Volume 3, Issue 1

Abstract. Suppose G$G$ is an abelian group and R$R$ is a commutative ring with identity of prime characteristic p$p$. In some special cases for R$R$ and G$G$ we compute only in their terms and sections the Warfield p$p$-invariants of the normed unit group VR[G]$VR[G]$ in a commutative group ring R[G]$R[G]$. This supplies our recent results in this way.

Abstract. Let G(,)$\operatorname{G}_\lambda ^{(\alpha ,\beta )}$ be the eigenfunctions of the Dunkl–Cherednik operator T(,)$\operatorname{T}^{(\alpha ,\beta )}$ on $\mathbb {R}$. In this paper we express the product G(,)(x)G(,)(y)$\operatorname{G}_\lambda ^{(\alpha ,\beta )}(x)\,\operatorname{G}_\lambda ^{(\alpha ,\beta )}(y)$ as an integral in terms of G(,)(z)$\operatorname{G}_\lambda ^{(\alpha ,\beta )}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. Rösler for the Dunkl kernel. We then define and study a convolution structure associated to G(,)$\operatorname{G}_\lambda ^{(\alpha ,\beta )}$.

Abstract. This work deals with nonlinear potential theory, particularly with the techniques of the construction of nonlinear resolvent associated with a given nonlinear operator. The first section is devoted to the theoretical case. After making some introductory remarks about the Dirichlet problem for the generalized Laplace equation, we define a nonlinear operator on the space of essentially bounded functions on an open bounded subset of k$k$-dimensional real space and we associate with it a sub-Markovian nonlinear resolvent.

Abstract. We study stationary solutions to the conserved Kuramoto–Sivashinsky equation th+y2f48(f+12)h+y2h+f12(yh)2=0.$ \partial _th+\partial ^2_{y}\left( \frac{f\alpha }{48}(f\alpha +12\delta )h+\partial ^2_{y}h+\frac{f}{12}(\partial _yh)^2\right)=0. $ This equation has recently been proposed to describe the step meandering instability on a vicinal surface. Attention is focussed on stationary periodic solutions which are the key for the coarsening process.

Abstract. We study the nonlinear boundary value problem - div ((|u(x)|p1(x)-2+|u(x)|p2(x)-2u(x)$ -\mathop {\rm div}\Bigl (\bigl ( |\nabla u(x)|^{p_{1}(x)-2}+|\nabla u(x)|^{p_{2}(x)-2}\bigl )\nabla u(x)\Bigr )\vspace*{-5.69054pt} $=V1(x)|u|q(x)-2u-V2(x)|u|(x)-2u$ \qquad \qquad = \lambda V_1(x)|u|^{q(x)-2}u-\mu V_2(x) |u|^{\alpha (x)-2}u\vspace*{5.69054pt} $ in $\Omega $, u=0$u=0$ on $\partial \Omega $, where $\Omega $ is a bounded domain in N$\mathbb {R}^{N}$ with smooth boundary, $\lambda $, $\mu $ are positive real numbers, p1$p_{1}$, p2$p_{2}$, q$q$, $\alpha $ are continuous functions on $\overline{\Omega }$, V1$V_1$ and V2$V_2$ are weight functions in generalized Lebesgue spaces Ls1(x)()$L^{s_1(x)}(\Omega )$ and Ls2(x)()$L^{s_2(x)}(\Omega )$, respectively, such that V1>0$V_1>0$ in an open set 0$\Omega _0\subset \Omega $ with |0|>0$|\Omega _{0}|>0$ and V20$V_2\ge 0$ on $\Omega $. We prove, under appropriate conditions that for any >0$\mu >0$ there exists *$\lambda _{*}$ sufficiently small such that for any (0,*)${\lambda \in (0,\lambda _{*})}$ the above nonhomogeneous quasilinear problem has a nontrivial positive weak solution. The proof relies on some variational arguments based on Ekeland's variational principle.

Abstract. Random variables corresponding to weight functions of the Askey scheme for continuous polynomials are introduced. Beside normal, Beta and Gamma distributions four new families of real random variables corresponding to Wilson, continuous dual Hahn, continuous Hahn and Meixner–Pollaczek polynomials are discussed. Formulas for the moments and asymptotic behavior of the tails of the distributions are provided.

Abstract. A theorem of Alesker et al. says that the Fourier transform on n$\mathbb {R}^{n}$ is essentially the only transform on the space of tempered distributions which interchanges convolutions and products. In this note we obtain a similar characterisation for the Weyl transform.