• Honam Mathematical Journal
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• v.40 no.3
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• pp.549-559
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• 2018

In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {$U_{kn}$} and {$V_{kn}$}. For example, for an odd prime number p, $${\sum\limits_{i=1}^{p-1}}H_i{\frac{U_{k(i+m)}}{V^i_k}}{\equiv}{\frac{(-1)^kU_{k(m+1)}}{_pV^{p-1}_k}}(V^p_k-V_{kp})(mod\;p)$$, where $m{\in}{\mathbb{Z}}$ and $k{\in}{\mathbb{Z}}$ with $p{\nmid}V_k$.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.703-722
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• 2021

Let G = (V, E) be a connected locally finite and weighted graph, ∆_p be the p-th graph Laplacian. Consider the p-th nonlinear equation -∆_pu + h|u|^p-2u = f(x, u) on G, where p > 2, h, f satisfy certain assumptions. Grigor'yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V. In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m-order differential operator 𝓛_m,p, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.

• Microbiology and Biotechnology Letters
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• v.32 no.3
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• pp.206-211
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• 2004

Levansucrase gene(lscA) from Pseudomonas aurantiaca was subcloned downstream of GAL1 promoter in pYES 2.0 and pYInu-AT [GAL10 promoter ＋ exoinulinase signal sequence of Kluyveromyces marxianus], resulting pYES-lscA and p YInu-lscA, respectively. The two expression plasmids were introduced into an invertase-deficient strain, Sacchromayces cerevisiae SEY2102, and transformants with high activity of levansucrase were selected. When each yeast transform ants was cultivated in medium containing galactose, the extracellular and intracellular activities of levansucrase reached about 8.62 U/ml with the strain harboring pYES-lscA and 5.43 U/ml with the strain harboring pYInu-lscA. The levansucrase activity of 80% was detected in the periplasmic space and cytoplasm. The levansucrase activity in the medium of SEY2102/pYInu-lscA was 0.87 U/ml whereas that of SEY2102/pYES-lscA was 0.47 U/ml, which implying the exoinulinase signal sequence didn't enhance the secretion efficiency of levansucrase. Furthermore, the recombinant levansucrase expressed in yeast seems to be produced as a hyper-glycosylated form.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.241-263
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• 2022

We consider a nonlinear Schrödinger equation with critical frequency, (P_𝜀) : 𝜀²∆v(x) - V(x)v(x) + |v(x)|^p-1v(x) = 0, x ∈ ℝ^N, and v(x) → 0 as |x| → +∞, for the infinite case as described by Byeon and Wang. Critical means that 0 ≤ V ∈ C(ℝ^N) verifies Ƶ = {V = 0} ≠ ∅. Infinite means that Ƶ = {x₀} and that, grossly speaking, the potential V decays at an exponential rate as x → x₀. For the semiclassical limit, 𝜀 → 0, the infinite case has a characteristic limit problem, (P_inf) : ∆u(x)-P(x)u(x) + |u(x)|^p-1u(x) = 0, x ∈ Ω, with u(x) = 0 as x ∈ Ω, where Ω ⊆ ℝ^N is a smooth bounded strictly star-shaped region related to the potential V. We prove the existence of an infinite number of solutions for both the original and the limit problem via a Ljusternik-Schnirelman scheme for even functionals. Fixed a topological level k we show that v_k,𝜀, a solution of (P_𝜀), subconverges, up to a scaling, to a corresponding solution of (P_inf ), and that v_k,𝜀 exponentially decays out of Ω. Finally, uniform estimates on ∂Ω for scaled solutions of (P_𝜀) are obtained.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.483-492
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• 1995

In this paper we discuss the existence and uniqueness of singular solutions for equations of the form $$ (F) u_t = u{xx} - $\mid$u$\mid$^{q-1} u_x - $\mid$u$\mid$^{p-1}u, p,q > 1, $$ in the domain $Q = {(x,t) : x \in R, t > 0}$. This equation represents a model of diffusion-convection with absorption.

• The Bulletin of The Korean Astronomical Society
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• v.20
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• pp.21.2-21.2
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• 1995

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1003-1031
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• 2020

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source $$\frac{{\partial}u}{{\partial}t}-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)={\lambda}u^k{\displaystyle\smashmargin{2}{\int\nolimits_{\Omega}}}u^sdx-{\mu}u^l{\mid}{\nabla}u{\mid}^q$$ in a bounded domain Ω ⊂ ℝ^N, where p > 1, the parameters k, s, l, q, λ > 0 and µ ≥ 0. Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on |∇u|. Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.

• Honam Mathematical Journal
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• v.17 no.1
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• pp.107-118
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• 1995

We study the asymptotic behavior of nonnegative singular solutions of semilinear parabolic equations of the type $$u_t={\Delta}u-(u^q)_y-u^p$$ defined in the whole space $x=(x,y){\in}R^{N-1}{\times}R$ for t>0, with initial data a Dirac mass, ${\delta}(x)$. The exponents q, p satisfy $$1 where $q^*=max\{q,(N+1)/N\}$.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.369-378
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• 2005

In this paper we will consider the weighted composition operator $W=uC_{\varphi}$ between two different $L^p(X,\;\Sigma,\;\mu)$ spaces, generated by measurable and non-singular transformations $\varphi$ from X into itself and measurable functions u on X. We characterize the functions u and transformations $\varphi$ that induce weighted composition operators between $L^p-spaces$ by using some properties of conditional expectation operator, pair $(u,\;\varphi)$ and the measure space $(X,\;\Sigma,\;\mu)$. Also, Fredholmness of these type operators will be investigated.

• The Pure and Applied Mathematics
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• v.21 no.3
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• pp.147-163
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• 2014

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions $$\{u^{{\prime}{\prime}}(t)+q(t)f(t,u(t),u^{\prime}(t))=0,\;t{\in}\mathbb{J}^{\prime},\\{\Delta}u(t_k)=I_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\{\Delta}u^{\prime}(t_k)=-L_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\u=(0)={\int}_{0}^{1}g(t)u(t)dt,\;u^{\prime}=0,$$) where the nonlinearity f(t, u, v) may be singular at v = 0. The proof is based on the theory of Leray-Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved.