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On the Validity of the Direct Sum Conjecture
SIAM Journal on Computing, 1986 The direct sum conjecture states that the multiplicative complexity of disjoint sets of bilinear computations is the sum of their separate multiplicative complexities. This conjecture is known to hold for only a few specialized cases. In this paper, we establish its validity for large classes of computations.
Jean Takche
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Sum and direct sum of frame sequences
Linear and Multilinear Algebra, 2013Casazza, Han and Larson characterized various properties of the direct sum of two frame sequences. We add characterizations of other properties and study the relationship between the direct sum and the sum of frame sequences. In particular, we find a necessary and sufficient condition for the sum of two strongly disjoint (orthogonal) frame sequences ...
Yoo Young Koo, Jae Kun Lim
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Modules with the Direct Summand Sum Property [PDF]
Czechoslovak Mathematical Journal, 2003 summary:The present work gives some characterizations of $R$-modules with the direct summand sum property (in short DSSP), that is of those $R$-modules for which the sum of any two direct summands, so the submodule generated by their union, is a direct ...
Dumitru Valcan, Valcan Dumitru
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The direct sum of universal relations
Information Processing Letters, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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A note on direct sums of Friedbergnumberings
Journal of Symbolic Logic, 1989We show that a translator ƒ: ω → ω from a Gödelnumbering φ into a direct sum η of a r.e. family of Friedbergnumberings satisfies ƒ ≰T0′. In particular, η cannot be a Gödelnumbering.In the following we use standard notation (cf.[3]): for i ≥ 1, Pi (respectively, Ri) is the set of partial (total) recursive i-place functions; φ is a Gödel numbering of P1.
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On a direct sum of irreducible groups
Mathematical Notes, 2015zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Boletín de la Sociedad Matemática Mexicana, 2015
Let \(R\) be an associative ring with an identity element. A unital right \(R\)-module \(M\) is called ADS* if for any direct summand \(N\) of \(M\) and any supplement \(K\) of \(N\) in \(M\) one has \(M=N\oplus K\). The aim of this paper is to investigate direct sums of ADS* modules. First, the authors provide a bunch of examples of ADS* modules whose
Tribak, Rachid +2 more
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Let \(R\) be an associative ring with an identity element. A unital right \(R\)-module \(M\) is called ADS* if for any direct summand \(N\) of \(M\) and any supplement \(K\) of \(N\) in \(M\) one has \(M=N\oplus K\). The aim of this paper is to investigate direct sums of ADS* modules. First, the authors provide a bunch of examples of ADS* modules whose
Tribak, Rachid +2 more
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Advances in Operator Theory, 2023
Let \(A\) be a square matrix partitioned as follows: \[ A = \left[ \begin{array}{cc} B & C \\ D & E \end{array} \right]. \] The authors study several sufficient conditions on \(A\) to assure that \(A\) is a direct sum of \(B\) and \(C\), i.e., \(C=0\) and \(D=0\).
Hwa-Long Gau, Pei Yuan Wu
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Let \(A\) be a square matrix partitioned as follows: \[ A = \left[ \begin{array}{cc} B & C \\ D & E \end{array} \right]. \] The authors study several sufficient conditions on \(A\) to assure that \(A\) is a direct sum of \(B\) and \(C\), i.e., \(C=0\) and \(D=0\).
Hwa-Long Gau, Pei Yuan Wu
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Communications in Algebra, 1974
It is a well-known result that the direct sum of any family of injective modules over a Noetherian ring is injective. Conversely, if A is a ring with the property that the direct sum of any family of injective modules is injective H. Bass [1] has shown that A is Noetherian.
B. Sarath, K. Varadarajan
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It is a well-known result that the direct sum of any family of injective modules over a Noetherian ring is injective. Conversely, if A is a ring with the property that the direct sum of any family of injective modules is injective H. Bass [1] has shown that A is Noetherian.
B. Sarath, K. Varadarajan
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