• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.907-913
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• 2010

Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D, $N_*\;=\;\{f\;{\in}\;D[X]|\;(c_D(f))^*\;=\;D\}$ and $R\;=\;D[X]_{N_*}$. Let b be the b-operation on R, and let $*_c$ be the star operation on D defined by $I^{*_c}\;=\;(ID[X]_{N_*})^b\;{\cap}\;K$. Finally, let Kr(R, b) (resp., Kr(D, $*_c$)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) $\subseteq$ Kr(D, $*_c$) and Kr(R, b) is a kfr with respect to K(Y) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, $*_c$) if and only if D is a $P{\ast}MD$. As a corollary, we have that if D is not a $P{\ast}MD$, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y) and X.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1023-1035
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• 2018

We strengthen a result of Michiel Kosters by proving the following theorems: (*) Let ${\phi}:W{\rightarrow}V$ be a finite surjective morphism of algebraic varieties over an ample field K. Suppose V has a simple K-rational point a such that $a{\not\in}{\phi}(W(K_{ins}))$. Then, card($V(K){\backslash}{\phi}(W(K))$ = card(K). (**) Let K be an infinite field of positive characteristic and let $f{\in}K[X]$ be a non-constant monic polynomial. Suppose all zeros of f in $\tilde{K}$ belong to $K_{ins}{\backslash}K$. Then, card(K \ f(K)) = card(K).

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1431-1443
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• 2016

Let k be a global field of characteristic unequal to two. Let $C:y^2=f(x)$ be a nonsingular projective curve over k, where f(x) is a quartic polynomial over k with nonzero discriminant, and K = k(C) be the function field of C. For each prime spot p on k, let ${\hat{k}}_p$ denote the corresponding completion of k and ${\hat{k}}_p(C)$ the function field of $C{\times}_k{\hat{k}}_p$. Consider the map $$h:Br(K){\rightarrow}{\prod\limits_{\mathfrak{p}}}Br({\hat{k}}_p(C))$$, where p ranges over all the prime spots of k. In this paper, we explicitly describe all the constant classes (coming from Br(k)) lying in the kernel of the map h, which is an obstruction to the Hasse principle for the Brauer groups of the curve. The kernel of h can be expressed in terms of quaternion algebras with their prime spots. We also provide specific examples over ${\mathbb{Q}}$, the rationals, for this kernel.

• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.1087-1107
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• 2023

Let E be an elliptic curve defined over ℚ of conductor N, c the Manin constant of E, and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K, P_K the Heegner point in E(K), and III(E/K) the Shafarevich-Tate group of E over K. Let 2u_K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P_K has infinite order in E(K), then the integer c · m · u_K · |III(E/K)| $\frac{1}{2}$ is divisible by |E(ℚ)_tor|. In this paper, we prove that this conjecture is true if E(ℚ)_tor ≅ ℤ/2ℤ or ℤ/4ℤ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein-Watkins conjecture.

• Journal of Sensor Science and Technology
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• v.23 no.3
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• pp.197-201
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• 2014

TEPC (Tissue Equivalent Proportional Counter) was usually used for high LET radiation dosimetry. We developed a prototype TEPC for micro-dosimetry in the range of $0.2{\sim}300 keV/{\mu}m$. And, the simulated site diameter of the TEPC is $2{\mu}m$, of similar size to a cell nucleus. For purposes of characterization the response for high LET radiation of the TEPC has been investigated under 135MeV/u Carbon ions in HIMAC (Heavy Ion Medical Accelerator). We determined the gas multiplication factor and measured the lineal energy spectrum [yd(y)] of 135 MeV/u Carbon ions. The value of the gas multiplication factor was 315 at 700 V bias voltage. As a result of the experiment, we could more understand the performance of the TEPC for high LET (Linear Energy Transfer) radiation. And the procedure of high LET radiation dosimetry using TEPC is established.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.889-895
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• 1995

Let $H$ be a separable, infinite dimensional, complex Hilbert space and let $L(H)$ denote the algebra of all bounded linear operators on $H$.

• Journal of the Korean Mathematical Society
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• v.33 no.2
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• pp.387-397
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• 1996

Let G be a finite group and F be a field of characteristic $p \geq 0$. Let $\Gamma = F^f G$ be a twisted group algebra corresponding to a 2-cocycle $f \in Z^2(G,F^*), where F^* = F - {0}$ is the multiplicative subgroup of F.

• Journal of the Chungcheong Mathematical Society
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• v.6 no.1
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• pp.131-137
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• 1993

Let R he a commutative ring with identity and let M be a nonzero multiplication R-module. In this note we prove that M is finitely generated if M is a faithful multiplication R-module.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.341-350
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• 1995

Let (M,g) be a compact manifold of dimension n with metric tensor g. Let $\Delta^p = d\delta + \deltad$ be the Laplace-Beltrami operator acting on the space of smooth p-forms.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1385-1394
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• 2016

Let (R, m) be a Noetherian local ring, I, J two ideals of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{>k}(I,A)$ the supremum of lengths of A-cosequences in dimension > k in I defined by Nhan-Hoang [9]. It is first shown that for each $t{\leq}r$ and each sequence $x_1,{\cdots},x_t$ which is an A-cosequence in dimension > k, the set $$\Large(\bigcup^{t}_{i=0}Att_R(0:_A(x_1^{n_1},{\ldots},x_i^{n_i})))_{{\geq}k}$$ is independent of the choice of $n_1,{\ldots},n_t$. Let r be the eventual value of $Width_{>k}(0:_AJ^n)$. Then our second result says that for each $t{\leq}r$ the set $\large(\bigcup\limits_{i=0}^{t}Att_R(Tor_i^R(R/I,\;(0:_AJ^n))))_{{\geq}k}$ is stable for large n.