On some ‘duality’ of the Nikodym property and the Hahn property

Drewnowski and Paúl proved in [L. Drewnowski, P.J. Paúl, The Nikodým property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 (2000) 485-503] that for any strongly nonatomic submeasure η on the power set P(N) of N the ideal Z(η)={N∈P(N)|η(N)=0} has the Nikodym property (NP); in particular, this result applies to densities dA defined by strongly regular matrices A. Grahame Bennett and the authors stated in [G. Bennett, J. Boos, T. Leiger, Sequences of 0's and 1's, Studia Math. 149 (2002) 75-99] that the strong null domain ₀|A| of any strongly regular matrix A has the Hahn property (HP). Moreover, Stuart and Abraham [C.E. Stuart, P. Abraham, Generalizations of the Nikodym boundedness and Vitali-Hahn-Saks theorems, J. Math. Anal. Appl. 300 (2) (2004) 351-361] pointed out that the said results are in some sense dual and that the last one follows from the first one by considering the density dA (defined by A) as submeasure on P(N) and the ideal Z(dA) as well by identifying P(N) with the set χ of sequences of 0's and 1's. In this paper we aim at a better understanding of the intimated duality and at a characterization of those members of special classes of matrices A such that Z(dA) has NP (equivalently, ₀|A| has HP).     

‘Duality’ of the Nikodym property and the Hahn property: Densities defined by sequences of matrices

The authors investigated in Boos and Leiger (2008) [5] the ‘duality’ of the Nikodym property (NP) of the set of all null sets of the density defined by any nonnegative matrix and the Hahn property (HP) of the strong null domain of it. In this paper, the investigation of the intimated duality is continued by considering densities defined by sequences of nonnegative matrices. These considerations are motivated by the known result that the ideal of the null sets of the uniform density has NP. In this context the general notion of S-convergence of double sequences (cf. Drewnowski, 2002 [8]) containing Pringsheim convergence, Hardy convergence and uniform convergence of double sequences is used.     

A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

This paper first presents a characterization of three classes of negligible closed convex sets (i.e., Gauss null sets, Aronszajn null sets and cube null sets) in terms of non-support points; then gives a generalization of Gâteaux differentiability theorems of Lipschitz mapping from open sets to those closed convex sets admitting non-support points; and as their application, finally shows that a closed convex set in a separable Banach space X can be Lipschitz embedded into a Banach space Y with the Radon–Nikodym property if and only if the closure of its linear span is linearly isomorphic to a closed subspace of Y.     

On the finiteness property for rational matrices

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This “finiteness conjecture” is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2×2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.     

On Efimov spaces and Radon measures

We give a construction under CH of an infinite Hausdorff compact space having no converging sequences and carrying no Radon measure of uncountable type. Under ? we obtain another example of a compact space with no convergent sequences, which in addition has the stronger property that every nonatomic Radon measure on it is uniformly regular. This example refutes a conjecture of Mercourakis from 1996 stating that if every measure on a compact space K is uniformly regular then K is necessarily sequentially compact.     

Decomposition of the Space of Regular Operators

Let E be a directed vector space with the Riesz decomposition property and F be a complete v?ctor lattice. It is proved that if G is a solid subspace of E then the subspace of all regular operators annihilating on G is a component in the space of all regular operators from E into F.¹²     

Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms

A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article, we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set, but also for all sufficiently nearby sets of equal cardinality. We also prove a theorem giving conditions under which the Barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar, and show that in certain cases these conditions are also persistent under small perturbations.     

Structure of vertex-transitive graphs

Every vertex-transitive graph has a characteristic structure. The specific details of structure of a vertex-transitive graph are closely related to the configuration of the parts of any minimum separating set of that graph. It is shown for a vertex-transitive graph G that (i) the number n of isomorphic atomic parts admitted by any minimum separating set S is unique for that G; (ii) G is then isomorphic to a disjoint union of m(≥2) copies of such a set of n atomic parts together with some additional edges joining them; (iii) any minimum separating set S of G consists of the vertex set of the union of some k(≥1) copies of the set of n atomic parts; (iv) at most one nonatomic part will be admitted in conjunction with one or more atomic parts by any minimum separating set of G.This configuration of structure extends to nonatomic parts. A vertex-transitive graph G may contain a not necessarily unique minimum separating set S which admits t(≥3) nonatomic parts. Then it is shown that (i) some (t ? 1)-set of these nonatomic parts are pairwise isomorphic; (ii) if the remaining nonatomic part is nonisomorphic to the others then it contains more vertices than the other parts; (iii) G is isomorphic to a disjoint union of d(≥2) copies of the set of q(≥t ? 1) isomorphic nonatomic parts together with some additional edges joining them; (iv) a minimum separating set S consists of the vertex set of the union of some y(≥1) copies of the set of q isomorphic nonatomic parts. For the case of t = 2 non-atomic parts admitted by a minimum separating set S, counterexamples of (iii) and (iv) are given.     

The Riesz Basis Property of an Indefinite Sturm–Liouville Problem with Non-Separated Boundary Conditions

We consider a regular indefinite Sturm–Liouville eigenvalue problem ?f′′ + q f = λ r f on [a, b] subject to general self-adjoint boundary conditions and with a weight function r which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions r for which the Riesz basis property can be completely characterized in terms of the local behavior of r in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of r in a neighborhood of the turning points, local conditions on r near the boundary are needed for the Riesz basis property. As an application, it is shown that the Riesz basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.     

Equilibrium points of nonatomic games over a nonreflexive Banach space

Schmeidler's results on the equilibrium points of nonatomic games with strategy sets in Euclidean n-space are generalized to nonatomic games with stategy sets in a separable Banach space whose dual possesses the Radon-Nikodým property.     

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.     

Weak regularity and consecutive topologizations and regularizations of pretopologies

L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n<ω there is a (regular) pretopology ρ (on a set of cardinality c) such that k(RT)ρ>n(RT)ρ for each k<n and n(RT)ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ?ω₀. Moreover, all these pretopologies have the property that all the points except one are topological and regular.     

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ₁), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ₁) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω₁.     

Finiteness property of pairs of 2×2 sign-matrices via real extremal polytope norms

This paper deals with the joint spectral radius of a finite set of matrices. We say that a set of matrices has the finiteness property if the maximal rate of growth, in the multiplicative semigroup it generates, is given by the powers of a finite product.Here we address the problem of establishing the finiteness property of pairs of $2\times 2$ sign-matrices. Such problem is related to the conjecture that pairs of sign-matrices fulfil the finiteness property for any dimension. This would imply, by a recent result by Blondel and Jungers, that finite sets of rational matrices fulfil the finiteness property, which would be very important in terms of the computation of the joint spectral radius. The technique used in this paper could suggest an extension of the analysis to $n\times n$ sign-matrices, which still remains an open problem.As a main tool of our proof we make use of a procedure to find a so-called real extremal polytope norm for the set. In particular, we present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing.For pairs of sign-matrices we develop the computations exactly and hence are able to prove analytically the finiteness property. On the other hand, the algorithm can be used in a floating point arithmetic and provide a general tool for approximating the joint spectral radius of a set of matrices.     

A cooperative system which does not satisfy the limit set dichotomy

The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x<y, then either ω(x)<ω(y), or ω(x)=ω(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.     

Riesz Basis Generation of Abstract Second-Order Partial Differential Equation Systems with General Non-Separated Boundary Conditions

Riesz basis analysis for a class of general second-order partial differential equation systems with nonseparated boundary conditions is conducted. Using the modern spectral analysis approach for parameterized ordinary differential operators, it is shown that the Riesz basis property holds for the general system if its associated characteristic equation is strongly regular. The Riesz basis property can then be readily established in a unified manner for many one-dimensional second-order systems such as linear string and beam equations with collocated or noncollocated boundary feedbacks and tip mass attached systems. Three demonstrative examples are presented.     

The Average Range Characterization of the Radon–Nikodym Property

A characterization of Banach spaces possessing the Radon—Nikodym property is given in terms of the average range of additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued additive interval function possessing absolutely continuous McShane (or Henstock) variational measure has nonempty average range almost everywhere on [0, 1].     

Weakly regular rings with ACC on annihilators and maximality of strongly prime ideals of weakly regular rings

Ramamurthi proved that weak regularity is equivalent to regularity and biregularity for left Artinian rings. We observe this result under a generalized condition. For a ring R satisfying the ACC on right annihilators, we actually prove that if R is left weakly regular then R is biregular, and that R is left weakly regular if and only if R is a direct sum of a finite number of simple rings. Next we study maximality of strongly prime ideals, showing that a reduced ring R is weakly regular if and only if R is left weakly regular if and only if R is left weakly π-regular if and only if every strongly prime ideal of R is maximal.     

On certain hyperbolic sets

Two invariant sets F of certain diffeomorphisms S that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of hyperbolic sets with the property (unexpected at that time) that, in some neighborhood of such an F, there is no locally maximal set containing F are considered. It is proved that this property, although referring to the behavior of the orbits of S near F, is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of S on F with the hyperbolicity of F. (This means that if a hyperbolic set F′ for a diffeomorphism S′ is equivariantly homeomorphic to a Fathi-Crovisier or a Fisher set, then F′ has a neighborhood in which S′ has no locally maximal set containing F′.)     

Riesz potentials associated with a composite power function on the space of rectangular matrices

On the space of real rectangular matrices, Riesz potentials depending on a multidimensional complex parameter are studied. These potentials are in a relationship with the composite power function of a matrix argument. For the potentials indicated, analogs of classical equalities are established. In particular, the semigroup property for the Riesz potentials with multidimensional complex parameter is proved under less restrictive limitations on the parameters of a rectangular matrix than the corresponding semigroup property for the Riesz potentials of one complex parameter. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 207–225.