• 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.

• Journal of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1461-1484
• /
• 2021

We deal with the following elliptic equations: $\{-div({\varphi}^{\prime}(\left|{\nabla}z\right|^2){\nabla}z)+V(x)\left|z\right|^{{\alpha}-2}z={\lambda}{\rho}(x)\left|z\right|^{r-2}z+h(x,z),\\z(x){\rightarrow}0,\;as\;\left|x\right|{\rightarrow}{\infty},$ in ℝ^N , where N ≥ 2, 1 < p < q < N, 1 < α ≤ p^*q'/p', α < q, 1 < r < min{p, α}, φ(t) behaves like t^q/2 for small t and t^p/2 for large t, and p' and q' the conjugate exponents of p and q, respectively. Here, V : ℝ^N → (0, ∞) is a potential function and h : ℝ^N × ℝ → ℝ is a Carathéodory function. The present paper is devoted to the existence of at least two distinct nontrivial solutions to quasilinear elliptic problems of Schrödinger type, which provides a concave-convex nature to the problem. The primary tools are the well-known mountain pass theorem and a variant of Ekeland's variational principle.

• 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 $L^1(R^n) into L^{1, \infty}(R^n)$.

• The Korean Journal of Ecology
• /
• v.21 no.3
• /
• pp.283-290
• /
• 1998

The vegetation pattern of Mt. Odae based on the soil humidity gradient showed 3 types: (1) the forest of Pinus densiflora under the mesic or xeric conditions of the low altitudinal area, (2) the forest of Acer including A. mono, A. pseudo-sieboldianum and Tilia amurensis under the submesic or subxeric conditions and (3) the forest of Quercus including Q. mongolica of the higher elevational area and Q. variabilis of the lower elevational area under the xeric condition. Water content, organic matter and total nitrogen of soil were relatively low in Pinus densiflora and Quercus variabilis communities while they were relatively high in Betula platyphylla var. japonica and Quercus mongolica communities. According to the result of cluster analysis based on similarity indices of the communities, the proposed successional sere in the forest vegetation of Mt. Odae was as follows. P. densiflora community $\longrightarrow$ P. densiflore + Q. mongolica community $\longrightarrow$ Q. mongolica + A. pseudo-sieboldianum community. P. densiflora community $\longrightarrow$ P. densiflora + Q. variabilis community $\longrightarrow$ Q. variabilis community $\longrightarrow$ Q. mongolica + Q. variabilis community $\longrightarrow$ Q. mongolica + A. pseudo-sieboldianum community.

• Honam Mathematical Journal
• /
• v.34 no.1
• /
• pp.1-9
• /
• 2012

The purpose of this paper is to give a new construction of the extended q-Euler numbers and polynomials of higher-order with weight by using p-adic q-integral on $\mathbb{Z}_p$.

• Korean Journal of Environment and Ecology
• /
• v.4 no.1
• /
• pp.1-15
• /
• 1990

To investigate the actual vegetation in Sokri mountain area, which consists of Sokri Mt. National Park, forest communities were classified by phytosocialogical analysis according to species composition and physiognomy. Actual vegetation map was drawn on the basis of classified forest communities. The forest communities in Mt. Sokri were classified as follows : 1. Pinus densiflora community 1) P. densiflora-Rhododedron mucronulatum subcommunity 2) P. densiflora-Rhododendron schippenbachii subcomminity 3) P. densiflora-Lespedeza maximowiczii subcommunity 4) P. densiflora-Sasa borealis subcommunity 2. P. densiflora-Quercus serrata community 3. P. densiflora-Q. variabilis community 4. Quercus serrata community 5. Q. serrata-Q. variabilis community 6. Carpinus laxiflora community 7. Quercus mongolica community 1) Typical subcommunity 2) Q. mongolica-Lespedeza maximowiczii subcommunity 3) Q. mongolica-Rhododendron schlippenbachii subcommunity 8. Fraxinus rhynchophylla community 9. Symplocos chinensis for. pilosa community 10. Quercus variabilis community

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.2
• /
• pp.319-339
• /
• 2010

In this paper, we prove that if $f^nf'\;-\;P$ and $g^ng'\;-\;P$ share 0 CM, where f and g are two distinct transcendental meromorphic functions, $n\;{\geq}\;11$ is a positive integer, and P is a nonzero polynomial such that its degree ${\gamma}p\;{\leq}\;11$, then either $f\;=\;c_1e^{cQ}$ and $g\;=\;c_2e^{-cQ}$, where $c_1$, $c_2$ and c are three nonzero complex numbers satisfying $(c_1c_2)^{n+1}c^2\;=\;-1$, Q is a polynomial such that $Q\;=\;\int_o^z\;P(\eta)d{\eta}$, or f = tg for a complex number t such that $t^{n+1}\;=\;1$. The results in this paper improve those given by M. L. Fang and H. L. Qiu, C. C. Yang and X. H. Hua, and other authors.

• Journal of the Korea Society of Computer and Information
• /
• v.19 no.7
• /
• pp.71-76
• /
• 2014

There is to be virtually impossible to solve the very large digits of prime number p and q from composite number n=pq using integer factorization in typical public-key cryptosystems, RSA. When the public key e and the composite number n are known but the private key d remains unknown in an asymmetric-key RSA, message decryption is carried out by first obtaining ${\phi}(n)=(p-1)(q-1)=n+1-(p+q)$ and then using a reverse function of $d=e^{-1}(mod{\phi}(n))$. Integer factorization from n to p,q is most widely used to produce ${\phi}(n)$, which has been regarded as mathematically hard. Among various integer factorization methods, the most popularly used is the congruence of squares of $a^2{\equiv}b^2(mod\;n)$, a=(p+q)/2,b=(q-p)/2 which is more commonly used then n/p=q trial division. Despite the availability of a number of congruence of scares methods, however, many of the RSA numbers remain unfactorable. This paper thus proposes an algorithm that directly and immediately obtains ${\phi}(n)$. The proposed algorithm computes $2^k{\beta}_j{\equiv}2^i(mod\;n)$, $0{\leq}i{\leq}{\gamma}-1$, $k=1,2,{\ldots}$ or $2^k{\beta}_j=2{\beta}_j$ for $2^j{\equiv}{\beta}_j(mod\;n)$, $2^{{\gamma}-1}$ < n < $2^{\gamma}$, $j={\gamma}-1,{\gamma},{\gamma}+1$ to obtain the solution. It has been found to be capable of finding an arbitrarily located ${\phi}(n)$ in a range of $n-10{\lfloor}{\sqrt{n}}{\rfloor}$ < ${\phi}(n){\leq}n-2{\lfloor}{\sqrt{n}}{\rfloor}$ much more efficiently than conventional algorithms.

• Journal of the Korean Statistical Society
• /
• v.21 no.1
• /
• pp.47-58
• /
• 1992

The autocorrelation function structures of bilinear time series model BL(p, q, r, s), $r \geq s$ are obtained and shown to be analogous to those of ARMA(p, l), l=max(q, s). Simulation studies are performed to investigate the adequacy of Akaike information criteria for identification between ARMA(p, l) and BL(p, q, r, s) models and for determination of orders of BL(p, q, r, s) models. It is suggested that the model of having minimum Akaike information criteria is selected for a suitable model.

• Korean journal of applied entomology
• /
• v.4
• /
• pp.11-18
• /
• 1965

In order to investigate the fungicidal activities against various plant pathogenes, diminishing effect of plant transpiration, phytotoxicities, vapor effect and the rate of reduction by ultraviolet rays of phenylmercuric 8-oxyquinolinate(P.M.Q), this experiments were undertaken under various laboratory conditions. 1. Inhibitory activity on the spore germination of this chemical was shown less effective than that of P.M.A..(Table 2, Table 3, Table 4, Table 5 and Table 6) Also, P.M.Q. was resulted a somewhat higher inhibitory activity on the hyphae growth than P.M.A. (Table 7). 2. In the diminishing effect of plant transpiration, 8-hydroxyquinoline sulfate(oxine sulfate) was more strong inhibitory at first than P.M.Q., while, at last, P.M.Q. was more strong inhibitory in comparison with oxine sulfate(Table 8, Fig. 1 and Table 9). 3. P.M.Q. was shown less injury on the germination of rice plant seeds and the emergence of their roots than P.M. A.(Table 10). Injuries was not observed on the rice seedlings and soy-bean seedlings sprayed with 40 ppm of this chemical. 4. P.M.A. had more inhibitory effects on the mycelial growth of phytopathogenes than P.M.Q. on the vapor effect (Table 11, Fig. 2). 5. Biological activity and chemical decomposition rate of P.M.A. were greatly reduced by exposure of this compound to ultraviolet rays. But, P.M.Q. was only slightly affected by similar treatment(Table 12, Fig. 3, Table 13 and Fig. 4). From the above results, this chemical will be a promising fungicide adding fungitoxicities against various phytopatho genes, diminishing effect of plant transpiration and physico-stability.