Query: BOUNDS FOR MULTIPLICITIES

title: Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters
abstract: This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second order self adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e. the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schrodinger operator with periodic boundary conditions of the form V$ discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

title: Bounds for multiplicities of automorphic representations

title: Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms
abstract: Let Goo be a semisimple real Lie group with unitary dual Goo We produce new upper bounds for the multiplicities with which representations e of cohomological type appear in certain spaces of cusp forms on Goo* The main new idea is to apply noncommutative Iwasawa theory to certain p adic completions of the cohomology of locally symmetric spaces.

title: Bounds for multiplicities

title: Bounds for multiplicities of automorphic forms of cohomological type on GL_2
abstract: We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological automorphic forms and a conditional power saving for their multiplicities when the associated locally symmetric space is a hyperbolic 3 manifold.

title: Note on bounds for multiplicities
abstract: Let S K x 1 x n be a polynomial ring and R S I be a graded K algebra where I S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S We prove the conjecture in the case that codim( R =2 which generalizes results in ( (2003) 201;  (1998) 2879) We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.

title: Integral closure and bounds for quotients of multiplicities of monomial ideals
abstract: Given a pair of monomial ideals I and J of finite colength of the ring of analytic function germs C n 0 C we prove that some power of I admits a reduction formed by homogeneous polynomials with respect to the Newton filtration induced by J if and only if the quotient of multiplicities e I e J attains a suitable upper bound expressed in terms of the Newton polyhedra of I and J. We also explore other connections between mixed multiplicities, Newton filtrations and the integral closure of ideals.

title: Lower bounds on Hilbert Kunz multiplicities and maximal F signatures
abstract: Hilbert Kunz multiplicity and F signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e. an upper bound, for F signature and revisit lower bounds on Hilbert Kunz multiplicity.

title: Upper bounds for multiplicities in a tensor product of representations and in a restriction of group representations to a subgroup
abstract: Upper bounds for multiplicities of irreducible representations of a semisimple compact Lie group in the tensor product of representations and multiplicities of irreducible representations of a subgroup in an irreducible representation of a group are derived. Pairs of groups and subgroups appear from the representation theory of non compact groups.

title: On lower bounds for s multiplicities
abstract: A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert Samuel and Hilbert Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe Yoshida minimality conjectures for unmixed singular rings.