• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1489-1492
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• 2009

Nagata[3] and Higman[1] showed that nil-algebra of the nilindex n is nilpotent of finite index. In this paper we show that the bounded degree of the nilpotency is less than or equal to $2^n-1$. Our proof needs only some elementary fact about Vandermonde determinant, which is much simpler than Nagata's or Higman's proof.

• Journal of the Korean Mathematical Society
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• v.48 no.3
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• pp.499-512
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• 2011

The purpose of this paper is twofold. The first is to establish a uniqueness theorem for entire function sharing two polynomials with its ${\kappa}$-th derivative, by using the theory of normal families. Meanwhile, the theorem generalizes some related results of Rubel and Yang and of Li and Yi. Several examples are provided to show the conditions are necessary. The second is to generalize the Br$\"{u}$-ck conjecture with the idea of sharing polynomial.

• ETRI Journal
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• v.38 no.2
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• pp.291-301
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• 2016

In this paper, we study an optimized block-diagonal zero-forcing (BD-ZF) precoder in a two-tiered cognitive network consisting of a macro cell (MC) and a small cell (SC). By exploiting multiuser multiple-input and multiple-output Vandermonde-subspace frequency-division multiplexing (VFDM) transmission, a cognitive SC can coexist with an MC. We first devise a cross-tier precoder based on the idea of VFDM to cancel the interference from the SC to the MC. Then, we propose an optimized BD-ZF intra-tier precoder (ITP) to suppress multiuser interference and maximize the throughput in the SC. In the case where the dimension of a provided null space is larger than that required by the BD-ZF ITP, the optimized BD-ZF ITP can collect all limited channel gain by optimizing rotating and selecting matrices. Otherwise, the optimized BD-ZF ITP is validated to be equivalent to the conventional BD-ZF ITP in terms of throughput. Numerical results are presented to demonstrate the throughput improvement of the proposed optimized BD-ZF ITP and to discover the impact of imperfect channel state information.