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.     

Constant angle surfaces in ℍ<Superscript>2</Superscript> × ℝ

We classify all surfaces in ℍ² × ℝ for which the unit normal makes a constant angle with the ℝ-direction. Here ℍ² is the hyperbolic plane. The author was supported by grants CEEX ET 5883/2006-2008 and PNII ID_ 398/2007-2010 ANCS (Romania).     

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.     

“Minimal Surfaces in ℍ<Superscript>2</Superscript> × ℝ”

In ² × ℝ one has catenoids, helicoids and Scherk-type surfaces. A Jenkins-Serrin type theorem holds here. Moreover there exist complete minimal graphs in ² with arbitrary continuous asymptotic values. Finally, a graph on a domain of ² cannot have an isolated singularity. Received: 20 June 2002     

Analytic properties for holomorphic matrix-valued maps in ℂ<Superscript>2×2</Superscript>

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on C^2×2 and Julia set of one dimensional complex dynamic system.     

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.     

Tight Lagrangian surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

We determine all tight Lagrangian surfaces in S ² × S ². In particular, globally tight Lagrangian surfaces in S ² × S ² are nothing but real forms of this symmetric space.     

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}.     

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).     

A Jenkins-Serrin theorem in <Emphasis Type="Italic">M</Emphasis><Superscript>2</Superscript> × ℝ

In this paper we study minimal surfaces in M × ℝ, where M is a complete surface. Our main result is a Jenkins-Serrin type theorem which establishes necessary and sufficient conditions for the existence of certain minimal vertical graphs in M × ℝ. We also prove that there exists a unique solution of the Plateau’s problem in M × ℝ whoseboundaryisaNitschegraphandweconstructaScherk-typesurfaceinthisspace. Thanks to CNPq Agency for financial support.     

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.     

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.     

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     

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.     

Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

A Hilbert-type theorem for spacelike surfaces with constant Gaussian curvature in ℍ<Superscript>2</Superscript> × ℝ<Subscript>1</Subscript>

There are examples of complete spacelike surfaces in the Lorentzian product ℍ² × ℝ₁ with constant Gaussian curvature K ≤ −1. In this paper, we show that there exists no complete spacelike surface in ℍ² × ℝ₁ with constant Gaussian curvature K > −1.     

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.     

A Ring Theoretic Construction of Hadamard Difference Sets in ℤ<Subscript>8</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let be the Galois ring of characteristic 2³ and rank n and let . We give an explicit construction of Hadamard difference sets in .}Research supported by NSA grant MDA 904-02-1-0080.     

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