38,156 research outputs found
Lorenz, G\"{o}del and Penrose: New perspectives on determinism and causality in fundamental physics
Despite being known for his pioneering work on chaotic unpredictability, the
key discovery at the core of meteorologist Ed Lorenz's work is the link between
space-time calculus and state-space fractal geometry. Indeed, properties of
Lorenz's fractal invariant set relate space-time calculus to deep areas of
mathematics such as G\"{o}del's Incompleteness Theorem. These properties,
combined with some recent developments in theoretical and observational
cosmology, motivate what is referred to as the `cosmological invariant set
postulate': that the universe can be considered a deterministic dynamical
system evolving on a causal measure-zero fractal invariant set in its
state space. Symbolic representations of are constructed explicitly based
on permutation representations of quaternions. The resulting `invariant set
theory' provides some new perspectives on determinism and causality in
fundamental physics. For example, whilst the cosmological invariant set appears
to have a rich enough structure to allow a description of quantum probability,
its measure-zero character ensures it is sparse enough to prevent invariant set
theory being constrained by the Bell inequality (consistent with a partial
violation of the so-called measurement independence postulate). The primacy of
geometry as embodied in the proposed theory extends the principles underpinning
general relativity. As a result, the physical basis for contemporary programmes
which apply standard field quantisation to some putative gravitational
lagrangian is questioned. Consistent with Penrose's suggestion of a
deterministic but non-computable theory of fundamental physics, a
`gravitational theory of the quantum' is proposed based on the geometry of
, with potential observational consequences for the dark universe.Comment: This manuscript has been accepted for publication in Contemporary
Physics and is based on the author's 9th Dennis Sciama Lecture, given in
Oxford and Triest
Formal Analysis of CRT-RSA Vigilant's Countermeasure Against the BellCoRe Attack: A Pledge for Formal Methods in the Field of Implementation Security
In our paper at PROOFS 2013, we formally studied a few known countermeasures
to protect CRT-RSA against the BellCoRe fault injection attack. However, we
left Vigilant's countermeasure and its alleged repaired version by Coron et al.
as future work, because the arithmetical framework of our tool was not
sufficiently powerful. In this paper we bridge this gap and then use the same
methodology to formally study both versions of the countermeasure. We obtain
surprising results, which we believe demonstrate the importance of formal
analysis in the field of implementation security. Indeed, the original version
of Vigilant's countermeasure is actually broken, but not as much as Coron et
al. thought it was. As a consequence, the repaired version they proposed can be
simplified. It can actually be simplified even further as two of the nine
modular verifications happen to be unnecessary. Fortunately, we could formally
prove the simplified repaired version to be resistant to the BellCoRe attack,
which was considered a "challenging issue" by the authors of the countermeasure
themselves.Comment: arXiv admin note: substantial text overlap with arXiv:1401.817
The largest singletons of set partitions
Recently, Deutsch and Elizalde studied the largest and the smallest fixed
points of permutations. Motivated by their work, we consider the analogous
problems in set partitions. Let denote the number of partitions of
with the largest singleton for .
In this paper, several explicit formulas for , involving a
Dobinski-type analog, are obtained by algebraic and combinatorial methods, many
combinatorial identities involving and Bell numbers are presented by
operator methods, and congruence properties of are also investigated.
It will been showed that the sequences and
(mod ) are periodic for any prime , and contain a
string of consecutive zeroes. Moreover their minimum periods are
conjectured to be for any prime .Comment: 14page
Surface Entanglement in Quantum Spin Networks
We study the ground-state entanglement in systems of spins forming the
boundary of a quantum spin network in arbitrary geometries and dimensionality.
We show that as long as they are weakly coupled to the bulk of the network, the
surface spins are strongly entangled, even when distant and non directly
interacting, thereby generalizing the phenomenon of long-distance entanglement
occurring in quantum spin chains. Depending on the structure of the couplings
between surface and bulk spins, we discuss in detail how the patterns of
surface entanglement can range from multi-pair bipartite to fully multipartite.
In the context of quantum information and communication, these results find
immediate application to the implementation of quantum routers, that is devices
able to distribute quantum correlations on demand among multiple network nodes.Comment: 8 pages, 8 figure
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