• Communications for Statistical Applications and Methods
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• v.17 no.4
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• pp.507-513
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• 2010

Let {$X_i,-{\infty}$ < 1 < $\infty$} be a doubly infinite sequence of identically distributed and negatively associated random variables with mean zero and finite variance and {$a_i,\;-{\infty}$ < i < ${\infty}$} be an absolutely summable sequence of real numbers. Define a moving average process as $Y_n={\sum}_{i=-\infty}^{\infty}a_{i+n}X_i$, n $\geq$ 1 and $S_n=Y_1+{\cdots}+Y_n$. In this paper we prove that E|$X_1$|$^rh$($|X_1|^p$) < $\infty$ implies ${\sum}_{n=1}^{\infty}n^{r/p-2-q/p}h(n)E{max_{1{\leq}k{\leq}n}|S_k|-{\epsilon}n^{1/p}}{_+^q}<{\infty}$ and ${\sum}_{n=1}^{\infty}n^{r/p-2}h(n)E{sup_{k{\leq}n}|k^{-1/p}S_k|-{\epsilon}}{_+^q}<{\infty}$ for all ${\epsilon}$ > 0 and all q > 0, where h(x) > 0 (x > 0) is a slowly varying function, 1 ${\leq}$ p < 2 and r > 1 + p/2.

• Journal of Microbiology and Biotechnology
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• v.7 no.6
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• pp.373-377
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• 1997

The gene for isopenicillin N synthase (cyclase; IPNS) was cloned from Lysobacter lactamgenus using DNA probe amplified with primers based on the consensus sequences of isopenicillin N synthase genes of other ${\beta}$-lactam-producing microorganisms. The genomic library of L. lactamgenus using pUC18 plasmid cloned at the SacI site were screened with the PCR-generated DNA probe and three positive clones were isolated. Enzyme activities in E. coli clones were confirmed by bioassay and HPLC assay. Throughout the functional mapping, it was observed that the gene for isopenicillin N synthase is located at the 1.3-kb XhoI-BamHI fragment of insert of positive clones. Nucleotide sequencing at both ends of the XhoI-BamHI fragment revealed that IPNS of L. lactamgenus has the common amino acid sequences at amino- and carboxy-termini.

• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1041-1054
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• 2014

Let $P{\geq}3$ be an integer and let ($U_n$) and ($V_n$) denote generalized Fibonacci and Lucas sequences defined by $U_0=0$, $U_1=1$; $V_0= 2$, $V_1=P$, and $U_{n+1}=PU_n-U_{n-1}$, $V_{n+1}=PV_n-V_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equations $V_n=kx^2$ and $V_n=2kx^2$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $U_n=kx^2$, $U_n=2kx^2$, $U_n=3kx^2$, $V_n=kx^2{\mp}1$, $V_n=2kx^2{\mp}1$, and $U_n=kx^2{\mp}1$. Moreover, when P is odd, we solve the equations $V_n=wx^2+1$ and $V_n=wx^2-1$ for w = 2, 3, 6. After that, we solve some Diophantine equations.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.915-931
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• 2023

Let F_n be the Farey sequence of order n. For S ⊆ F_n, let 𝓠(S) be the set of rational numbers x/y with x, y ∈ S, x ≤ y and y ≠ 0. Recently, Wang found all subsets S of F_n with |S| = n + 1 for which 𝓠(S) ⊆ F_n. Motivated by this work, we try to determine the structure of S ⊆ F_n such that |S| = n and 𝓠(S) ⊆ F_n. In this paper, we determine all sets S ⊆ F_n satisfying these conditions for n ∈ {p, 2p}, where p is prime.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.983-991
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• 2013

For each real number $n$ > 6, we prove that there is a sequence $\{pk(n,z)\}^{\infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${\alpha}{\beta}$ where ${\alpha}$ and ${\beta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{\prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$ for $m{\geq}1$, with the usual empty product conventions, i.e., ${\prod}_{j=0}^{-1}\;b_j=1$.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.25 no.6B
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• pp.1136-1142
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• 2000

In this paper, we present a construction for binary sequences {s(t)} of period N= \ulcorner-1 for an odd prime \ulcornerbased on the polynomial (z+1)d+az d +b, and discuss them in some cases of parameters p, m, d, a and b. We show that new sequences from our construction are balanced or almost balanced, and have optimal three-level autocorrelation. We also derive the distribution of autocorrelation values they take on. The sequences satisfy constant-on-the-coset property, and we will show that there are more than one characteristic phases with constant-on-the-coset property. Some other interesting properties of these sequences will be presented. Results of an extensive computer search are summarized.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.437-448
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• 2007

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let $T_1,\;T_2\;and\;T_3\;:\;K{\rightarrow}E$ be nonexpansive mappings with nonempty common fixed points set. Let $\{\alpha_n\},\;\{\beta_n\},\;\{\gamma_n\},\;\{\alpha'_n\},\;\{\beta'_n\},\;\{\gamma'_n\},\;\{\alpha'_n\},\;\{\beta'_n\}\;and\;\{\gamma'_n\}$ be real sequences in [0, 1] such that ${\alpha}_n+{\beta}_n+{\gamma}_n={\alpha}'_n+{\beta'_n+\gamma}'_n={\alpha}'_n+{\beta}'_n+{\gamma}'_n=1$, starting from arbitrary $x_1{\in}K$, define the sequence $\{x_n\}$ by $$\{zn=P({\alpha}'_nT_1x_n+{\beta}'_nx_n+{\gamma}'_nw_n)\;yn=P({\alpha}'_nT_2z_n+{\beta}'_nx_n+{\gamma}'_nv_n)\;x_{n+1}=P({\alpha}_nT_3y_n+{\beta}_nx_n+{\gamma}_nu_n)$$ with the restrictions $\sum^\infty_{n=1}{\gamma}_n<\infty,\;\sum^\infty_{n=1}{\gamma}'_n<\infty,\; \sum^\infty_{n=1}{\gamma}'_n<\infty$. (i) If the dual $E^*$ of E has the Kadec-Klee property, then weak convergence of a $\{x_n\}$ to some $x^*{\in}F(T_1){\cap}{F}(T_2){\cap}(T_3)$ is proved; (ii) If $T_1,\;T_2\;and\;T_3$ satisfy condition(A'), then strong convergence of $\{x_n\}$ to some $x^*{\in}F(T_1){\cap}{F}(T_2){\cap}(T_3)$ is obtained.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1467-1480
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• 2015

Generalized Fibonacci and Lucas sequences ($U_n$) and ($V_n$) are defined by the recurrence relations $U_{n+1}=PU_n+QU_{n-1}$ and $V_{n+1}=PV_n+QV_{n-1}$, $n{\geq}1$, with initial conditions $U_0=0$, $U_1=1$ and $V_0=2$, $V_1=P$. This paper deals with Fibonacci and Lucas numbers of the form $U_n$(P, Q) and $V_n$(P, Q) with the special consideration that $P{\geq}3$ is odd and Q = -1. Under these consideration, we solve the equations $V_n=5kx^2$, $V_n=7kx^2$, $V_n=5kx^2{\pm}1$, and $V_n=7kx^2{\pm}1$ when $k{\mid}P$ with k > 1. Moreover, we solve the equations $V_n=5x^2{\pm}1$ and $V_n=7x^2{\pm}1$.

• Journal of applied mathematics & informatics
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• v.7 no.3
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• pp.833-842
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• 2000

Consider the even-order neutral difference equation (*) ${\delta}^m(x_n{-}p_ng(x_{n-k}))-q_nh(x_{n-1})=0$, n=0,1,2,... where $\Delta$ is the forward difference operator, m is even, ${-p_n},{q_n}$ are sequences of nonnegative real numbers, k, l are nonnegative integers, g(x), h(x) ${\in}$ C(R, R) with xg(x) > 0 for $x\;{\neq}\;0$. In this paper, we obtain some linearized oscillation theorems of (*) for $p_n\;{\in}\;(-{\infty},0)$ which are discrete results of the open problem by Gyori and Ladas.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.709-715
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• 1997

For a fibration (E,B,p) with fiber F and a fiber map f, we show that if the inclusion $i : F \to E$ has a left homotopy inverse, then $G^f_n(E,F)$ is isomorphic to $G^f_n(F,E) \oplus \pi_n(B)$. In particular, by taking f as the identity map on E we have $G_n(E,F)$ is isomorphic to $G_n(F) \oplus \pi_n(B)$.