Since A is g-Drazin invertible, X = R(I-P)[direct sum] N(I-P), A = [A.sub.1] [direct sum] [A.sub.2], where [A.sub.1] is closed and invertible and [A.sub.2] is bounded and quasinilpotent with respect to the direct sum
. Therefore (3) has a solution if and only if each of the following two initial value problems has a solution on R(I - P) and R(P), respectively
Let [B.sub.i] be the direct sum
of uniserial modules of length i and x, y [member of] [B.sub.i].
Proof: One can check these identities directly from the definitions of direct sum
, restriction and deletion for matroids (see again the previous section).
This is an indefinite inner product space which has the structure of the direct sum
of a Hilbert space and a negative Hilbert space.
1 and 2] give recurrences that reduce the computation of the Mobius function [mu]([sigma], [tau]) to Mobius function calculations of the form [mu]([sigma]', [tau]') where [tau]' is a single component of [tau] and [sigma]' is a direct sum
of consecutive components of [sigma].
Faticoni (mathematics, Fordham U.) explores advanced topics in direct sum
decompositions of abelian groups and their consequences.
The direct sum
notation ([direct sum]) was introduced in the introduction.
The (canonical) isotypic decomposition of V is the direct sum
of its kG-homogeneous components.
We define their direct sum
M [direct sum] M' on the ground set E [??] E' by
Hereditary Noetherian prime rings may be the only non-commutative Noetherian rings whose projective modules, both finitely and infinitely generated, have nontrivial direct sum
behavior and a structure theorem describing that behavior, say mathematicians Levy (U.
We may assume without loss of generality that the norm of the direct sum
Y [direct sum] X is in fact equal to the maximum norm Y [[direct sum].sub.[infinity]] X.
We show that such a polytope is lattice equivalent to a direct sum
of del Pezzo polytopes, pseudo del Pezzo polytopes, or a (possibly skew) bipyramid over (pseudo) del Pezzo polytopes.