An Effective Version of Hall's Theorem

Kierstead, H. A.

doi:10.2307/2045123

Cited by 3 publications

(6 citation statements)

References 2 publications

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“…In this section we generalize the work of Kierstead [15] concerning an effective version of the Hall's Theorem. These results will be applied in the next section to effective paradoxical decompositions.…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 96%

“…These results will be applied in the next section to effective paradoxical decompositions. Below we follow the presentation of [15].…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

“…Proof. We extend the proof of Theorem 3 of the Kierstead's paper [15]. At the first step of the algorithm we choose a 0 , the first element of the set A.…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

“…| for all n ∈ N. This definition and the three definitions below are due to Kierstead [15]. Definition 6.4.…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

“…Proof. We extend the proof of Theorem 3 of the Kierstead's paper [15]. We fix a computable enumeration of A and B.…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

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Karol Borsuk 1905–1982

Duda¹

2016

Portrety Uczonych. Profesorowie Uniwersytetu Warszawskiego Po 1945, A−K

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In this paper we extend the approach of M. Cavaleri to effective amenability to the class of computably enumerable groups, i.e. in particular we do not assume that groups are finitely generated. The main results of the paper concern some new directions in this approach. In the case of computable groups we study decidability of amenability of finitely generated subgroups, complexity of the set of all effective Følner sequences and effective paradoxical decomposition.

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Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 96%

“…These results will be applied in the next section to effective paradoxical decompositions. Below we follow the presentation of [15].…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

“…Proof. We extend the proof of Theorem 3 of the Kierstead's paper [15]. At the first step of the algorithm we choose a 0 , the first element of the set A.…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

“…| for all n ∈ N. This definition and the three definitions below are due to Kierstead [15]. Definition 6.4.…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

“…Proof. We extend the proof of Theorem 3 of the Kierstead's paper [15]. We fix a computable enumeration of A and B.…”

Section: An Effective Version Of Hall's Harem Theoremmentioning

confidence: 99%

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Karol Borsuk 1905–1982

Duda¹

2016

Portrety Uczonych. Profesorowie Uniwersytetu Warszawskiego Po 1945, A−K

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Hyperfinite transversal theory

Živaljević¹

1992

Trans. Amer. Math. Soc.

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Abstract. A measure theoretic version of a well-known P. Hall's theorem, about the existence of a system of distinct representatives of a finite family of finite sets, has been proved for the case of the Loeb space of an internal, uniformly distributed, hyperfinite counting space. We first prove Hall's theorem for UP^k) graphs after which we develop the version of discrete Transversal Theory. We then prove a new version of Hall's theorem in the case of 2^(k) monotone graphs and give an example of a Z^ graph which satisfies Hall's condition and which does not possess an internal a.e. matching.

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Deterministic and Probabilistic Conditions for Finite Completability of Low-Tucker-Rank Tensor

Ashraphijuo

Aggarwal

Wang

2019

IEEE Trans. Inform. Theory

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In this paper, we analyze the fundamental conditions for low-rank tensor completion given the separation or tensor-train (TT) rank, i.e., ranks of unfoldings. We exploit the algebraic structure of the TT decomposition to obtain the deterministic necessary and sufficient conditions on the locations of the samples to ensure finite completability. Specifically, we propose an algebraic geometric analysis on the TT manifold that can incorporate the whole rank vector simultaneously in contrast to the existing approach based on the Grassmannian manifold that can only incorporate one rank component. Our proposed technique characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern and the TT decomposition, which is instrumental to obtaining the deterministic condition on the sampling pattern for finite completability. In addition, based on the proposed analysis, assuming that the entries of the tensor are sampled independently with probability p, we derive a lower bound on the sampling probability p, or equivalently, the number of sampled entries that ensures finite completability with high probability. Moreover, we also provide the deterministic and probabilistic conditions for unique completability.

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An Effective Version of Hall's Theorem

Cited by 3 publications

References 2 publications

Karol Borsuk 1905–1982

Karol Borsuk 1905–1982

Hyperfinite transversal theory

Deterministic and Probabilistic Conditions for Finite Completability of Low-Tucker-Rank Tensor

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