The stationary set splitting game

The stationary set splitting game is a game of perfect information of length ω₁ between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ²₂ maximality with a predicate for the nonstationary ideal on ω₁, and an example of a consistently undetermined game of length ω₁ with payoff de.nable in the second‐order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a₀ and on the two first off‐diagonals the constants a₁(lower) and a₋₁(upper), which are all complex values, there exist closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. For example, for the 1D discrete Laplacian, this triple is (a₀,a₁,a₋₁)=(2,−1,−1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form (a₀,a_ω,a_−ω), but where the two off‐diagonals are positioned ω steps from the main diagonal instead of only one. We show that its eigenvalues and eigenvectors can also be identified in closed form and that interesting connections with the standard Toeplitz symbol are identified. Furthermore, as numerical evidences clearly suggest, it turns out that the eigenvalue behavior of a general banded symmetric Toeplitz matrix with real entries can be described qualitatively in terms of the symmetrically sparse tridiagonal case with real a₀, a_ω=a_−ω, ω=2,3,…, and also quantitatively in terms of those having monotone symbols. A discussion on the use of such results and on possible extensions complements the paper.     

A local normal form theorem for infinitary logic with unary quantifiers

We prove a local normal form theorem of the Gaifman type for the infinitary logic L_∞ω( Q _u)^ω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht‐Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L_∞ω( Q _u)^ω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form (?^≥iy)ψ(y), where ψ(y) has counting quantifiers restricted to the (2^n–1 – 1)‐neighborhood of y. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reflection of Long Game Formulas

We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II¹₁ reflection (or ∑₁-compactness). We show that admissible sets such as H(ω₂) and L_ω2 which fail to have strict-II¹₁ reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms. Mathematics Subject Classification : 03C70, 03C75.     

Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω₂. This implies that ω₁ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω₂. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Proper and piecewise proper families of reals

I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ ω₁. Here, I investigate the question of the existence of proper and piecewise proper families of reals of different cardinalities. I show that it is consistent relative to ZFC to have continuum many proper families of cardinality ω₁ and continuum many piecewise proper families of cardinality ω₂ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous Linear Extension of Ultradifferentiable Functions of Beurling Type

For a weight function ω and a closed set A ? ?^N let ?_(ω)(A) denote the space of all ω-Whitney jets of Beurling type on A. It is shown that for each closed set A ? ?^N there exists an ω-extension operator EA: ?_(ω)(A) → ?_(ω)(?^N) if and only if ω is a (DN)-function (see MEISE and TAYLOR [18], 3.3). Moreover for a fixed compact set K ? ?^N there exists an ω-extension operator E_K: ?_(ω)(K) → ?_(ω)(?^N) if and only if the Fréchet space ?_(ω)(K) satisfies the property (DN) (see Vogt [29], 1.1.).     

A Modeltheoretic Solution to a Problem of Tarski

Let 1 < n < ω and β > n. We show that the class Nr_nCA_β of n‐dimensional neat reducts of β‐dimensional cylindric algebras is not closed under forming elementary subalgebras. This solves a long‐standing open problem of Tarski and his co‐authors Andréka, Henkin, Monk and Németi. The proof uses genuine model‐theoretic arguments.     

Spaces which contain a copy of Sω or S2 and their applications

Let S_ω and S₂ denote the sequential fan and Arens' space, respectively. In this paper, we shall prove the following main results. (1) If Π^∞_i=1 X_i contains a copy of S_ω (S₂), then some Πⁿ_i=1 X_i contains a copy of S_ω (S_ω or S₂, respectively). (2) Let f: X → Y be a closed map such that any f^-1(y) contains no closed copy of S_ω (S₂). If X contains a closed copy of S_ω (S₂), then Y contains a closed copy of S_ω (S_ω or S₂, respectively).As applications of (1) and (2), we shall consider the Fréchet or strongly Fréchetness, or sequentiality of products of finitely or countably many spaces.     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cut‐elimination Theorems for Some Infinitary Modal Logics

In this article, a cut‐free system TLM_ω1 for infinitary propositional modal logic is proposed which is complete with respect to the class of all Kripke frames.The system TLM_ω1 is a kind of Gentzen style sequent calculus, but a sequent of TLM_ω1 is defined as a finite tree of sequents in a standard sense. We prove the cut‐elimination theorem for TLM_ω1 via its Kripke completeness.     

Criteria for farthest points on convex surfaces

We provide a sharp, sufficient condition to decide if a point y on a convex surface S is a farthest point (i.e., is at maximal intrinsic distance from some point) on S, involving a lower bound π on the total curvature ω_y at y, ω_y ≥ π. Further consequences are obtained when ω_y > π, and sufficient conditions are derived to guarantee that a convex cap contains at least one farthest point. A connection between simple closed quasigeodesics O of S, points y ∈ S\O with ω_y > π, and the set ?? of all farthest points on S, is also investigated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

Retracts of box products with odd‐angulated factors

Let G be a connected graph with odd girth 2κ+1. Then G is a (2κ+1)‐angulated graph if every two vertices of G are connected by a path such that each edge of the path is in some (2κ+1)‐cycle. We prove that if G is (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+3, then any retract of the box (or Cartesian) product G□ H is S□T where S is a retract of G and T is a connected subgraph of H. A graph G is strongly (2κ+1)‐angulated if any two vertices of G are connected by a sequence of (2κ+1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+1, then any retract of G□ H is S□T where S is a retract of G and T is a connected subgraph of H or |V(S)|=1 and T is a retract of H. These two results improve theorems on weakly and strongly triangulated graphs by Nowakowski and Rival [Disc Math 70 ( 13 ), 169–184]. As a corollary, we get that the core of the box product of two strongly (2κ+1)‐angulated cores must be either one of the factors or the box product itself. Furthermore, if G is a strongly (2κ+1)‐angulated core, then either Gⁿ is a core for all positive integers n, or the core of Gⁿ is G for all positive integers n. In the latter case, G is homomorphically equivalent to a normal Cayley graph [Larose, Laviolette, Tardiff, European J Combin 19 ( 12 ), 867–881]. In particular, let G be a strongly (2κ+1)‐angulated core such that either G is not vertex‐transitive, or G is vertex‐transitive and any two maximum independent sets have non‐empty intersection. Then Gⁿ is a core for any positive integer n. On the other hand, let G_i be a (2κ_i+1)‐angulated core for 1 ≤ i≤ n where κ₁ < κ₂ < … < κ_n. If G_i has a vertex that is fixed under any automorphism for 1 ≤ i ≤ n‐1, or G_i is vertex‐transitive such that any two maximum independent sets have non‐empty intersection for 1 ≤ i ≤ n‐1, then □_i=1ⁿ G_i is a core. We then apply the results to construct cores that are box products with Mycielski construction factors or with odd graph factors. We also show that K(r,2r+1) □ C_2l+1 is a core for any integers l ≥ r ≥ 2. It is open whether K(r,2r+1) □ C_2l+1 is a core for r > l ≥ 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

The alternative Daugavet property of C *‐algebras and JB *‐triples

A Banach space X is said to have the alternative Daugavet property if for every (bounded and linear) rank‐one operator T: X → X there exists a modulus one scalar ω such that ∥Id+ωT ∥ = 1 + ∥T ∥. We give geometric characterizations of this property in the setting of C *‐algebras, JB *‐triples, and of their isometric preduals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The strong partition relation on ω1 revisited

We give a new proof of the strong partition relation on ω₁, assuming the axiom of determinacy, which uses only a general argument not involving the complete analysis of a measure on ω₁. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ω‐categorical weakly o‐minimal expansions of Boolean lattices

We study ω‐categorical weakly o‐minimal expansions of Boolean lattices. We show that a structure ?? = (A,≤, ?) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o‐minimal if and only if it is ω‐categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ?. We propose other related examples of weakly o‐minimal ω‐categorical models in this framework, and we examine the internal structure of these models.     

On neat embeddings of cylindric algebras

We show that certain properties of dimension complemented cylindric algebras, concerning neat embeddings, do not generalize much further. Let α ≥ ω. There are non‐isomorphic representable cylindric algebras of dimension α each of which is a generating subreduct of the same β dimensional cylindric algebra. We also show that there exists a representable cylindric algebra ?? of dimension α, such that ?? is a generating subreduct of ?? and ??′, both in CA_{α +ω} , however ?? and ??′ are not isomorphic. This settle questions raised by Henkin, Monk and Tarski (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)