• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1063-1074
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• 2023

Let S be a semigroup. We determine the complex-valued solutions of the following functional equation f(xy) + 𝜇(y)f(𝜎(y)x) = 2f(x)g(y), x, y ∈ S, where 𝜎 : S → S is an automorphism, and 𝜇 : S → ℂ is a multiplicative function such that 𝜇(x𝜎(x)) = 1 for all x ∈ S.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.607-617
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• 2014

Let S = C(r,m), the finite cyclic semigroup with index r and period m. Each subsemigroup of S is cyclic if and only if either r = 1; r = 2; or r = 3 with m odd. For $r{\neq}1$, the maximum value of the minimum number of elements in a (minimal) generating set of a subsemigroup of S is 1 if r = 3 and m is odd; 2 if r = 3 and m is even; (r-1)/2 if r is odd and unequal to 3; and r/2 if r is even. The number of cyclic subsemigroups of S is $r-1+{\tau}(m)$. Formulas are also given for the number of 2-generated subsemigroups of S and the total number of subsemigroups of S. The minimal generating sets of subsemigroups of S are characterized, and the problem of counting them is analyzed.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.55-82
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• 2019

Given ${\sigma}:G{\rightarrow}G$ an involutive automorphism of a semigroup G, we study the solutions and stability of the following functional equations $$f(x{\sigma}(y))=f(x)g(y)+g(x)f(y),\;x,y{\in}G,\\f(x{\sigma}(y))=f(x)f(y)-g(x)g(y),\;x,y{\in}G$$ and $$f(x{\sigma}(y))=f(x)g(y)-g(x)f(y),\;x,y{\in}G$$, from the theory of trigonometric functional equations. (1) We determine the solutions when G is a semigroup generated by its squares. (2) We obtain the stability results for these equations, when G is an amenable group.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.1
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• pp.47-51
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• 2019

The purpose of this paper is to present approximation of $C_0$-sequentially equicontinuous semigroups on a sequentially complete locally convex space X.

• Korean Journal of Mathematics
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• v.9 no.2
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• pp.115-121
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• 2001

In this paper, we establish the conditions that a mild C-existence family yields a solution to the abstract Cauchy problem. And we show the relation between mild C-existence family and C-regularized semigroup if the family of linear operators is exponentially bounded and C is a bounded injective linear operator.

• East Asian mathematical journal
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• v.7 no.2
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• pp.171-178
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• 1992

• Korean Journal of Mathematics
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• v.22 no.2
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• pp.349-354
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• 2014

In this paper, we establish convergence of contraction $C_0$ semigroups in the weak topology on a general Banach space. We remove the restriction on a Banach space X and weaken the condition on resolvents of generators in the previous results [4, 5].

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.177-182
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• 2018

The purpose of this paper is to present approximation of $C_0-sequentially$ equicontinuous semigroups on a sequentially complete locally convex space X.

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.869-889
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• 2015

Let ${\Gamma}^+$ be the positive cone in a totally ordered abelian group ${\Gamma}$, and ${\alpha}$ an action of ${\Gamma}^+$ by extendible endomorphisms of a $C^*$-algebra A. Suppose I is an extendible ${\alpha}$-invariant ideal of A. We prove that the partial-isometric crossed product $\mathcal{I}:=I{\times}^{piso}_{\alpha}{\Gamma}^+$ embeds naturally as an ideal of $A{\times}^{piso}_{\alpha}{\Gamma}^+$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\mathcal{I}$ together with the kernel of a natural homomorphism $\phi:A{\times}^{piso}_{\alpha}{\Gamma}^+{\rightarrow}A{\times}^{iso}_{\alpha}{\Gamma}^+$ gives a composition series of ideals of $A{\times}^{piso}_{\alpha}{\Gamma}^+$ studied by Lindiarni and Raeburn.

• East Asian mathematical journal
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• v.13 no.1
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• pp.63-70
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• 1997