Type G Distributions on {\mathbb{R}} d

This paper presents a systematic study of the class of multivariate distributions obtained by a Gaussian randomization of jumps of a Lévy process. This class, called the class of type G distributions, constitutes a closed convolution semigroup of the family of symmetric infinitely divisible probability measures. Spectral form of Lévy measures of type G distributions is obtained and it is shown that type G property can not be determined by one dimensional projections. Conditionally Gaussian structure of type G random vectors is exhibited via series representations.     

Compactness and local compactness for {\mathbb{R}}-trees

We establish (geometric) criteria for an -tree to be compact and to be locally compact. It follows that locally compact -trees are separable. Received: 10 September 2007     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Geodesic Curves on {\mathbb{R}}-Complex Finsler Spaces

The main propose of this paper is to investigate the geodesic curves on a strongly convex \({\mathbb{R}}\)-complex Finsler space (M, F). We survey the first variation of the length integral associated to F and use this to give the equation of geodesic curves on such spaces. We prove the local existence and uniqueness of geodesic curves, under the weakly Kähler assumption. As an application, we characterize the critical points of the displacement function of a holomorphic isometry on M.     

Non-harmonic Cones are Heisenberg Uniqueness Pairs for the Fourier Transform on $${\mathbb {R}}^n$$

In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on \({\mathbb {R}}^n\). We derive that \(\left( S^2, \text { paraboloid}\right) \) and \(\left( S^2, \text { geodesic of } S_r(o)\right) \) are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in \({\mathbb {R}}^3\). Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

Maximal q-Plurisubharmonic Functions in {\mathbb{C}^{n}}

In this paper we will study maximal q-plurisubharmonic functions in ${\mathbb{C}^n}$ . At the same time, we define a notion above weakly q-plurisubharmonic functions and describe the relation between these functions and maximal q-plurisubharmonic functions.     

Invariant surfaces with constant mean curvature in $${\mathbb{H}}^2 \times {\mathbb{R}}$$

We classify the profile curves of all surfaces with constant mean curvature in the product space , which are invariant under the action of a 1-parameter subgroup of isometries. The author was supported by INdAM (Italy) and Fapesp (Brazil).     

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ and $${\mathbb {GL}}_N$$GLN

This paper studies the empirical laws of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups \({\mathbb {U}}_N\) and the general linear groups \({\mathbb {GL}}_N\), for \(N\in {\mathbb {N}}\). It establishes the strongest known convergence results for the empirical eigenvalues in the \({\mathbb {U}}_N\) case, and the first known almost sure convergence results for the eigenvalues and singular values in the \({\mathbb {GL}}_N\) case. The limit noncommutative distribution associated with the heat kernel measure on \({\mathbb {GL}}_N\) is identified as the projection of a flow on an infinite-dimensional polynomial space. These results are then strengthened from variance estimates to \(L^p\) estimates for even integers p.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.     

Geometric complexity of embeddings in {\mathbb{R}^{d}}

Given a simplicial complex K, we consider several notions of geometric complexity of embeddings of K in a Euclidean space \({\mathbb{R}^d}\) : thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any n-complex with N simplices which topologically embeds in \({\mathbb{R}^{2n}, n > 2}\) , can be PL embedded in \({\mathbb{R}^{2n}}\) with refinement complexity \({O(e^{N^{4+{\epsilon}}})}\) . Families of simplicial n-complexes K are constructed such that any embedding of K into \({\mathbb{R}^{2n}}\) has an exponential lower bound on thickness and refinement complexity as a function of the number of simplices of K. This contrasts embeddings in the stable range, \({K\subset \mathbb{R}^{2n+k}, k > 0}\) , where all known bounds on geometric complexity functions are polynomial. In addition, we give a geometric argument for a bound on distortion of expander graphs in Euclidean spaces. Several related open problems are discussed, including questions about the growth rate of complexity functions of embeddings, and about the crossing number and the ropelength of classical links.     

On a variant of the Kakeya problem in {\mathbb{R}}

We prove that the sharp lower bounds of the Minkowski and Hausdorff dimensions of circular Kakeya sets in ${\mathbb{R}}$ are 1/2 and 0 respectively.     

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

Extended quasi-homogeneous polynomial system in {\mathbb{R}^{3}}

In this paper we define an extended quasi-homogeneous polynomial system d x/dt = Q = Q ₁ + Q ₂ + ... + Q _δ , where Q _i are some 3-dimensional quasi-homogeneous vectors with weight α and degree i, i = 1, . . . ,δ. Firstly we investigate the limit set of trajectory of this system. Secondly let Q _T be the projective vector field of Q. We show that if δ ≤ 3 and the number of closed orbits of Q _T is known, then an upper bound for the number of isolated closed orbits of the system is obtained. Moreover this upper bound is sharp for δ = 3. As an application, we show that a 3-dimensional polynomial system of degree 3 (resp. 5) admits 26 (resp. 112) isolated closed orbits. Finally, we prove that a 3-dimensional Lotka-Volterra system has no isolated closed orbits in the first octant if it is extended quasi-homogeneous.     

Absolute Continuity of the Spectrum for Periodically Modulated Leaky Wires in $${\mathbb{R}^{3}}$$

We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction. Submitted: August 29, 2005. Accepted: June 9, 2006.     

Lagrangian Klein Bottles in {\mathbb{R}^{2n}}

It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into \mathbbR²ⁿ{\mathbb{R}^{2n}} if and only if n is odd.     

On nonatomic submeasures on {\mathbb{N}}

A submeasure μ defined on the subsets of is nonatomic if for every ℓ ≥ 1 there exists a partition of into a finite number of parts on which μ is bounded from above by 1/ℓ. In this paper we answer several natural questions concerning nonatomic submeasures d _F that are determined (like the standard density) by a family F of finite subsets of . We first show that if the number of n-element sets in F grows at most exponentially with n, then d _F is nonatomic; but if this growth condition fails, then d _F need not be nonatomic in general. We next prove that, for a nonatomic submeasure d _F , the minimal number of sets in a 1/ℓ-small partition of can grow arbitrarily fast with ℓ. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type d _F . The second author acknowledges a generous support of the Foundation for Polish Science.