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  1. Kenshi Miyabe (2011). Truth-Table Schnorr Randomness and Truth-Table Reducible Randomness. Mathematical Logic Quarterly 57 (3):323-338.
    Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's (...)
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  2. Kenshi Miyabe (2010). An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals. Notre Dame Journal of Formal Logic 51 (3):337-349.
    Van Lambalgen's Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen's Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi'}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$ . We also prove that there exists A such that, for each n (...)
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