5,103 research outputs found
Discrete Variational Derivative Methods for the EPDiff equation
The aim of this paper is the derivation of structure preserving schemes for
the solution of the EPDiff equation, with particular emphasis on the two
dimensional case. We develop three different schemes based on the Discrete
Variational Derivative Method (DVDM) on a rectangular domain discretized with a
regular, structured, orthogonal grid.
We present numerical experiments to support our claims: we investigate the
preservation of energy and linear momenta, the reversibility, and the empirical
convergence of the schemes. The quality of our schemes is finally tested by
simulating the interaction of singular wave fronts.Comment: 41 pages, 41 figure
Reversibility and the structure of the local state space
The richness of quantum theory's reversible dynamics is one of its unique
operational characteristics, with recent results suggesting deep links between
the theory's reversible dynamics, its local state space and the degree of
non-locality it permits. We explore the delicate interplay between these
features, demonstrating that reversibility places strong constraints on both
the local and global state space. Firstly, we show that reversible dynamics are
trivial (composed of local transformations and permutations of subsytems) in
maximally non-local theories whose local state spaces satisfy a dichotomy
criterion; this applies to a range of operational models that have previously
been studied, such as d-dimensional "hyperballs" and almost all regular
polytope systems. By separately deriving a similar result for odd-sided
polygons, we show that classical systems are the only regular polytope state
spaces whose maximally non-local composites allow for non-trivial reversible
dynamics. Secondly, we show that non-trivial reversible dynamics do exist in
maximally non-local theories whose state spaces are reducible into two or more
smaller spaces. We conjecture that this is a necessary condition for the
existence of such dynamics, but that reversible entanglement generation remains
impossible even in this scenario.Comment: 12+epsilon pages, 2 figure
Fredkin Gates for Finite-valued Reversible and Conservative Logics
The basic principles and results of Conservative Logic introduced by Fredkin
and Toffoli on the basis of a seminal paper of Landauer are extended to
d-valued logics, with a special attention to three-valued logics. Different
approaches to d-valued logics are examined in order to determine some possible
universal sets of logic primitives. In particular, we consider the typical
connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As
a result, some possible three-valued and d-valued universal gates are described
which realize a functionally complete set of fundamental connectives.Comment: 57 pages, 10 figures, 16 tables, 2 diagram
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