• The Pure and Applied Mathematics
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• v.5 no.1
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• pp.73-78
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• 1998

Let $H_1$ ($\Delta$, M) be the family of all 1-1 holomorphic mappings of the unit disk $\Delta\; \subset\; C$ into a complex manifold M. Following the method of Royden, Hahn introduces a new pseudo-differential metric $S_{M}$ on M. The present paper is to study the product property of the metric $S_{M}$ when M is given by the product of two domains $D_1$ and $D_2$ in the complex plane C, thus investigating the hyperbolicity of the product domain $D_1 \;\times\; D_2$ with respect to $S_{M}$ metric.

• The Pure and Applied Mathematics
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• v.18 no.1
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• pp.13-29
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• 2011

In this paper, we introduce the new concept of ${\omega}-D^*$-distance on a $D^*$-metric space and prove a non-convex minimization theorem which improves the result of Caristi[1], ${\'{C}}iri{\'{c}}$[2], Ekeland[4], Kada et al.[5] and Ume[8, 9].

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.233-247
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• 2022

In the present manuscript, we employ the concepts of Θ-map and Φ-map to define a strong (𝜃, 𝜙)s-contraction of a map f in a b-metric space (M, d_b). Then we prove and derive many fixed point theorems as well as we provide an example to support our main result. Moreover, we utilize our results to obtain many results in the settings of metric and G-metric spaces. Our results improve and modify many results in the literature.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.231-240
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• 2005

The paper shows that a hypersurface with constant curvature with the condition $hD{\subset}D$, of a contact metric manifold with a nullity condition and ${\varphi}-constant$ sectional curvature, has the curvature equals to k and ${\mu}=0$.

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.123-129
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• 2023

For an ordered set W = {w₁, w₂, . . . , w_k} of vertices and a vertex v in a connected graph G, the k-vector r(v|W) = (d(v, w₁), d(v, w₂), . . . , d(v, w_k)) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, G ⊙ H of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(G ⊙ K₁), characterize all graphs G with dim(G ⊙ K₁) = dim(G), and prove that dim(G ⊙ K₁) = n - 1 if and only if G is the complete graph K_n or the star graph K_1,n-1.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.63-82
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• 2011

On the hyperbolic space $D^n$, codimension-one totally geodesic foliations of class $C^k$ are classified. Except for the unique parabolic homogeneous foliation, the set of all such foliations is in one-one correspondence (up to isometry) with the set of all functions z : [0, $\pi$] $\rightarrow$ $S^{n-1}$ of class $C^{k-1}$ with z(0) = $e_1$ = z($\pi$) satisfying |z'(r)| ${\leq}1$ for all r, modulo an isometric action by O(n-1) ${\times}\mathbb{R}{\times}\mathbb{Z}_2$. Since 1-dimensional metric foliations on $D^n$ are always either homogeneous or flat (that is, their orthogonal distributions are integrable), this classifies all 1-dimensional metric foliations as well. Equations of leaves for a non-trivial family of metric foliations on $D^2$ (called "fifth-line") are found.

• Korean Journal of Computational Design and Engineering
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• v.3 no.2
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• pp.140-144
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• 1998

We propose the area between a pair of non-self-intersecting 2-D parametric curves with same endpoints as an alternative distance metric between the curves. This metric is used when d curve is approximated with another in a simpler form to evaluate how good the approximation is. The traditional set-theoretic Hausdorff distance can he defined for any pair of curves but requires expensive calculations. Our proposed metric is not only intuitively appealing but also very easy to numerically compute. We present the numerical schemes and test it on some examples to show that our proposed metric converges in a few steps within a high accuracy.

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1127-1142
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• 2020

In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the Poincaré's hyperbolic metric when the domains are simply connected. In this paper, we provide an alternate definition of the Hurwitz metric through which we could define a generalized Hurwitz metric in arbitrary subdomains of the complex plane. This paper mainly highlights various important properties of the Hurwitz metric and the generalized metric including the situations where they coincide with each other.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.1
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• pp.1-8
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• 2006

In this paper, we obtain some unique common fixed point theorems for four self-maps and pair of maps in D-metric space using orbitally lower semi continuity conditions.

• Journal of Korean Society for Geospatial Information Science
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• v.11 no.3 s.26
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• pp.3-11
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• 2003

This study has made photographing respectively by changing the photographic distance, converging angle, picturing direction by use of Rollei d7 metric and d7 $metric^{5}$ that is a measurement digital camera. And also in order to minimize the errors happened at the relative orientation, we have sorted out the round target that the relative orientation is automatically on the programming and have calculated RMSE by carrying out the bundle adjustment. We think that such a study could be used as very important basic data necessary in deriving the optimal photographic conditions by the close-range digital photogrammetry and in judging such a degree.