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CONTINUUM-WISE EXPANSIVENESS FOR C1 GENERIC VECTOR FIELDS

  • Manseob Lee
    • 대한수학회지
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    • 제60권5호
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    • pp.987-998
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    • 2023
  • It is shown that every continuum-wise expansive C1 generic vector field X on a compact connected smooth manifold M satisfies Axiom A and has no cycles, and every continuum-wise expansive homoclinic class of a C1 generic vector field X on a compact connected smooth manifold M is hyperbolic. Moreover, every continuum-wise expansive C1 generic divergence-free vector field X on a compact connected smooth manifold M is Anosov.

STRUCTURE JACOBI OPERATOR OF SEMI-INVARINAT SUBMANIFOLDS IN COMPLEX SPACE FORMS

  • KI, U-HANG;KIM, SOO JIN
    • East Asian mathematical journal
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    • 제36권3호
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    • pp.389-415
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    • 2020
  • Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, ξ, η, g) in a complex space form Mn+1(c), c ≠ 0. We denote by Rξ and R'X be the structure Jacobi operator with respect to the structure vector ξ and be R'X = (∇XR)(·, X)X for any unit vector field X on M, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a scalar 𝜃(≠ 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies Rξ𝜙 = 𝜙Rξ and at the same time R'ξ = 0, then M is a Hopf real hypersurfaces of type (A), provided that the scalar curvature ${\bar{r}}$ of M holds ${\bar{r}}-2(n-1)c{\leq}0$.

컴퓨터 통신 네트워크의 보안성을 위한 공개키 배낭 암호시스템에 대한 연구 (A Study on Public Key Knapsack Cryptosystem for Security in Computer Communication Networks)

  • 양태규
    • 정보학연구
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    • 제5권4호
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    • pp.129-137
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    • 2002
  • 본 논문에서는 컴퓨터 통신 네트워크의 데이터 안전을 위해서 다항식을 인수분해 하는 데 어려움이 있는 공개키 다항식 배낭 암호시스템 알고리즘을 제안하였다. 제안된 공개키 다항식 배낭 암호시스템은 먼저, 초증가 벡터 P를 변환하여 다항식 벡터 Q(x,y,z)를 형성하고, 다항식 g(x,y,z)를 선택한다. 이러한 두개의 다항식 Q(x,y,z)와 g(x,y,z)를 공개키로 한다. 공개키 다항식 Q(x,y,z), g(x,y,z)와 난수 $\alpha$를 사용하여 평문을 암호화하여 암호문 R(x,y,z)을 수신자에게 보낸다. 수신자는 암호문 R(x,y,z)을 g(x,y,z)=0의 근, x, y와 z 그리고 비밀키 벡터의 초증가성을 사용하여 평문을 구하게 된다. 따라서 해독과정에서 3변수 다항식 g(x,y,z)=0의 인수분해의 어려움 때문에 안전성을 갖는 공개키 다항식 배낭 암호시스템으로 된다. 제안된 공개키 다항식 배낭 암호시스템의 타당성을 컴퓨터 시뮬레이션을 통하여 입증하였다.

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SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS

  • CHEN, BANG-YEN
    • 대한수학회보
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    • 제52권5호
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    • pp.1535-1547
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    • 2015
  • A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ${\nabla}X^v={\mu}X$ for any vector X tangent to M, where ${\nabla}$ is the Levi-Civita connection and ${\mu}$ is a non-trivial function on M. A smooth vector field ${\xi}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $$\frac{1}{2}L_{\xi}g+Ric={\lambda}g$$, where $L_{\xi}g$ is the Lie-derivative of the metric tensor g with respect to ${\xi}$, Ric is the Ricci tensor of (M, g) and ${\lambda}$ is a constant. A Ricci soliton (M, g, ${\xi}$, ${\lambda}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.

ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C1 VECTOR FIELDS

  • Lee, Keonhee;Le, Huy Tien;Wen, Xiao
    • 대한수학회지
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    • 제51권1호
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    • pp.17-53
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    • 2014
  • Let ${\gamma}$ be a hyperbolic closed orbit of a $C^1$ vector field X on a compact boundaryless Riemannian manifold M, and let $C_X({\gamma})$ be the chain component of X which contains ${\gamma}$. We say that $C_X({\gamma})$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $\mathcal{U}$ of X such that for any $Y{\in}\mathcal{U}$, $C_Y({\gamma}_Y)$ is shadowable for $Y_t$, where ${\gamma}_Y$ denotes the continuation of ${\gamma}$ with respect to Y. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X({\gamma})$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X({\gamma})$ has no non-hyperbolic singularity.

STRICT TOPOLOGIES AND OPERATORS ON SPACES OF VECTOR-VALUED CONTINUOUS FUNCTIONS

  • Nowak, Marian
    • 대한수학회지
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    • 제52권1호
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    • pp.177-190
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    • 2015
  • Let X be a completely regular Hausdorff space, and E and F be Banach spaces. Let $C_{rc}(X,E)$ be the Banach space of all continuous functions $f:X{\rightarrow}E$ such that f(X) is a relatively compact set in E. We establish an integral representation theorem for bounded linear operators $T:C_{rc}(X,E){\rightarrow}F$. We characterize continuous operators from $C_{rc}(X,E)$, provided with the strict topologies ${\beta}_z(X,E)$ ($z={\sigma},{\tau}$) to F, in terms of their representing operator-valued measures.

An Empiricla Bayes Estimation of Multivariate nNormal Mean Vector

  • Kim, Hea-Jung
    • Journal of the Korean Statistical Society
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    • 제15권2호
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    • pp.97-106
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    • 1986
  • Assume that $X_1, X_2, \cdots, X_N$ are iid p-dimensional normal random vectors ($p \geq 3$) with unknown covariance matrix. The problem of estimating multivariate normal mean vector in an empirical Bayes situation is considered. Empirical Bayes estimators, obtained by Bayes treatmetn of the covariance matrix, are presented. It is shown that the estimators are minimax, each of which domainates teh maximum likelihood estimator (MLE), when the loss is nonsingular quadratic loss. We also derive approximate credibility region for the mean vector that takes advantage of the fact that the MLE is not the best estimator.

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STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY NORMED SPACES

  • Bae, Jae-Hyeong;Park, Won-Gil
    • 대한수학회논문집
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    • 제26권2호
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    • pp.237-251
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    • 2011
  • In this paper, we determine some stability results concerning the 2-dimensional vector variable quadratic functional equation f(x+y, z+w) + f(x-y, z-w) = 2f(x, z) + 2f(y, w) in intuitionistic fuzzy normed spaces (IFNS). We dene the intuitionistic fuzzy continuity of the 2-dimensional vector variable quadratic mappings and prove that the existence of a solution for any approximately 2-dimensional vector variable quadratic mapping implies the completeness of IFNS.