3,484 research outputs found
A variation principle for ground spaces
The ground spaces of a vector space of hermitian matrices, partially ordered
by inclusion, form a lattice constructible from top to bottom in terms of
intersections of maximal ground spaces. In this paper we characterize the
lattice elements and the maximal lattice elements within the set of all
subspaces using constraints on operator cones. Our results contribute to the
geometry of quantum marginals, as their lattices of exposed faces are
isomorphic to the lattices of ground spaces of local Hamiltonians.Comment: 18 pages, 2 figures, version v3 has an improved exposition, v4 has a
new non-commutative example and catches a glimpse of three qubit
Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions
We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite
dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka
and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of
coideal subalgebras. We reduce the study to KD(n) by proving that the others
are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We
derive many examples of lattices of intermediate subfactors of the inclusions
of depth 2 associated to those Kac algebras, as well as the corresponding
principal graphs, which is the original motivation.
Along the way, we extend some general results on the Galois correspondence
for depth 2 inclusions, and develop some tools and algorithms for the study of
twisted group algebras and their lattices of coideal subalgebras. This research
was driven by heavy computer exploration, whose tools and methodology we
further describe.Comment: v1: 84 pages, 13 figures, submitted. v2: 94 pages, 15 figures, added
connections with Masuoka's families KA and KB, description of K3 in KD(n),
lattices for KD(8) and KD(15). v3: 93 pages, 15 figures, proven lattice for
KD(6), misc improvements, accepted for publication in Journal of Algebra and
Its Application
Space Efficient Breadth-First and Level Traversals of Consistent Global States of Parallel Programs
Enumerating consistent global states of a computation is a fundamental
problem in parallel computing with applications to debug- ging, testing and
runtime verification of parallel programs. Breadth-first search (BFS)
enumeration is especially useful for these applications as it finds an
erroneous consistent global state with the least number of events possible. The
total number of executed events in a global state is called its rank. BFS also
allows enumeration of all global states of a given rank or within a range of
ranks. If a computation on n processes has m events per process on average,
then the traditional BFS (Cooper-Marzullo and its variants) requires
space in the worst case, whereas ou r
algorithm performs the BFS requires space. Thus, we
reduce the space complexity for BFS enumeration of consistent global states
exponentially. and give the first polynomial space algorithm for this task. In
our experimental evaluation of seven benchmarks, traditional BFS fails in many
cases by exhausting the 2 GB heap space allowed to the JVM. In contrast, our
implementation uses less than 60 MB memory and is also faster in many cases
Complexity Measures from Interaction Structures
We evaluate new complexity measures on the symbolic dynamics of coupled tent
maps and cellular automata. These measures quantify complexity in terms of
-th order statistical dependencies that cannot be reduced to interactions
between units. We demonstrate that these measures are able to identify
complex dynamical regimes.Comment: 11 pages, figures improved, minor changes to the tex
The power of linear programming for general-valued CSPs
Let , called the domain, be a fixed finite set and let , called
the valued constraint language, be a fixed set of functions of the form
, where different functions might have
different arity . We study the valued constraint satisfaction problem
parametrised by , denoted by VCSP. These are minimisation
problems given by variables and the objective function given by a sum of
functions from , each depending on a subset of the variables.
Finite-valued constraint languages contain functions that take on only rational
values and not infinite values.
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation (BLP). For a valued constraint language , BLP is a decision
procedure for if and only if admits a symmetric fractional
polymorphism of every arity. For a finite-valued constraint language ,
BLP is a decision procedure if and only if admits a symmetric
fractional polymorphism of some arity, or equivalently, if admits a
symmetric fractional polymorphism of arity 2.
Using these results, we obtain tractability of several novel classes of
problems, including problems over valued constraint languages that are: (1)
submodular on arbitrary lattices; (2) -submodular on arbitrary finite
domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors
(arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213)
to appear in SIAM Journal on Computing (SICOMP
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