• The Pure and Applied Mathematics
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• v.20 no.3
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• pp.199-206
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• 2013

We present a 4-dimensional nil-manifold as the first example of a closed non-K$\ddot{a}$hlerian symplectic manifold with the following property: a function is the scalar curvature of some almost K$\ddot{a}$hler metric iff it is negative somewhere. This is motivated by the Kazdan-Warner's work on classifying smooth closed manifolds according to the possible scalar curvature functions.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.445-453
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• 2021

The purpose of the paper is to study of Para-Kenmotsu metric as a 𝜂-Ricci soliton. The paper is organized as follows: • If an 𝜂-Einstein para-Kenmotsu metric represents an 𝜂-Ricci soliton with flow vector field V, then it is Einstein with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with the flow vector field V being an infinitesimal paracontact transformation, then V is strict and the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with non-zero flow vector field V being collinear with 𝜉, then the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). Finally, we cited few examples to illustrate the results obtained.

• International Journal of Naval Architecture and Ocean Engineering
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• v.4 no.3
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• pp.291-301
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• 2012

A scalar metric for the assessment of hull surface producibility was known to be useful in estimating the complexity of a hull form of ships or large offshore structures by looking at their shape. However, it could not serve as a comprehensive measuring tool due to its lack of important components of the hull form such as longitudinals, stiffeners, and web frames attached to the hull surface. To have a complete metric for cost estimation, these structural members must be included. In this paper, major inner structural members are considered by measuring the complexity of their geometric shape. The final scalar metric thus consists of the classes containing inner members with various curvature magnitudes as well as the classes containing curved plates with single and double curvature distribution. Those two distinct metrics are merged into a complete scalar metric that accounts for the total cost estimation of complex structural bodies.

• The Pure and Applied Mathematics
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• v.20 no.4
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• pp.269-276
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• 2013

We find an explicit $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on the 4-d hyperbolic space $\mathbb{H}^4$, for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ > 0 with the following property: $g_0$ is the hyperbolic metric on $\mathbb{H}^4$, the scalar curvatures of $g_t$ are strictly decreasing in t in an open ball and $g_t$ is isometric to the hyperbolic metric in the complement of the ball.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1087-1098
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• 2013

We find a $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on $\mathbb{R}^k$, $k{\geq}3$, for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ > 0 with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^k$, the scalar curvatures of $g_t$ are strictly decreasing in $t$ in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.

• Journal of applied mathematics & informatics
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• v.8 no.2
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• pp.551-560
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• 2001

In this paper, we investigate the continuity if addition and scalar multiplication on the space F(R/sup p/) of upper semicontinuous fuzzy sets in R/sup p/ under the Shorokhod metric.

• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.629-632
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• 1994

Let (M, g) be a Riemannian manifold. The relation between a (pointwise) conformal metric of the metric g and the scalar curvature of this new metrics is investigated by Kazdan, Warner and Schoen (cf. [1, 4]).

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.145-149
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• 2006

We demonstrate a novel almost Kahler deformation on $\mathbb{R}^4$ which diffuses the scalar curvature.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1087-1094
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• 2016

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

• The KIPS Transactions:PartD
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• v.8D no.3
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• pp.233-240
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• 2001

As software technology is in progress and software quantification is getting more important, many metrics have been proposed to quantify a variety of system entities. These metrics can be classified into two different forms : scalar metric and metric vector. Though some recent studies pointed out the composition problem of the scalar metric form, many scalar metrics are successfully used in software development organizations due to their practical applications. In this paper, it is concluded that hybrid metric form weighting external complexity is most suitable for scalar metric form. With this concept, a general framework for hybrid metrics construction independent of the development methodologies and target system type is proposed. This framework was successfully used in two projects that quantify the analysis phase of the structured methodology and the design phase of the object oriented real-time system, respectively. Any organization can quantify system entities in a short time using this framework.