Colombeau products of distributions
In this paper, some products of distributions are derived. The results are obtained in Colombeau algebra of generalized functions, which is the most relevant algebraic construction for dealing with Schwartz distributions. Colombeau algebra $${\mathcal{G}}({\mathbf{R}})$$G(R) contains the space $${\mathcal{D}'}({\mathbf{R}})$$D′(R) of Schwartz distributions as a subspace, and has a notion of 'association’ that allows us to evaluate the results in terms of distributions.
On Products of Distributions in Colombeau Algebra
Results on products of distributions and are derived. They are obtained in Colombeau differential algebra of generalized functions that contains the space of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.
Super-additive sequences and algebras of polynomials
. If K is a ﬁeld with discrete valuation ν and D = { a ∈ K : ν ( a ) ≥ 0 } , then an algebra D [ x ] ⊆ A ⊆ K [ x ] has associated to it a sequence of fractional ideals {I n : n = 0 , 1 , 2 ,... } with I n consisting of 0 and the leading coeﬃcients of elements of A of degree no more than n and a sequence of integers { a ( n ) : n = 0 , 1 , 2 ,... } with a ( n ) = − ν ( I n ). Combinatorial properties of this integer sequence reﬂect algebraic properties of A , and these are used to identify the degrees of generators of A and to characterize ﬁnitely generated algebras A by a periodicity property of this sequence.