• The Pure and Applied Mathematics
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• v.22 no.3
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• pp.275-283
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• 2015

Let T be a bounded linear operator on a complex Hilbert space H. For a positive integer k, an operator T is said to be a k-quasi-2-isometric operator if T^∗k(T^∗2T² − 2T^∗T + I)T^k = 0, which is a generalization of 2-isometric operator. In this paper, we consider basic structural properties of k-quasi-2-isometric operators. Moreover, we give some examples of k-quasi-2-isometric operators. Finally, we prove that generalized Weyl’s theorem holds for polynomially k-quasi-2-isometric operators.

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.985-1000
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• 2019

Let k be an integer with $k{\geq}3$. Define $h(k)=[{\frac{k+1}{2}}]$, ${\sigma}(k)={\min}\(2^{h(k)-1},\;{\frac{1}{2}}h(k)(h(k)+1)\)$. Suppose that ${\lambda}_1,{\ldots},{\lambda}_5$ are non-zero real numbers, not all of the same sign, satisfying that ${\frac{{\lambda}_1}{{\lambda}_2}}$ is irrational. Then for any given real number ${\eta}$ and ${\varepsilon}>0$, the inequality $${\mid}{\lambda}_1p^2_1+{\lambda}_2p^2_2+{\lambda}_3p^2_3+{\lambda}_4p^2_4+{\lambda}_5p^k_5+{\eta}{\mid}<({\max_{1{\leq}j{\leq}5}}p_j)^{-{\frac{3}{20{\sigma}(k)}}+{\varepsilon}}$$ has infinitely many solutions in prime variables $p_1,{\ldots},p_5$. This gives an improvement of the recent results.

• 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.

• Honam Mathematical Journal
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• v.32 no.2
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• pp.255-260
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• 2010

Given vectors x and y in a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate compact interpolation problems for vectors in a tridiagonal algebra. We show the following : Let Alg$\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let x = $(x_i)$ and y = $(y_i)$ be vectors in H. Then the following are equivalent: (1) There exists a compact operator A = $(a_{ij})$ in Alg$\mathcal{L}$ such that Ax = y. (2) There is a sequence ${{\alpha}_n}$ in $\mathbb{C}$ such that ${{\alpha}_n}$ converges to zero and for all k ${\in}$ $\mathbb{N}$, $y_1 = {\alpha}_1x_1 + {\alpha}_2x_2$ $y_{2k} = {\alpha}_{4k-1}x_{2k}$ $y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}+x_{2k+2}$.

• Korean Journal of Food Science and Technology
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• v.35 no.5
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• pp.865-870
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• 2003

The effects of storage atmosphere on the physicochemical and organoleptic properties of Asian pears (Pyrus pyrifolia Nakai cv. Niitaka) were investigated. The pears were stored at $0^{\circ}C$ for up to 8 months under CA conditions (3 kPa $O_2$＋1 kPa $CO_2$, 1 kPa $O_2$+5 kPa $CO_2$, 1 kPa $O_2$+1 kPa $CO_2$) controlled by a static system or air (21 kPa $O_2$+0 kPa $CO_2$). CA conditions retarded the loss of weight, flesh firmness, organoleptic properties (appearance, aroma, sweetness, texture), and the development of injuries in the fruits during storage. Low $O_2$ plus high $CO_2$ conditions maintained the highest flesh firmness. In addition, low $O_2$ reduced injuries more than the other conditions. Skin color (lightness, chroma, hue), soluble solids, titratable acidity, pH and vitamin C were unaffected by storage atmosphere.

• Transactions of the Korean hydrogen and new energy society
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• v.27 no.5
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• pp.517-525
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• 2016

The present work investigates the effect of $K_2CO_3$ catalyst on steam gasification of Kideco coal and the deactivation of the catalyst due to thermal exposure and interaction with coal ash. The gasification reactivity at $700^{\circ}C$ is highly enhanced by $K_2CO_3$, which is not deactivated by the heat treatment at $T{\leq}800^{\circ}C$. TGA and XRD results prove minor decomposition of $K_2CO_3$ after the calcination at $800^{\circ}C$. $K_2CO_3$ is, however, evaporated at the higher temperature. Assuming the conversion of $K_2CO_3$ into $K_2O$ by the decomposition and into $K_2O{\cdot}2.5SiO_2$ and $KAlO_2$ by the interaction with coal ash, the reactivity of the gasification is evaluated in the presence of $K_2O$, $K_2O{\cdot}2.5SiO_2$ and $KAlO_2$. Among them, $K_2O$ is the most active, but much lower in the activity than $K_2CO_3$. XRD results show that $K_2CO_3$ could react readily with the ash above $700^{\circ}C$.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.255-260
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• 2006

Let $m$ and $k$ be any positive integers, let $\mathbb{Z}_k$ the ring of integers of modulo $k$, let $G_m(\mathbb{Z}_k)$ the group of all $m$ by $m$ nonsingular matrices over $\mathbb{Z}_k$ and let ${\phi}_m(k)$ the order of $G_m(\mathbb{Z}_k)$. In this paper, ${\phi}_m(k)$ can be computed by the following investigation: First, for any relatively prime positive integers $s$ and $t$, $G_m(\mathbb{Z}_{st})$ is isomorphic to $G_m(\mathbb{Z}_s){\times}G_m(\mathbb{Z}_t)$. Secondly, for any positive integer $n$ and any prime $p$, ${\phi}_m(p^n)=p^{m^2}{\cdot}{\phi}_m(p^{n-1})=p{^{2m}}^2{\cdot}{\phi}_m(p^{n-2})={\cdots}=p^{{(n-1)m}^2}{\cdot}{\phi}_m(p)$, and so ${\phi}_m(k)={\phi}_m(p_1^n1){\cdot}{\phi}_m(p_2^{n2}){\cdots}{\phi}_m(p_s^{ns})$ for the prime factorization of $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.109-123
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• 2003

A tree is called starlike if it has exactly one vertex of degree greate. than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) : {1,2, …, n} and let H = {$H_1$, $H_2$, …, $H_{n}$} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of $H_{i}$ with the i-th vertex of G. In particular, if H is the family of the paths $P_k_1,P_k_2,...P_k_2$ with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G($k_1,k_2,...k_n$). For any $(x_1,x_2,...,x_n)\;\in\;{I_*}^n$, where $I_{*}$ = : {0,1}, let G$(x_1,x_2,...,x_n)$ be the subgraph of G which is obtained by deleting the vertices $i_1,i_2,...i_j\;\in\;V(G)\;(O\leq j\leq n)$, provided that $x_i_1=x_i_2=...=x_i_j=o.\;Let \;G[x_1,x_2,...x_n]$ be characteristic polynomial of G$(x_1,x_2,...,x_n)$, understanding that G[0,0,...,0] $\equiv$1. We prove that $G[k_1,k_2,...,k_n]-\sum_{x\in In}[{\prod_{\imath=1}}^n\;P_k_i+x_i-2(\lambda)](-1)...G[x_1,x_2,...,X_n]$ where x=($x_1,x_2,...,x_n$);G[$k_1,k_2,...,k_n$] and $P_n(\lambda)$ denote the characteristic polynomial of G($k_1,k_2,...,k_n$) and $P_n$, respectively. Besides, if G is a graph with $\lambda_1(G)\;\geq1$ we show that $\lambda_1(G)\;\leq\;\lambda_1(G(k_1,k_2,...,k_n))<\lambda_1(G)_{\lambda_1}^{-1}(G}$ for all positive integers $k_1,k_2,...,k_n$, where $\lambda_1$ denotes the largest eigenvalue.

• Honam Mathematical Journal
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• v.41 no.4
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• pp.813-821
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• 2019

In this paper, we investigate Hyers-Ulam-Rassias stability of a functional equation f(x + ky) + f(x - ky) - k²f(x + y) - k²f(x - y) + 2(k² - 1)f(x) + (k² + k)f(y) + (k² - k)f(-y) - 2f(ky) = 0.

• Bulletin of the Korean Chemical Society
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• v.28 no.6
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• pp.921-928
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• 2007

Three different phosphonated trisbipyridine ruthenium complexes, [(4-CH3-4'-CH2PO(OH)2-2,2'-bipyridine)- (bpy)2Ru]·(PF6)2 (Ru-P1), [(4-CH3-4'-CH2PO(OH)2-2,2'-bipyridine)3Ru]·(PF6)2 (Ru-P2), and [(4,4'-CH2PO- (OH)2-2,2'-bipyridine)3Ru]·(PF6)2 (Ru-P3) were synthesized and their photochemical and electrochemical properties were studied. These ruthenium complexes were strongly adsorbed on the surface of the layered metal oxide semiconductor K4Nb6O17 that was partially acid-exchanged and sensitized up to pH 10, while the carboxylated ruthenium complex, (4,4'-COOH-2,2'-bipyridine)3Ru·Cl2 (Ru-C) that was previously studied was sensitized only below pH 4. The visible light water reduction at K4Nb6O17 that was internally platinized and sensitized by these phosphonated Ru-complexes was comparatively studied using a reversible electron donor iodide.