• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.539-545
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• 1994

Let $q = p^r$, where p is a prime. A polynomial $f(x) \in GF(q)[x]$ is called a permutation polynomial (PP) over GF(q) if the numbers f(a) where $a \in GF(Q)$ are a permutation of the a's. In other words, the equation f(x) = a has a unique solution in GF(q) for each $a \in GF(q)$. More generally, $f(x_1, \cdots, x_n)$ is a PP in n variables if $f(x_1,\cdots,x_n) = \alpha$ has exactly $q^{n-1}$ solutions in $GF(q)^n$ for each $\alpha \in GF(q)$. Mullen ([3], [4], [5]) has studied the concepts of local permutation polynomials (LPP's) over finite fields. A polynomial $f(x_i, x_2, \cdots, x_n) \in GF(q)[x_i, \codts,x_n]$ is called a LPP if for each i = 1,\cdots, n, f(a_i,\cdots,x_n]$ is a PP in $x_i$ for all $a_j \in GF(q), j \neq 1$.Mullen ([3],[4]) found a set of necessary and three variables over GF(q) in order that f be a LPP. As examples, there are 12 LPP's over GF(3) in two indeterminates ; $f(x_1, x_2) = a_{10}x_1 + a_{10}x_2 + a_{00}$ where $a_{10} = 1$ or 2, $a_{01} = 1$ or x, $a_{00} = 0,1$, or 2. There are 24 LPP's over GF(3) of three indeterminates ; $F(x_1, x_2, x_3) = ax_1 + bx_2 +cx_3 +d$ where a,b and c = 1 or 2, d = 0,1, or 2.

• Bulletin of the Korean Mathematical Society
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• v.21 no.2
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• pp.67-75
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• 1984

Let X be a topological space. We consider a groupoid G over X and the quotient groupoid G/N for any normal subgroupoid N of G. The concept of groupoid (topological groupoid) is a natural generalization of the group(topological group). An useful example of a groupoid over X is the foundamental groupoid .pi.X whose object group at x.mem.X is the fundamental group .pi.(X, x). It is known [5] that if X is locally simply connected, then the topology of X determines a topology on .pi.X so that is becomes a topological groupoid over X, and a covering space of the product space X*X. In this paper the concept of the locally simple connectivity of a topological space X is applied to the groupoid G over X. That concept is defined as a term '1-connected local subgroupoid' of G. Using this concept we topologize the groupoid G so that it becomes a topological groupoid over X. With this topology the connected groupoid G is a covering space of the product space X*X. Further-more, if ob(.overbar.G)=.overbar.X is a covering space of X, then the groupoid .overbar.G is also a covering space of the groupoid G. Since the fundamental groupoid .pi.X of X satisfying a certain condition has an 1-connected local subgroupoid, .pi.X can always be topologized. In this case the topology on .pi.X is the same as that of [5]. In section 4 the results on the groupoid G are generalized to the quotient groupoid G/N. For any topological groupoid G over X and normal subgroupoid N of G, the abstract quotient groupoid G/N can be given the identification topology, but with this topology G/N need not be a topological groupoid over X [4]. However the induced topology (H) on G makes G/N (with the identification topology) a topological groupoid over X. A final section is related to the covering morphism. Let G$_{1}$ and G$_{2}$ be groupoids over the sets X$_{1}$ and X$_{2}$, respectively, and .phi.:G$_{1}$.rarw.G$_{2}$ be a covering spimorphism. If X$_{2}$ is a topological space and G$_{2}$ has an 1-connected local subgroupoid, then we can topologize X$_{1}$ so that ob(.phi.):X$_{1}$.rarw.X$_{2}$ is a covering map and .phi.: G$_{1}$.rarw.G$_{2}$ is a topological covering morphism.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.45-51
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• 2010

Let A be an abelian variety defined over a number field K and let L be a cyclic extension of K with Galois group G = <${\sigma}$> of order n. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and of A over L. Assume III(A/L) is finite. Let M(x) be a companion matrix of 1+x+${\cdots}$+$x^{n-1}$ and let $A^x$ be the twist of $A^{n-1}$ defined by $f^{-1}{\circ}f^{\sigma}$ = M(x) where $f:A^{n-1}{\rightarrow}A^x$ is an isomorphism defined over L. In this paper we compute [III(A/K)][III($A^x$/K)]/[III(A/L)] in terms of cohomology, where [X] is the order of an finite abelian group X.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.119-126
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• 1989

Throughout this paper, X will always denote a Banach space over the complex numbers C, and L(X) will denote the Banach algebra of all continuous linear operators on X. Operator will always mean continuous linear operator. An n-tuple of operators T$_{1}$,..,T$_{n}$ on X will be denoted by over ^ T=(T$_{1}$,..,T$_{n}$ ). Let L$^{n}$ (X) be the set of all n-tuples of operators on X. X' will denote the dual space of X, S(X) its unit sphere and .PI.(X) the subset of X*X' defined by .PI.(X)={(x,f).mem.X*X': ∥x∥=∥f∥=f(x)=1}.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.265-277
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• 1995

In [9], we gave a very explicit description of the injective envelope of an arbitray simple module over a polynomial ring $K[X_1, \ldots, X_n]$ over a field K in indeterminates $X_1, \ldots, X_n$. This paper presents another approach to give a description.

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.247-257
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• 1999

In this article, we give an example of local derivation, that is not derivation, on the algebra F(x1,…, xn) of rational functions in x1, …, xn over an infinite field F, and show that if X is a set of symbols and {x1,…, xn} is a finite subset of X, n$\geq$1, then each local derivation of F[x1,…, xn] into F[X] is a F-derivation and each local derivation of F[X] into itself is also a F-derivation.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.277-283
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• 1992

Let GF(q) be a finite field with q elements where q=p$^{n}$ for a prime number p and a positive integer n. Consider an arbitrary function .phi. from GF(q) into GF(q). By using the Largrange's Interpolation formula for the given function .phi., .phi. can be represented by a polynomial which is congruent (mod x$^{q}$ -x) to a unique polynomial over GF(q) with the degree < q. In [3], Wells characterized all polynomial over a finite field which commute with translations. Mullen [2] generalized the characterization to linear polynomials over the finite fields, i.e., he characterized all polynomials f(x) over GF(q) for which deg(f) < q and f(bx+a)=b.f(x) + a for fixed elements a and b of GF(q) with a.neq.0. From those papers, a natural question (though difficult to answer to ask is: what are the explicit form of f(x) with zero terms\ulcorner In this paper we obtain the exact form (together with zero terms) of a polynomial f(x) over GF(q) for which satisfies deg(f) < p$^{2}$ and (1) f(x+a)=f(x)+c for the fixed nonzero elements a and c in GF(q).

• Kyungpook Mathematical Journal
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• v.13 no.2
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• pp.235-240
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• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• The Korean Journal of Zoology
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• v.6 no.2
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• pp.11-15
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• 1963

A total of 220 adult male mice (18-20g) of the S.M. strain were divided into ten experimental and control groups. The total-body X-ray irradiation doses used were 50 r, 100r, 200r, 400r, 600r, 800r, 1,000r, 1,200r, 1,400r, and1,600r. The respiratory arrest (mortality) caused by each irradiation doses were observed for 30 days. Relationships between irradiation doses and survival time and percentage of response were examined. From this experiment, a formula was obtained to express the relationship among three factors, which may be presented as follows : {{{{{{{{P= { 10} over { SQRT { 2 pi } } INT _{ - INF }^{ p'} e-{(p'-50)^2 } over {200 }dp···(a) p'=100 LEFT { t^0.3- LEFT ( { { 16.9965} over {D-60 } } RIGHT ) ^{ { 1} over {2.5 } } } / LEFT { LEFT ( { 26372.43} over {D-81.86 } RIGHT ) ^{ { 1} over {2.5 } } -( { { 16.9965} over {D-60 } } RIGHT ) ^{ { 1} over {2.5 } } ···(b) p= { (D-60) t^0.75-16.9965} over {0.2186 t^0.75 +263.55434 }····(c) }} {{{{P= { 10} over { SQRT { 2 pi } } INT _{ - INF }^{ p'} e-{(p'-50)^2 } over {200 }dp···(a) p'=100 LEFT { t^0.3- LEFT ( { { 16.9965} over {D-60 } } RIGHT ) ^{ { 1} over {2.5 } } } / LEFT { LEFT ( { 26372.43} over {D-81.86 } RIGHT ) ^{ { 1} over {2.5 } } -( { { 16.9965} over {D-60 } } RIGHT ) ^{ { 1} over {2.5 } } ···(b) p= { (D-60) t^0.75-16.9965} over {0.2186 t^0.75 +263.55434 }····(c)

• Journal of the Korean Dietetic Association
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• v.12 no.2
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• pp.172-184
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• 2006

This study was aimed to provide safety management guidance by evaluating the microbial quality of cooked dried-seafoods in school foodservice operations. Nineteen seafood items were collected from six elementary schools, those were dried-anchovy, dried-seaweed and dried-fish, which were classified as cooking process. The temperatures at receiving and after cooking were measured and the analyses of cooking processes and microbial quality were performed. The temperatures of all foods after cooking were higher than the temperature limit of $74\^circC$. The number of total aerobic bacteria and S. aureus in dried-anchovy over the limit of $10^5$ and even the level of S. aureus was found to be unsatisfactory. The count of total aerobic bacteria was 2.1x$10^8$ CFU/g and the number of total aerobic bacteria after cooking was over the limit in one school. The level of E. coli (3.1x$10^3$ CFU/g) was over the limit at one school and the number of S. aureus (1.2×$10^4$ CFU/g) was considered as unacceptable. Dried- tangle and green laver were contaminated with total aerobic bacteria showing the over the limit. The numbers of total aerobic bacteria in dried- filefish, pollack and squid were 4.3x$10^6$, 3.4x$10^6$-3.9x$10^7$ and 4.6x$10^5$-4.1x$10^7$ CFU/g, respectively, which were in acceptable or unsatisfactory level. The E. coli in dried- filefish and pollack were over the limit. The total aerobic bacteria levels, 4.6x$10^5$-1.5x$10^6$ CFU/g in dried-pollack and 8.0x$10^5$-2.2x$10^7$ CFU/g in dried-squid, were over the limit after cooking except dried-filefish. The E. coli levels, 4.3x$10^3$ CFU/g in dried-filefish and 2.5x$10^2$ CFU/g in dried-pollack, were over the limit of $10^2$ CFU/g. The numbers of Enterobacteriaceae were either acceptable (3.3x$10^3$ CFU/g) or unsatisfactory (1.6x$10^4$ CFU/g) level in dried-pollack. S. aureus was unsatisfactory level (6.5x$10^4$ CFU/g) in dried-filefish while unacceptable in dried-pollack both before and after cooking. Unacceptable levels of S. aureus, 2.4x$10^4$ and 1.3x$10^5$ CFU/g were found from two schools, respectively. These results suggest that the contamination of raw materials and the seasonings added after cooking should be controlled to manage the microbial safety of cooked dried-seafoods.