45,846 research outputs found
Symmetry witnesses
A symmetry witness is a suitable subset of the space of selfadjoint trace
class operators that allows one to determine whether a linear map is a symmetry
transformation, in the sense of Wigner. More precisely, such a set is invariant
with respect to an injective densely defined linear operator in the Banach
space of selfadjoint trace class operators (if and) only if this operator is a
symmetry transformation. According to a linear version of Wigner's theorem, the
set of pure states, the rank-one projections, is a symmetry witness. We show
that an analogous result holds for the set of projections with a fixed rank
(with some mild constraint on this rank, in the finite-dimensional case). It
turns out that this result provides a complete classification of the set of
projections with a fixed rank that are symmetry witnesses. These particular
symmetry witnesses are projectable; i.e., reasoning in terms of quantum states,
the sets of uniform density operators of corresponding fixed rank are symmetry
witnesses too.Comment: 15 page
Symmetry witnesses
A symmetry witness is a suitable subset of the space of selfadjoint trace
class operators that allows one to determine whether a linear map is a symmetry
transformation, in the sense of Wigner. More precisely, such a set is invariant
with respect to an injective densely defined linear operator in the Banach
space of selfadjoint trace class operators (if and) only if this operator is a
symmetry transformation. According to a linear version of Wigner's theorem, the
set of pure states, the rank-one projections, is a symmetry witness. We show
that an analogous result holds for the set of projections with a fixed rank
(with some mild constraint on this rank, in the finite-dimensional case). It
turns out that this result provides a complete classification of the set of
projections with a fixed rank that are symmetry witnesses. These particular
symmetry witnesses are projectable; i.e., reasoning in terms of quantum states,
the sets of uniform density operators of corresponding fixed rank are symmetry
witnesses too.Comment: 15 page
Hierarchical hyperbolicity of graphs of multicurves
We show that many graphs naturally associated to a connected, compact,
orientable surface are hierarchically hyperbolic spaces in the sense of
Behrstock, Hagen and Sisto. They also automatically have the coarse median
property defined by Bowditch. Consequences for such graphs include a distance
formula analogous to Masur and Minsky's distance formula for the mapping class
group, an upper bound on the maximal dimension of quasiflats, and the existence
of a quadratic isoperimetric inequality. The hierarchically hyperbolic
structure also gives rise to a simple criterion for when such graphs are Gromov
hyperbolic.Comment: 27 pages, 4 figures. Minor changes from previous version. Addition of
appendix describing a hierarchically hyperbolic structure on the arc grap
Partially Punctual Metric Temporal Logic is Decidable
Metric Temporal Logic \mathsf{MTL}[\until_I,\since_I] is one of the most
studied real time logics. It exhibits considerable diversity in expressiveness
and decidability properties based on the permitted set of modalities and the
nature of time interval constraints . Henzinger et al., in their seminal
paper showed that the non-punctual fragment of called
is decidable. In this paper, we sharpen this decidability
result by showing that the partially punctual fragment of
(denoted ) is decidable over strictly monotonic finite point
wise time. In this fragment, we allow either punctual future modalities, or
punctual past modalities, but never both together. We give two satisfiability
preserving reductions from to the decidable logic
\mathsf{MTL}[\until_I]. The first reduction uses simple projections, while
the second reduction uses a novel technique of temporal projections with
oversampling. We study the trade-off between the two reductions: while the
second reduction allows the introduction of extra action points in the
underlying model, the equisatisfiable \mathsf{MTL}[\until_I] formula obtained
is exponentially succinct than the one obtained via the first reduction, where
no oversampling of the underlying model is needed. We also show that
is strictly more expressive than the fragments
\mathsf{MTL}[\until_I,\since] and \mathsf{MTL}[\until,\since_I]
Tensor decomposition and homotopy continuation
A computationally challenging classical elimination theory problem is to
compute polynomials which vanish on the set of tensors of a given rank. By
moving away from computing polynomials via elimination theory to computing
pseudowitness sets via numerical elimination theory, we develop computational
methods for computing ranks and border ranks of tensors along with
decompositions. More generally, we present our approach using joins of any
collection of irreducible and nondegenerate projective varieties
defined over . After computing
ranks over , we also explore computing real ranks. Various examples
are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix
multiplication with zeros. (26 pages, 1 figure
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