592 research outputs found
Nonstandard Hulls of Locally Exponential Lie Algebras
We show how to construct the nonstandard hull of certain infinite-dimensional
Lie algebras in order to generalize a theorem of Pestov on the enlargeability
of Banach-Lie algebras. In the process, we consider a nonstandard smoothness
condition on functions between locally convex spaces to ensure that the induced
function between the nonstandard hulls is smooth. We also discuss some
conditions on a function between locally convex spaces which guarantee that its
nonstandard extension maps finite points to finite points
Parameter test ideals of Cohen Macaulay rings
We describe an algorithm for computing parameter-test-ideals in certain local
Cohen-Macaulay rings. The algorithm is based on the study of a Frobenius map on
the injective hull of the residue field of the ring and on the application of
Rodney Sharp's notion of ``special ideals''.
Our techniques also provide an algorithm for computing indices of nilpotency
of Frobenius actions on top local cohomology modules of the ring and on the
injective hull of its residue field. The study of nilpotent elements on
injective hulls of residue fields also yields a great simplification of the
proof of the fact that for a power series ring of prime characteristic, for
all nonzero , generates as a -module.Comment: 16 pages To appear in Compositio Mathematic
Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata
We investigate the low-noise regime of a large class of probabilistic
cellular automata, including the North-East-Center model of Toom. They are
defined as stochastic perturbations of cellular automata belonging to the
category of monotonic binary tessellations and possessing a property of
erosion. We prove, for a set of initial conditions, exponential convergence of
the induced processes toward an extremal invariant measure with a highly
predominant spin value. We also show that this invariant measure presents
exponential decay of correlations in space and in time and is therefore
strongly mixing.Comment: 21 pages, 0 figure, revised version including a generalization to a
larger class of models, structure of the arguments unchanged, minor changes
suggested by reviewers, added reference
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Given a set of spheres in , with and
odd, having a fixed number of distinct radii , we
show that the worst-case combinatorial complexity of the convex hull
of is
, where
is the number of spheres in with radius .
To prove the lower bound, we construct a set of spheres in
, with odd, where spheres have radius ,
, and , such that their convex hull has combinatorial
complexity
.
Our construction is then generalized to the case where the spheres have
distinct radii.
For the upper bound, we reduce the sphere convex hull problem to the problem
of computing the worst-case combinatorial complexity of the convex hull of a
set of -dimensional convex polytopes lying on parallel hyperplanes
in , where odd, a problem which is of independent
interest. More precisely, we show that the worst-case combinatorial complexity
of the convex hull of a set
of -dimensional convex polytopes lying on parallel hyperplanes of
is
, where
is the number of vertices of .
We end with algorithmic considerations, and we show how our tight bounds for
the parallel polytope convex hull problem, yield tight bounds on the
combinatorial complexity of the Minkowski sum of two convex polytopes in
.Comment: 22 pages, 5 figures, new proof of upper bound for the complexity of
the convex hull of parallel polytopes (the new proof gives upper bounds for
all face numbers of the convex hull of the parallel polytopes
Affine hom-complexes
For two general polytopal complexes the set of face-wise affine maps between
them is shown to be a polytopal complex in an algorithmic way. The resulting
algorithm for the affine hom-complex is analyzed in detail. There is also a
natural tensor product of polytopal complexes, which is the left adjoint
functor for Hom. This extends the corresponding facts from single polytopes,
systematic study of which was initiated in [6,12]. Explicit examples of
computations of the resulting structures are included. In the special case of
simplicial complexes, the affine hom-complex is a functorial subcomplex of
Kozlov's combinatorial hom-complex [14], which generalizes Lovasz' well-known
construction [15] for graphs.Comment: final version, to appear in Portugaliae Mathematic
Abstract commensurators of lattices in Lie groups
Let Gamma be a lattice in a simply-connected solvable Lie group. We construct
a Q-defined algebraic group A such that the abstract commensurator of Gamma is
isomorphic to A(Q) and Aut(Gamma) is commensurable with A(Z). Our proof uses
the algebraic hull construction, due to Mostow, to define an algebraic group H
so that commensurations of Gamma extend to Q-defined automorphisms of H. We
prove an analogous result for lattices in connected linear Lie groups whose
semisimple quotient satisfies superrigidity.Comment: 35 pages. v4 added references and improved exposition based on
referee's comments. Added results 6.4 and 6.5 relating abstract commensurator
of lattice to automorphism group of some ambient Lie group. Final version, to
appear in Comm. Math. Hel
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