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Q-measures on the dual unit ball of a JB${}^*$-triple
C. Martin Edwards,Lina Oliveira 대한수학회 2019 대한수학회지 Vol.56 No.1
Let $A$ be a JB$^*$-triple with Banach dual space $A^*$ and bi-dual the JBW$^*$-triple $A^{**}$. Elements $x$ of $A^*$ of norm one may be regarded as normalised `Q-measures' defined on the complete ortho-lattice $\tilde{\mathcal U}(A^{**})$ of tripotents in $A^{**}$. A Q-measure $x$ possesses a support $e(x)$ in $\tilde{\mathcal U}(A^{**})$ and a compact support $e_c(x)$ in the complete atomic lattice $\tilde{\mathcal U}_c(A)$ of elements of $\tilde{\mathcal U}(A^{**})$ compact relative to $A$. Necessary and sufficient conditions for an element $v$ of $\tilde{\mathcal U}_c(A)$ to be a compact support tripotent $e_c(x)$ are given, one of which is related to the Q-covering numbers of $v$ by families of elements of $\tilde{\mathcal U}_c(A)$.
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On some type elements of zero-symmetric near-ring of polynomials
Ebrahim Hashemi,Fatemeh Shokuhifar 대한수학회 2019 대한수학회지 Vol.56 No.1
Let $R$ be a commutative ring with unity. In this paper, we characterize the unit elements, the regular elements, the $\pi$-regular elements and the clean elements of zero-symmetric near-ring of polynomials $R_{0}[x]$, when $ \mathrm{nil}(R)^{2}=0 $. Moreover, it is shown that the set of $\pi$-regular elements of $R_{0}[x]$ forms a semigroup. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its ``multiplication'' operation.
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신용수 대한수학회 2019 대한수학회지 Vol.56 No.1
In \cite{GHM}, Geramita, Harbourne, and Migliore find a graded minimal free resolution of the $2$nd order symbolic power of the ideal of a linear star configuration in $\mathbb P^n$ of any codimension $r$. In \cite{GGSV}, Geramita, Galetto, Shin, and Van Tuyl extend the result on a general star configuration in $\mathbb P^n$ but for codimension $2$. In this paper, we find a graded minimal free resolution of the $2$nd order symbolic power of the ideal of a general star configuration in $\mathbb P^n$ of any codimension $r$ using a matroid configuration in \cite{GHMN}. This generalizes both the result on a {\em linear} star configuration in $\mathbb P^n$ of codimension $r$ in \cite{GHM} and the result on a general star configuration in $\mathbb P^n$ of {\em codimension $2$} in \cite{GGSV}.
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Elliptic obstacle problems with measurable nonlinearities in non-smooth domains
김유찬,유승진 대한수학회 2019 대한수학회지 Vol.56 No.1
The Calder\'{o}n-Zygmund type estimate is proved for elliptic obstacle problems in bounded non-smooth domains. The problems are related to divergence form nonlinear elliptic equation with measurable nonlinearities. Precisely, nonlinearity $\ba(\xi,x_1,x')$ is assumed to be only measurable in one spatial variable $x_1$ and has locally small BMO semi-norm in the other spatial variables $x'$, uniformly in $\xi$ variable. Regarding non-smooth domains, we assume that the boundaries are locally flat in the sense of Reifenberg. We also investigate global regularity in the settings of weighted Orlicz spaces for the weak solutions to the problems considered here.
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최동순,신준국 대한수학회 2019 대한수학회지 Vol.56 No.1
Corrigendum to ``Free actions of finite abelian groups on 3-dimensional nilmanifolds'' [J. Korean Math. Soc. 42 (2005), No. 4, pp. 795--826]
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The Ricci curvature on directed graphs
Taiki Yamada 대한수학회 2019 대한수학회지 Vol.56 No.1
In this paper, we consider the Ricci curvature of a {\em directed} graph, based on Lin-Lu-Yau's definition. We give some properties of the Ricci curvature, including conditions for a directed regular graph to be Ricci-flat. Moreover, we calculate the Ricci curvature of the cartesian product of directed graphs.
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Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation
Peng Chen,Renming Song,Longjie Xie,Yingchao Xie 대한수학회 2019 대한수학회지 Vol.56 No.1
We give a direct proof of the sharp two-sided estimates, recently established in \cite{C-K-S-1, P-R}, for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel's formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require $D$ to be $C^{1,\theta}$ for some $\theta\in (\alpha/2, 1]$.
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Characterization of certain types of $r$-plateaued functions
현종윤,이정연,이윤진 대한수학회 2018 대한수학회지 Vol.55 No.6
We study a subclass of $p$-ary functions in $n$ variables, denoted by ${\mathcal A}_n$, which is a collection of $p$-ary functions in $n$ variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all $p$-ary $(n-1)$-plateaued functions in $n$ variables by proving that every $(n-1)$-plateaued function should be contained in $\mathcal{A}_n$. Secondly, we prove that if $f$ is a $p$-ary $r$-plateaued function contained in ${\mathcal A}_n$ with $\deg{f} > 1+\frac{n-r}{4}(p-1)$, then the highest degree term of $f$ is only a single term. Furthermore, we prove that there is no $p$-ary $r$-plateaued function in ${\mathcal A}_n$ with maximum degree $(p-1)\frac{n-r}{2}+1$. As application, we partially classify all $(n-2)$-plateaued functions in ${\mathcal A}_n$ when $p=3,5,$ and $7$, and $p$-ary bent functions in ${\mathcal A}_2$ are completely classified for the cases $p=3$ and $5$.
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Stability properties in impulsive differential systems of non-integer order
강보원,구남집 대한수학회 2019 대한수학회지 Vol.56 No.1
In this paper we establish some new explicit solutions for impulsive linear fractional differential equations with impulses at fixed times, which provides a handy tool in deriving singular integral-sum inequalities and an impulsive fractional comparison principle. Thus we study the Mittag-Leffler stability of impulsive differential equations with the Caputo fractional derivative by using the impulsive fractional comparison principle and piecewise continuous functions of Lyapunov's method. Also, we give some examples to illustrate our results.
