direct sum


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direct sum

[də¦rekt ′səm]
(mathematics)
If each of the sets in a finite direct product of sets has a group structure, this structure may be imposed on the direct product by defining the composition “componentwise”; the resulting group is called the direct sum.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
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.

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