The distance between the second and first eigenvalues for the Dirichlet Laplacian of a domain is called its (spectral) gap. We show that the gap of a convex planar domain$D$symmetric about both the$x$and$y$axes is no smaller than the gap of an oriented rectangle which contains$D$.

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In this paper we give a complete description of the power weighted inequalities, of strong, weak and restricted weak type for the pair of Riesz transforms associated with the Laguerre function system $\{\mathcal{L}_k^{\alpha}\}$ , for any given α>-1. We achieve these results by a careful estimate of the kernels: near the diagonal we show that they are local Calderón–Zygmund operators while in the complement they are majorized by Hardy type operators and the maximal heat-diffusion operator. We also show that in all the cases our results are sharp.

This paper introduces a new efficient algorithm, called MXL_3, for computing Gröbner bases of zero-dimensional ideals. The MXL_3 is based on XL algorithm, mutant strategy, and a new sufficient condition for a set of polynomials to be a Gröbner basis. We present experimental results comparing the behavior of MXL_3 to F_4 on HFE and random generated instances of the MQ problem. In both cases the first implementation of the MXL_3 algorithm succeeds faster and uses less memory than Magma’s implementation of F_4.

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