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.