• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.127-133
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• 2018

We look at Toeplitz arrays on ${\mathbb{Z}}^d$ and study a characterizing property for pure discrete spectrum in terms of the periodic structures of the Toeplitz arrays.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.9 no.10
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• pp.3867-3886
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• 2015

This article investigates the problem of opportunistic spectrum access in dynamic environment, in which the signal-to-noise ratio (SNR) is time-varying. Different from existing work on continuous feedback, we consider more practical scenarios in which the transmitter receives an Acknowledgment (ACK) if the received SNR is larger than the required threshold, and otherwise a Non-Acknowledgment (NACK). That is, the feedback is discrete. Several applications with different threshold values are also considered in this work. The channel selection problem is formulated as a non-cooperative game, and subsequently it is proved to be a potential game, which has at least one pure strategy Nash equilibrium. Following this, a multi-agent Q-learning algorithm is proposed to converge to Nash equilibria of the game. Furthermore, opportunistic spectrum access with multiple discrete feedbacks is also investigated. Finally, the simulation results verify that the proposed multi-agent Q-learning algorithm is applicable to both situations with binary feedback and multiple discrete feedbacks.

• Kyungpook Mathematical Journal
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• v.60 no.2
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• pp.297-305
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• 2020

We show that selfadjoint operator extensions of minimal second order difference operators have only discrete spectrum when the odd order coefficient is unbounded but grows or decays according to specific conditions. Selfadjoint operator extensions of minimal differential operator under similar growth and decay conditions on the coefficients have a absolutely continuous spectrum of multiplicity one.