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Given a 2N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6 , we first observe that the multiplication table of its imaginary units ea , 1 ≤ a ≤ 2N - 1 , is encoded in the properties of the projective space PG(N - 1,2) if these imaginary units are regarded as points and distinguished triads of them {ea, eb , ec} , 1 ≤ a < b < c ≤ 2N - 1 and eaeb = ±ec , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b ≠ c . Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG(N - 1,2) , the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration CN; in particular, C3 (octonions) is isomorphic to the Pasch (62, 43) -configuration, C4 (sedenions) is the famous Desargues (103) -configuration, C5 (32-nions) coincides with the Cayley-Salmon (154, 203) -configuration found in the well-known Pascal mystic hexagram and C6 (64-nions) is identical with a particular (215, 353) -configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic CN leads to a conjecture that CN is isomorphic to a combinatorial Grassmannian of type G2(N + 1).

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that two of these sets/configurations, namely the 18_2−12_3 and 2_4 14_2−4_3 6_4 ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types V_{22}(37;0,12,15,10) and V_4(49;0,0,21,28) in the classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797]. Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.

The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions $R_{diamondsuit}$ (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of $R_{diamondsuit}$. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left $R_{diamondsuit}$-module $R_{diamondsuit}^3$ is shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding ''ternion-induced'' factorization of the lines of the Fano plane sitting in the middle of the Fano-Snowflake is found to differ fundamentally from the natural one, i.e., from that with respect to the Jacobson radical of the Galois field of two elements.

Given a finite associative ring with unity, R, any free (left) cyclic submodule (FCS) generated by a unimodular (n + 1)-tuple of elements of R represents a point of the n-dimensional projective space over R. Suppose that R also features FCSs generated by (n + 1)-tuples that are not unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for n = 2 when R is the (unique) non-commutative ring of order eight. The corresponding geometry is dubbed a ''Fano-Snowflake'' due to its diagrammatic appearance and the fact that it contains the Fano plane in its center. There exist, in fact, two such configurations – each being tied to either of the two maximal ideals of the ring – which have the Fano plane in common and can, therefore, be viewed as twins. Potential relevance of these noteworthy configurations to quantum information theory and stringy black holes is also outlined.

Given a remarkable representation of the generalized Pauli operators of two-qubits in terms of the points of the generalized quadrangle of order two, W(2), it is shown that specific subsets of these operators can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements - the so-called Veldkamp space of W(2). An intriguing novelty is the recognition of (uni- and tri-centric) triads and specific pentads of the Pauli operators in addition to the ''classical'' subsets answering to geometric hyperplanes of W(2).

The parallel vibration−rotation band ν4of methyl isocyanide (CH3NC), with a band center at 944.9 cm−1, was studied by FTIR spectroscopy between 890 and 980 cm−1in order to improve the ground-state rotational constants. Such improvement is essential for the scheduled studies of excited vibrational levels and their mutual anharmonic resonances occurring at higher values of the Krotational number. Ground-state combination differences generated from this band, spanning values of J/Kfrom 0 to 85/13, were combined with rotational data from the literature and newly measured rotational transitions, extending the J/Krange from 3/0 up to 31/14, and fitted simultaneously with a fully quantitative reproduction of the data. The infrared data of the ν4band were analyzed together with rotational data of the ν4= 1 level, spanning values of J/Kfrom 4/0 to 14/12. The fit in the approximation of an isolated vibrational state, with the transitions perturbed by weak local resonances excluded, yields reproduction of the data within experimental uncertainties. [ABSTRACT FROM AUTHOR]