25 research outputs found
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 -dimensional qudit, 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 () and clock () 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 AND INFRARED BANDS OF
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 and (E) fundamental bands of isotopically pure Br, centered at 929.752 and respectively have been analyzed from a high resolution Fourier transform spectrum in order to improve a former . These two bands strongly interact through a X-Y Coriolis resonance. We also included in the model the weak Coriolis resonance between and , with a crossing near and K = 14, the r-type resonance between and of , with a crossing at J = 58 and the accidental resonance between the levels of and of . We made a global fit of 4200 wavenumbers for and and 630 upper state energies of , deduced from previously transitions of and of . A few perturbation allowed transitions were also included in the fit, which gives a s. d. on residuals better than . 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
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 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 tetrad of the strongly prolate symmetric rotor propyne, , had been recorded in order to analyze some weaker The analysis, in particular for higher K transitions, indicated new problems: Interactions of members of the tetrad (, and ) with those of the dyad ( and ). More than 300 rotational transitions within have been recorded with and 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 . Inclusion of IR transitions from a reanalysis of the spectrum that was used in the study of the confirmed these findings. It turned out that the first order energies of the levels coincide to within with those of the overtone of the substate. A weaker Coriolis interaction occurs between and Meanwhile, the investigation of has been completed, and those of and have begun. Fermi-type interactions occur between the states and 3 and between and . Among the Coriolis resonances, the one between and permitted intervibrational transitions to be detected in the submillimeter region. These interactions are expected to improve , etc. for . 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