Dark Energy and Projective Symmetry

Nurowski [arXiv:1003.1503] has recently suggested a link between the observation of Dark Energy in cosmology and the projective equivalence of certain Friedman-Lemaitre-Robertson-Walker (FLRW) metrics. Specifically, he points out that two FLRW metrics with the same unparameterized geodesics have their energy densities differing by a constant. From this he queries whether the existence of dark energy is meaningful. We point out that physical observables in cosmology are not projectively invariant and we relate the projective symmetry uncovered by Nurowski to some previous work on projective equivalence in cosmology

Les bibliothèques et la deuxième naissance Loin de porter du cinéma

Loin de porter concurrence aux exploitants de cinéma en salle, l’offre développée par les médiathèques a souvent pallié les déficiences du réseau commercial. La preuve en Nord – Pas-de-Calais où, dans l’urgence, des formations ont même émergé de façon informelle

Crystallization in a model glass: influence of the boundary conditions

Using molecular dynamics calculations and the Voronoi tessellation, we study the evolution of the local structure of a soft-sphere glass versus temperature starting from the liquid phase at different quenching rates. This study is done for different sizes and for two different boundary conditions namely the usual cubic periodic boundary conditions and the isotropic hyperspherical boundary conditions for which the particles evolve on the surface of a hypersphere in four dimensions. Our results show that for small system sizes, crystallization can indeed be induced by the cubic boundary conditions. On the other hand we show that finite size effects are more pronounced on the hypersphere and that crystallization is artificially inhibited even for large system sizes.Comment: 11 pages, 2 figure

Affine Wa(A4), Quaternions, and Decagonal Quasicrystals

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I2 (h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh of order 2h as a maximal subgroup. As a simple application we demonstrate projections of the root and weight lattices of A4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine Wa(A4) can be decomposed into two orthogonal spaces whose point groups is the dihedral group D5 which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group Wa(A3), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group Wa(An), n>=4 . We also note that the projection of the Voronoi cell of the root lattice of Wa(A4) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.Comment: 26 pages, 17 figure

Quaternionic Root Systems and Subgroups of the Aut(F4)Aut(F_{4})

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)= SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E_{8} which form a closed octonion algebra. The automorphism group Aut(F_{4}) of the Dynkin diagram of F_{4} of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F_{4}.The Weyl groups of many Lie algebras, such as, G_{2},SO(7),SO(8),SO(9),SU(3)XSU(3) and SP(3)X SU(2) have been constructed as the subgroups of Aut(F_{4}). We have also classified the other non-parabolic subgroups of Aut(F_{4}) which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group $G_{2}$ of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e_{1},e_{2},e_{3} is used to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and F_{4} respectively

Quaterionic Construction of the W(F_4) Polytopes with Their Dual Polytopes and Branching under the Subgroups B(B_4) and W(B_3)*W(A_1)

4-dimensional $F_{4}$ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(F_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an arbitrary \textbf{$W(F_{4})$} orbit under the Coxeter groups $W(B_{4}$ and $W(B_{3}) \times W(A_{1})$ have been presented. The role of group theoretical technique and the use of quaternions have been emphasizedComment: 26 pages, 10 figure

Borcherds symmetries in M-theory

It is well known but rather mysterious that root spaces of the $E_k$ Lie groups appear in the second integral cohomology of regular, complex, compact, del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms) of toroidal compactifications of M theory. Their Borel subgroups are actually subgroups of supergroups of finite dimension over the Grassmann algebra of differential forms on spacetime that have been shown to preserve the self-duality equation obeyed by all bosonic form-fields of the theory. We show here that the corresponding duality superalgebras are nothing but Borcherds superalgebras truncated by the above choice of Grassmann coefficients. The full Borcherds' root lattices are the second integral cohomology of the del Pezzo surfaces. Our choice of simple roots uses the anti-canonical form and its known orthogonal complement. Another result is the determination of del Pezzo surfaces associated to other string and field theory models. Dimensional reduction on $T^k$ corresponds to blow-up of $k$ points in general position with respect to each other. All theories of the Magic triangle that reduce to the $E_n$ sigma model in three dimensions correspond to singular del Pezzo surfaces with $A_{8-n}$ (normal) singularity at a point. The case of type I and heterotic theories if one drops their gauge sector corresponds to non-normal (singular along a curve) del Pezzo's. We comment on previous encounters with Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real fermionic simple roots when they would naively aris

Practical recommendations for radium-223 treatment of metastatic castration-resistant prostate cancer

is the first targeted alpha therapy for patients with castration resistant prostate cancer and symptomatic bone metastases. Radium-223 provides a new treatment option for this setting, but also necessitates a new treatment management approach. We provide straightforward and practical recommendations for European nuclear medicine centres to optimize radium-223 service provision. Methods An independent research consultancy agency observed radium-223 procedures and conducted interviews with all key staff members involved in radium-223 treatment delivery in 11 nuclear medicine centres across six countries (Germany, Italy, the Netherlands, Spain, Switzerland and the UK) experienced in administering radium-223. The findings were collated and discussed at a meeting of experts from these centres, during which key consensus recommendations were defined. Results The recommendations cover centre organization and preparation; patient referral; radium-223 ordering, preparation and disposal; radium-223 treatment delivery/administration; and patient experience. Guidance includes structured coordination and communication within centres and multidisciplinary teams, focusing on sharing best practice to provide high quality, patient-centred care throughout the treatment pathway. Conclusions These expert recommendations are intended to complement existing management guidelines. Sharing best practice and experience will help nuclear medicine centres to optimize radium-223 service provision and improve patient care