• Bulletin of the Korean Mathematical Society
• /
• v.54 no.6
• /
• pp.2165-2182
• /
• 2017

Let G be a group and $\mathbb{C}$ the field of complex numbers. Suppose ${\sigma}:G{\rightarrow}G$ is an endomorphism satisfying ${{\sigma}}({{\sigma}}(x))=x$ for all x in G. In this paper, we first determine the central solution, f : G or $G{\times}G{\rightarrow}\mathbb{C}$, of the functional equation $f(xy)+f({\sigma}(y)x)=2f(x)+2f(y)$ for all $x,y{\in}G$, which is a variant of the quadratic functional equation. Using the central solution of this functional equation, we determine the general solution of the functional equation f(pr, qs) + f(sp, rq) = 2f(p, q) + 2f(r, s) for all $p,q,r,s{\in}G$, which is a variant of the equation f(pr, qs) + f(ps, qr) = 2f(p, q) + 2f(r, s) studied by Chung, Kannappan, Ng and Sahoo in [3] (see also [16]). Finally, we determine the solutions of this equation on the free groups generated by one element, the cyclic groups of order m, the symmetric groups of order m, and the dihedral groups of order 2m for $m{\geq}2$.

• Journal of the Korean Chemical Society
• /
• v.24 no.6
• /
• pp.444-451
• /
• 1980

In order to further elucidate the process of electron solvation in liquids, the medium effect, as the difference between the free energy of $H^+$ in aqueous and non-aqueous states (${\Delta}{\Delta}G_0$), of THF-water mixtures has been investigated. (${\Delta}{\Delta}G_0$) were determined by electromotive force masurements of the cell Pt$H_2Q$, Q, HCI, THF, $H_2O$|KC1 | $Hg_2Cl_2$|Hg(Pt), where $H_2Q$ and Q are hydroquinone and quinone respectively. The effect of dielectric constant on the difference of free energy and the absorption energy of solvated electrons have been studied. For the consideration of these effect the polymerization of water in THF has been studied. Near infrared spectrum of O-H stetching energy has been used to measure the extent of water aggregates. The expermental results indicate that at least in some composition of binary mixtures the electrons or other ions are solvated preferentially with one component of solvents.

• 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)$$.

• Korean Journal of Food Science and Technology
• /
• v.47 no.3
• /
• pp.409-412
• /
• 2015

Changes in the contents of mevinolins (natural statins, $0-568.18{\mu}g$), coenzyme Q10 (CoQ10, $26.41-65.59{\mu}g$), and tocopherols ($232.80-312.87{\mu}g/g$ dry weight) in Monascus-fermented soybean were determined using HPLC. Significant increases (p<0.05) in mevinolins and CoQ10 were obtained in Monascus-fermented soybean after 20 days of fermentation compared with unfermented soybean (0 days), whereas no significant change (p>0.05), or a slight decrease, in tocopherols was observed. The results indicate that Monascus-fermentation has great potential for enriching mevinolin and CoQ10 in soybeans.

• Applied Chemistry for Engineering
• /
• v.7 no.2
• /
• pp.379-386
• /
• 1996

Photosensitive compound, 1-methyl-4-[2-(4-diethylacetylphenyl)ethenyl] pridinium methosulfate(SbQ-A salt), was synthesized from dimethyl sulfate, terephthalaldehyde mono-(diethylacetal) and 4-picoline. SbQ-A salts were reacted with poly(vinyl alcohol)s, (PVA) in aqueous solution with phosphoric acid as catalyst to give photosensitive PVA-SbQ with different SbQ content and molecular weight. Relative photosensitivity of PVA-SbQ was determined by gray scale(GS) method. The rotative sensitivity of PVA-SbQ increased with increasing amount of bound SbQ in the case of high molecular weight(MW=77,000-79,000g/mol) as substrate and decreased with decreasing molecular weight of PVA with about constant(1.3mol%) amount of bound SbQ. The most sensitive polymer was obtained when SbQ group content in PVA-SbQ reached about 2.63mol% in the case of high molecular weight(77,000-79,000g/mol) PVA. This sample showed 90 times greater sensitivity than dichromated PVA as reference photosensitive system. PVA-SbQ photosensitive polymer synthesized was applied to the photolithographic screening process of phosphor on the panel of cathode ray tube(CRT). Phosphor slurry was made with PVA-SbQ, phosphor, a small amount of surfactant and other additives using water as medium. The slurry was coated onto panel, dried by heater, exposed to UV light and then developed by distilled water. When a small amount of cationic surfactant such as cetyltrimethylammonium chloride was used in the slurry formulation, the sharpness of phosphor pattern was equal to or better than that of dichromated PVA photosensitive polymer system used currently.

• Journal of applied mathematics & informatics
• /
• v.35 no.1_2
• /
• pp.121-130
• /
• 2017

This study is concerned with the existence of positive solution for the following nonlinear elliptic system $$\{-M_1(\int_{\Omega}{\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^pdx)div({\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)\\{\hfill{120}}={\mid}x{\mid}^{-(a+1)p+c_1}\({\alpha}_1A_1(x)f(v)+{\beta}_1B_1(x)h(u)\),\;x{\in}{\Omega},\\-M_2(\int_{\Omega}{\mid}x{\mid}^{-bq}{\mid}{\nabla}v{\mid}^qdx)div({\mid}x{\mid}^{-bq}{\mid}{\nabla}v{\mid}^{q-2}{\nabla}v)\\{\hfill{120}}={\mid}x{\mid}^{-(b+1)q+c_2}\({\alpha}_2A_2(x)g(u)+{\beta}_2B_2(x)k(v)\),\;x{\in}{\Omega},\\{u=v=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}$ is a bounded smooth domain of ${\mathbb{R}}^N$ with $0{\in}{\Omega}$, 1 < p, q < N, $0{\leq}a$ < $\frac{N-p}{p}$, $0{\leq}b$ < $\frac{N-q}{q}$ and ${\alpha}_i,{\beta}_i,c_i$ are positive parameters. Here $M_i,A_i,B_i,f,g,h,k$ are continuous functions and we discuss the existence of positive solution when they satisfy certain additional conditions. Our approach is based on the sub and super solutions method.

• Korean Journal of Mathematics
• /
• v.27 no.2
• /
• pp.505-514
• /
• 2019

In this paper we study the approximation properties of a continuous function by the sequence of (p, q)-Bernstein operators for q > p > 1. We obtain bounds of (p, q)-Bernstein operators. Further we prove that if a continuous function admits an analytic continuation into the disk $\{z:{\mid}z{\mid}{\leq}{\rho}\}$, then $B^n_{p,q}(g;z){\rightarrow}g(z)(n{\rightarrow}{\infty})$ uniformly on any compact set in the given disk $\{z:{\mid}z{\mid}{\leq}{\rho}\}$, ${\rho}>0$.

• Journal of the Korean Mathematical Society
• /
• v.8 no.1
• /
• pp.25-30
• /
• 1971

Put ($$^*$$) $$G[x,y]={\sum}\limits^{p+q=n}_{p,q=0}[-n]_{p+q}c_{p,q}x^py^q$$, where $[{\lambda}]_m$ is the Pocbhammer symbol and the $c_{p,q}$ are arbitrary constants. Making use of the specialized forms of some of his earlier results (see [8] and [9] the author derives here bilateral generating functions of the type ($$^{**}$$) $${\sum}\limits^{\infty}_{n=0}{\frac{[\lambda]_n}{n!}}_2F_1[\array{{\rho}-n,\;{\alpha};\\{\lambda}+{\rho};}x]\;G[y,z]t^n$$ where ${\alpha}$, ${\rho}$ and ${\lambda}$ are arbitrary complex numbers. In particular, it is shown that when G[y, z] is a double hypergeometric polynomial, the right-band member of ($^{**}$) belongs to a class of general triple hypergeometric functions introduced by the author [7]. An interesting special case of ($^{**}$) when ${\rho}=-m,\;m$ being a nonnegative integer, yields a class of bilateral generating functions for the Jacobi polynomials $\{P_n{^{{\alpha},{\beta}}}(x)\}$ in the form ($$^{***}$$) $${\sum\limits^{\infty}_{n=0}}\(\array{m+n\\n}\)P{^{({\alpha}-n,{\beta}-n)}_{m+n}(x)\;G[y,z]{\frac{t^n}{n!}}$$, which provides a unification of several known results. Further extensions of ($^{**}$) and ($^{***}$) with G[y, z] replaced by an analogous multiple sum $H\[y_1,{\cdots},y_m\]$ are also discussed.

• Bulletin of the Korean Mathematical Society
• /
• v.36 no.2
• /
• pp.287-303
• /
• 1999

The general solution of the functional equation f1(pr, qs) + f2(ps, qr) = g(p,q) + h(r,s) for p, q, r, s $\in$] 0, 1[will be investigated without any regularity assumptions on the unknown functions f1, f2, g, h:]0.1[->R.

• Journal of the Korean Mathematical Society
• /
• v.42 no.2
• /
• pp.365-386
• /
• 2005

Let L be the free Lie superalgebra generated by a $Z_2$-graded vector space V over C. Suppose that g is a Lie superalgebra gl(m, n) or q(n). We study the g-module structure on the kth homogeneous component Lk of L when V is the natural representation of g. We give the multiplicities of irreducible representations of g in Lk by using the character of Lk. The multiplicities are given in terms of the character values of irreducible (projective) representations of the symmetric groups.