14,313 research outputs found
Racks and blocked braids
In the paper Blocked-braid Groups, submitted to Applied Categorical
Structures, the present authors together with Davide Maglia introduced the
blocked-braid groups BB_n on n strands, and proved that a blocked torsion has
order either 2 or 4. We conjectured that the order was actually 4 but our
methods in that paper, which involved introducing for any group G a braided
monoidal category of tangled relations, were inadequate to demonstrate this
fact. Subsequently Davide Maglia in unpublished work investigated exactly what
part of the structure and properties of a group G are needed to permit the
construction of a braided monoidal category with a tangle algebra and was able
to distinguish blocked two-torsions from the identity.
In this paper we present a simplification of his answer, which turns out to
be related to the notion of rack. We show that if G is a rack then there is a
braided monoidal category TRel_G generalizing that of the above paper. Further
we introduce a variation of the notion of rack which we call irack which yields
a tangle algebra in TRel_G. Iracks are in particular racks but have in addition
to the operations abstracting group conjugation also a unary operation
abstracting group inverse. Using iracks we obtain new invariants for tangles
and blocked braids permitting us to present a proof of Maglia's result that a
blocked double torsion is not the identity.
This work was presented at the Conference in Memory of Aurelio Carboni,
Milan, 24-26 June 2013
The AF structure of non commutative toroidal Z/4Z orbifolds
For any irrational theta and rational number p/q such that q|qtheta-p|<1, a
projection e of trace q|qtheta-p| is constructed in the the irrational rotation
algebra A_theta that is invariant under the Fourier transform. (The latter is
the order four automorphism U mapped to V, V mapped to U^{-1}, where U, V are
the canonical unitaries generating A_theta.) Further, the projection e is
approximately central, the cut down algebra eA_theta e contains a Fourier
invariant q x q matrix algebra whose unit is e, and the cut downs eUe, eVe are
approximately inside the matrix algebra. (In particular, there are Fourier
invariant projections of trace k|qtheta-p| for k=1,...,q.) It is also shown
that for all theta the crossed product A_theta rtimes Z_4 satisfies the
Universal Coefficient Theorem. (Z_4 := Z/4Z.) As a consequence, using the
Classification Theorem of G. Elliott and G. Gong for AH-algebras, a theorem of
M. Rieffel, and by recent results of H. Lin, we show that A_theta rtimes Z_4 is
an AF-algebra for all irrational theta in a dense G_delta.Comment: 35 page
Tangled Circuits
The theme of the paper is the use of commutative Frobenius algebras in
braided strict monoidal categories in the study of varieties of circuits and
communicating systems which occur in Computer Science, including circuits in
which the wires are tangled. We indicate also some possible novel geometric
interest in such algebras
Comparison of hydrogen and methane as coolants in regeneratively cooled panels
Comparison of hydrogen and methane as coolants in regeneratively cooled panel
Blocked-braid Groups
We introduce and study a family of groups , called the
blocked-braid groups, which are quotients of Artin's braid groups
, and have the corresponding symmetric groups as
quotients. They are defined by adding a certain class of geometrical
modifications to braids. They arise in the study of commutative Frobenius
algebras and tangle algebras in braided strict monoidal categories. A
fundamental equation true in is Dirac's Belt Trick; that
torsion through is equal to the identity. We show that
is finite for and 3 but infinite for
Bicategories of spans as cartesian bicategories
Bicategories of spans are characterized as cartesian bicategories in which
every comonad has an Eilenberg-Moore ob ject and every left adjoint arrow is
comonadic
Quantum dynamics of the avian compass
The ability of migratory birds to orient relative to the Earth's magnetic
field is believed to involve a coherent superposition of two spin states of a
radical electron pair. However, the mechanism by which this coherence can be
maintained in the face of strong interactions with the cellular environment has
remained unclear. This Letter addresses the problem of decoherence between two
electron spins due to hyperfine interaction with a bath of spin 1/2 nuclei.
Dynamics of the radical pair density matrix are derived and shown to yield a
simple mechanism for sensing magnetic field orientation. Rates of dephasing and
decoherence are calculated ab initio and found to yield millisecond coherence
times, consistent with behavioral experiments
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