised. The calculus is proved to be refutation sound and complete for the countable case. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic.

UNIFORM PROCEDURES IN UNCOUNTABLE STRUCTURES

AbstractThis article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of ${\Cal R}^n $ contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.

Symmetry and measurability

In the Hewitt–Savage 0-1 law, symmetric measurable sets are considered in a countable product of a σ-algebra Σ with itself. Therefore, it may be of interest to find simple generators for these sets in terms of the generators of Σ. We put the problem into a more general framework of action groups and as an application find simple generators for the finite product case. We also have a partial result in the countable product case.

Köhler theory for countable quadruple systems

From the late 1970s to the early 1980s, Köhler developed a theory for constructing finite quadruple systems with point-transitive Dihedral automorphism groups by introducing a certain algebraic graph, now widely known as the (first) Köhler graph in finite combinatorics. In this paper, we define the countable Köhler graph and discuss countable extensions of a series of Köhler's works, with emphasis on various gaps between the finite and countable cases. We show that there is a simple 2-fold quadruple system over Z with a point-transitive Dihedral automorphism group if the countable Köhler graph has a so-called [1, 2]-factor originally introduced by Kano (1986) in the study of finite graphs. We prove that a simple Dihedral $\ell$-fold quadruple system over Z exists if and only if $\ell = 2$. The paper also covers some related remarks about Hrushovski's constructions of countable projective planes.