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The rotational spectrum of NSF3 in the ground and v5 = 1 vibrational states has been investigated in the centimeter- and millimeter-wave ranges. R-branch (J + 1-- J) transitions for J = 0, 1 and Q-branch rotational transitions for the v5 = 1 vibrational state have been measured by waveguide Fourier transform microwave spectroscopy in the range 8-26.5 GHz. The Q-branch transitions include 28 direct l-type doubling transitions (kl = +1, A1)--(kl = +1, A2) with Jor = 62, and 108 direct l-type resonance transitions following the selection rule delta k = delta l = +/-2 with Jor = 60 and G = |k - l|or = 3. A process called "regional resonance" was observed in which a cluster of levels interacted strongly over a large range in J. This process led to the observation of 55 perturbation-allowed transitions following the selection rules delta(k - l) = +/-3, +/-6. In particular, (kl = +1, A+)--(kl = -2, A-), (kl = +4, A+)--(kl = +1, A-), (kl = +2)--(kl = -1), (kl = +3)--(kl = 0), (kl = +2)--(kl = -3), and (kl = +3)--(kl = -3). The various aspects of the regional resonances are discussed in detail. An accidental near-degeneracy of the kl = 0 and kl = -4 levels at J = 26/27 led to the observation of perturbation-allowed transitions following the selection rule delta(k-l) = +/-6 with (kl = +2)--(kl = -4). A corresponding near-degeneracy between kl = -1 and kl = -3 levels at J = 30/31 led to the detection of similar transitions, but with (kl = +3)--(kl = -3). In the range 230-480 GHz, R-branch rotational transitions have been measured by absorption spectroscopy up to J = 49 in the ground-state and up to J = 50 in the v5 = 1 vibrational state. The transition frequencies have been analyzed using various reduced forms of the effective Hamiltonians. The data for the v5 = 1 vibrational state have been fitted successfully using two models up to seventh order with delta k = +/-3 interaction parameters constrained (dt constrained to zero, and epsilon to zero or to the ground-state value). On the other hand, reductions with the (delta k = +/-1, deltal = -/+2) interaction parameter q12 fixed to zero failed to reproduce the experimental data since the parameters defining the reduction transformation do not arise in the correct order of magnitude. The ground-state data have been analyzed including parameters up to fourth order constraining either parameters of the delta k = +/-3 interactions to zero (reduction A), or of the delta k = +/-6 interactions to zero (reduction B). The unitary equivalence of the different parameter sets obtained is demonstrated for both vibrational states.

The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators - generalized Pauli matrices - characterizing two-qubit systems. The relevant sub-configuration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/non-commuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF(2) x GF(2) (the "Mermin" part) and two lines over GF(4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually non-commuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and give their numerous applications a wholly new perspective., Comment: 8 pages, three tables; Version 2 - a few typos and one discrepancy corrected; Version 3: substantial extension of the paper - two-qubits are generalized quadrangles of order two; Version 4: self-dual picture completed; Version 5: intriguing triality found -- three kinds of geometric hyperplanes within GQ and three distinguished subsets of Pauli operators

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)., 7 pages, 3 figures, 2 tables; Version 2 - math nomenclature fine-tuned and two references added; Version 3 - published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of points on the line, the number of points represented by coordinates with at least one entry being a unit, the cardinality of the neighbourhood of a generic point of the line as well as those of the intersections between the neighbourhoods of two and three mutually distant points, the number of `Jacobson' points per a neighbourhood, the maximum number of pairwise distant points and, finally, a list of representative/base rings. The classification is presented in form of a table in order to see readily not only the fine traits of the hierarchy, but also the changes in the structure of the lines as one goes from one type to the other. We hope this study will serve as an impetus to a search for possible applications of these remarkable geometries in physics, chemistry, biology and other natural sciences as well., 7 pages, 1 figure; Version 2: classification extended up to order 20, references updated; Version 3: classification extended up to order 31, two more references added; Version 4: references updated, minor correction

High-resolution Fourier-transform infrared spectra between 1235 and 1680 cm −1 and subterahertz spectra between 250 and 630 GHz of monoisotopic 13 CH 3 35 Cl have been recorded and analyzed simultaneously, with all Coriolis, α -resonance, and l -type interactions in the polyad of the v 2 = 1, v 5 = 1, and v 3 = 2 levels taken into account. Several α -resonances (Δ k = ±2, Δ l = ∓1) generating perturbation-allowed transitions have been assigned in the rovibrational spectra. These resonances enabled us to determine accurately and independently the ground state rotational and centrifugal distortion parameters A 0 = 5.205 746 9 (55) cm −1 and D K 0 = 8.4404 ( 84 ) × 10 - 5 cm - 1 . Even H K 0 = 8.527 ( 39 ) × 10 - 9 cm - 1 , which is, however, correlated to higher-order α -resonance terms, was determined. With 51 upper state parameters varied, about 5800 rovibrational wavenumbers and more than 550 rotational frequencies pertaining to the excited vibrational states were fitted within their experimental accuracy.

Four vibrational levels in the energy region around 1000 cm(-1) were studied. These were the v(5)=1 and v(8)=1 fundamental levels, both components of the v(9)=v(10)=1 combination level (l(9)=l(10)=+/-1 and l(9)=-l(10)=+/-1), and both components of the v(10)=3 overtone level (l(10)=+/-1 and +/-3). New FTIR spectra with a synchrotron radiation source were recorded in the region of the "superhot" v(10)=3--2 bands, which made possible the first assignment of levels of the v(10)=3(+/-1) sublevel. More than 330 new rotational transitions in the combination and overtone levels were measured by millimeter-wave spectroscopy betwen 50 and 360 GHz. The new data were analyzed simultaneously together with the previously assigned rovibrational data for the fundamental and combination levels and rotational data for the fundamental levels using a global model with all anharmonic, Coriolis, l-type, and alpha-resonances. Significant improvement of data reproduction and very good consistency with the Hamiltonian parameters of the lower vibrational levels v(9)=1 and v(10)=1, 2 were achieved. A strong dependence of the A(v) constant on the l(10) quantum number is found for propyne: this is shown to be characteristic of skeleton C-C identical withC or C-C identical withN bending modes in H(3)CCCH, H(3)CCN, and their fully deuterated species. Copyright 2001 Academic Press.

The nu(5) fundamental band of trifluorosilane-d (SiDF(3)) at 627 cm(-1) was studied for the first time by high-resolution FTIR spectroscopy at a resolution of 2.4 x 10(-3) cm(-1). The analysis was performed simultaneously with available microwave and newly measured submillimeter-wave data in the approximation of an isolated degenerate fundamental level of a C(3 Kv) symmetric top molecule leading to a standard deviation of 0.22 x 10(-3) cm(-1) for the reproduction of the infrared wavenumbers, 36 kHz for the microwave, and 198 kHz for the submillimeter-wave frequencies, respectively. The unitary equivalence between the two reductions (Q and D) of the effective Hamiltonian applied in the analysis is demonstrated. Copyright 2000 Academic Press.

The nu(6) (555.453 cm(-1)), the nu(3) (888.899 cm(-1)), and the very weak 2nu(6) infrared bands (2nu(-/+2)(6) 1101.734 cm(-1), 2nu(0)(6) 1100.102 cm(-1)) for the (28)Si species of D(3)SiF have been recorded with a resolution of 3.3, 2.4, and 5.0 x 10(-3) cm(-1), respectively. Millimeter-wave spectra up to 640 GHz of D(3)SiF in the ground, v(3) = 1, and v(6) = 1 and 2 states were measured. Ground state constants complete up to H constants including the K-dependent parameters A(0), D(0)(K), and H(0)(K) as obtained by the nu(+/-1)(6)/2nu(+/-2)(6)-nu(+/-1)(6)/2nu(-/+2)(6) loop method were determined by a merge of 2388 ground state combination differences with 59 rotational data. The v(3) = 1 and v(6) = 1 and 2 levels appear to be unperturbed intervibrationally for the J and K values that could be accessed. However, Deltal = Deltak = +/-2 and Deltal = +/-2, Deltak = -/+4 interactions affect the v(6) = 1 level while the v(6) = 2 levels undergo three interactions of Deltal = Deltak = +/-2, Deltal = +/-2, Deltak = -/+1 and Deltal = +/-4, Deltak = -/+2 type. Typically, for the different bands, 2000-4000 pieces of infrared data augmented by 36-120 rotational data were fitted together. Owing to the weakness of the 2nu(6) band, the body of v(6) = 2 data was enlarged by energies that are deduced from the 2nu(6)-nu(6) and nu(6) bands and which span in particular high K values. Comparison with available ab initio data derived from the harmonic and anharmonic force fields is made. Copyright 1999 Academic Press.

The Fourier transform infrared spectrum of monoisotopic H3Si79Br (resolution 7.7 x 10(-3) cm-1) was studied from 4200 to 4520 cm-1, in the region of the first overtones of the Si-H stretching vibration. The investigation of the spectrum revealed the presence of two band systems, the first consisting of one parallel (nu0 = 4340.2002 cm-1) and one perpendicular (nu0 = 4342.1432 cm-1) strong component, and the second of one parallel (nu0 = 4405.789 cm-1) and one perpendicular (nu0 = 4416.233 cm-1) weak component. The rovibrational analysis shows strong local perturbations for both strong and weak systems. Seven hundred eighty-one nonzero-weighted transitions belonging to the strong system [the (200) manifold in the local mode picture] were fitted to a simple model involving a perpendicular component interacting by a weak Coriolis resonance with a parallel component. The most severely perturbed transitions (whose ||obs-calc || values exceeded 3 x 10(-3) cm-1) were given zero weights. The standard deviations of the fit were 1.0 x 10(-3) and 0.69 x 10(-3) cm-1 for the parallel and the perpendicular components, respectively. The weak band system, severely perturbed by many "dark" perturbers, was fitted to a model involving one parallel and one perpendicular band, connected by a Coriolis-type resonance. The K" . DeltaK = +10 to +18 subbands of the perpendicular component, which showed very high observed - calculated values ( approximately 0.5 cm-1), were excluded from this calculation. The standard deviations of the fit were 11 x 10(-3) and 13 x 10(-3) cm-1 for the parallel and the perpendicular components, respectively. Copyright 1998 Academic Press.

Ab initio SCF electric dipole moment functions of the electronic ground states of ammonia and the oxonium ion are determined over a wide range of vibrational coordinates. The dipole moment functions are used within the framework of the nonrigid invertor Hamiltonian to evaluate transition moments for various types of vibrational bands. Reasonable agreement with experimental data for ammonia is achieved by scaling the dipole moment function. Several new transition moments for combination and hot bands are predicted.

The Fano-Snowflake, a specific $non$-unimodular projective lattice configuration associated with the smallest ring of ternions $R_{\diamondsuit}$ (arXiv:0803.4436 and 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 $R_{\diamondsuit}^{3}$ are 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 (6 -- 7 -- 3) 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 (3 -- 3 -- 1)., 7 pages, 3 figures

10 pages, 3 figures; International audience; Recently (arXiv:1205.5015), Waegell and Aravind 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 ${\cal V}_{22}(37; 0, 12, 15, 10)$ and ${\cal V}_{4}(49; 0, 0, 21, 28)$ in the classification of Frohardt and Johnson (Communications in Algebra 22 (1994) 773). Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.