• Journal of applied mathematics & informatics
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• 제40권3_4호
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• pp.729-740
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• 2022

Let ℤ be the ring of integers and let ℤ_n be the ring of integers modulo n. Let ℤ(+)ℤ_n be the idealization of ℤ_n in ℤ and let (ℤ(+)ℤ_n)[X]] be either (ℤ(+)ℤ_n)[X] or (ℤ(+)ℤ_n)[[X]]. In this article, we study the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. More precisely, we completely characterize the diameter and the girth of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. We also calculate the chromatic number of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]].

• 대한수학회지
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• 제53권4호
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• pp.929-967
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• 2016

Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.

• Kyungpook Mathematical Journal
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• 제60권4호
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• pp.723-729
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• 2020

Let ℤ_n be the ring of integers modulo n and let ℤ_n[X]] be either ℤ_n[X] or ℤ_n[[X]]. Let 𝚪(Z_n[X]]) be the zero-divisor graph of ℤ_n[X]]. In this paper, we study some properties of 𝚪(ℤ_n[X]]). More precisely, we completely characterize the diameter and the girth of 𝚪(ℤ_n[X]]). We also calculate the chromatic number of 𝚪(ℤ_n[X]]).

• 대한수학회지
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• 제47권5호
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• pp.1097-1106
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• 2010

Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering $\Gamma$(R), the zero-divisor graph of R, under the regular action on X by G as follows: (1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of $\Gamma$(R) which is adjacent to every other vertex in $\Gamma$(R) if and only if R is a local ring or $R\;{\simeq}\;\mathbb{Z}_2\;{\times}\;F$ where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, $J^2$, $\ldots$, $J^n$, R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.

• 대한수학회지
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• 제52권5호
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• pp.945-954
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• 2015

Let ${\lambda}_f(n)$ denote the n-th normalized Fourier coefficient of a primitive holomorphic form f for the full modular group ${\Gamma}=SL_2({\mathbb{Z}})$. In this paper, we are concerned with ${\Omega}$-result on the summatory function ${\sum}_{n{\leqslant}x}{\lambda}^2_f(n^2)$, and establish the following result ${\sum}_{\leqslant}{\lambda}^2_f(n^2)=c_1x+{\Omega}(x^{\frac{4}{9}})$, where $c_1$ is a suitable constant.