In this article we consider elliptic partial differential equations with random coefficients and/or random forcing terms. In the current treatment of such problems by stochastic Galerkin methods it is standard to assume that the random diffusion coefficient is bounded by positive deterministic constants or modeled as lognormal random field. In contrast, we make the significantly weaker assumption that the non-negative random coefficients can be bounded strictly away from zero and infinity by random...

This paper deals with the exact distribution of L1(vc) of Votaw. The results are given in terms of Meijer's G-function as well as in series form suitable for computation of percentage points.

We give a complete description of the boundary behaviour of the generalized hypergeometric functions, introduced by Faraut and Koranyi, on Cartan domains of rank 2. The main tool is a new integral representation for some spherical polynomials, which may be of independent interest.

We deal with several classes of integral transformations of the form $$f\left(x\right)\to D{\int }_{{ℝ}_{+}^2}\frac{1}{u}({\mathrm{e}}^{-ucosh(x+v)}+{\mathrm{e}}^{-ucosh(x-v)})h\left(u\right)f\left(v\right)\mathrm{d}u\mathrm{d}v,$$ where $D$ is an operator. In case $D$ is the identity operator, we obtain several operator properties on $L_p\left({ℝ}_{+}\right)$ with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on $L_2\left({ℝ}_{+}\right)$ and define the inversion formula. Further, for an other class of differential operators of finite...

Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure.As a byproduct, we also give...

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed $\alpha \in \mathbb{Q}-{\mathbb{Z}}_{\<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{\left(\alpha \right)}\left(x\right)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+\alpha }{n-j}\right){(-x)}^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha =0,1,\pm \frac{1}{2},-1-n$.