17,747 research outputs found
Distributive Lattices, Polyhedra, and Generalized Flow
A D-polyhedron is a polyhedron such that if are in then so are
their componentwise max and min. In other words, the point set of a
D-polyhedron forms a distributive lattice with the dominance order. We provide
a full characterization of the bounding hyperplanes of D-polyhedra.
Aside from being a nice combination of geometric and order theoretic
concepts, D-polyhedra are a unifying generalization of several distributive
lattices which arise from graphs. In fact every D-polyhedron corresponds to a
directed graph with arc-parameters, such that every point in the polyhedron
corresponds to a vertex potential on the graph. Alternatively, an edge-based
description of the point set can be given. The objects in this model are dual
to generalized flows, i.e., dual to flows with gains and losses.
These models can be specialized to yield some cases of distributive lattices
that have been studied previously. Particular specializations are: lattices of
flows of planar digraphs (Khuller, Naor and Klein), of -orientations of
planar graphs (Felsner), of c-orientations (Propp) and of -bonds of
digraphs (Felsner and Knauer). As an additional application we exhibit a
distributive lattice structure on generalized flow of breakeven planar
digraphs.Comment: 17 pages, 3 figure
Distributive Laws and Decidable Properties of SOS Specifications
Some formats of well-behaved operational specifications, correspond to
natural transformations of certain types (for example, GSOS and coGSOS laws).
These transformations have a common generalization: distributive laws of monads
over comonads. We prove that this elegant theoretical generalization has
limited practical benefits: it does not translate to any concrete rule format
that would be complete for specifications that contain both GSOS and coGSOS
rules. This is shown for the case of labeled transition systems and
deterministic stream systems.Comment: In Proceedings EXPRESS/SOS 2014, arXiv:1408.127
The projective geometry of a group
We show that the pair given by the power set and by the "Grassmannian"(set of
all subgroups) of an arbitrary group behaves very much like the pair given by a
projective space and its dual projective space. More precisely, we generalize
several results from preceding joint work with M. Kinyon (arXiv:0903.5441),
which concerned abelian groups, to the case of general non-abelian groups. Most
notably, pairs of subgroups parametrize torsor and semitorsor structures on the
power set. The r\^ole of associative algebras and -pairs from loc. cit. is now
taken by analogs of near-rings
A discussion on the origin of quantum probabilities
We study the origin of quantum probabilities as arising from non-boolean
propositional-operational structures. We apply the method developed by Cox to
non distributive lattices and develop an alternative formulation of
non-Kolmogorvian probability measures for quantum mechanics. By generalizing
the method presented in previous works, we outline a general framework for the
deduction of probabilities in general propositional structures represented by
lattices (including the non-distributive case).Comment: Improved versio
Coalgebraic Behavioral Metrics
We study different behavioral metrics, such as those arising from both
branching and linear-time semantics, in a coalgebraic setting. Given a
coalgebra for a functor , we define a framework for deriving pseudometrics on which
measure the behavioral distance of states.
A crucial step is the lifting of the functor on to a
functor on the category of pseudometric spaces.
We present two different approaches which can be viewed as generalizations of
the Kantorovich and Wasserstein pseudometrics for probability measures. We show
that the pseudometrics provided by the two approaches coincide on several
natural examples, but in general they differ.
If has a final coalgebra, every lifting yields in a
canonical way a behavioral distance which is usually branching-time, i.e., it
generalizes bisimilarity. In order to model linear-time metrics (generalizing
trace equivalences), we show sufficient conditions for lifting distributive
laws and monads. These results enable us to employ the generalized powerset
construction
Distributivity and deformation of the reals from Tsallis entropy
We propose a one-parameter family \ \ of deformations of the
reals, which is motivated by the generalized additivity of the Tsallis entropy.
We introduce a generalized multiplication which is distributive with respect to
the generalized addition of the Tsallis entropy. These operations establish a
one-parameter family of field isomorphisms \ \ between \
\ and \ \ through which an absolute value on \ \
is introduced. This turns out to be a quasisymmetric map, whose metric and
measure-theoretical implications are pointed out.Comment: 16 pages, Standard LaTeX2e, To be published in Physica
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