• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권4호
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• pp.403-412
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• 2015

Kato[6] and Torres[9] characterized the Weierstrass semigroup of ramification points on h-hyperelliptic curves. Also they showed the converse results that if the Weierstrass semigroup of a point P on a curve C satisfies certain numerical condition then C can be a double cover of some curve and P is a ramification point of that double covering map. In this paper we expand their results on the Weierstrass semigroup of a ramification point of a double covering map to the Weierstrass semigroup of a pair (P, Q). We characterized the Weierstrass semigroup of a pair (P, Q) which lie on the same fiber of a double covering map to a curve with relatively small genus. Also we proved the converse: if the Weierstrass semigroup of a pair (P, Q) satisfies certain numerical condition then C can be a double cover of some curve and P, Q map to the same point under that double covering map.

• 전자공학회논문지C
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• 제34C권7호
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• pp.82-91
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• 1997

In this paper, we analyze the dynamics of langevine competitive learning neural network based on its fokker-plank equation. From the viewpont of the stochastic differential equation (SDE), langevine competitive learning equation is one of langevine stochastic differential equation and has the diffusin equation on the topological space (.ohm., F, P) with probability measure. We derive the fokker-plank equation from the proposed algorithm and prove by introducing a infinitestimal operator for markov semigroups, that the weight vector in the particular simplex can converge to the globally optimal point under the condition of some convex or pseudo-convex performance measure function. Experimental resutls for pattern recognition of the remote sensing data indicate the superiority of langevine competitive learning neural network in comparison to the conventional competitive learning neural network.

• Kyungpook Mathematical Journal
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• 제49권3호
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• pp.457-471
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• 2009

Let S = {a, b, c, ...} and ${\Gamma}$ = {${\alpha}$, ${\beta}$, ${\gamma}$, ...} be two nonempty sets. S is called a ${\Gamma}$-semigroup if $a{\alpha}b{\in}S$, for all ${\alpha}{\in}{\Gamma}$ and a, b ${\in}$ S and $(a{\alpha}b){\beta}c=a{\alpha}(b{\beta}c)$, for all a, b, c ${\in}$ S and for all ${\alpha}$, ${\beta}$ ${\in}$ ${\Gamma}$. An element $e{\in}S$ is said to be an ${\alpha}$-idempotent for some ${\alpha}{\in}{\Gamma}$ if $e{\alpha}e$ = e. A ${\Gamma}$-semigroup S is called an E-inversive ${\Gamma}$-semigroup if for each $a{\in}S$ there exist $x{\in}S$ and ${\alpha}{\in}{\Gamma}$ such that a${\alpha}$x is a ${\beta}$-idempotent for some ${\beta}{\in}{\Gamma}$. A ${\Gamma}$-semigroup is called a right E-${\Gamma}$-semigroup if for each ${\alpha}$-idempotent e and ${\beta}$-idempotent f, $e{\alpha}$ is a ${\beta}$-idempotent. In this paper we investigate different properties of E-inversive ${\Gamma}$-semigroup and right E-${\Gamma}$-semigroup.

• Korean Journal of Mathematics
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• 제19권3호
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• pp.279-289
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• 2011

Let E be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, C be a nonempty closed convex subset of E and f : $C{\rightarrow}C$ be a fixed bounded continuous strong pseudocontraction with the coefficient ${\alpha}{\in}(0,1)$. Let $\{{\lambda}_t\}_{0<t<1}$ be a net of positive real numbers such that ${\lim}_{t{\rightarrow}0}{\lambda}_t={\infty}$ and S = {$T(s)$ : $0{\leq}s$ < ${\infty}$} be a nonexpansive semigroup on C such that $F(S){\neq}{\emptyset}$, where F(S) denotes the set of fixed points of the semigroup. Then sequence {$x_t$} defined by $x_t=tf(x_t)+(1-t)\frac{1}{{\lambda}_t}{\int_{0}}^{{\lambda}_t}T(s)x{_t}ds$ converges strongly as $t{\rightarrow}0$ to $\bar{x}{\in}F(S)$, which solves the following variational inequality ${\langle}(f-I)\bar{x},\;p-\bar{x}{\rangle}{\leq}0$ for all $p{\in}F(S)$.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.185-220
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• 2016

We study the Gauss and Poisson semigroups connected with the Riemann-Liouville operator defined on the half plane. Next, we establish a principle of maximum for the singular partial differential operator $${\Delta}_{\alpha}={\frac{{\partial}^2}{{\partial}r^2}+{\frac{2{\alpha}+1}{r}{\frac{\partial}{{\partial}r}}+{\frac{{\partial}^2}{{\partial}x^2}}+{\frac{{\partial}^2}{{\partial}t^2}}};\;(r,x,t){\in}]0,+{\infty}[{\times}{\mathbb{R}}{\times}]0,+{\infty}[$$. Later, we define the Littlewood-Paley g-function and using the principle of maximum, we prove that for every $p{\in}]1,+{\infty}[$, there exists a positive constant $C_p$ such that for every $f{\in}L^p(d{\nu}_{\alpha})$, $${\frac{1}{C_p}}{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}{\leqslant}{\parallel}g(f){\parallel}_{p,{\nu}_{\alpha}}{\leqslant}C_p{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}$$.

• Kyungpook Mathematical Journal
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• 제48권4호
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• pp.559-575
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• 2008

A commutative hypercomplex system $L_1$(Q,m) is, roughly speaking, a space which is defined by a structure measure (c(A,B, r), (A,$B{\in}{\beta}$(Q)). Such space has bee studied by Berezanskii and Krein. Our main purpose is to establish a generalization of convolution semigroups and to discuss the role of the L$\'{e}$vy measure in the L$\'{e}$vy-Khinchin representation in terms of continuous negative definite functions on the dual hypercomplex system.

• 대한수학회보
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• 제24권1호
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• pp.23-26
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• 1987

The aim of this note is to study some properties of Schrodinger operators, the magnetic case, $H_{0}$ (a)=1/2(-i.del.-a)$^{2}$; H(a)= $H_{0}$ (a)+V, where a=( $a_{1}$,.., $a_{n}$ ).mem. $L^{2}$$_{loc}$ and V is a potential energy. Also, we are interested in solutions, .psi., of H(a).psi.=E.psi. in the sense that (.psi., $e^{-tH}$(a).PSI.)= $e^{-tE}$(.psi.,.PSI.) for all .PSI..mem. $C_{0}$ $^{\infty}$( $R^{n}$ ) (see B. Simon [1]). In section 2, under some conditions, we find that a semibounded quadratic form of H9a) exists and that the Schrodinger operator H(a) with Re V.geq.0 is accretive on a form domain Q( $H_{0}$ (a)). But, it is well-known that the Schrodinger operator H=1/2.DELTA.+V with Re V.geq.0 is accretive on $C_{0}$ $^{\infty}$( $R^{n}$ ) in N Okazawa [4]. In section 3, we want to discuss $L^{p}$ estimates of Schrodinger semigroups.ups.

• 대한수학회논문집
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• 제34권1호
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• pp.185-202
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• 2019

In this paper, we investigate many types of stability, like (uniform stability, exponential stability and h-stability) of the first order dynamic equations of the form $$\{u^{\Delta}(t)=Au(t)+f(t),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ and $$\{u^{\Delta}(t)=Au(t)+f(t,u),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ in terms of the stability of the homogeneous equation $$\{u^{\Delta}(t)=Au(t),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ where f is rd-continuous in $t{\in}{\mathbb{T}}$ and with values in a Banach space X, with f(t, 0) = 0, and A is the generator of a $C_0$-semigroup $\{T(t):t{\in}{\mathbb{T}}\}{\subset}L(X)$, the space of all bounded linear operators from X into itself. Here D(A) is the domain of A and ${\mathbb{T}}{\subseteq}{\mathbb{R}}^{{\geq}0}$ is a time scale which is an additive semigroup with property that $a-b{\in}{\mathbb{T}}$ for any $a,b{\in}{\mathbb{T}}$ such that a > b. Finally, we give illustrative examples.