Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.     

A homogeneous extremally disconnected countably compact space

It is well known that no infinite homogeneous space is both compact and extremally disconnected. (Since there are infinite compact homogeneous spaces and infinite extremally disconnected homogeneous spaces, it is the combination of compactness and extremal disconnectedness that brings about this result.) The following question then arises naturally: How “close to compact” can a homogeneous, extremally disconnected space be? The aim of this paper is to show that a homogeneous extremally disconnected space can be countably compact. It is shown also, assuming Martin's Axiom, that there exist countably compact, homogeneous, extremally disconnected spaces whose product is not countably compact.     

乘积σ—代数的一种生成方法

本文得到了有限及可数无限维乘积σ—代数的一种新的生成方法，并证明了可数无限维Ｂｏｒｅｌ域具有连续统的势．     

Homomorphisms between Stone-Cech Homomorphisms into Stone-Cech Remainders of Countable Groups

We study properties of continuous homomorphisms from β S into T^* and from S^* into T^*, where S denotes a countably infinite semigroup and T denotes a countably infinite group. We show that they have striking algebraic properties if they do not arise as continuous extensions of homomorphisms from S to T.     

The lattice complexity of quasivarieties of graphs and endographs

We construct a countably infinite descending chain of Q-universal quasivarieties of graphs, starting with D, and show that its intersection has a distributive countably infinite lattice of subquasivarieties. It is also proved that the quasivariety of symmetric endographs is Q-universal. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 273–281, May–June, 1997.     

The existence of multiple positive solutions to boundary value problems of nonlinear delay differential equations with countably many singularities on infinite interval

In this paper, we consider the existence of countably many positive solutions to a boundary value problem of a nonlinear delay differential equation with countably many singularities on infinite interval

     

Optimal control for infinite dimensional stochastic differential equations with infinite Markov jumps and multiplicative noise

In this paper we solve an infinite-horizon linear quadratic control problem for a class of differential equations with countably infinite Markov jumps and multiplicative noise. The global solvability of the associated differential Riccati-type equations is studied under detectability hypotheses. A nonstochastic, operatorial approach is used. Some properties of the linear stochastic systems, such as stability, stabilizability and detectability, are also discussed on the basis of a new solution representation result. A generalized Ito's formula which applies to infinite dimensional stochastic differential equations with countably infinite Markov jumps is also provided.     

Connected components of compact matrix quantum groups and finiteness conditions

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1.     

Maximal clones on uncountable sets that include all permutations

We first determine the maximal clones on a set X of infinite regular cardinality κ which contain all permutations but not all unary functions, extending a result of Heindorf’s for countably infinite X. If κ is countably infinite or weakly compact, this yields a list of all maximal clones containing the permutations, since in that case the maximal clones above the unary functions are known. We then generalize a result of Gavrilov’s to obtain on all infinite X a list of all maximal submonoids of the monoid of unary functions which contain the permutations. Received January 8, 2004; accepted in final form December 22, 2004.     

Infinite,highly connected digraphs with no two arc-disjoint spanning trees

We present a construction of countably infinite, highly connected graphs and digraphs, which shows that several basic connectivity results on finite graphs, including Edmondsapos;s branching theorem, cannot be extended to the infinite case.     

Countably compact groups from a selective ultrafilter

We prove that the existence of a selective ultrafilter on implies the existence of a countably compact group without non-trivial convergent sequences all of whose powers are countably compact. Hence, by using a selective ultrafilter on , it is possible to construct two countably compact groups without non-trivial convergent sequences whose product is not countably compact.

     

On the number of automorphisms of structures

The number of automorphisms of a structure with no uncountable orbits and at most countably many infinite orbits is either less than or equal to ?₀ or else 2' for same infinite cardinal κ.     

Countably categorical theories

A series of countably categorical theories are constructed based on the Fra?sse method. In particular, an example of a decidable countably categorical theory in a finite signature is given for which no decidable model has an infinite computable set of order indiscernibles. Such a theory is used to refute Ershov’s conjecture on the representability of models of c-simple theories over linear orders.     

Lyapunov coupled equations for continuous-time infinite Markov jump linear systems

This paper deals with Lyapunov equations for continuous-time Markov jump linear systems (MJLS). Out of the bent which wends most of the literature on MJLS, we focus here on the case in which the Markov chain has a countably infinite state space. It is shown that the infinite MJLS is stochastically stabilizable (SS) if and only if the associated countably infinite coupled Lyapunov equations have a unique norm bounded strictly positive solution. It is worth mentioning here that this result do not hold for mean square stabilizability (MSS), since SS and MSS are no longer equivalent in our set up (see, e.g., [J. Baczynski, Optimal control for continuous time LQ-problems with infinite Markov jump parameters, Ph.D. Thesis, Federal University of Rio de Janeiro, UFRJ/COPPE, 2000]). To some extent, a decomplexification technique and tools from operator theory in Banach space and, in particular, from semigroup theory are the very technical underpinning of the paper.     

More Sublattices of the Lattice of Local Clones

We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that: For example, the lattice M_2^wM_{2^\omega} is a sublattice of the local clone lattice.     

Complete groups are complete co-analytic

The set of complete groups is a complete co-analytic subset of the standard Borel space of countably infinite groups.     

First order autoregressive markov processes

This paper is concerned with establishing conditions under which finite (and then countably infinite) stationary Markov chains have first order autoregressive representations.     

Infinite compact sets of idempotents in $$\beta S$$

We show that if S is a countably infinite right cancellative semigroup and T is an infinite compact set of idempotents in the Stone–?ech compactification \(\beta S\) of S, then T contains an infinite compact left zero semigroup.