Minimal submanifolds in ℝ<Superscript>4</Superscript> with a G.C.K. structure

In this paper we obtain all invariant, anti-invariant and CR submanifolds in (?⁴, g, J) endowed with a globally conformal Kähler structure which are minimal and tangent or normal to the Lee vector field of the g.c.K. structure.     

Surfaces with constant curvature in S<Superscript>2</Superscript>×R and H<Superscript>2</Superscript>×R. Height estimates and representation

We obtain optimal height estimates for surfaces in ℍ² × ℝ and × ℝ with constant Gaussian curvature K(I) and positive extrinsic curvature, characterizing the extreme cases as the revolution ones. Moreover, we get a representation for surfaces with constant Gaussian curvature in such ambient spaces, paying special attention to the cases of K(I) = 1 in × ℝ and K(I) = −1 in ℍ² × ℝ. The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant No. PAI-05-034. The authors are partially supported by MEC-FEDER, Grant No. MTM2007-65249.     

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.     

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.     

Isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S<Superscript>3</Superscript> × S<Superscript>3</Superscript>

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S³ × S³ is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((?h)(v, v, v), Jv) = λ holds for all unit tangent vector v.     

On the F. and M. Riesz theorem on wedges with edges of class <Emphasis Type="Italic">C</Emphasis><Superscript>1,α</Superscript>

This work extends the classical F. and M. Riesz theorem to measures that are boundary values of holomorphic functions defined on wedges in with edges that are in the class C ^1,α.This work was supported in part by NSF INT-0203005, CNPq and FAPESP     

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.     

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).     

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.     

Polysubharmonic Functions Near Infinity in ℜ<Superscript>n</Superscript>

For polyharmonic functions u of finite order defined outside a compact set in ⁿ, representations by means of -subharmonic functions defined on the whole of ⁿ are given. As a consequence, certain analogues of the classical Bôcher theorem are proved for u. Some of these results are extended to the case of polysubharmonic functions defined near infinity in ⁿ. Mathematics Subject Classification (2000) 31B30.     

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.     

Homogeneous Hypersurfaces in ℂ<Superscript>3</Superscript>, Associated with a Model CR-Cubic

The model 4-dimensional CR-cubic in ℂ³ has the following “model” property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in ℂ³ with finite-dimensional infinitesimal automorphism algebra \mathfrakg\mathfrak{g} and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all \mathfrakg\mathfrak{g}-homogeneous hypersurfaces in ℂ³ and also classify the corresponding local transitive actions of the model algebra \mathfrakg\mathfrak{g} on hypersurfaces in ℂ³.     

On nice equivalence relations on <Superscript>λ</Superscript>2

Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets. I would like to thank Alice Leonhardt for the beautiful typing.This research was supported by The Israel Science Foundation. Publication 724. Mathematics Subject Classification (2000): 03E47, 03E35; 20K20, 20K35     

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.     

A trajectory in ℝ<Superscript>3</Superscript> concealed from observers

In the problem of tracking an object moving in ?³ by observers, the most concealed trajectory is characterized under the condition that the object is at any time visible to at most two observers.     

A long ℂ<Superscript>2</Superscript> which is not Stein

We construct a 2-dimensional complex manifold X which is the increasing union of proper subdomains that are biholomorphic to ℂ², but X is not Stein.     

Weighted inequalities for real-analytic functions in ℝ<Superscript>2</Superscript>

Let ƒ and g be real-analytic functions near the origin in ℝ². Given 1 < p < ∞, we obtain a characterization of the set of positive numbers ∈ and δ that ensures