• Kyungpook Mathematical Journal
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• 제62권3호
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• pp.467-484
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• 2022

In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.

• 대한수학회논문집
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• 제23권3호
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• pp.421-426
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• 2008

Let $M^n$ be a complete immersed super stable minimal submanifold in $\mathbb{R}^{n+p}$ with fiat normal bundle. We prove that if M has finite total $L^2$ norm of its second fundamental form, then M is an affine n-plane. We also prove that any complete immersed super stable minimal submanifold with flat normal bundle has only one end.

• 대한수학회보
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• 제26권1호
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• pp.75-82
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• 1989

It is recently proved by Aiyama and the authors [2] that a complete space-like complex submanifold of a complex space form $M^{n+p}$$_{p}$ (c') (c'.geq.0) is to totally geodesic. This is a complex version of the Bernstein-type theorem in the Minkowski space due to Calabi [4] and Cheng and Yau [5], which is generalized by Nishikawa[7] in the Lorentz manifold satisfying the strong energy condition. The purpose of this paper is to consider his result in the complex Lorentz manifold and is to prove the following.e following.

• 대한수학회보
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• 제53권5호
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• pp.1457-1469
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• 2016

In this paper, we study a class of Finsler metric. First, we find some rigidity results of the dually flat (${\alpha}$, ${\beta}$)-metric where the underline Riemannian metric ${\alpha}$ satisfies nonnegative curvature properties. We give a new geometric approach of the Monge-$Amp{\acute{e}}re$ type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.