132 research outputs found
Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics
Before we dive into the accessibility stream of nowadays indicatory
applications of octonions to computer and other sciences and to quantum physics
let us focus for a while on the crucially relevant events for todays revival on
interest to nonassociativity. Our reflections keep wandering back to the
two square identity and then via the four
square identity up to the eight square identity.
These glimpses of history incline and invite us to retell the story on how
about one month after quaternions have been carved on the bridge
octonions were discovered by , jurist and
mathematician, a friend of . As for today we just
mention en passant quaternionic and octonionic quantum mechanics,
generalization of equations for octonions and triality
principle and group in spinor language in a descriptive way in order not
to daunt non specialists. Relation to finite geometries is recalled and the
links to the 7stones of seven sphere, seven imaginary octonions units in out of
the cave reality applications are appointed . This way we are welcomed
back to primary ideas of , and other distinguished
fathers of quantum mechanics and quantum gravity foundations.Comment: 26 pages, 7 figure
MUBs inequivalence and affine planes
There are fairly large families of unitarily inequivalent complete sets of
N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The
number of such sets is not bounded above by any polynomial as a function of N.
While it is standard that there is a superficial similarity between complete
sets of MUBs and finite affine planes, there is an intimate relationship
between these large families and affine planes. This note briefly summarizes
"old" results that do not appear to be well-known concerning known families of
complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical
Physics 53, 032204 (2012) except for format changes due to the journal's
style policie
Fast-Decodable Asymmetric Space-Time Codes from Division Algebras
Multiple-input double-output (MIDO) codes are important in the near-future
wireless communications, where the portable end-user device is physically small
and will typically contain at most two receive antennas. Especially tempting is
the 4 x 2 channel due to its immediate applicability in the digital video
broadcasting (DVB). Such channels optimally employ rate-two space-time (ST)
codes consisting of (4 x 4) matrices. Unfortunately, such codes are in general
very complex to decode, hence setting forth a call for constructions with
reduced complexity.
Recently, some reduced complexity constructions have been proposed, but they
have mainly been based on different ad hoc methods and have resulted in
isolated examples rather than in a more general class of codes. In this paper,
it will be shown that a family of division algebra based MIDO codes will always
result in at least 37.5% worst-case complexity reduction, while maintaining
full diversity and, for the first time, the non-vanishing determinant (NVD)
property. The reduction follows from the fact that, similarly to the Alamouti
code, the codes will be subsets of matrix rings of the Hamiltonian quaternions,
hence allowing simplified decoding. At the moment, such reductions are among
the best known for rate-two MIDO codes. Several explicit constructions are
presented and shown to have excellent performance through computer simulations.Comment: 26 pages, 1 figure, submitted to IEEE Trans. Inf. Theory, October
201
Multipole analysis in cosmic topology
Low multipole amplitudes in the Cosmic Microwave Background CMB radiation can
be explained by selection rules from the underlying multiply-connected
homotopy. We apply a multipole analysis to the harmonic bases and introduce
point symmetry.We give explicit results for two cubic 3-spherical manifolds and
lowest polynomial degrees, and derive three new spherical 3-manifolds.Comment: 15 pages, with figure
Perfect Space–Time Block Codes
In this paper, we introduce the notion of perfect space–time block codes (STBCs). These codes have full-rate, full-diversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas
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