Composition-Diamond lemma for λ\lambda-differential associative algebras with multiple operators

In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K$ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential Rota-Baxter algebras. In particular, linear bases of free $\lambda$-differential Rota-Baxter algebras are obtained and consequently, the free $\lambda$-differential Rota-Baxter algebras are constructed by words

Part-products of SS-restricted integer compositions

If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda}=(\lambda_1,\lambda_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the random variable ${\bf B}(\vec{\lambda})=\prod_i \lambda_i$ is asymptotically lognormal. The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.Comment: 18 page

Modelling interstellar extinction and polarization with spheroidal grains

We calculate the wavelength dependence of the ratio of the linear polarization degree to extinction (polarizing efficiency) $P(\lambda)/A(\lambda)$ from the ultraviolet to near-infrared. The prolate and oblate particles with aspect ratios from $a/b=1.1$ up to 10 are assumed to be rotating and partially aligned with the mechanism of paramagnetic relaxation (Davis--Greenstein). Size/shape/orientation effects are analyzed. It is found that the wavelength dependence of $P(\lambda)/A(\lambda)$ is mainly determined by the particle composition and size whereas the values of $P(\lambda)/A(\lambda)$ depend on the particle shape, degree and direction of alignment.Comment: 13 pages, 9 figures, aacepted for publication in Journal of Quantitative Spectroscopy & Radiative Transfer (special issue, X Conference on Electromagnetic & Light Scattering