325 research outputs found
Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups
Generalizing a result of Conway, Sloane, and Wilkes for real reflection
groups, we show the Cayley graph of an imprimitive complex reflection group
with respect to standard generating reflections has a Hamiltonian cycle. This
is consistent with the long-standing conjecture that for every finite group, G,
and every set of generators, S, of G the undirected Cayley graph of G with
respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments,
to appear in Discrete Mathematic
On Cayley digraphs that do not have hamiltonian paths
We construct an infinite family of connected, 2-generated Cayley digraphs
Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the
generators a and b are arbitrarily large. We also prove that if G is any finite
group with |[G,G]| < 4, then every connected Cayley digraph on G has a
hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or
5).Comment: 10 pages, plus 14-page appendix of notes to aid the refere
Physics Without Physics: The Power of Information-theoretical Principles
David Finkelstein was very fond of the new information-theoretic paradigm of
physics advocated by John Archibald Wheeler and Richard Feynman. Only recently,
however, the paradigm has concretely shown its full power, with the derivation
of quantum theory (Chiribella et al., Phys. Rev. A 84:012311, 2011; D'Ariano et
al., 2017) and of free quantum field theory (D'Ariano and Perinotti, Phys. Rev.
A 90:062106, 2014; Bisio et al., Phys. Rev. A 88:032301, 2013; Bisio et al.,
Ann. Phys. 354:244, 2015; Bisio et al., Ann. Phys. 368:177, 2016) from
informational principles. The paradigm has opened for the first time the
possibility of avoiding physical primitives in the axioms of the physical
theory, allowing a refoundation of the whole physics over logically solid
grounds. In addition to such methodological value, the new
information-theoretic derivation of quantum field theory is particularly
interesting for establishing a theoretical framework for quantum gravity, with
the idea of obtaining gravity itself as emergent from the quantum information
processing, as also suggested by the role played by information in the
holographic principle (Susskind, J. Math. Phys. 36:6377, 1995; Bousso, Rev.
Mod. Phys. 74:825, 2002). In this paper I review how free quantum field theory
is derived without using mechanical primitives, including space-time, special
relativity, Hamiltonians, and quantization rules. The theory is simply provided
by the simplest quantum algorithm encompassing a countable set of quantum
systems whose network of interactions satisfies the three following simple
principles: homogeneity, locality, and isotropy. The inherent discrete nature
of the informational derivation leads to an extension of quantum field theory
in terms of a quantum cellular automata and quantum walks. A simple heuristic
argument sets the scale to the Planck one, and the observed regime is that of
small wavevectors ...Comment: 34 pages, 8 figures. Paper for in memoriam of David Finkelstei
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
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