• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1381-1388
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• 2018

Let ${\Delta}$ be a simplicial complex, $I_{\Delta}$ its Stanley-Reisner ideal and $K[{\Delta}]$ its Stanley-Reisner ring over a field K. Assume that ${\Gamma}(R)$ denotes the zero-divisor graph of a commutative ring R. Here, first we present a condition on two reduced Noetherian rings R and R', equivalent to ${\Gamma}(R){\cong}{\Gamma}(R{^{\prime}})$. In particular, we show that ${\Gamma}(K[{\Delta}]){\cong}{\Gamma}(K^{\prime}[{\Delta}^{\prime}])$ if and only if ${\mid}Ass(I_{\Delta}){\mid}={\mid}Ass(I_{{{\Delta}^{\prime}}}){\mid}$ and either ${\mid}K{\mid}$, ${\mid}K^{\prime}{\mid}{\leq}{\aleph}_0$ or ${\mid}K{\mid}={\mid}K^{\prime}{\mid}$. This shows that ${\Gamma}(K[{\Delta}])$ contains little information about $K[{\Delta}]$. Then, we define the squarefree zero-divisor graph of $K[{\Delta}]$, denoted by ${\Gamma}_{sf}(K[{\Delta}])$, and prove that ${\Gamma}_{sf}(K[{\Delta}){\cong}{\Gamma}_{sf}(K[{\Delta}^{\prime}])$ if and only if $K[{\Delta}]{\cong}K[{\Delta}^{\prime}]$. Moreover, we show how to find dim $K[{\Delta}]$ and ${\mid}Ass(K[{\Delta}]){\mid}$ from ${\Gamma}_{sf}(K[{\Delta}])$.

• Journal of applied mathematics & informatics
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• v.42 no.2
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• pp.367-377
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• 2024

Let G = (V, E) be a graph with p-vertices and q-edges and let R^◦ be a finite zero ring of order n. An injective function f : V (G) → {r₁, r₂, _…, r_k}, where r_i ∈ R^◦ is called vertex pair sum k-zero ring labeling, if it is possible to label the vertices x ∈ V with distinct labels from R^◦ such that each edge e = uv is labeled with f(e = uv) = [f(u) + f(v)] (mod n) and the edge labels are distinct. A graph admits such labeling is called vertex pair sum k-zero ring graph. The minimum value of positive integer k for a graph G which admits a vertex pair sum k-zero ring labeling is called the vertex pair sum k-zero ring index denoted by 𝜓_pz(G). In this paper, we defined the vertex pair sum k-zero ring labeling and applied to some graphs.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.331-342
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• 2017

Let R be a commutative ring with nonzero identity and G be a nontrivial finite group. Also, let Z(R) be the set of zero-divisors of R and, for $a{\in}Z(R)$, let $ann(a)=\{r{\in}R{\mid}ra=0\}$. The annihilator graph of the group ring RG is defined as the graph AG(RG), whose vertex set consists of the set of nonzero zero-divisors, and two distinct vertices x and y are adjacent if and only if $ann(xy){\neq}ann(x){\cup}ann(y)$. In this paper, we study the annihilator graph associated to a group ring RG.

• The Pure and Applied Mathematics
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• v.24 no.2
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• pp.69-77
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• 2017

If q(z) is a polynomial of degree n with all zeros in the unit circle, then the self-reciprocal polynomial $q(z)+x^nq(1/z)$ has all its zeros on the unit circle. One might naturally ask: where are the zeros of $q(z)+x^nq(1/z)$ located if q(z) has different zero distribution from the unit circle? In this paper, we study this question when $q(z)=(z-1)^{n-k}(z-1-c_1){\cdots}(z-1-c_k)+(z+1)^{n-k}(z+1+c_1){\cdots}(z+1+c_k)$, where $c_j$ > 0 for each j, and q(z) is a 'zeros dragged' polynomial from $(z-1)^n+(z+1)^n$ whose all zeros lie on the imaginary axis.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.267-278
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• 2013

We consider zero range processes with density dependent jump rates g given by $g=g(n,k)=g_1(n)g_2(k/n)$ with $g_1(x)=x^{-\alpha}$ and $$g_2(x)=\{^{x^{-\alpha}\;if\;a&lt;x}_{Mx^{-\alpha}\;if\;x{\leq}a}$$. (0.1) In this case, with 1/2 < a < 1 and ${\alpha}$ > 0, we show that non-complete condensation occurs with maximum cluster size an. More precisely, for any ${\epsilon}$ > 0, there exists $M^*$ > 0 such that, for any 0 < M ${\leq}M^*$, the maximum cluster size is between (a - ${\epsilon}$)n and (a + ${\epsilon}$)n for large n. This provides a simple example of non-complete condensation under perturbation of rates which are deep in the range of perfect condensation (e.g. ${\alpha}$ >> 1) and supports the instability of the condensation transition.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.3
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• pp.137-154
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• 2018

We investigate the condensation transition of a zero range process with jump rate g given by $g(k)=\left\{\frac{M}{k^{\alpha}},\;if\;k{\leq}an\\{\frac{1}{k^{\alpha}}},\;if\;k>an,$ (0.1), where ${\alpha}$ > 0 and a(0 < a < 1/2) is a rational number. We show that for any ${\epsilon}$ > 0, there exists $M^*$ > 0 such that, for any 0<$M{\leq}M^*$, the maximum cluster size is between ($a-{\epsilon}$)n and ($a+{\epsilon}$)n for large n.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.697-707
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• 2013

In this paper, we introduce the notion of permuting $n$-derivations in near-ring N and investigate commutativity of addition and multiplication of N. Further, under certain constrants on a $n!$-torsion free prime near-ring N, it is shown that a permuting $n$-additive mapping D on N is zero if the trace $d$ of D is zero. Finally, some more related results are also obtained.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.653-655
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• 2006

We give a simple proof of the statement that if T is a bounded linear operator on a complex HIlbert space then T is Fredholm if and only if ${\pi}(T)$ is not a TDZ, where ${\pi}(\cdot)$ is the Catkin homomorphism.

• Journal of Multimedia Information System
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• v.9 no.3
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• pp.233-244
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• 2022

A MATLAB program was developed to calculate the half-wavelength of a sine-curve baseband signal with white noise by using an autocorrelation function, a SG filter, and zero-crossing detection. The frequency of the input signal can be estimated from 1) the first zero-crossing (corresponding to ¼λ) and 2) the R value (the Y axis of the correlogram) at the center of the segment. Thereby, the frequency information of the preceding segment can be obtained. If the segment size were optimized, and a portion with a large zero-crossing dynamic range were obtained, the frequency discrimination ability would improve. Furthermore, if the values of the correlogram for each frequency prepared on the CPU side were prepared in a table, the volume of calculations can be reduced by 98%. As background, period detection by autocorrelation coefficients requires an integer multiple of 1/2λ (when using a sine wave as the object of the autocorrelation function), otherwise the correlogram drawn by R value will not exhibit orthogonality. Therefore, it has not been used in bio-telemetry where the frequencies move around.