Volume 9, Issue 2

In this paper, we define various kinds of transitivity of maps on G -spaces. We obtain conditions on G -spaces and on maps for one type of transitivity to imply another type of transitivity. Giving several examples and proving various equivalences, we provide a complete description of the relationships among the different types of transitivities defined for maps on G -spaces.

This article deals with Lp{L^{p}}-L2{L^{2}} estimates for solutions of the wave equation associated to the Grushin operator G:=-Δ-|x|2⁢∂2∂⁡t2{G:=-\Delta-|x|^{2}\frac{\partial^{2}}{\partial t^{2}}} on ℝn+1{\mathbb{R}^{n+1}}. We make use of the Lp{L^{p}} boundedness of Riesz transforms for the Grushin operator in studying the wave equation.

The main objective of the present work is to provide a procedure to construct Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets. Given two full-rank lattices of the same volume, we investigate conditions under which there exists a common fundamental domain which is the image of a unit cube under an invertible linear operator. More precisely, we provide a characterization of pairs of full-rank lattices in ℝd{\mathbb{R}^{d}} admitting common connected fundamental domains of the type N⁢[0,1)d{N[0,1)^{d}}, where N is an invertible matrix. As a byproduct of our results, we are able to construct a large class of Gabor windows which are indicator functions of sets of the type N⁢[0,1)d{N[0,1)^{d}}. We also apply our results to construct multivariate Gabor frames generated by smooth windows of compact support. Finally, we prove in the two-dimensional case that there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type N⁢[0,1)2{N[0,1)^{2}}, where N is an invertible matrix.

For a root system R on ℝd{\mathbb{R}^{d}} and a nonnegative multiplicity function k on R , we consider the heat kernel pk⁢(t,x,y){p_{k}(t,x,y)} associated to the Dunkl Laplacian operator Δk{\Delta_{k}}. For β∈]0,d+2γ[{\beta\in{]0,d+2\gamma[}}, where γ=12⁢∑α∈Rk⁢(α){\gamma=\frac{1}{2}\sum_{\alpha\in R}k(\alpha)}, we study the Δk{\Delta_{k}}-Riesz kernel of index β, defined by Rk,β⁢(x,y)=1Γ⁢(β/2)⁢∫0+∞tβ/2-1⁢pk⁢(t,x,y)⁢𝑑t{R_{k,\beta}(x,y)=\frac{1}{\Gamma(\beta/2)}\int_{0}^{+\infty}t^{{\beta/2}-1}p_% {k}(t,x,y)\,dt}, and the corresponding Δk{\Delta_{k}}-Riesz potential Ik,β⁢[μ]{I_{k,\beta}[\mu]} of a Radon measure μ on ℝd{\mathbb{R}^{d}}. According to the values of β, we study the Δk{\Delta_{k}}-superharmonicity of these functions, and we give some applications like the Δk{\Delta_{k}}-Riesz measure of Ik,β⁢[μ]{I_{k,\beta}[\mu]}, the uniqueness principle and a pointwise Hedberg inequality.

In the present work, the harmonic vector field is defined on closed Finsler measure spaces through different approaches. At first, the weighted harmonic vector field is obtained as the solution space of a PDE system. Then a suitable Dirichlet energy functional is introduced. A σ-harmonic vector field is considered as the critical point of related action. It is proved that a σ-harmonic vector field on a closed Finsler space with an extra unit norm condition is an eigenvector of the defined Laplacian operator on vector fields. Moreover, we prove that a unit weighted harmonic vector field on a closed generalized Einstein manifold is a σ-harmonic vector field.

A new Schulz-type method to compute the Moore–Penrose inverse of a matrix is proposed. Every iteration of the method involves four matrix multiplications. It is proved that this method always converge with fourth-order. A wide set of numerical comparisons of the proposed method with nine higher order methods shows that the average number of matrix multiplications and the average CPU time of our method are considerably less than those of other methods. For each of sizes n×n{n\times n} and n×(n+10){n\times(n+10)}, n=200,400,600,800,1000,1200{n=200,400,600,800,1000,1200}, ten random matrices were chosen to make these comparisons.

We consider the Cauchy problem for the Thirring model in the Gevrey spaces Gσ,s{G^{\sigma,s}}. In particular, we prove that the analyticity of solutions persists for a short time. In addition, we derive a sufficient condition to ensure that solutions will continue to be analytic for all time.