9,908 research outputs found
The Multiplicity Polar Theorem, collections of 1-forms and Chern numbers
In this work we show how the Multiplicity Polar Theorem can be used to
calculate Chern numbers for a collection of 1-forms.Comment: 27 page
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
The affine rank minimization problem consists of finding a matrix of minimum
rank that satisfies a given system of linear equality constraints. Such
problems have appeared in the literature of a diverse set of fields including
system identification and control, Euclidean embedding, and collaborative
filtering. Although specific instances can often be solved with specialized
algorithms, the general affine rank minimization problem is NP-hard. In this
paper, we show that if a certain restricted isometry property holds for the
linear transformation defining the constraints, the minimum rank solution can
be recovered by solving a convex optimization problem, namely the minimization
of the nuclear norm over the given affine space. We present several random
ensembles of equations where the restricted isometry property holds with
overwhelming probability. The techniques used in our analysis have strong
parallels in the compressed sensing framework. We discuss how affine rank
minimization generalizes this pre-existing concept and outline a dictionary
relating concepts from cardinality minimization to those of rank minimization
Lifting free divisors
Let be a morphism between smooth complex analytic spaces,
and let define a free divisor on . We prove that if the deformation
space of is a Cohen-Macaulay -module of
codimension 2, and all of the logarithmic vector fields for lift via
, then defines a free divisor on ; this is
generalized in several directions.
Among applications we recover a result of Mond-van Straten, generalize a
construction of Buchweitz-Conca, and show that a map
with critical set of codimension
has a with the desired properties. Finally, if is a
representation of a reductive complex algebraic group and is the
algebraic quotient with smooth, we describe sufficient
conditions for to be Cohen-Macaulay of codimension . In one such
case, a free divisor on lifts under the operation of
"castling" to a free divisor on , partially generalizing
work of Granger-Mond-Schulze on linear free divisors. We give several other
examples of such representations.Comment: 30 pages. Many minor changes from v1 in response to a thorough review
process. To appear in Proc. London Math. Soc. This version differs from the
final published versio
Cokernels of random matrices satisfy the Cohen-Lenstra heuristics
Let A be an n by n random matrix with iid entries taken from the p-adic
integers or Z/NZ. Then under mild non-degeneracy conditions the cokernel of A
has a universal probability distribution. In particular, the p-part of an iid
random matrix over the integers has cokernel distributed according to the
Cohen-Lenstra measure up to an exponentially small error.Comment: 21 pages; submitte
Bruhat Order in the Full Symmetric Toda Lattice on Partial Flag Space
In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described
the asymptotic behaviour of trajectories of the full symmetric
Toda lattice in the case of distinct eigenvalues of the Lax
matrix. It turned out that it is completely determined by the Bruhat order on
the permutation group. In the present paper we extend this result to the case
when some eigenvalues of the Lax matrix coincide. In that case the trajectories
are described in terms of the projection to a partial flag space where the
induced dynamical system verifies the same properties as before: we show that
when the trajectories of the induced dynamical system converge
to a finite set of points in the partial flag space indexed by the Schubert
cells so that any two points of this set are connected by a trajectory if and
only if the corresponding cells are adjacent. This relation can be explained in
terms of the Bruhat order on multiset permutations
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