• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.471-478
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• 2024

The aim of this paper is to investigate some properties of the critical points equations on the statistical manifolds. We obtain some geometric equations on the statistical manifolds which admit critical point equations. We give a relation only between potential function and difference tensor for a CPE metric on the statistical manifolds to be Einstein.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.735-747
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• 2018

Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e., the induced metric g on M satisfies the equation: $-({\Delta}_g{\lambda})g+{\nabla}^2_g{\lambda}-{\lambda}Ric=g$, where ${\lambda}$ is a smooth function on M. At first, for the case where M is Hopf, c = 0 and $c{\neq}0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space admitting Miao-Tam critical metric with ${\lambda}$ > 0 or ${\lambda}$ < 0 is a sphere and a compact hypersurface of a non-flat complex space form does not exist such a critical metric.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.587-595
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• 2008

It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is isometric to a standard sphere. In this paper we investigate the relationship between the first Betti number and stable minimal surfaces, and study the analytic properties of stable minimal surfaces in M for n = 3.

• Nuclear Engineering and Technology
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• v.44 no.6
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• pp.663-672
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• 2012

As software based digital I&C (Instrumentation and Control) systems are used more prevalently in nuclear plants, enhancement of software dependability has become an important issue in the area of nuclear I&C systems. Critical attributes of software dependability are safety and reliability. These attributes are tightly related to software failures caused by faults. Software testing and V&V (Verification and Validation) activities are hence important for enhancing software dependability. If the risky modules of safety-critical software can be predicted, it will be possible to focus on testing and V&V activities more efficiently and effectively. It should also make it possible to better allocate resources for regulation activities. We propose a prediction technique to estimate risky software modules by adopting machine learning models based on software complexity metrics. An empirical study with various machine learning algorithms was executed for comparing the prediction performance. Experimental results show SVMs (Support Vector Machines) perform as well or better than the other methods.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.6 no.1
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• pp.98-115
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• 2012

New trend called ubiquitous leads the recent business by standardization and integration. It should be the main issue how to guarantee the integration and accountability on each business, especially in mission critical system which is mainly supported by M2M (Machine to Machine) control mechanism. This study is from the analysis of digital forensics case study that is from the M2M Sensing Control Mechanism problem of the "Imjin River" case in 2009, where a group of family is swept away to death by water due to M2M control error. The ubiquitous surroundings bring the changes in the field of criminal investigation to real time controls such as M2M systems. The needs of digital forensics on M2M control are increasing on every crime scene but we suffer from the lack of control metrics to get this done efficiently. The court asks for more accurately analyzed results accounting high quality product development design. Investigators in the crime scene need real-time analysis against the crime caused by poor quality of mission critical systems. It seems to be every need of Real-Time-Enterprise, so called ubiquitous society on the case. We try to find the efficiency and productivity in discovering non-functional design defects in M2M convergence products focusing on three metrics in study model with quick implementation. Digital forensics system in present status depends on know-how of each investigator and is hard to expect professional analysis on every field. This study set up a hypothesis "Co-working of professional investigators on each field will qualify Performance and Integrity" especially in mission critical system such as M2M and suggests "Online co-work analysis model" to efficiently detect and prevent mission critical errors in advance. At the conclusion, this study proved the statistical research that was surveyed by digital forensics specialists around M2M crime scene cases with quick implementation of dash board.

• Bulletin of the Korean Mathematical Society
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• v.38 no.3
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• pp.551-564
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• 2001

We view Weyl structures as generalizations of Riemannian metrics and study the critical points of geometric functional which involve scalar curvature, defined on the space of Weyl structures on a closed 4-manifold. The main goal here is to provide a framework to analyze critical Weyl structures by defining functionals, discussing function spaces and writing down basic formulas for the equations of critical points.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.655-667
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• 2012

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold $M$. We prove that if the critical point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an $n$-dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.775-779
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• 2005

On a compact n-dimensional manifold $M^n$, a critical point of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satifies the critical point equation (CPE), given by $Z_g\;=\;s_g^{1\ast}(f)$. It has been conjectured that a solution (g, f) of CPE is Einstein. The purpose of the present paper is to prove that every compact stable minimal hypersurface is in a certain hypersurface of $M^n$ under an assumption that Ker($s_g^{1\ast}{\neq}0$).

• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.13-18
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• 1994

The study of the integral of the scalar curvature, $A(g)\;=\;{\int}_M\;RdV_9$ as a functional on the set M of all Riemannian metrics of the same total volume on a compact orient able manifold M is now classical, dating back to Hilbert [6] (see also Nagano [8]). Riemannian metric g is a critical point of A(g) if and only if g is an Einstein metric.(omitted)

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.867-871
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• 2013

It has been conjectured that, on a compact 3-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that ker $s^{\prime}^*_g{\neq}0$, which generalizes the previous partial results.