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     

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.      

Accuracy of approximation in the Poisson theorem in terms of the χ<Superscript>2</Superscript>-distance

We study the limit behavior of the χ²-distance between the distributions of the nth partial sum of independent not necessarily identically distributed Bernoulli random variables and the accompanying Poisson law. As a consequence in the i.i.d. case we make the multiplicative constant preciser in the available upper bound for the rate of convergence in the Poisson limit theorem.     

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.     

The hypertree poset and the ℓ<Superscript>2</Superscript>-Betti numbers of the motion group of the trivial link

We give explicit formulae for the Euler characteristic and ²-cohomology of the group of motions of the trivial link, or isomorphically the group of free group automorphisms that send each standard generator to a conjugate of itself. The method is primarily combinatorial and ultimately relies on a computation of the Möbius function for the poset of labelled hypertrees.Partially supported by NSF grant no. DMS-0101506Partially supported by an AMS Centennial Research Fellowship     

On the complexity of the family of convex sets in ℝ<Superscript> <Emphasis Type="Italic">d</Emphasis> </Superscript>

Estimates of quantities characterizing the complexity of the family of convex subsets of the d-dimensional cube [1, n]^d as n→∞ are given. The geometric properties of spaces with norm generated by the generalized majorant of partial sums are studied.     

Elementary transformations and the rationality of the Moduli Spaces of Vector Bundles on ℙ<Superscript>2</Superscript>

In this work, using elementary transformations and prioritary sheaves, we establish birational maps between certain moduli spaces of stable vector bundles over ² with the same rank and different Chern classes. As an application we give a simple proof of the rationality of the moduli spaces M(r; c ₁, c ₂) of rank r stable vector bundles over ² with given Chern classes for a huge families of the triples (r; c ₁, c ₂).Partially supported by BFM2001-3584 Mathematics Subject Classification (2000):Primary 14D20, 14D05; Secondary 14F05     

On the definition of the Monge-Ampère operator in ℂ<Superscript>2</Superscript>

We show that if u is a plurisubharmonic function defined on an open subset of ² then the Monge-Ampère measure (dd^cu)² can be well defined if and only if u belongs to the Sobolev space W^1,2_loc().Partially supported by KBN Grant #2 P03A 028 19     

Exact Calculation of Sums of the Lorentz Spaces Λ<Superscript>α</Superscript> and Applications

The norm on the sum of Lorentz spaces endowed with norms equal to the products of the classical norm by some numbers is exactly calculated. The obtained result makes it possible to prove an extrapolation theorem for collections of Lorentz, Lebesgue, and Marcinkiewicz spaces with a sharp constant.     

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.     

The narrow class groups of the ℤ<Subscript>17</Subscript>- and ℤ<Subscript>19</Subscript>-extensions over the rational field

Let p be either 17 or 19, let ℤ _p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p ². We shall prove, with the help of a computer, that the l-class group of the ℤ _p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤ _p -extension.     

On the measurability of the conic sections’ family in the projective space<Emphasis FontCategory="NonProportional">P</Emphasis><Subscript>3</Subscript><Subscript>3</Subscript>

In this paper we prove that the non degenerate conic sections’ family in the projective spaceP ₃ is measurable.     

On the heritability of the property <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">π</Emphasis></Subscript> by subgroups

We consider the so-called Jordan-Pochhammer systems, a special class of linear Pfaffian systems of Fuchsian type on complex linear (or projective) spaces. These systems appeared as systems of differential equations for hypergeometric type integrals in which the integrand is a product of powers of linear functions. These systems also arise in some reductions of the Knizhnik-Zamolodchikov equations. The main advantage of these systems is the possibility of presenting a basis in the solution space of such systems in an explicit integral form and, as a consequence, of describing their monodromy representation. The main focus in the paper is placed on the applications of Jordan-Pochhammer systems. We describe the relationship of Jordan-Pochhammer systems to isomonodromic deformations of Fuchsian systems that are described by the Schlesinger equations, as well as to the linearization of the dynamical system of bending spatial polygons. We also describe the application of Jordan-Pochhammer systems to constructing Kohno systems on the Manin-Schechtman configuration spaces.     

Some properties of the <Emphasis Type="Italic">ɛ</Emphasis><Subscript>∞</Subscript>-product of quotient bornological spaces

Some properties of the ? _∞-product defined in [4] are obtained by a study of a kind of isomorphism between the computation of this ? _∞-product and the ordinary ?-product of L. Schwartz [9]. The paper contains several corollaries.     

Fibers of the <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> algebra and disintegration of measures

It is shown that Gelfand transforms of elements \({f\in L^{\infty} (\mu)}\) are almost constant at almost every fiber \({\Pi^{-1}(\{x\})}\) of the spectrum of L ^∞(μ) in the following sense: for each \({f\in L^{\infty} (\mu)}\) there is an open dense subset U = U(f) of this spectrum having full measure and such that the Gelfand transform of f is constant on the intersection \({\Pi^{-1}(\{x\})\cap U}\). As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption.     

<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.     

Poincaré series of the multigraded algebras of <Emphasis Type="Italic">SL</Emphasis><Subscript>2</Subscript>-invariants

We deduce formulas for finding the Poincaré multiseries P( C_d,z₁,z₂, ?, z_n,t ) \mathcal{P}\left( {{\mathcal{C}_d},{z_1},{z_2}, \ldots, {z_n},t} \right) and P( I_d,z₁,z₂, ?, z_n ) \mathcal{P}\left( {{\mathcal{I}_d},{z_1},{z_2}, \ldots, {z_n}} \right) , where C_d {\mathcal{C}_d} and I_d {\mathcal{I}_d} , d = (d ₁, d ₂, . . . , d _n ), are multigraded algebras of joint covariants and joint invariants for n binary forms of degrees d ₁, d ₂, . . . , d _n .     

Approximation in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> by Partial Integrals of the Multidimensional Fourier Transform over the Eigenfunctions of the Sturm–Liouville Operator

For approximations in the space L₂(?₊ ^d ) by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove the Jackson inequality with sharp constant and optimal argument in the modulus of continuity. The multidimensional weight that defines the Sturm–Liouville operator is the product of onedimensional weights. The one-dimensional weights can be, in particular, power and hyperbolic weights with various parameters. The optimality of the argument in the modulus of continuity is established by means of the multidimensional Gauss quadrature formula over zeros of an eigenfunction of the Sturm–Liouville operator. The obtained results are complete; they generalize a number of known results.     

Approximation in <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator

For approximations in the space L₂(?₊) by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove Jackson’s inequality with exact constant and optimal argument in the modulus of continuity. The optimality of the argument in the modulus of continuity is established using the Gauss quadrature formula on the half-line over the zeros of the eigenfunction of the Sturm–Liouville operator.     

Comparing the <Emphasis Type="Italic">ℤ</Emphasis><Subscript>2</Subscript>-Graded Identities of Two Minimal Superalgebras with the Same Superexponent

Let F be a field of characteristic zero. We study two minimal superalgebras A and B having the same superexponent but such that T ₂ (A) ? T ₂ (B), thus providing the first example of a minimal superalgebra generating a non minimal supervariety. We compare the structures and codimension sequences of A and B.