29,615 research outputs found
Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals
An ideal of a local polynomial ring can be described by calculating a
standard basis with respect to a local monomial ordering. However standard
basis algorithms are not numerically stable. Instead we can describe the ideal
numerically by finding the space of dual functionals that annihilate it,
reducing the problem to one of linear algebra. There are several known
algorithms for finding the truncated dual up to any specified degree, which is
useful for describing zero-dimensional ideals. We present a stopping criterion
for positive-dimensional cases based on homogenization that guarantees all
generators of the initial monomial ideal are found. This has applications for
calculating Hilbert functions.Comment: 19 pages, 4 figure
An algebraic approach to the scattering equations
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism
Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support
Let be an infinite cardinal number and let
denote the subspace of consisting of all functions that
assume at most countably many non-zero values. We classify all infinite
dimensional complemented subspaces of , proving that
they are isomorphic to for some cardinal number
. Then we show that the Banach algebra of all bounded linear operators
on or has the unique maximal
ideal consisting of operators through which the identity operator does not
factor. Using similar techniques, we obtain an alternative to Daws' approach
description of the lattice of all closed ideals of , where or for some , and we
classify the closed ideals of that
contains the ideal of weakly compact operators.Comment: 15 pp., to appear in Proc. Amer. Math. So
Inner Ideals of Simple Locally Finite Lie Algebras
Inner ideals of simple locally finite dimensional Lie algebras over an
algebraically closed field of characteristic 0 are described. In particular, it
is shown that a simple locally finite dimensional Lie algebra has a non-zero
proper inner ideal if and only if it is of diagonal type. Regular inner ideals
of diagonal type Lie algebras are characterized in terms of left and right
ideals of the enveloping algebra. Regular inner ideals of finitary simple Lie
algebras are described
Identities of finitely generated graded algebras with involution
We consider associative algebras with involution graded by a finite abelian
group G over a field of characteristic zero. Suppose that the involution is
compatible with the grading. We represent conditions permitting
PI-representability of such algebras. Particularly, it is proved that a
finitely generated (Z/qZ)-graded associative PI-algebra with involution
satisfies exactly the same graded identities with involution as some finite
dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4.
This is an analogue of the theorem of A.Kemer for ordinary identities, and an
extension of the result of the author for identities with involution. The
similar results were proved also recentely for graded identities
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