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Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, 2015
Summary: The \textit{\(k\)-fold direct sum encoding} of a string \(a\in\{0,1\}^n\) is a function \(f_a\) that takes as input sets \(S\subseteq [n]\) of size \(k\) and outputs \(f_a(S)=\sum_{i\in S}a_i\pmod 2\). In this paper we prove a direct sum testing theorem. We describe a three query test that accepts with probability one any function of the form \
Roee David +4 more
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Summary: The \textit{\(k\)-fold direct sum encoding} of a string \(a\in\{0,1\}^n\) is a function \(f_a\) that takes as input sets \(S\subseteq [n]\) of size \(k\) and outputs \(f_a(S)=\sum_{i\in S}a_i\pmod 2\). In this paper we prove a direct sum testing theorem. We describe a three query test that accepts with probability one any function of the form \
Roee David +4 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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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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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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2002
In Theorem 2.6 we obtained, for an inner product space V and a finite-dimensional subspace W of V, a direct sum decomposition of the form V = W ⊕W⊥. We now consider the following general notion.
T. S. Blyth, E. F. Robertson
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In Theorem 2.6 we obtained, for an inner product space V and a finite-dimensional subspace W of V, a direct sum decomposition of the form V = W ⊕W⊥. We now consider the following general notion.
T. S. Blyth, E. F. Robertson
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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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DIRECT SUMS OF OPERATOR SPACES
Journal of the London Mathematical Society, 2001It is proved that if X and Y are operator spaces such that every completely bounded operator from X into Y is completely compact and Z is a completely complemented subspace of X [oplus ] Y, then there exists a completely bounded automorphism τ: X [oplus ] Y → X [oplus ] Y with completely bounded inverse such that τZ = X0 [oplus ] Y0, where ...
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Primary direct sum decomposition
Communications in Algebra, 1976(1976). Primary direct sum decomposition. Communications in Algebra: Vol. 4, No. 3, pp. 285-304.
John Fuelberth, James Kuzmanovich
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Direct Sums and Direct Products
2015The concept of direct sum is of utmost importance for the theory. This is mostly due to two facts: first, if we succeed in decomposing a group into a direct sum, then it can be studied by investigating the summands separately, which are, in numerous cases, simpler to deal with.
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