Polyhedrality in Orlicz spaces

We present a construction of an Orlicz space admitting a C ^∞-smooth bump which depends locally on finitely many coordinates, and which is not isomorphic to a subspace of any C(K), K scattered. In view of the related results this space is possibly not isomorphic to a polyhedral space. Supported by grants: Institutional Research Plan AV0Z10190503, GAČR 201/04/0090, GAČR 201/07/0394, the research project MSM 0021620839, GAČR 201/05/P582.     

Locally flat Banach spaces

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher order smoothness is involved. In this note we investigate the structural properties of Banach spaces admitting (arbitrary) bump functions depending locally on finitely many coordinates. This work was supported by the research project MSM 0021620839 and by the grant of the Grant Agency of the Czech Republic No. 201/05/P582 and No. 201/06/0018.     

The group of isometries of a locally compact metric space with one end

In this note we study the dynamics of the natural evaluation action of the group of isometries G of a locally compact metric space (X,d) with one end. Using the notion of pseudo-components introduced by S. Gao and A.S. Kechris we show that X has only finitely many pseudo-components exactly one of which is not compact and G acts properly on this pseudo-component. The complement of the non-compact component is a compact subset of X and G may fail to act properly on it.     

On the recognition problem of quasi-homogeneous locally nilpotent derivations in dimension three

We give a criterion to decide if a given w-homogeneous derivation on A?k[X₁,X₂,X₃] is locally nilpotent. We deduce an algorithm which decides if a k-subalgebra of A, which is finitely generated by w-homogeneous elements, is the kernel of some locally nilpotent derivation.     

Counting Cocomponents of a Topological Space

Dualizing the statement about a number of components of a topological space X, we say that, for a natural number n, the space X has at most n cocomponents if every continuous map f:X ⁿ⁺¹X factorizes through X ⁿ , i.e., f depends on at most n coordinates. We construct metrizable spaces X ₁,X ₂,X ₃ such that(1) X ₁ does not have finitely many cocomponents but every continuous map X ₁ X ₁ depends only on finitely many coordinates;(2) X ₂=A×B for rigid spaces A,B and there is a continuous map X ₂ X ₂ depending on all coordinates;(3) X ₃(n) has precisely n cocomponents but it cannot be expressed as A×B with |A|>1 and |B|>1.     

Decidability of Equational Theories of Coverings of Semigroup Varieties

For every proper semigroup variety X, there exists a semigroup variety Y satisfying the following three conditions: (1) Y covers X, (2) if X is finitely based then so is Y, and (3) the equational theory of X is decidable if and only if so is the equational theory of Y. If X is an arbitrary semigroup variety defined by identities depending on finitely many variables and such that all periodic groups of X are locally finite, then one of the following two conditions holds: (1) all nilsemigroups of X are locally finite and (2) X includes a subvariety Y whose equational theory is undecidable and which has infinitely many covering varieties with undecidable equational theories.     

Graph with given achromatic number

An irreducible (point-determining) graph is one in which distinct vertices have distinct neighbourhoods. Every graph X can be reduced to an irreducible graph X¹ by identifying all vertices with the same neighbourhood; the colourability properties of X¹ carry over to X. Hence irreducible graphs are instrumental in the study of achromatic number. We prove that there are only finitely many irreducible graphs with a given achromatic number, and describe all graphs with achromatic number less than four. We deduce certain bounds on the achromatic number X in terms of the number of vertices of X¹. In the course of the proofs we calculate the achromatic numbers of paths and cycles. Generalizations of the main theorem to homomorphisms other than colourings are discussed.     

The Beilinson equivalence for differential operators

Let D be the ring of differential operators on a smooth irreducible affine variety X over C, or, more generally, the enveloping algebra of any locally free Lie algebroid on X. The category of finitely generated graded modules of the Rees algebra has a natural quotient category P_D which imitates the category of modules on Proj of a graded commutative ring. We show that the derived category D^b(P_D) is equivalent to the derived category of finitely generated modules of a sheaf of algebras E on X which is coherent over X. This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins in [6] to describe the moduli space of left ideals in D.     

Topological complexity, fibrations and symmetry

We show how locally smooth actions of compact Lie groups on a manifold X can be used to obtain new upper bounds for the topological complexity TC(X), in the sense of Farber. We also obtain new bounds for the topological complexity of finitely generated torsion-free nilpotent groups.     

Poisson Traces and D-Modules on Poisson Varieties

To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism ${\phi : X \to Y}To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism f: X ? Y{\phi : X \to Y} and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M_f(X,N){M_\phi(X,N)} on X, and prove that it is holonomic if X has finitely many symplectic leaves, f{\phi} is finite, and N is coherent.     

Torsion points on families of squares of elliptic curves

In a recent paper we proved that there are at most finitely many complex numbers λ ≠  0,1 such that the points \({(2,\sqrt{2(2-\lambda)})}\) and \({(3, \sqrt{6(3-\lambda)})}\) are both torsion on the elliptic curve defined by Y ² = X(X ? 1)(X ? λ). Here we give a generalization to any two points with coordinates algebraic over the field Q(λ) and even over C(λ). This implies a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes.     

On extendibility of unavoidable sets

A subset X of a free monoid A¹ is said to be unavoidable if all but finitely many words in A¹ contain some word of X as a subword. A. Ehrenfeucht has conjectured that every unavoidable set X is extendible in the sense that there exist x ? X and a ? A such that (X ? {x}) ∪ {xa} is itself unavoidable. This problem remains open, we give some partial solutions and show how to efficiently test unavoidability, extendibility and other properties of X related to the problem.     

Cut points in some connected topological spaces

We prove that a connected topological space with endpoints has exactly two non-cut points and every cut point is a strong cut point; it follows that such a space is a COTS and the only two non-cut points turn out to be endpoints (in each of the two orders) of the COTS. A non-indiscrete connected topological space with exactly two non-cut points and having only finitely many closed points is proved homeomorphic to a finite subspace of the Khalimsky line. Further, it is shown, without assuming any separation axiom, that in a connected and locally connected topological space X, for a, b in X, S[a,b] is compact whenever it is closed. Using this result we show that an H(i) connected and locally connected topological space with exactly two non-cut points is a compact COTS with end points.     

Commutative Algebras In Which Polynomials Have Infinitely Many Roots

Let K be a field. Then there exists a commutative K-algebra A such that each polynomial in K[X] of degree at least 2 has infinitely many roots in A. If B is a finite-dimensional commutative K-algebra and char(K) ≠ 3 (resp., char(K ) = 3), then X ² + X + 1 (resp., X ² + X-1) has only finitely many roots in B. Relevant examples are also given, especially of K-algebras of the form K + N, where N is the nilradical.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

K-flows are forward deterministic,backward completely non-deterministic stationary point processes

Stationary processes with discrete time parameter and finitely many states are forward deterministic if and only if they are backward deterministic. In contrast to this we prove in the case of continuous time parameter: Every K-flow in a Lebesgue space is isomorphic to the flow of shifts of a stationary forward deterministic, backward completely non-deterministic process (X_t, t?R¹) with two states and with right-continuous paths having only finitely many jumps in any finite time-interval. The process may be considered as a point process. The result is obtained from a representation theorem for flows, describing increasing sub-σ-algebras of a flow up to “equivalence.”     

Finitely Presented Wreath Products and Double Coset Decompositions

We characterize which permutational wreath products $G \ltimes W^{(X)}We characterize which permutational wreath products are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X ². On the one hand, this extends a result of G. Baumslag about infinite presentation of standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.      

Lifting bounded approximation properties from Banach spaces to their dual spaces

Based on a new reformulation of the bounded approximation property, we develop a unified approach to the lifting of bounded approximation properties from a Banach space X to its dual X^*. This encompasses cases when X has the unique extension property or X is extendably locally reflexive. In particular, it is shown that the unique extension property of X permits to lift the metric A-approximation property from X to X^*, for any operator ideal A, and that there exists a Banach space X such that X,X**,… are extendably locally reflexive, but X^*,X***,… are not.     

On p-adic vector measure spaces

For a separating algebra R of subsets of a set X, E a complete Hausdorff non-Archimedean locally convex space and m:R→E a bounded finitely additive measure, we study some of the properties of the integrals with respect to m of scalar-valued functions on X. The concepts of convergence in measure, with respect to m, and of m-measurable functions are introduced and several results concerning these notions are given.     

A metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI

We show that a minimal metric flowX whose enveloping semigroupE(X), contains finitely many minimal ideals, is a PI-flow; i.e.X has a proximal extensionX' which can be built by iterating proximal and isometric extensions (starting with the trivial one point flow). An example is given which shows that the converse theorem does not hold. Finally, we show that ifX is a minimal, non-trivial, metric, weakly mixing flow and the group action is nilpotent thenE(X) contains infinitely many minimal ideals.