A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions Rà (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of Rà. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left Rà-module Rà³ 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

The Veldkamp Space of Two-Qubits

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)

Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions

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

Projective Ring Line of a Specific Qudit

A very particular connection between the commutation relations of the elements of the generalized Pauli group of a $d$-dimensional qudit, $d$ being a product of distinct primes, and the structure of the projective line over the (modular) ring \bZ_{d} is established, where the integer exponents of the generating shift ($X$) and clock ($Z$) operators are associated with submodules of \bZ^{2}_{d}. Under this correspondence, the set of operators commuting with a given one -- a perp-set -- represents a \bZ_{d}-submodule of \bZ^{2}_{d}. A crucial novel feature here is that the operators are also represented by {\it non}-admissible pairs of \bZ^{2}_{d}. This additional degree of freedom makes it possible to view any perp-set as a {\it set-theoretic} union of the corresponding points of the associated projective line

Projective Ring Line Encompassing Two-Qubits

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 operator

The Veldkamp space of multiple qubits

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections thereof. Among the geometric hyperplanes one can find the set of self-dual operators with respect to the Wootters spin-flip operation well-known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.Comment: 15 pages, 1 figure; added references, corrected typos; minor change

Geometric Hyperplanes of the Near Hexagon L_3 times GQ(2, 2)

Having in mind their potential quantum physical applications, we classify all geometric hyperplanes of the near hexagon that is a direct product of a line of size three and the generalized quadrangle of order two. There are eight different kinds of them, totalling to 1023 = 2^{10} - 1 = |PG(9, 2)|, and they form two distinct families intricately related with the points and lines of the Veldkamp space of the quadrangle in question.Comment: 10 pages, 5 figures and 2 tables; Version 2 - more detailed discussion of the properties of hyperplane

THE v2v_{2} AND v5v_{5} INFRARED BANDS OF H3Si79BrH_{3}Si^{79}Br

1. H. Bürger, H. Beckers, and J. Kauppinen. J. Mol. Spectrosc. 108, 215 (1984) 2. A. Ceausu, G. Graner, H. Bürger, E.B. Mkalhri, J. Ccaleou and A. G. Lesarri, J. Mol. Spectrosc, (1995) (in press).Author Institution: associé aux Universités P. et M. Curie et Paris. Sud, Bât 350, Campus d'Orsay. 91405 Orsay, France.; Universität Gesamthochschule, FB9, 42097 Wappertal, Germany.; J. Heyrovsky Institute of Physical Chemistry, Dolejskova 3, 18223 Praha 8. Czech Republic.The $v_{2} (A_{1})$ and $v_{5}$ (E) fundamental bands of isotopically pure $H_{3} Si^{79}$ Br, centered at 929.752 and $946.427 cm^{-1}$ respectively have been analyzed from a high resolution Fourier transform spectrum in order to improve a former $work^{1}$. These two bands strongly interact through a X-Y Coriolis resonance. We also included in the model the weak Coriolis resonance between $v_{5}$ and $2v_{3}$, with a crossing near $k\ell = -13$ and K = 14, the r-type $\ell(2. - 1)$ resonance between $k\ell = 2$ and $k\ell = -3$ of $v_{5}$, with a crossing at J = 58 and the accidental resonance $\ell (1. - 2)$ between the $(k\ell = 1.A^{2})$ levels of $v_{5}$ and $(K = 3, A^{2})$ of $v_{2}$. We made a global fit of 4200 wavenumbers for $v_{2}$ and $v_{5}$ and 630 upper state energies of $v_{3} = 2$, deduced from previously $determined^{2}$ transitions of $2v_{3} - v_{3}$ and of $v_{3}$. A few perturbation allowed transitions were also included in the fit, which gives a s. d. on residuals better than $0.00015 cm^{-1}$. A set of molecular parameters for all levels involved will be reported

STUDI KELAYAKAN SARANA DAN PRASARANA GUNA MENINGKATKAN KOMPETENSI PROGRAM KEAHLIAN TEKNIK GAMBAR BANGUNAN SMK NEGERI 1 MIRI SRAGEN

Penelitian  ini  bertujuan  untuk:  (1).  Mengetahui  kesesuaian  kondisi sarana dan prasarana Program Keahlian Teknik Gambar Bangunan SMK Negeri 1 Miri Sragen dengan Permendiknas No. 40 Tahun 2008. (2). Mengetahui tingkat kelayakan sarana dan prasarana Program Keahlian Teknik Gambar Bangunan SMK Negeri 1 Miri Sragen sesuai Permendiknas No. 40 Tahun 2008. (3). Mengetahui cara merencanakan atau mengembangkan sarana dan prasarana Program Keahlian Teknik Gambar Bangunan SMK Negeri 1 Miri Sragen yang layak sesuai Permendiknas No. 40 Tahun 2008.Penelitian yang digunakan merupakan jenis penelitian kualitatif evaluatif dengan pendekatan metode studi kasus. Data yang dikumpulkan dalam penelitian ini diperoleh dari hasil observasi, wawancara dan dokumentasi terhadap objek penelitian yaitu Sarana dan Prasarana Program Keahlian Teknik Gambar Bangunan SMK Negeri 1 Miri, Kab. Sragen. Instrumen penelitian menggunakan checklist dengan skala penilaian model Rating Scale. Data sarana dan prasarana kemudian  dibandingkan  dengan  standar  yang telah  ditentukan yang  berdasarPermendiknas No. 40 Tahun 2008 Tentang Standar Sarana dan Prasarana SMK/MAK dan BSNP No. 1023-P2-13/14 Mengenai Instrumen verifikasi SMK/MAK Tentang Penyelenggara Ujian Praktik Kejuruan. Hasil penelitian ini menunjukkan bahwa: (1). Kondisi rata-rata sarana dan prasarana pada Program Keahlian Teknik Gambar Bangunan SMK Negeri 1Miri Sragen adalah memenuhi syarat yang tercantum pada Lampiran Permendiknas No. 40 Tahun 2008 (2). Tingkat kelayakan rata-rata sarana adalah sebesar 65,54% masuk pada kisaran 51-75% yang berarti memenuhi syarat yang tercantum pada Lampiran Permendiknas No. 40 Tahun 2008, sedangkan tingkat kelayakan rata-rata prasarana adalah sebesar 66,69% masuk pada kisaran 51- 75%   yang    berarti   memenuhi    syarat    yang    tercantum    pada    LampiranPermendiknas No. 40 Tahun 2008 pada Program Keahlian Teknik Gambar Bangunan SMK Negeri 1 Miri Sragen. (3). Sarana dan prasarana pada Program Keahlian   Teknik   Gambar   Bangunan   SMK   Negeri   1   Miri   Sragen   perlu perencanaan ulang supaya memenuhi ketentuan dengan yang tercantum pada Permendiknas No. 40 Tahun 2008. Kata Kunci : kelayakan, sarana, prasarana

INTERACTIONS IN SYMMETRIC TOP MOLECULES BETWEEN VIBRATIONAL POLYADS: ROTATIONAL AND ROVIBRATIONAL SPECTROSCOPY OF LOW-LYING STATES OF PROPYNE

$^{a}$P. Pracna, G. Graner, J. Cosl\'eou, J. Demaison, G. Wlodarczak, V.-M. Horneman, and M. Koivusaari, J. Mol. Spectrosc. 206, (2001) 150-157; H. S. P. M\""uller, P. Pracna, et al., unpublished $^{b}$H. S. P. M\""uller, P. Pracna, and V.-M. Horneman, J. Mol. Spectrosc. 216, (2002) 397-407Author Institution: I. Physikalisches Institut, Universit\""at zu K\""oln; I. Physikalisches Institut, Academy of Sciences of the Czech Republic; I. Physikalisches Institut, 90014 University of Oulu; I. Physikalisches Institut, Justus-Liebig-Universit\""atA large body of very accurate (mostly 10-20 kHz) rotational transitions within the $10 {\mu}m$ tetrad of the strongly prolate symmetric rotor propyne, $H_{3}C-C\equiv CH$, had been recorded in order to analyze some weaker $resonances.^{a}$ The analysis, in particular for higher K transitions, indicated new problems: Interactions of members of the $10 {\mu}m$ tetrad ($v_{5} = 1, v_{9} = v_{10} = 1, v_{10} = 3$, and $v_{8} = 1$) with those of the $15 {\mu}m$ dyad ($v_{9} = 1$ and $v_{10} = 2$). More than 300 rotational transitions within $v_{10} = 1 (E_{vib} = 330.9 cm^{-1})$ have been recorded with $J^{\prime\prime} \leq 53$ and $-11 \leq K \cdot l_{10} \leq +16$ for a more systematic investigation into the lower excited vibrational states of propyne. Even the inclusion of many high order terms prevented the highest K transitions to be fit within experimental uncertainties, in particular those having $K \cdot l_{10} \leq 0$. Inclusion of $\nu_{10}$ IR transitions from a reanalysis of the spectrum that was used in the study of the $10 {\mu}m$ $tetrad^{a}$ confirmed these findings. It turned out that the first order energies of the $K \cdot l_{10} = -12$ levels coincide to within $1 cm^{-1}$ with those of the overtone $v_{10} = 2, K = 12$ of the $l_{10} = +2$ substate. A weaker Coriolis interaction occurs between $v_{10} = 1, K \cdot l_{10} = -11$ and $v_{9} = 1, K \cdot l_{9} = +10.^{b}$ Meanwhile, the investigation of $v_{10} = 1$ has been completed, and those of $v_{9} = 1$ and $v_{10} = 2$ have begun. Fermi-type interactions occur between the states $v_{10} = 2$ and 3 and between $v_{9} = 1$ and $v_{9} = v_{10} = 1$. Among the Coriolis resonances, the one between $v_{10} = 2, l_{10} = -2$ and $v_{5} = 1$ permitted intervibrational transitions to be detected in the submillimeter region. These interactions are expected to improve $A, D_{K}$, etc. for $v = 0$. Selected details of our ongoing analyses will be presented. In addition, the relevance of these types of interactions for other symmetric top molecules will be discussed