Revisiting the Direct Sum Theorem and Space Lower Bounds in Random Order Streams

AbstractWe prove two theorems on continuous modules:Decomposition Theorem. A continuous moduleMhas a decomposition,M=M1⊕M2, such thatM1is essential over a direct sumof indecomposable summandsAiofM, andM2has no uniform submodules; and these data are uniquely determined byMup to isomorphism.Direct Sum Theorem. A finite direct sumof indecomposable modulesAiis continuous if and only if eachAiis continuous andAj-injective for allj≠ i.

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A Direct Sum Theorem for Corruption and the Multiparty NOF Communication Complexity of Set Disjointness

20th Annual IEEE Conference on Computational Complexity (CCC'05) ◽

10.1109/ccc.2005.1 ◽

2005 ◽

Cited By ~ 7

Author(s):

P. Beame ◽

T. Pitassi ◽

N. Segerlind ◽

A. Wigderson

Keyword(s):

Communication Complexity ◽

Direct Sum ◽

Sum Theorem

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The quantum query complexity of certification

Quantum Information and Computation ◽

10.26421/qic10.3-4-1 ◽

2010 ◽

Vol 10 (3&4) ◽

pp. 181-189

Author(s):

A. Ambainis ◽

A.M. Childs ◽

F. Le Gall ◽

S. Tani

Keyword(s):

Lower Bound ◽

Query Complexity ◽

Direct Sum ◽

Quantum Query Complexity ◽

Sum Theorem ◽

Adversary Method ◽

Quantum Query

We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced \nand formula. We show that the query complexity is $\tilde\Theta(d^{(k+1)/2})$ for 0-certificates, and $\tilde\Theta(d^{k/2})$ for 1-certificates. In particular, this shows that the zero-error quantum query complexity of evaluating such formulas is $\tilde O(d^{(k+1)/2})$. Our lower bound relies on the fact that the quantum adversary method obeys a direct sum theorem.

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ON THE GALOIS STRUCTURE OF ARITHMETIC COHOMOLOGY I: COMPACTLY SUPPORTED -ADIC COHOMOLOGY

Nagoya Mathematical Journal ◽

10.1017/nmj.2018.41 ◽

2018 ◽

Vol 239 ◽

pp. 294-321

Author(s):

DAVID BURNS

Keyword(s):

Lower Bounds ◽

Projective Module ◽

Number Fields ◽

Upper And Lower Bounds ◽

Selmer Groups ◽

Cohomology Groups ◽

Field Extensions ◽

Direct Sum ◽

Galois Modules ◽

Galois Structure

We investigate the Galois structures of $p$-adic cohomology groups of general $p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded $p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.

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