direct sum

Since A is g-Drazin invertible, by Theorem (1), X = R (I - P) [direct sum] N(I - P), A = [A.sub.1] [direct sum] [A.sub.2], where [A.sub.1] is closed invertible and [A.sub.2] is bounded quasinilpotent with respect to the direct sum. Therefore Problem (2) has a unique solution if and only if each of the following two initial value problems has a unique solution on R(I - P) and R(P), respectively

If [B.sub.i] is the direct sum of uniserial modules of length i and x ([not equal to] 0) [member of] [B.sub.i,] then

If furthermore, the family M is closed under formation of direct sums, then k(M) is a Hopf algebra, with product induced by direct sum.

Therefore [G.sub.V] can be written as a direct sum of of its right cosets.

The direct sum structure [C.sup.nxn] = S [direct sum] A and the Pythagorean relationship (2.2) imply that

The author has organized this work into twenty-seven chapters covering direct sum decompositions, subfield structures, modules for alternating groups, and many other related topics.

The direct sum of two permutations [alpha] and [beta], denoted [alpha] [direct sum] [beta], is the concatenation of [alpha] and [beta]', where [beta]' is obtained from [beta] by adding to each of its letters the largest letter of [alpha].

Direct sum decompositions of torsion-free finite rank groups.

In fact an arbitrary matrix of finite order is, up to conjugation, a direct sum of primitives (Theorem 2.7).

His topics are admissible topological rings, the C-completion of an abstract module over a topological ring, the case of an admissible topological ring, the higher C-completions, the direct sum and direct limit of C-completion left A-modules, and ext and tor in the categories of C-complete left A-modules.

For C(K) equipped with the Whitney inner product, there is an orthogonal direct sum decomposition

If V is a direct sum of irreducible kG-submodules, then we call V (as well as G and [phi]) completely reducible.

We can prove that the Drazin spectrum satisfies the spectral mapping theorem, and the Drazin spectrum of a direct sum is the union of the Drazin spectrum of the components.

They achieve positive results by placing restrictive hypotheses on only a small subset of the complement submodules and explain why direct sum decomposition of various kinds occurs.

The module m decomposes into a direct sum of three Ad(K)-invariant irreducible modules pairwise orthogonal with respect to B, i.e.