• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.239-268
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• 2015

Let Q be a finite quiver and $G{\subseteq}Aut(\mathbb{k}Q)$ a finite abelian group. Assume that $\hat{Q}$ and ${\Gamma}$ are the generalized Mckay quiver and the valued graph corresponding to (Q, G) respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $g({\Gamma})$. Moreover, we may lift G to $\bar{G}{\subseteq}Aut(g(\hat{Q}))$ such that $g({\Gamma})$ embeds into the fixed point algebra $g(\hat{Q})^{\bar{G}}$ and $g(\hat{Q})^{\bar{G}}$ as a $g({\Gamma})$-module is integrable.

• Honam Mathematical Journal
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• v.41 no.3
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• pp.581-593
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• 2019

The notions of (${\in}$, ${\in}{\vee}q$)-permeable S-value and (${\in}$, ${\in}{\vee}q$)-permeable I-value are introduced, and related properties are investigated. Relations among (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra, (${\in}$, ${\in}{\vee}q$)-fuzzy ideal, (strong) lower and (strong) upper level sets, (${\in}$, ${\in}{\vee}q$)-permeable S-value, (${\in}$, ${\in}{\vee}q$)-permeable I-value, S-energetic set, I-energetic set, right stable set and right vanished set are discussed.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.407-419
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• 2009

The simple non-associative algebra $N(e^{A_S},q,n,t)_k$ and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$ and its antisymmetrized algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$. We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations $Der_{anti}(WN(e^{{\pm}x^r},0,2)_B^-)$ of te antisymmetrized algebra $WN(e^{{\pm}x^r}0,2)_B^-$ and every derivation of the algebra is outer in this paper.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.641-649
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• 2016

A linear functional T on a $Fr{\acute{e}}echet$ algebra (A, (pn)) is called almost multiplicative with respect to the sequence ($p_n$), if there exists ${\varepsilon}{\geq}0$ such that ${\mid}Tab-TaTb{\mid}{\leq}{\varepsilon}p_n(a)p_n(b)$ for all $n{\in}\mathbb{N}$ and for every $a,b{\in}A$. We show that an almost multiplicative linear functional on a $Fr{\acute{e}}echet$ algebra is either multiplicative or it is continuous, and hence every almost multiplicative linear functional on a functionally continuous $Fr{\acute{e}}echet$ algebra is continuous.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.397-408
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• 2016

Let R be a 2-torsion free prime ring with center Z, U be the Utumi quotient ring, Q be the Martindale quotient ring of R, d be a derivation of R and L be a Lie ideal of R. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v{\in}L$, where m, n, l are xed positive integers, then $L{\subseteq}Z$. We also examine the case when R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1141-1158
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• 2018

Hahn difference operator $D_{q,{\omega}}$ which is defined by $$D_{q,{\omega}}g(t)=\{{\frac{g(gt+{\omega})-g(t)}{t(g-1)+{\omega}}},{\hfill{20}}\text{if }t{\neq}{\theta}:={\frac{\omega}{1-q}},\\g^{\prime}({\theta}),{\hfill{83}}\text{if }t={\theta}$$ received a lot of interest from many researchers due to its applications in constructing families of orthogonal polynomials and in some approximation problems. In this paper, we investigate sufficient conditions for stability of the abstract linear Hahn difference equations of the form $$D_{q,{\omega}}x(t)=A(t)x(t)+f(t),\;t{\in}I$$, and $$D^2{q,{\omega}}x(t)+A(t)D_{q,{\omega}}x(t)+R(t)x(t)=f(t),\;t{\in}I$$, where $A,R:I{\rightarrow}{\mathbb{X}}$, and $f:I{\rightarrow}{\mathbb{X}}$. Here ${\mathbb{X}}$ is a Banach algebra with a unit element e and I is an interval of ${\mathbb{R}}$ containing ${\theta}$.

• Communications of the Korean Mathematical Society
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• v.27 no.4
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• pp.669-687
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• 2012

As a generalization of the concept of a fuzzy implicative filter which is introduced by Liu and Li [3], the notion of (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filters is introduced, and related properties are investigated. The relationship between (${\in}$, ${\in}{\vee}q_k$)-fuzzy filters and (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filters is established. Conditions for an (${\in}$, ${\in}{\vee}q_k$)-fuzzy filter to be an (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filter are considered. Characterizations of an (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filter are provided, and the implication-based fuzzy implicative filters of an $R_0$-algebra is discussed.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.443-454
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• 2017

We study a notion of q-frequent hypercyclicity of linear maps between the Banach algebras consisting of operators on a separable infinite dimensional Banach space. We derive a sufficient condition for a linear map to be q-frequently hypercyclic in the strong operator topology. Some properties are investigated regarding q-frequently hypercyclic subspaces as shown in [5], [6] and [7]. Finally, we study q-frequent hypercyclicity of tensor products and direct sums of operators.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.173-181
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• 2007

Using the belongs to relation ($\in$) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of ($\alpha,\;\beta$)-fuzzy subalgebras where $\alpha,\;\beta$ are any two of $\{{\in},\;q,\;{\in}\;{\vee}\;q,\;{\in}\;{\wedge}\;q\}$ with ${\alpha}\;{\neq}\;{\in}\;{\wedge}\;q$ was introduced, and related properties were investigated in [3]. As a continuation of the paper [3], in this paper, the notion of a fuzzy subalgebra with thresholds is introduced, and its characterizations are obtained. Relations between a fuzzy subalgebra with thresholds and an (${\in},\;{\in}\;{\vee}\;q$)-fuzzy subalgebra are provided.

• Journal of applied mathematics & informatics
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• v.4 no.2
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• pp.555-561
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• 1997

In this paper we show that if A is a Banach algebra with radical R and D is a left derivation on A then $D(A){\subset}R$ if and only if $Q_RD^n$ is continuous for all $n{\geq}1$, where $Q_R$ is the canonical quotient map from A onto A/R.