Irreducibility

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Hyun Jae Yoo - One of the best experts on this subject based on the ideXlab platform.

  • Quantum Markov chains associated with open quantum random walks.
    arXiv: Mathematical Physics, 2018
    Co-Authors: Ameur Dhahri, Chul Ki Ko, Hyun Jae Yoo
    Abstract:

    In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/Irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and Irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/Irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.

  • Reducibility and Irreducibility of open quantum random walks: A quantum Markov chain approach
    arXiv: Mathematical Physics, 2018
    Co-Authors: Ameur Dhahri, Chul Ki Ko, Hyun Jae Yoo
    Abstract:

    In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. We then discuss the reducibility and Irreducibility of open quantum random walks via the corresponding quantum Markov chains. The quantum Markov chain was introduced by Accardi by using transition expectation, and the construction of the quantum Markov chains for the open quantum random walks was introduced by Dhahri and Mukhamedov. Here we construct nonhomogeneous quantum Markov chains for the open quantum random walks so that we can naturally express the evolution of any open quantum random walk by the corresponding quantum Markov chain. We provide with some examples. In particular, we show that the classical Markov chains are reconstructed as quantum Markov chains as well and the reducibility and Irreducibility properties of classical Markov chains can be investigated in the language of quantum Markov chains.

David G Forney - One of the best experts on this subject based on the ideXlab platform.

  • local Irreducibility of tail biting trellises
    IEEE Transactions on Information Theory, 2013
    Co-Authors: Heide Gluesingluerssen, David G Forney
    Abstract:

    This paper investigates tail-biting trellis realizations for linear block codes. Intrinsic trellis properties are used to characterize Irreducibility on given intervals of the time axis. It proves beneficial to always consider the trellis and its dual simultaneously. A major role is played by trellis properties that amount to observability and controllability of trellis fragments of various lengths. For fragments of length less than the minimum span length of the code it is shown that fragment observability and fragment controllability are equivalent to Irreducibility. For reducible trellises, a constructive reduction procedure is presented. The considerations also lead to a characterization for when the dual of a trellis allows a product factorization into elementary (“atomic”) trellises.

  • local Irreducibility of tail biting trellises
    arXiv: Information Theory, 2012
    Co-Authors: Heide Gluesingluerssen, David G Forney
    Abstract:

    This paper investigates tail-biting trellis realizations for linear block codes. Intrinsic trellis properties are used to characterize Irreducibility on given intervals of the time axis. It proves beneficial to always consider the trellis and its dual simultaneously. A major role is played by trellis properties that amount to observability and controllability for fragments of the trellis of various lengths. For fragments of length less than the minimum span length of the code it is shown that fragment observability and fragment controllability are equivalent to Irreducibility. For reducible trellises, a constructive reduction procedure is presented. The considerations also lead to a characterization for when the dual of a trellis allows a product factorization into elementary ("atomic") trellises.

Christian Fleischhack - One of the best experts on this subject based on the ideXlab platform.

Philip Matchett Wood - One of the best experts on this subject based on the ideXlab platform.

  • Irreducibility of random polynomials
    Experimental Mathematics, 2018
    Co-Authors: Christian Borst, Evan Boyd, Claire Brekken, Samantha Solberg, Melanie Matchett Wood, Philip Matchett Wood
    Abstract:

    ABSTRACTWe study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data support conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert’s Irreducibility Theorem and work of van der Waerden. The data indicate that the probability that a random polynomial is reducible divided by the probability that there is a linear factor appears to approach a constant and, in the large-degree limit, this constant appears to approach 1. In cases where the model makes it impossible for the random polynomial to have a linear factor, the probability of reducibility appears to be close to the probability of having a nonlinear, low-degree factor. We also study characteristic polynomials of random matrices with + 1 and − 1 entries.

  • Irreducibility of random polynomials
    arXiv: Probability, 2017
    Co-Authors: Christian Borst, Evan Boyd, Claire Brekken, Samantha Solberg, Melanie Matchett Wood, Philip Matchett Wood
    Abstract:

    We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data supports conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert's Irreducibility Theorem and work of van der Waerden. The data indicates that the probability that a random polynomial is reducible divided by the probability that there is a linear factor appears to approach a constant and, in the large-degree limit, this constant appears to approach one. In cases where the model makes it impossible for the random polynomial to have a linear factor, the probability of reducibility appears to be close to the probability of having a non-linear, low-degree factor. We also study characteristic polynomials of random matrices with +1 and -1 entries.

Ameur Dhahri - One of the best experts on this subject based on the ideXlab platform.

  • Quantum Markov chains associated with open quantum random walks.
    arXiv: Mathematical Physics, 2018
    Co-Authors: Ameur Dhahri, Chul Ki Ko, Hyun Jae Yoo
    Abstract:

    In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/Irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and Irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/Irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.

  • Reducibility and Irreducibility of open quantum random walks: A quantum Markov chain approach
    arXiv: Mathematical Physics, 2018
    Co-Authors: Ameur Dhahri, Chul Ki Ko, Hyun Jae Yoo
    Abstract:

    In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. We then discuss the reducibility and Irreducibility of open quantum random walks via the corresponding quantum Markov chains. The quantum Markov chain was introduced by Accardi by using transition expectation, and the construction of the quantum Markov chains for the open quantum random walks was introduced by Dhahri and Mukhamedov. Here we construct nonhomogeneous quantum Markov chains for the open quantum random walks so that we can naturally express the evolution of any open quantum random walk by the corresponding quantum Markov chain. We provide with some examples. In particular, we show that the classical Markov chains are reconstructed as quantum Markov chains as well and the reducibility and Irreducibility properties of classical Markov chains can be investigated in the language of quantum Markov chains.