The set of 63 real generalized Pauli matrices of three-qubits can be factored into two subsets of 35 symmetric and 28 antisymmetric elements. This splitting is shown to be completely embodied in the properties of the Fano plane; the elements of the former set being in a bijective correspondence with the 7 points, 7 lines, and 21 flags, whereas those of the latter set having their counterparts in 28 antiflags of the plane. This representation naturally extends to the one in terms of the split Cayley hexagon of order two. Sixty three points of the hexagon split into 9 orbits of 7 points (operators) each under the action of an automorphism of order 7. Sixty three lines of the hexagon carry three points each and represent the triples of operators such that the product of any two gives, up to a sign, the third one. Since this hexagon admits a full embedding in a projective 5-space over $GF(2)$, the 35 symmetric operators are also found to answer to the points of a Klein quadric in such space. The 28 antisymmetric matrices can be associated with the 28 vertices of the Coxeter graph, one of two distinguished subgraphs of the hexagon. The $PS{L}_2(7)$ subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter subgeometry is found to be intricately related to the $E_7$-symmetric black-hole entropy formula in string theory. It is also conjectured that the full geometry/symmetry of the hexagon should manifest itself in the corresponding black-hole solutions. Finally, an intriguing analogy with the case of Hopf sphere fibrations and a link with coding theory are briefly mentioned.

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of $N$-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding {\it Pauli graph} are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) $N$-qubit Pauli graph is shown to be pseudo-geometric, i.\,e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle $Q(4,3)$, the dual of $W(3)$.

It is shown that the $E_{6(6)}$ symmetric entropy formula describing black holes and black strings in $D=5$ is intimately tied to the geometry of the generalized quadrangle GQ(2, 4) with automorphism group the Weyl group $W(E_6)$. The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ(2, 4). Different truncations with 15, 11 and 9 charges are represented by three distinguished subconfigurations of GQ(2, 4), well known to finite geometers; these are the “doily” [i.e. GQ(2, 2)] with 15, the “perp set” of a point with 11, and the “grid” [i.e. GQ(2, 1)] with nine points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a noncommutative labeling for the points of GQ(2, 4). For the 40 different possible truncations with nine charges this labeling yields 120 Mermin squares—objects well known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the $E_{7(7)}$ symmetric entropy formula in $D=4$ by observing that the structure of GQ(2, 4) is linked to a particular kind of geometric hyperplane of the split Cayley hexagon of order 2, featuring 27 points located on nine pairwise disjoint lines (a distance-3-spread). We conjecture that the different possibilities of describing the $D=5$ entropy formula using Jordan algebras, qubits and/or qutrits correspond to employing different coordinates for an underlying noncommutative geometric structure based on GQ(2, 4).

Context. The structure of the white-light and emission solar coronas and their MHD modelling are the context of our work.Aims. A comparison is made between the structure of the solar corona as observed during the 2008 August 1 total eclipse from Mongolia and that predicted by an MHD model. Methods. The model has an improved energy formulation, including the effect of coronal heating, conduction of heat parallel to the magnetic field, radiative losses, and acceleration by Alfvén waves. Results. The white-light corona, which was visible up to 20 solar radii, was of an intermediate type with well-pronounced helmet streamers situated above a chain of prominences at position angles of 48, 130, 241, and 322 degrees. Two polar coronal holes, filled with a plethora of thin polar plumes, were observed. High-quality pictures of the green (530.3 nm, Fe XIV) corona were obtained with the help of two narrow-passband filters (centered at the line itself and the vicinity of 529.1 nm background), with a FWHM of 0.15 nm. Conclusions. The large-scale shape of both the white-light and green corona was found to agree well with that predicted by the model. In this paper we describe the morphological properties of the observed corona, and how it compares with that predicted by the model. A more detailed analysis of the quantitative properties of the corona will be addressed in a future publication.

As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single $d$-dimensional qudit is made, with $d$ being {\it any} positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring $\bZ_{d}$. Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of $\bZ^{2}_{d}$. We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless $d$ is a product of distinct primes. The operators are also seen to be structured into disjoint `layers' according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made.