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.     

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     

Weierstrass representation for discrete isotropic surfaces in ℝ<Superscript>2,1</Superscript>, ℝ<Superscript>3,1</Superscript>, and ℝ<Superscript>2,2</Superscript>

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation for hyperbolic surfaces parameterized along isotropic directions in ℝ^2,1, ℝ^3,1, and ℝ^2,2. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.     

Projective Multi-Resolution Analyses for L<Superscript>2</Superscript>(ℝ<Superscript>2</Superscript>)

We define the notion of projective multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra C(Tⁿ) of continuous complex-valued functions on an n-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for L²(ⁿ). Then we show how to construct examples for the case of any diagonal 2 × 2 dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective C(T²)-module. We compute the isomorphism classes of the corresponding wavelet modules.     

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.     

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.     

Quincunx multiresolution analysis for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℚ<Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.     

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.     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).     

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 .     

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.     

Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

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.     

Lorentzian Sasaki-Einstein metrics on connected sums of <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We settle completely an open problem formulated by Boyer and Galicki in [5] which asks whether or not #kS ² × S ³ are negative Sasakian manifolds for all k. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki η-Einstein and Lorentzian Sasaki-Einstein metrics on these five-manifolds for all k and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces in weighted projective space that contain branch divisors \({\Delta=\sum_{i}\left(1-\frac{1}{m_{i}}\right)D_i}\) such that the D _i are rational curves.     

Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

Irregular wavelet frames on<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript>n</Superscript>)

In this paper, we present the conditions on dilation parameter {s _j}_j that ensure a discrete irregular wavelet system {s _j ^n/2ψ(s _j ·−bk)} _{j∈ℤ,k∈ℤ} ⁿ to be a frame on L²(ℝⁿ), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.     

<Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript> regularity of solutions of the Levi equation in<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">2n+1</Emphasis></Superscript>

We study the regularity of the solutions of the Levi equation in ?²ⁿ⁺¹. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C ². We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC ^∞ regularity of the solutions.