• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.565-583
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• 2007

We offer a refinement of the classical Clifford inequality about special linear series on smooth irreducible complex curves. Namely, we prove about curves of genus g and odd gonality at least 5 that for any linear series $g^r_d$ with $d{\leq}g+1$, the inequality $3r{\leq}d$ holds, except in a few sporadic cases. Further, we show that the dimension of the set of curves in the moduli space for which there exists a linear series $g^r_d$ with d<3r for $d{\leq}g+l,\;0{\leq}l{\leq}\frac{g}{2}-3$, is bounded by $2g-1+\frac{1}{3}(g+2l+1)$.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.1
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• pp.39-52
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• 2022

We introduce Clifford-valued neural networks with leakage delays. Furthermore, we study the uniqueness and existence of Clifford-valued Hopfield artificial neural networks having the Stepanov weighted pseudo almost periodic forcing terms on leakage delay terms. However the noncommutativity of the Clifford numbers' multiplication made our investigation diffcult, so our results are obtained by decomposing Clifford-valued neural networks into real-valued neural networks. Our analysis is based on the differential inequality techniques and the Banach contraction mapping principle.

• Korean Journal of Mathematics
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• v.30 no.3
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• pp.491-502
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• 2022

In this paper we investigate some sufficient conditions to guarantee the existence and uniqueness of Stepanov-like weighted pseudo almost periodic solutions of cellular neural networks on Clifford algebra for non-automomous cellular neural networks with multi-proportional delays. Our analysis is based on the differential inequality techniques and the Banach contraction mapping principle.

• Communications of the Korean Mathematical Society
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• v.23 no.1
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• pp.11-18
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• 2008

For a nef and big divisor D on a smooth projective surface S, the inequality $h^{0}$(S;$O_{s}(D)$) ${\leq}\;D^2\;+\;2$ is well known. For a nef and big canonical divisor KS, there is a better inequality $h^{0}$(S;$O_{s}(K_s)$) ${\leq}\;\frac{1}{2}{K_{s}}^{2}\;+\;2$ which is called the Noether inequality. We investigate an inequality $h^{0}$(S;$O_{s}(D)$) ${\leq}\;\frac{1}{2}D^{2}\;+\;2$ like Clifford theorem in the case of a curve. We show that this inequality holds except some cases. We show the existence of a counter example for this inequality. We prove also the base-locus freeness of the linear system in the exceptional cases.