A criterion for the <Emphasis Type="Italic">F</Emphasis> <Subscript> <Emphasis Type="Italic">π</Emphasis> </Subscript>-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks

Let G be the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroup H, H ≠ A and H ≠ B. Suppose that, for some set π of primes, the groups A and B are residually F_π, where F_π is the class of all finite p-groups. We prove that G is residually F_π if and only if H is F_π-separable in A and B.     

Interpreting arithmetics in the ideal lattice of a free vector lattice ℱ<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A vector space V over a real field R is a lattice under some partial order, which is referred to as a vector lattice if u + (v ∨ w) = (u + v) ∨ (u + w) and u + (v ∧ w) = (u + v) ∧ (u + w) for all u, v, w ∈ V. It is proved that a model N of positive integers with addition and multiplications is relatively elementarily interpreted in the ideal lattice ℱ _n of a free vector lattice ℱ _n on a set of n generators. This, in view of the fact that an elementary theory for N is hereditarily undecidable, implies that an elementary theory for ℱ _n is also hereditarily undecidable. __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 71–82, January–February, 2008.     

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

The Δ<Superscript>0</Superscript><Subscript><Emphasis Type="Italic">α</Emphasis></Subscript>-Computable Enumerations of the Classes of Projective Planes

Studying computable representations of projective planes, for the classes K of pappian, desarguesian, and all projective planes, we prove that K ^c /? admits no hyperarithmetical Friedberg enumeration and admits a Friedberg Δ⁰_α+3-computable enumeration up to a Δ⁰ _α -computable isomorphism.     

Non-<Emphasis Type="Italic">φ</Emphasis>-admissible normal subgroups of free burnside groups

In the present paper for arbitrary automorphism φ of the free Bunside group B(m, n) and for any odd number n ≥ 1003 a sufficient condition for existence of non-φ-admissible normal subgroup of B(m, n) was found. In particular, if automorphism φ is normal, then for any basis {a ₁, a ₂, …, a _m } of the group B(m, n) there is an integer k such that for each i the elements a _i and φ(a i) ^k are conjugates.     

Seas of squares with sizes from a Π<Stack><Subscript>1</Subscript><Superscript>0</Superscript></Stack> set

A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+o(1))( _n/2 ⁿ ). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number α(n) of such families, and they proved that \({2^{\left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)}} \leqslant \alpha \left( n \right) \leqslant {2^{2\sqrt 2 \left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)\left( {1 + o\left( 1 \right)} \right)}}\) They conjectured that the constant \(2\sqrt 2 \) can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.     

Local structure of Rogers semilattices of Σ<Subscript>n</Subscript><Superscript>0</Superscript>-computable numberings

We deal in specific features of the algebraic structure of Rogers semilattices of _n ⁰-computable numberings, for n 2. It is proved that any Lachlan semilattice is embeddable (as an ideal) in such every semilattice, and that over an arbitrary non 0-principal element of such a lattice, any Lachlan semilattice is embeddable (as an interval) in it.Supported by INTAS grant No. 00-499, by FP Universities of Russia grant UR.04.01.013, and by the Grant Center for Fundamental Research (GCFR), project PD02-1.1-475.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 148–172, March–April, 2005.     

A blend of methods of recursion theory and topology: A Π<Subscript>1</Subscript><Superscript>0</Superscript> tree of shadow points

This paper is a sequel to our [7]. In that paper we constructed a ₁⁰ tree of avoidable points. Here we construct a ₁⁰ tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2^<. One interesting fact is that while our ₁⁰ tree of sharp filters (a subtree of ^<) is isomorphic to the tree of addresses (a subtree of 2^<), the tree of addresses is recursively enumerable but not recursive. To achieve this end we use a finite injury priority argument.Mathematics Subject Classification (2000): 03D45, 03D80, 03C57, 54A20     

Biflatness of ℓ<Superscript>1</Superscript>-Semilattice Algebras

We show that if L is a semilattice then the ℓ¹-convolution algebra of L is biflat precisely when L is "uniformly locally finite". Our proof technique shows in passing that if this convolution algebra is biflat then it is isomorphic as a Banach algebra to the Banach space ℓ¹(L) equipped with pointwise multiplication. At the end we sketch how these techniques may be extended to prove an analogous characterisation of biflatness for Clifford semigroup algebras.     

Optimal boundary control by a displacement in <Emphasis Type="Italic">W</Emphasis>′<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> of a string with free end

We consider an optimal boundary control of a string with free end by a displacement of the other end in W′ _p (Q, T). For p ≠ 2, we prove that the optimal control depends on the initial and terminal conditions nonlinearly.     

<Emphasis Type="Italic">OD</Emphasis>-Characterization of the automorphism groups of <Emphasis Type="Italic">O</Emphasis><Stack><Subscript>10</Subscript><Superscript>±</Superscript></Stack>(2)

The prime graph of a finite group was introduced by Gruenberg and Kegel. The degree pattern of a finite group G associated to its prime graph was introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Till now a lot of finite simple groups were shown to be OD-characterizable, and also some finite groups especially the automorphism groups of some finite simple groups were shown not being OD-characterizable but k-fold OD-characterizable for some k > 1. In the present paper, the authors continue this topic and show that the automorphism groups of orthogonal groups O ₁₀⁺(2) and O ₁₀⁻(2) are OD-characterizable.     

On the <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript> limits of subsets of the real line

We point out that it is consistent with ZFC that 2 ^ω > ℵ₁ and every subset of ℝ is the ω ₁ limit of a sequence of G _δ sets in ℝ. We prove also that assuming cov ( ) > ℵ₁, not every set in ℝ is the ω ₁ limit of a sequence of measurable sets. This solves two problems of T. Natkaniec and J. Wesołowska.      

Dual Complexes of Cubical Subdivisions of ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We use a distortion to define the dual complex of a cubical subdivision of ℝ ⁿ as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated by the topological analysis of high-dimensional digital image data, we consider such subdivisions defined by generalizations of quad- and oct-trees to n dimensions. Assuming the subdivision is balanced, we show that mapping each vertex to the center of the corresponding n-cube gives a geometric realization of the dual complex in ℝ ⁿ .     

On complexity reduction of Σ<Subscript>1</Subscript> formulas

 For a fixed q  ℕ and a given Σ₁ definition φ(d,x), where d is a parameter, we construct a model M of 1 Δ₀ + ? exp and a non standard d  M such that in M either φ has no witness smaller than d or phgr; is equivalent to a formula ϕ(d,x) having no more than q alternations of blocks of quantifiers. Received: 29 September 1998 / Revised version: 7 November 2001 Published online: 10 October 2002 RID="⋆" ID="⋆" Research supported in part by The State Committee for Scientific Research (Poland), KBN, grant number 2 PO3A 018 13. RID="⋆" ID="⋆" Research supported in part by The State Committee for Scientific Research (Poland), KBN, grant number 2 PO3A 018 13.     

Embeddings into the Medvedev and Muchnik lattices of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

Let _w and _M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty ₁⁰ subsets of 2, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of _w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of _M.Simpsons research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.     

Factoriality of complete intersections in ℙ<Superscript>5</Superscript>

Let X be a complete intersection of two hypersurfaces F _n and F _k in ℙ⁵ of degree n and k, respectively, with n ≥ k, such that the singularities of X are nodal and F _k is smooth. We prove that if the threefold X has at most (n + k − 2)(n − 1) − 1 singular points, then it is factorial.     

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.     

The number of S-precomplete classes in the functional system <Emphasis Type="Italic">P</Emphasis><Stack><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript><Superscript><Emphasis Type="Italic">τ</Emphasis></Superscript></Stack>

The problem on the number of precomplete classes in the functional system P _κ ^τ is considered, elements of P _κ ^τ are deterministic S-functions defined on words of length τ composed from letters of an alphabet of cardinality κ. An asymptotics for the number of S-precomplete classes in P _κ ^τ is obtained for arbitrary fixed κ and τ tending to infinity.