• Journal of the Institute of Convergence Signal Processing
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• v.11 no.1
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• pp.47-52
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• 2010

For cryptographic systems, nonlinearity is crucial since most linear systems are easily decipherable. C.Bracken, Z.Zhaetc., propose the APN(Almost Perfect Nonlinear) functions with the properties similar to those of the bent functions with perfect nonlinearity. We design two kinds of new code sequence generating algorithms using the above APN functions. And we find that the out of phase ${\tau}\;{\neq}\;0$, autocorrelation functions, $R_{ii}(\tau)$ and the crosscorrelation functions, $R_{ik}(\tau)$ of the binary code sequences generated by two new algorithms over GF(2), have three values of {-1, $-1-2^{n/2}$, $-1+2^{n/2}$}. We also find that the out of phase ${\tau}\;{\neq}\;0$, autocorrelation functions, $R_{p,ii}(\tau)$ and the crosscorrelation functions, $R_{p,ik}(\tau)$ of the nonbinary code sequences generated by the modified algorithms over GF(p), $p\;{\geq}\;3$, have also three values of {$-1+p^{n-1}$, $-1-p^{(n-1)/2}+p^{n-1}$, $-1+p^{(n-1)/2}p^{n-1}$}. We show that these code sequences have the characteristics of the correlation functions similar to those of Gold code sequences.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.153-171
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• 2022

In 1960, Schatten studied operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(x_n{\otimes}{\bar{y_n}})$, where {x_n}n and {y_n}_n are orthonormal sequences in a Hilbert space, and {λ_n}n ∈ ℓ^∞(ℕ). Balazs generalized some of the results of Schatten in 2007. In this paper, we further generalize results of Balazs by studying the operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(A^*_nx_n{\otimes}{\bar{B^*_ny_n}})$, where {A_n}n and {B_n}_n are operator-valued Bessel sequences, {x_n}_n and {y_n}_n are sequences in the Hilbert space such that {║x_n║║y_n║}_n ∈ ℓ^∞(ℕ). We also generalize the class of Hilbert-Schmidt operators studied by Balazs.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.437-460
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• 2024

In this paper we present the weighted shift operators having the property of moment infinite divisibility. We first review the monotone theory and conditional positive definiteness. Next, we study the infinite divisibility of sequences. A sequence of real numbers γ is said to be infinitely divisible if for any p > 0, the sequence γ^p = {γ^p_n}^∞_n=0 is positive definite. For sequences α = {α_n}^∞_n=0 of positive real numbers, we consider the weighted shift operators W_α. It is also known that W_α is moment infinitely divisible if and only if the sequences {γ_n}^∞_n=0 and {γ_n+1}^∞_n=0 of W_α are infinitely divisible. Here γ is the moment sequence associated with α. We use conditional positive definiteness to establish a new criterion for moment infinite divisibility of W_α, which only requires infinite divisibility of the sequence {γ_n}^∞_n=0. Finally, we consider some examples and properties of weighted shift operators having the property of (k, 0)-CPD; that is, the moment matrix M_γ(n, k) is CPD for any n ≥ 0.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.33 no.9C
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• pp.669-675
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• 2008

For an odd prime p, n=4k and $d=((p^2k+1)/2)^2$, there are $(p^{2k}+1)/2$ distinct decimated sequences, s(dt+1), $0{\leq}l<(p^{2k}+1)/2$, of a p-ary m-sequence, s(t) of period $p^n-1$. In this paper, it is shown that the cross-correlation function between s(t) and s(dt+l) takes the values in $\{-1,-1{\pm}\sqrt{p^n},-1+2\sqrt{p^n}\}$ and their, cross-correlation distribution is also derived.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.245-256
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• 2005

We consider the second-order nonlinear difference equation (1) $$\Delta(a_nh(x_{n+1}){\Delta}x_n)+p_{n+1}f(x_{n+1})=0,\;n{\geq}n_0$$ where ${a_n},\;{p_n}$ are sequences of integers with $a_n\;>\;0,\;\{P_n\}$ is a real sequence without any restriction on its sign. hand fare real-valued functions. We obtain some necessary conditions for (1) existing nonoscillatory solutions and sufficient conditions for (1) being oscillatory.

• Journal of applied mathematics & informatics
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• v.41 no.1
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• pp.1-9
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• 2023

In the present paper, two theorems on absolute matrix summability of infinite series are generalized for the 𝜑 - |A, p_n|_k summability method using quasi 𝛽-power increasing sequences instead of almost increasing sequences.

• Journal of Plant Biotechnology
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• v.2 no.2
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• pp.101-108
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• 2000

The liverwort Pellia borealis is a diploid, monoecious, allopolypliod species (n=18) that as it was postulated, originated after hybridization and duplication of chromosome sets of two cryptic species: Pellia epiphylta-species N (n=9) and Pellia epiphylla-species 5 (n=9). Our recent results have supported the allopolyploid origin of P.borealis. We have shown that the nuclear genome of P.borealis consists of two nuclear genomes: one derived from P.epiphylla-species N and the other from P.epiphylla-species 5. In this paper we show the origin of chloroplast and mitochondrial genomes in an allopolyploid species P.borealis. To our knowledge there is no information concerning the way of mitochondria and chloroplast inheritance in Brophyta. Using an allopolyploid species of p. borealis as a model species we have decided to look into chloroplast and mitochondrial genomes of P.borealis, P.epiphylla-species N and P.epiphylla-species S for nucleotide sequences that would allow us to differentiate between both cryptic species and to identify the origin of organelle genomes in the alloploid species. We have amplified and sequenced a chloroplast $tRNA^{Leu}$ gene (anticodon UAA) containing an intron that has shown to be highly variable in a nucleotide sequence and used for plant population genetics. Unfortunately these sequences were identical in all three liverwort species tested. The analysis of the nucleotide sequence of chloroplast, an intron containing $tRNA^{Gly}$ (anticodon UCC) genes, gave expected results: the intron nucleotide sequence was identical in the case of both P.borealis and P.epiphyllaspecies N, while the sequence obtained from P.epiphyllasperies S was different in several nucleotide positions. These results were confirmed by the nucleotide sequence of another chloroplast molecular marker the chloroplast, an intron-contaning $tRNA^{Lys}$ gene (anticodon UUU). We have also sequenced mitochondrial, an intron-containing $tRNA^{Ser}$ gene (anticodon GCU) in all three liverwort species. In this case we found that, as in the case of the chloroplast genome, P.borealis mitochondrial genome was inherited from P.epiphylla-species N. On the basis of our results we claim that both organelle genomes of P.borealis derived from P.epiphylla-species N.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.1-16
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• 2019

Let a, b, P, and Q be real numbers with $PQ{\neq}0$ and $(a,b){\neq}(0,0)$. The Horadam sequence $\{W_n\}$ is defined by $W_0=a$, $W_1=b$ and $W_n=PW_{n-1}+QW_{n-2}$ for $n{\geq}2$. Let the sequence $\{X_n\}$ be defined by $X_n=W_{n+1}+QW_{n-1}$. In this study, we obtain some new identities between the Horadam sequence $\{W_n\}$ and the sequence $\{X_n\}$. By the help of these identities, we show that Diophantine equations such as $$x^2-Pxy-y^2={\pm}(b^2-Pab-a^2)(P^2+4),\\x^2-Pxy+y^2=-(b^2-Pab+a^2)(P^2-4),\\x^2-(P^2+4)y^2={\pm}4(b^2-Pab-a^2),$$ and $$x^2-(P^2-4)y^2=4(b^2-Pab+a^2)$$ have infinitely many integer solutions x and y, where a, b, and P are integers. Lastly, we make an application of the sequences $\{W_n\}$ and $\{X_n\}$ to trigonometric functions and get some new angle addition formulas such as $${\sin}\;r{\theta}\;{\sin}(m+n+r){\theta}={\sin}(m+r){\theta}\;{\sin}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},\\{\cos}\;r{\theta}\;{\cos}(m+n+r){\theta}={\cos}(m+r){\theta}\;{\cos}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},$$ and $${\cos}\;r{\theta}\;{\sin}(m+n){\theta}={\cos}(n+r){\theta}\;{\sin}\;m{\theta}+{\cos}(m-r){\theta}\;{\sin}\;n{\theta}$$.

• Honam Mathematical Journal
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• v.30 no.3
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• pp.479-486
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• 2008

Let {${\Omega}$, F, P} be a probability space and {$X_n{\mid}n{\geq}1$} be a sequence of random variables defined on it. We study the Hajeck-Renyi-type inequality for p..mixing random variable sequences and obtain the strong law of large numbers by using this inequality. We also consider the strong law of large numbers for weighted sums of ${\tilde{\rho}}$-mixing sequences.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.12 no.1
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• pp.21-32
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• 2002

In this paper, for any prime q, new cyclic difference sets with Singer parameter equation omitted are constructed by using the q-ary sequences (d-homogeneous functions) of period $q_n$-1. When q is a power of 3, new cyclic difference sets with Singer parameter equation omitted are constructed from the ternary sequences of period $q_n$-1 with ideal autocorrealtion found by Helleseth, Kumar and Martinsen.