• Journal of applied mathematics & informatics
• /
• v.35 no.3_4
• /
• pp.261-265
• /
• 2017

Let X and Y be independent identically distributed nondegenerate random variables with common absolutely continuous probability distribution function F(x) and the corresponding probability density function f(x) and $E(X^2)$<${\infty}$. Put Z = max(X, Y) and W = min(X, Y). In this paper, it is proved that Z - W and Z + W or$(X-Y)^2$ and X + Y are independent if and only if X and Y have normal distribution.

• Proceedings of the Optical Society of Korea Conference
• /
• 2001.02a
• /
• pp.16-17
• /
• 2001

광결정(photonic crystal)은 서로 다른 유전체가 규칙적으로 배열되어 있는 구조로서, 빛이 진행할 수 없는 진동수 영역인 광밴드갭(photonic bandgap)이 존재한다. 광밴드갭 특성으로 빛의 자발 방출과 진행 방향이 조절될 수 있기 때문에, 광결정은 나노 레이저, 광도파관, LED(Light Emitting Diode) 등의 광소자 개발에 응용되고 있다. 지금까지 2차원, 3차원의 광결정에 대한 많은 연구가 수행되어 왔으며, 현재에는 2차원의 슬랩(slab) 구조에 대해 활발하게 연구되고 있다. (중략)

• East Asian mathematical journal
• /
• v.38 no.3
• /
• pp.347-355
• /
• 2022

For a nondegenerate projective variety, it is a classical problem to study its defining equations with respect to a given embedding. In this paper, we precisely determine minimal sets of generators of the defining ideals of some curves of maximal regularity in ℙ⁵.

• East Asian mathematical journal
• /
• v.40 no.3
• /
• pp.287-293
• /
• 2024

For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations and the syzygies among them. In this paper, we precisely determine a minimal generating set and the minimal free resolution of defining ideals of some rational curves of maximal genus in ℙ³.

• Journal of the Korean Mathematical Society
• /
• v.46 no.2
• /
• pp.363-447
• /
• 2009

The author previously defined the spectral invariants, denoted by $\rho(H;\;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${\cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $\rho(H;\;a)$ states that the mini-max value is a critical value of the action functional ${\cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $\omega$). We also prove that the spectral invariant function ${\rho}_a$ : $H\;{\mapsto}\;\rho(H;\;a)$ can be pushed down to a continuous function defined on the universal (${\acute{e}}tale$) covering space $\widetilde{HAM}$(M, $\omega$) of the group Ham((M, $\omega$) of Hamiltonian diffeomorphisms on general (M, $\omega$). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$ of Hamiltonians, the function ${\rho}_a\;{\circ}\;{\cal{H}}$ : $s\;{\mapsto}\;{\rho}(H^s;\;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.

• Journal of Powder Materials
• /
• v.31 no.2
• /
• pp.119-125
• /
• 2024

The n-type Bi_2-xSb_xTe₃ compounds have been of great interest due to its potential to achieve a high thermoelectric performance, comparable to that of p-type Bi_2-xSb_xTe₃. However, a comprehensive understanding on the thermoelectric properties remains lacking. Here, we investigate the thermoelectric transport properties and band characteristics of n-type Bi_2-xSb_xTe₃ (x = 0.1 - 1.1) based on experimental and theoretical considerations. We find that the higher power factor at lower Sb content results from the optimized balance between the density of state effective mass and nondegenerate mobility. Additionally, a higher carrier concentration at lower x suppresses bipolar conduction, thereby reducing thermal conductivity at elevated temperatures. Consequently, the highest zT of ~ 0.5 is observed at 450 K for x = 0.1 and, according to the single parabolic band model, it could be further improved by ~70 % through carrier concentration tuning.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1707-1714
• /
• 2016

Let $C{\subset}{\mathbb{P}}^r$ be a nondegenerate projective curve of degree d > r + 1 and of maximal regularity. Such curves are always contained in the threefold scroll S(0, 0, r - 2). Also some of such curves are even contained in a rational normal surface scroll. In this paper we study the minimal free resolution of the homogeneous coordinate ring of C in the case where $d{\leq}2r-2$ and C is contained in a rational normal surface scroll. Our main result provides all the graded Betti numbers of C explicitly.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.6
• /
• pp.1137-1146
• /
• 2011

Let {$X_i$, $i{\geq}1$} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote $S_n={\sum}_{i=1}^n\;X_i$, $M_n=max_{1{\leq}i{\leq}n}\;{\mid}S_i{\mid}$ and $V_n^2={\sum}_{i=1}^n\;X_i^2$. Then for d > -1, we showed that under some regularity conditions, $$\lim_{{\varepsilon}{\searrow}0}{\varepsilon}^2^{d+1}\sum_{n=1}^{\infty}\frac{(loglogn)^d}{nlogn}I\{M_n/V_n{\geq}\sqrt{2loglogn}({\varepsilon}+{\alpha}_n)\}=\frac{2}{\sqrt{\pi}(1+d)}{\Gamma}(d+3/2)\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2d+2}}\;a.s.$$ holds in this paper, where If g denotes the indicator function.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.4
• /
• pp.1323-1330
• /
• 2017

For a nondegenerate projective irreducible variety $X{\subset}{\mathbb{P}}^r$, it is a natural problem to find an upper bound for the value of $${\ell}_{\beta}(X)=max\{length(X{\cap}L){\mid}L={\mathbb{P}}^{\beta}{\subset}{\mathbb{P}}^r,\;{\dim}(X{\cap}L)=0\}$$ for each $1{\leq}{\beta}{\leq}e$. When X is locally Cohen-Macaulay, A. Noma in [10] proves that ${\ell}_{\beta}(X)$ is at most $d-e+{\beta}$ where d and e are respectively the degree and the codimension of X. In this paper, we construct some surfaces $S{\subset}{\mathbb{P}}^5$ of degree $d{\in}\{7,{\ldots},12\}$ which satisfies the inequality $${\ell}_2(S){\geq}d-3+{\lfloor}{\frac{d}{2}}{\rfloor}$$. This shows that Noma's bound is no more valid for locally non-Cohen-Macaulay varieties.

• Journal of the Chungcheong Mathematical Society
• /
• v.28 no.2
• /
• pp.251-259
• /
• 2015

Let $\mathbb{F}^d_q$ be a d-dimensional vector space over a finite field $\mathbb{F}^d_q$ with q elements. We endow the space $\mathbb{F}^d_q$ with a normalized counting measure dx. Let ${\sigma}$ be a normalized surface measure on an algebraic variety V contained in the space ($\mathbb{F}^d_q$, dx). We define the restricted averaging operator AV by $A_Vf(X)=f*{\sigma}(x)$ for $x{\in}V$, where $f:(\mathbb{F}^d_q,dx){\rightarrow}\mathbb{C}$: In this paper, we initially investigate $L^p{\rightarrow}L^r$ estimates of the restricted averaging operator AV. As a main result, we obtain the optimal results on this problem in the case when the varieties V are any nondegenerate algebraic curves in two dimensional vector spaces over finite fields. The Fourier restriction estimates for curves on $\mathbb{F}^2_q$ play a crucial role in proving our results.