The possibility that our space is multi rather than singly connected has gained a renewed interest after the discovery of the low power for the first multipoles of the CMB by WMAP. To test the possibility that our space is a multi-connected spherical space, it is necessary to know the eigenmodes of such spaces. Excepted for lens and prism space, and in some extent for dodecahedral space, this r...

Local structure analysis of Al-containing magnesium silicate perovskite has been carried out with X-ray absorption spectra recorded at the Mg, Al, and Si K-edges using the SA32 beam-line of SuperAco (Orsay, France). The Al-XAFS spectrum of (MgSi)0.85Al0.3O3 perovskite (synthesized in a multi-anvil apparatus) cannot be explained by assuming that Al31 occurs in octahedral or dodecahedral sites on...

A Platonic map is a regular map M on the sphere S. Following [CMo] we say that M has type {n,m} if it has n-gonal faces and the vertices have valency m; since these parameters determine a Platonic map uniquely, one can unambiguously write M = {n,m}. As one must have 0 ≤ (m − 2)(n − 2) < 4, there are precisely the possibilities M = {n, 2} (dihedron), M = {2,m} (hosohedron), M = {3, 3} (tetrahedr...

The chromatic polynomials of certain families of graphs can be calculai ed by a transfer matrix method. The transfer matrix commutes with an action of the symmetric group on the colours. Using représentation theory, it is shown that the matrix is équivalent to a block-diagonal matrix. The multiplicities and the sizes of the blocks are obtained. Using a repeated inclusion-exclusion argument the ...

Uniform M3PMo12O40·nH2O (M = NH , K, Cs) rhombic dodecahedral nanocrystals are successfully obtained using mild hydrothermal conditions or room temperature chemical precipitation method for their self-organization. The reaction temperatures and times play important roles in the possible growth proceeding of these nanostructures. Interestingly, from antibacterial testing, the nanostructured prod...

The development of liquid, efficient securities markets has been a major goal for policy makers in the third world. Casual empiricism suggests that small countries are unlikely to obtain liquid securities markets. There are only 16 countries in the world where the annual turnover on the stock market exceeds 75% of the market capitalisation of the stock market. All but one of these countries has...

The only finite nonabelian simple group acting on a homology 3-sphere necessarily non-freely is the dodecahedral group A5 ∼= PSL(2, 5) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group A∗5 ∼= SL(2, 5)). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, a...

In studying a large number of carefully polished surfaces of pyrite it has been found that this mineral is, in tie general case, optically anisotropic. Interference colours are blue-green to orange-red, and bear a constant relationship to crystal morphoJogy. On cube faces maximum green or red is developed when cube edges are inclined at approximately 45o to the planes of polarisation of the nic...

What is the shape of space is a long-standing question in cosmology. I review recent advances in cosmic topology since it has entered a new era of experimental tests. High redshift surveys of astronomical sources and accurate maps of the Cosmic Microwave Background radiation (CMB) are beginning to hint at the shape of the universe, or at least to limit the wide range of possibilities. Among tho...

The structural environment of substitutional Cr ion in a natural pyrope Mg3Al2Si3O12 has been investigated by Cr K-edge Extended X-ray Absorption Fine Structure (EXAFS) and X-ray Absorption Near Edge Structure (XANES) coupled with first-principles computations. The Cr-O distance is close to that in knorringite Mg3Cr2Si3O12, indicating a full relaxation of the first neighbors. The local C3i symm...