• Bulletin of the Korean Chemical Society
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• 제25권8호
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• pp.1241-1243
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• 2004

• East Asian mathematical journal
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• 제38권1호
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• pp.43-63
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• 2022

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (φ, ξ, η, g) in a complex space form M_n+1(c). We denote by R_ξ the structure Jacobi operator with respect to the structure vector field ξ and by ${\bar{r}}$ the scalar curvature of M. Suppose that R_ξ is φ∇_ξξ-parallel and at the same time 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 R_ξφ = φR_ξ, then M is a Hopf hypersurface of type (A) in M_n+1(c) provided that ${\bar{r}-2(n-1)c}$ ≤ 0.

• 한국전기전자재료학회논문지
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• 제17권7호
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• pp.718-723
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• 2004

Crystal structure of $(Ba-{1-x}La_x)[Mg_\frac{1+x}{3}}Nb_\frac{{2-x}{3}}]O_3$ (BLMN) ceramics with 0\leq1x \geq was investigated using synchrotron X-ray powder diffraction (XRD) and high reso(B $a_{l-x}$L $a_{x}$)[M $g_{(1+x)}$3/N $b_{(2-x)/3}$］ $O_3$lution transmission electron microscopy (HRTEM). When the La content, x, is above 0.1, the 1:2 ordered hexagonal structure found in Ba($Mg_\frac{1}{3}Nb_\frac{2}{3}})O_3$(BMN) was transformed into 1:1 ordered cubic structure. The 1:1 ordered cubic structure was maintained up to x=0.7. When x exceeded 0.7, however, BLMN was transformed into 1:1 ordered structure which has cation displacement and in-phase and anti-phase tilt of octahedra.

• 대한수학회논문집
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• 제37권1호
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• pp.229-257
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• 2022

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, 𝜉, 𝜂, g) in a complex space form M_n+1(c), c ≠ 0. We denote by A and R_𝜉 the shape operator in the direction of distinguished normal vector field and the structure Jacobi operator with respect to the structure vector 𝜉, 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 R_𝜉A = AR_𝜉 and at the same time ∇_𝜉R_𝜉 = 0 on M, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.

• Kyungpook Mathematical Journal
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• 제61권4호
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• pp.791-803
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• 2021

Earlier investigators have made detailed studies of geometric properties such as integrability, partial integrability, and invariants, such as the fundamental 2-form, of some canonical f-structures, such as f³ ± f = 0, on the frame bundle FM. Our aim is to study metallic structures on the frame bundle: polynomial structures of degree 2 satisfying F² = pF +qI where p, q are positive integers. We introduce a tensor field F_α, α = 1, 2…, n on FM show that it is a metallic structure. Theorems on Nijenhuis tensor and integrability of metallic structure F_α on FM are also proved. Furthermore, the diagonal lifts g^D and the fundamental 2-form Ω_α of a metallic structure F_α on FM are established. Moreover, the integrability condition for horizontal lift F_α^H of a metallic structure F_α on FM is determined as an application. Finally, the golden structure that is a particular case of a metallic structure on FM is discussed as an example.

• 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 M_n+1(c), c ≠ 0. We denote by R_ξ 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 R_ξ𝜙 = 𝜙R_ξ 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 ${\bar{r}}-2(n-1)c{\leq}0$.

• Bulletin of the Korean Chemical Society
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• 제30권2호
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• pp.464-466
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• 2009

• 대한수학회지
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• 제58권4호
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• pp.895-920
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• 2021

In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface M in the complex quadric Q^m from the equation of Gauss and some important formulas for the structure Jacobi operator R_ξ and its derivatives ∇R_ξ under the Levi-Civita connection ∇ of M. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, ∇_ξR_ξ = 0, in the complex quadric Q^m for m ≥ 3. In addition, we also consider a new notion of 𝒞-parallel structure Jacobi operator of M and give a nonexistence theorem for Hopf real hypersurfaces with 𝒞-parallel structure Jacobi operator in Q^m, for m ≥ 3.

• Journal of Applied and Pure Mathematics
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• 제5권3_4호
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• pp.187-196
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• 2023

This study is about spin half add operations in 𝓑₂ and 𝓑₃. The burden of technological structures has increased due to the increase in the use of today's technological applications or the processes in the digital systems used. This has increased the importance of fast transactions and storage areas. For this, less transactions, more gain and storage space are foreseen. We have handle tit (triple digit) system instead of bit (binary digit). 729 is reached in 3⁶ in 𝓑₃ while 256 is reached with 2⁸ in 𝓑₂. The volume and number of transactions are shortened in 𝓑₃. The limited storage space at the maximum level is storaged. The logic connectors and the complement of an element in 𝓑₂ and the course of the connectors and the complements of the elements in 𝓑₃ are examined. "Carry" calculations in calculating addition and "borrow" in calculating difference are given in 𝓑₃. The logic structure 𝓑₂ is seen to embedded in the logic structure 𝓑₃. This situation enriches the logic structure. Some theorems and lemmas and properties in logic structure 𝓑₂ are extended to logic structure 𝓑₃.

• Bulletin of the Korean Chemical Society
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• 제34권12호
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• pp.3847-3850
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• 2013