• Kyungpook Mathematical Journal
• /
• v.54 no.1
• /
• pp.95-101
• /
• 2014

An n-tuple ($a_1,a_2,{\ldots},a_n$) is symmetric, if $a_k$ = $a_{n-k+1}$, $1{\leq}k{\leq}n$. Let $H_n$ = {$(a_1,a_2,{\ldots},a_n)$ ; $a_k$ ${\in}$ {+,-}, $a_k$ = $a_{n-k+1}$, $1{\leq}k{\leq}n$} be the set of all symmetric n-tuples. A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair $S_n$ = (G,${\sigma}$) ($S_n$ = (G,${\mu}$)), where G = (V,E) is a graph called the underlying graph of $S_n$ and ${\sigma}$:E ${\rightarrow}H_n({\mu}:V{\rightarrow}H_n)$ is a function. The restricted super line graph of index r of a graph G, denoted by $\mathcal{R}\mathcal{L}_r$(G). The vertices of $\mathcal{R}\mathcal{L}_r$(G) are the r-subsets of E(G) and two vertices P = ${p_1,p_2,{\ldots},p_r}$ and Q = ${q_1,q_2,{\ldots},q_r}$ are adjacent if there exists exactly one pair of edges, say $p_i$ and $q_j$, where $1{\leq}i$, $j{\leq}r$, that are adjacent edges in G. Analogously, one can define the restricted super line symmetric n-sigraph of index r of a symmetric n-sigraph $S_n$ = (G,${\sigma}$) as a symmetric n-sigraph $\mathcal{R}\mathcal{L}_r$($S_n$) = ($\mathcal{R}\mathcal{L}_r(G)$, ${\sigma}$'), where $\mathcal{R}\mathcal{L}_r(G)$ is the underlying graph of $\mathcal{R}\mathcal{L}_r(S_n)$, where for any edge PQ in $\mathcal{R}\mathcal{L}_r(S_n)$, ${\sigma}^{\prime}(PQ)$=${\sigma}(P){\sigma}(Q)$. It is shown that for any symmetric n-sigraph $S_n$, its $\mathcal{R}\mathcal{L}_r(S_n)$ is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs $S_n$ for which $\mathcal{R}\mathcal{L}_r(S_n)$~$\mathcal{L}_r(S_n)$ and $$\mathcal{R}\mathcal{L}_r(S_n){\sim_=}\mathcal{L}_r(S_n)$$, where ~ and $$\sim_=$$ denotes switching equivalence and isomorphism and $\mathcal{R}\mathcal{L}_r(S_n)$ and $\mathcal{L}_r(S_n)$ are denotes the restricted super line symmetric n-sigraph of index r and super line symmetric n-sigraph of index r of $S_n$ respectively.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.1
• /
• pp.21-28
• /
• 2016

It is known that no two of the roots of the polynomial equation (1) $$\prod\limits_{l=1}^{n}(x-r_l)+\prod\limits_{l=1}^{n}(x+r_l)=0$$, where 0 < $r_1{\leq}r_2{\leq}{\cdots}{\leq}r_n$, can be equal and all of its roots lie on the imaginary axis. In this paper we show that for 0 < h < $r_k$, the roots of $$(x-r_k+h)\prod\limits_{{l=1}\\{l{\neq}k}}^{n}(x-r_l)+(x+r_k-h)\prod\limits_{{l=1}\\{l{\neq}k}}^{n}(x+r_l)=0$$ and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis.

• Journal of the Korean Mathematical Society
• /
• v.39 no.1
• /
• pp.31-49
• /
• 2002

An R$^{n}$ -geometry (P$^{n}$ , L) is a generalization of the Euclidean geometry on R$^{n}$ (see Def. 1.1). We can consider some topologies (see Def. 2.2) on the line set L such that the join operation V : P$^{n}$ $\times$ P$^{n}$ \ $\Delta$ longrightarrow L is continuous. It is a notable fact that in the case n = 2 the introduced topologies on L are same and the join operation V : P$^2$ $\times$ P$^2$ \ $\Delta$ longrightarrow L is continuous and open [10, 11]. It is a fundamental topological property of plane geometry, but in the cases n $\geq$ 3, it is no longer true. There are counter examples [2]. Hence, it is a fundamental problem to find suitable topologies on the line set L in an R$^{n}$ -geometry (P$^{n}$ , L) such that these topologies are compatible with the incidence structure of (P$^{n}$ , L). Therefore, we need to study the topologies of the line set L in an R$^{n}$ -geometry (P$^{n}$ , L). In this paper, the relations of such topologies on the line set L are studied.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.1
• /
• pp.35-43
• /
• 2009

We say that an isometric immersed hypersurface x : $M^n\;{\rightarrow}\;{\mathbb{R}}^{n+1}$ is of $L_k$-finite type ($L_k$-f.t.) if $x\;=\;{\sum}^p_{i=0}x_i$ for some positive integer p < $\infty$, $x_i$ : $M{\rightarrow}{\mathbb{R}}^{n+1}$ is smooth and $L_kx_i={\lambda}_ix_i$, ${\lambda}_i\;{\in}\;{\mathbb{R}}$, $0{\leq}i{\leq}p$, $L_kf=trP_k\;{\circ}\;{\nabla}^2f$ for $f\;{\in}\'C^{\infty}(M)$, where $P_k$ is the kth Newton transformation, ${\nabla}^2f$ is the Hessian of f, $L_kx\;=\;(L_kx^1,\;{\ldots},\;L_kx^{n+1})$, $x=(x^1,\;{\ldots},\;x^{n+1})$. In this article we study the following(hyper)surfaces in ${\mathbb{R}}^{n+1}$ from the view point of $L_1$-finiteness type: totally umbilic ones, generalized cylinders $S^m(r){\times}{\mathbb{R}}^{n-m}$, ruled surfaces in ${\mathbb{R}}^{n+1}$ and some revolution surfaces in ${\mathbb{R}}^3$.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.4
• /
• pp.1071-1085
• /
• 2016

For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.

• Communications of the Korean Mathematical Society
• /
• v.16 no.4
• /
• pp.621-626
• /
• 2001

Let { $A_{>o}$ t= exp(M log t)} $_{t}$ be a dilation group where M is a real n$\times$n matrix whose eigenvalues has strictly positive real part, and let $\rho$be an $A_{t}$ -homogeneous distance function defined on ( $R^{n}$ ). Suppose that K is a function defined on ( $R^{n}$ ) such that /K(x)/$\leq$ (No Abstract.see full/text) for a decreasing function defined on (t) on R+ satisfying where wo(x)=│log│log (x)ll. For f$\in$ $L_{1}$ ( $R^{n}$ ), define f(x)=sup t>0 Kt*f(x)=t-v K(Al/tx) and v is the trace of M. Then we show that \ulcorner is a bounded operator of $L_{-{1}( $R^{n}$ ) into $L^1$,$\infty$( $R^{n}$).

• Journal of the Korean Mathematical Society
• /
• v.48 no.2
• /
• pp.431-447
• /
• 2011

We investigate the asymptotic behavior of positive solutions to the elliptic equation (0.1) ${\Delta}u+|x|^{l_1}u^p+|x|^{l_2}u^q=0$ in $\mathbb{R}^n$. We obtain a conclusion that, for n $\geq$ 3, -2 < $l_2$ < $l_1$ $\leq$ 0 and q > p > 1, any positive radial solution to (0.1) has the following properties: $lim_{r{\rightarrow}{\infty}}r^{\frac{2+l_1}{p-1}}\;u$ and $lim_{r{\rightarrow}0}r^{\frac{2+l_2}{q-1}}\;u$ always exist if $\frac{n+1_1}{n-2}$ < p < q, $p\;{\neq}\;\frac{n+2+2l_1}{n-2}$, $q\;{\neq}\;\frac{n+2+2l_2}{n-2}$. In addition, we prove that the singular positive solution of (0.1) is unique under some conditions.

• Journal of the Korean Society of Fisheries and Ocean Technology
• /
• v.9 no.1
• /
• pp.1-18
• /
• 1973

For regulating the depth of midwater trawl nets towed at the optimum constant speed, the changes in the shape of warps caused by adding a weight on an arbitrary point of the warp of catenary shape is studied. The shape of a warp may be approximated by a catenary. The resultant inferences under this assumption were experimented. Accordingly feasibilities for the application of the result of this study to the midwater trawl nets were also discussed. A series of experiments for basic midwater trawl gear models in water tank and a couple of experiments of a commercial scale gears at sea which involve the properly designed depth control devices having a variable attitude horizontal wing were carried out. The results are summarized as follows: 1. According to the dimension analysis the depth y of a midwater trawl net is introduced by $$y=kLf(\frac{W_r}{R_r},\;\frac{W_o}{R_o},\;\frac{W_n}{R_n})$$) where k is a constant, L the warp length, f the function, and $W_r,\;W_o$ and $W_n$ the apparent weights of warp, otter board and the net, respectively, 2. When a boat is towing a body of apparent weight $W_n$ and its drag $D_n$ by means of a warp whose length L and apparent weight $W_r$ per unit length, the depth y of the body is given by the following equation, provided that the shape of a warp is a catenary and drag of the warp is neglected in comparison with the drag of the body: $$y=\frac{1}{W_r}\{\sqrt{{D_n^2}+{(W_n+W_rL)^2}}-\sqrt{{D_n^2+W_n}^2\}$$ 3. The changes ${\Delta}y$ of the depth of the midwater trawl net caused by changing the warp length or adding a weight ${\Delta}W_n$_n to the net, are given by the following equations: $${\Delta}y{\approx}\frac{W_n+W_{r}L}{\sqrt{D_n^2+(W_n+W_{r}L)^2}}{\Delta}L$$ $${\Delta}y{\approx}\frac{1}{W_r}\{\frac{W_n+W_rL}{\sqrt{D_n^2+(W_n+W_{r}L)^2}}-{\frac{W_n}{\sqrt{D_n^2+W_n^2}}\}{\Delta}W_n$$ 4. A change ${\Delta}y$ of the depth of the midwater trawl net by adding a weight $W_s$ to an arbitrary point of the warp takes an equation of the form $${\Delta}y=\frac{1}{W_r}\{(T_{ur}'-T_{ur})-T_u'-T_u)\}$$ Where $$T_{ur}^l=\sqrt{T_u^2+(W_s+W_{r}L)^2+2T_u(W_s+W_{r}L)sin{\theta}_u$$ $$T_{ur}=\sqrt{T_u^2+(W_{r}L)^2+2T_uW_{r}L\;sin{\theta}_u$$ $$T_{u}^l=\sqrt{T_u^2+W_s^2+2T_uW_{s}\;sin{\theta}_u$$ and $T_u$ represents the tension at the point on the warp, ${\theta}_u$ the angle between the direction of $T_u$ and horizontal axis, $T_u^2$ the tension at that point when a weights $W_s$ adds to the point where $T_u$ is acted on. 5. If otter boards were constructed lighter and adequate weights were added at their bottom to stabilize them, even they were the same shapes as those of bottom trawls, they were definitely applicable to the midwater trawl gears as the result of the experiments. 6. As the results of water tank tests the relationship between net height of H cm velocity of v m/sec, and that between hydrodynamic resistance of R kg and the velocity of a model net as shown in figure 6 are respectively given by $$H=8+\frac{10}{0.4+v}$$ $$R=3+9v^2$$ 7. It was found that the cross-wing type depth control devices were more stable in operation than that of the H-wing type as the results of the experiments at sea. 8. The hydrodynamic resistance of the net gear in midwater trawling is so large, and regarded as nearly the drag, that sweeping depth of the gear was very stable in spite of types of the depth control devices. 9. An area of the horizontal wing of the H-wing type depth control device was $1.2{\times}2.4m^2$. A midwater trawl net of 2 ton hydrodynamic resistance was connected to the devices and towed with the velocity of 2.3 kts. Under these conditions the depth change of about 20m of the trawl net was obtained by controlling an angle or attack of $30^{\circ}$.

• The Pure and Applied Mathematics
• /
• v.8 no.1
• /
• pp.71-78
• /
• 2001

In 1984, Johnson[A bounded convergence theorem for the Feynman in-tegral, J, Math. Phys, 25(1984), 1323-1326] proved a bounded convergence theorem for hte Feynman integral. This is the first stability theorem of the Feynman integral as an $L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$ theory. Johnson and Lapidus [Generalized Dyson series, generalized Feynman digrams, the Feynman integral and Feynmans operational calculus. Mem, Amer, Math, Soc. 62(1986), no 351] studied stability theorems for the Feynman integral as an $L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$ theory for the functional with arbitrary Borel measure. These papers treat functionals which involve only a single integral. In this paper, we obtain the stability theorems for the Feynman integral as an $L(L_1 (\mathbb{R}^N), L_{\infty}(\mathbb{R}^{N}))$theory for the functionals which involve double integral with some Borel measures.

• Journal of the Korean Mathematical Society
• /
• v.34 no.4
• /
• pp.1029-1036
• /
• 1997

Let ${A_t)}_{t>0}$ be a dilation group given by $A_t = exp(-P log t)$, where P is a real $n \times n$ matrix whose eigenvalues has strictly positive real part. Let $\nu$ be the trace of P and $P^*$ denote the adjoint of pp. Suppose that $K$ is a function defined on $R^n$ such that $$\mid$K(x)$\mid$ \leq k($\mid$x$\mid$_Q)$ for a bounded and decreasing function $k(t) on R_+$ satisfying $k \diamond $\mid$\cdot$\mid$_Q \in \cup_{\varepsilon >0}L^1((1 + $\mid$x$\mid$)^\varepsilon dx)$ where $Q = \int_{0}^{\infty} exp(-tP^*) exp(-tP)$ dt and the norm $$\mid$\cdot$\mid$_Q$ stands for $$\mid$x$\mid$_Q = \sqrt{}, x \in R^n$. For $f \in L^1(R^n)$, define $mf(x) = sup_{t>0}$\mid$K_t * f(x)$\mid$$ where $K_t(X) = t^{-\nu}K(A_{1/t}^* x)$. Then we show that $m$ is a bounded operator of $L^1(R^n) into L^{1, \infty}(R^n)$.