Upper Bounds for the Davenport Constant

We prove that for all but a certain number of abelian groups of order n its Davenport constant is atmost n/k+k-1 for k=1,2,..,7. For groups of order three we improve on the existing bound involving the Alon-Dubiner constant.Comment: article soumis, decembre 200

Lattice expansion and non-collinear to collinear ferrimagnetic order in MnCr2_2O4_4 nanoparticle

We report magnetic behaviour of MnCr$_2$O$_4$, which belongs to a special class of spinel, known as chromite. Bulk MnCr$_2$O$_4$ shows a sequence of magnetic states, which follows paramagnetic (PM) to collinear ferrimagnetic (FM) state below T$_C$ $\sim$ 45 K and collinear FM state to non-collinear FM state below T$_S$ $\sim$ 18 K. The non-collinear spin structure has been modified on decreasing the particle size, and magnetic transition at T$_S$ decreases in nanoparticle samples. However, ferrimagnetic order is still dominating in nanoparticles, except the observation of superparamagnetic like blocking and decrease of spontaneous magnetization for nanoparticle. This may, according to the core-shell model of ferrimagnetic nanoparticle, be the surface disorder effect of nanoparticle. The system also show the increase of T$_C$ in nanoparticle samples, which is not consistent with the core-shell model. The analysis of the M(T) data, applying spin wave theory, has shown an unusual Bloch exponent value 3.35 for bulk MnCr$_2$O$_4$, which decreases and approaches to 1.5, a typical value for any standard ferromagnet, with decreasing the particle size. MnCr$_2$O$_4$ has shown a few more unusual behaviour. For example, lattice expansion in nanoparticle samples. The present work demonstrates the correlation between a systematic increase of lattice parameter and the gradual decrease of B site non-collinear spin structure in the light of magnetism of MnCr$_2$O$_4$ nanoparticles