• Bulletin of the Korean Mathematical Society
• /
• v.55 no.5
• /
• pp.1441-1462
• /
• 2018

In this paper, we investigate the fractional p&q-Kirchhoff type system $$\{M_1([u]^p_{s,p})(-{\Delta})^s_pu+V_1(x){\mid}u{\mid}^{p-2}u\\{\hfill{10}}={\ell}k^{-1}F_u(x,\;u,\;v)+{\lambda}{\alpha}(x){\mid}u{\mid}^{m-2}u,\;x{\in}{\Omega}\\M_2([u]^q_{s,q})(-{\Delta})^s_qv+V_2(x){\mid}v{\mid}^{q-2}v\\{\hfill{10}}={\ell}k^{-1}F_v(x,u,v)+{\mu}{\alpha}(x){\mid}v{\mid}^{m-2}v,\;x{\in}{\Omega},\\u=v=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}{\subset}{\mathbb{R}}^N$ is an unbounded domain with smooth boundary ${\partial}{\Omega}$, and $0<s<1<p{\leq}q$ and sq < N, ${\lambda},{\mu}>0$, $1<m{\leq}k<p^*_s$, ${\ell}{\in}R$, while $[u]^t_{s,t}$ denotes the Gagliardo semi-norm given in (1.2) below. $V_1(x)$, $V_2(x)$, $a(x):{\mathbb{R}}^N{\rightarrow}(0,\;{\infty})$ are three positive weights, $M_1$, $M_2$ are continuous and positive functions in ${\mathbb{R}}^+$. Using variational methods, we prove existence of infinitely many high-energy solutions for the above system.

• Kyungpook Mathematical Journal
• /
• v.59 no.3
• /
• pp.415-431
• /
• 2019

In this paper we introduce a new class of operators which will be called the class of k-quasi-class A(s, t) operators. An operator $T{\in}B(H)$ is said to be k-quasi-class A(s, t) if $$T^{*k}(({\mid}T^*{\mid}^t{\mid}T{\mid}^{2s}{\mid}T^*{\mid}^t)^{\frac{1}{t+s}}-{\mid}T^*{\mid}^{2t})T^k{\geq}0$$, where s > 0, t > 0 and k is a natural number. We show that an algebraically k-quasi-class A(s, t) operator T is polaroid, has Bishop's property ${\beta}$ and we prove that Weyl type theorems for k-quasi-class A(s, t) operators. In particular, we prove that if $T^*$ is algebraically k-quasi-class A(s, t), then the generalized a-Weyl's theorem holds for T. Using these results we show that $T^*$ satisfies generalized the Weyl's theorem if and only if T satisfies the generalized Weyl's theorem if and only if T satisfies Weyl's theorem. We also examine the hyperinvariant subspace problem for k-quasi-class A(s, t) operators.

• Journal of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.843-867
• /
• 2022

Maynard proved that there exists an effectively computable constant q1 such that if q ≥ q1, then $\frac{{\log}\;q}{\sqrt{q}{\phi}(q)}Li(x){\ll}{\pi}(x;\;q,\;m)<\frac{2}{{\phi}(q)}Li(x)$ for x ≥ q8. In this paper, we will show the following. Let 𝛿1 and 𝛿2 be positive constants with 0 < 𝛿1, 𝛿2 < 1 and 𝛿1 + 𝛿2 > 1. Assume that L ≠ ℚ is a number field. Then there exist effectively computable constants c0 and d1 such that for dL ≥ d1 and x ≥ exp (326n𝛿1L(log dL)1+𝛿2), we have $$\|{\pi}_C(x)-\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)\|\;{\leq}\;\(1-c_0\frac{1og\;d_L}{d^{7.072}_L}\)\;\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)$$.

• Journal of the Korean Mathematical Society
• /
• v.49 no.6
• /
• pp.1259-1271
• /
• 2012

We consider a tournament T = (V,A). For each subset X of V is associated the subtournament T(X) = (X,$A{\cap}(X{\times}X)$) of T induced by X. We say that a tournament T' embeds into a tournament T when T' is isomorphic to a subtournament of T. Otherwise, we say that T omits T'. A subset X of V is a clan of T provided that for a, $b{\in}X$ and $x{\in}V{\backslash}X$, $(a,x){\in}A$ if and only if $(b,x){\in}A$. For example, ${\emptyset}$, $\{x\}(x{\in}V)$ and V are clans of T, called trivial clans. A tournament is indecomposable if all its clans are trivial. In 2003, B. J. Latka characterized the class ${\tau}$ of indecomposable tournaments omitting a certain tournament $W_5$ on 5 vertices. In the case of an indecomposable tournament T, we will study the set $W_5$(T) of vertices $x{\in}V$ for which there exists a subset X of V such that $x{\in}X$ and T(X) is isomorphic to $W_5$. We prove the following: for any indecomposable tournament T, if $T{\notin}{\tau}$, then ${\mid}W_5(T){\mid}{\geq}{\mid}V{\mid}$ -2 and ${\mid}W_5(T){\mid}{\geq}{\mid}V{\mid}$ -1 if ${\mid}V{\mid}$ is even. By giving examples, we also verify that this statement is optimal.

• Journal of the Korean Mathematical Society
• /
• v.39 no.5
• /
• pp.783-800
• /
• 2002

Let B denote the unit ball in $C^n$, and ν the normalized Lebesgue measure on B. For $\alpha$ > -1, define $dv_\alpha$(z) ＝ $c_\alpha$$(1-\midz\mid^2)^{\alpha}$dν(z), z $\in$ B. Here $c_\alpha$ is a positive constant such that $v_\alpha$(B) ＝ 1. Let H(B) denote the space of all holomorphic functions in B. For $p\geq1$, define the Bergman-Privalov space $(AN)^{p}(v_\alpha)$ by $(AN)^{p}(v_\alpha)$ = ${f\inH(B)$ : $\int_B{log(1+\midf\mid)}^pdv_\alpha\;<\;\infty}$ In this paper we prove that a function $f\inH(B)$ is in $(AN)^{p}$$(v_\alpha)$ if and only if $(1+\midf\mid)^{-2}{log(1+\midf\mid)}^{p-2}\mid\nablaf\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case 1＜p＜$\infty$, or $(1+\midf\mid)^{-2}\midf\mid^{-1}\mid{\nabla}f\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case p ＝ 1, where $nabla$f is the gradient of f with respect to the Bergman metric on B. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].

• The Journal of the Korea institute of electronic communication sciences
• /
• v.11 no.2
• /
• pp.165-170
• /
• 2016

The characteristic and application of a new duality structure in the $NCV-{\mid}v_1$ > library is studied in this paper. All unitary operations on arbitrarily many qudit's n can be expressed as composition of one- and two-qudit $NCV-{\mid}v_1$ > libraries if their state vectors are eigenvectors. This research provides an extended realization from Barenco's many bits n operator(U(2n)) which is applicable to only all positive polarity statevectors to whole polarity ones. At the control gate synthesis of a unitary operator, such an enhanced expansion is possible due to their symmetric duality property in the case of using both $NCV-{\mid}v_1$ > and $NCV^{\dag}-{\mid}v_1$ > libraries which make the AND predominantly dependent cascade synthesis possible.

• Journal of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.747-775
• /
• 2020

In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows, $$(P)\;\{(-{\Delta}_p)^su={\lambda}{\mid}u{\mid}^{q-2}u+{\frac{{\mid}u{\mid}^{p{^*_s}(t)-2}u}{{\mid}x{\mid}^t}}{\hspace{10}}in\;{\Omega},\\u=0{\hspace{217}}in\;{\mathbb{R}}^N{\backslash}{\Omega},$$ where Ω ⊂ ℝ^N is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < t < sp < N, 1 < q < p < p^∗_s where $p^*_s={\frac{N_p}{N-sp}}$, $p^*_s(t)={\frac{p(N-t)}{N-sp}}$, are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional p-laplacian (-∆_p)^su with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by $\displaystyle(-{\Delta}_p)^su(x)=2{\lim_{{\epsilon}{\searrow}0}}\int{_{{\mathbb{R}}^N{\backslash}{B_{\epsilon}}}}\;\frac{{\mid}u(x)-u(y){\mid}^{p-2}(u(x)-u(y))}{{\mid}x-y{\mid}^{N+ps}}dy$, x ∈ ℝ^N. The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some α ∈ (0, 1), the weak solution to the problem (P) is in C^1,α(${\bar{\Omega}}$).

• Journal of Korean Association for Spatial Structures
• /
• v.19 no.3
• /
• pp.51-59
• /
• 2019

A hybrid mid-story seismic isolation system with a smart damper has been proposed to mitigate seismic responses of tall buildings. Based on previous research, a hybrid mid-story seismic isolation system can provide effective control performance for reduction of seismic responses of tall buildings. Structural design of the hybrid mid-story seismic isolation system is generally performed after completion of structural design of a building structure. This design concept is called as an iterative design which is a general design process for structures and control devices. In the iterative design process, optimal design solution for the structure and control system is changed at each design stage. To solve this problem, the integrated optimal design method for the hybrid mid-story seismic isolation system and building structure was proposed in this study. An existing building with mid-story isolation system, i.e. Shiodome Sumitomo Building, was selected as an example structure for more realistic study. The hybrid mid-story isolation system in this study was composed of MR (magnetorheological) dampers. The stiffnessess and damping coefficients of the example building, maximum capacity of MR damper, and stiffness of isolation bearing were simultaneously optimized. Multi-objective genetic optimization method was employed for the simultaneous optimization of the example structure and the mid-story seismic isolation system. The optimization results show that the simultaneous optimization method can provide better control performance than the passive mid-story isolation system with reduction of structural materials.

• Journal of Astronomy and Space Sciences
• /
• v.22 no.4
• /
• pp.409-418
• /
• 2005

Magnetospheric substorms occur frequently during magnetic storms, suggesting that the two phenomena are closely associated. We can investigate the relation between magnetospheric substorms and magnetic storms by examining the correlation between AE and Dst indices. For this purpose, we calculated the monthly cumulative AU, $\mid{AL}\mid$ and $\mid{Dst}\mid$ indices. The correlation coefficient between the monthly cumulative $\mid{AL}\mid$ and $\mid{Dst}\mid$ index is found to be 0.60, while that between monthly cumulative AU and $\mid{Dst}\mid$ index is 0.28. This result indicates that substorms seem to contribute to the development of magnetic storms. On the other hand, it has been reported that the interplanetary electric field associated with southward IMF intensifies the magnetospheric convection, which injects charged particles into the inner magnetosphere, thus developing the ring current. To evaluate the contribution of the interplanetary electric field to the development of the storm time ring current belt, we compared the monthly cumulative interplanetary electric field and the monthly cumulative Dst index. The correlation coefficient between the two cumulative indices is 0.83 for southward IMP and 0.39 for northward IMF. It indicates that magnetospheric convection induced by southward IMF is also important in developing magnetic storms. Therefore, both magnetospheric substorm and enhanced magnetospheric convection seem to contribute to the buildup of magnetic storm.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.1
• /
• pp.105-116
• /
• 2013

In this paper, we consider the positive solution to a Cauchy problem in $\mathbb{B}^N$ of the fast diffusive equation: ${\mid}x{\mid}^mu_t={div}(\mid{\nabla}u{\mid}^{p-2}{\nabla}u)+{\mid}x{\mid}^nu^q$, with nontrivial, nonnegative initial data. Here $\frac{2N+m}{N+m+1}$ < $p$ < 2, $q$ > 1 and 0 < $m{\leq}n$ < $qm+N(q-1)$. We prove that $q_c=p-1{\frac{p+n}{N+m}}$ is the critical Fujita exponent. That is, if 1 < $q{\leq}q_c$, then every positive solution blows up in finite time, but for $q$ > $q_c$, there exist both global and non-global solutions to the problem.