• Nonlinear Functional Analysis and Applications
• /
• v.26 no.5
• /
• pp.903-915
• /
• 2021

In this paper, we introduce the concepts of an orthogonal generalized F-contraction mapping and prove some fixed point theorems for a self mapping in an orthogonal metric space. The given results are generalization and extension some of the well-known results in the literature. An example to support our result is presented.

• Communications of the Korean Mathematical Society
• /
• v.37 no.4
• /
• pp.1147-1170
• /
• 2022

In this paper, we introduce a new class of mappings called the generalized orthogonal F-Suzuki contraction for a family of multivalued mappings in the setup of orthogonal b-metric spaces. We established the existence of some common fixed point results without using any commutativity condition for this new class of mappings in orthogonal b-metric spaces. Moreover, we illustrate and support these common fixed point results with example. The results obtained in this work generalize and extend some recent and classical related results in the existing literature.

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

• Kyungpook Mathematical Journal
• /
• v.53 no.4
• /
• pp.639-646
• /
• 2013

We investigate the stability of orthogonally additive set-valued functional equation $$F(x+y)=F(x)+F(y)(x{\perp}y)$$ in Hausdorff topology on closed convex subsets of a Banach space.

• Structural Engineering and Mechanics
• /
• v.69 no.6
• /
• pp.615-626
• /
• 2019

The 3-noded metric Timoshenko beam element with an offset of the internal node from the element centre is used here to demonstrate the best-fit paradigm using function space formulation under locking and mesh distortion. The best-fit paradigm follows from the projection theorem describing finite element analysis which shows that the stresses computed by the displacement finite element procedure are the best approximation of the true stresses at an element level as well as global level. In this paper, closed form best-fit solutions are arrived for the 3-noded Timoshenko beam element through function space formulation by combining field consistency requirements and distortion effects for the element modelled in metric Cartesian coordinates. It is demonstrated through projection theorems how lock-free best-fit solutions are arrived even under mesh distortion by using a consistent definition for the shear strain field. It is shown how the field consistency enforced finite element solution differ from the best-fit solution by an extraneous response resulting from an additional spurious force vector. However, it can be observed that when the extraneous forces vanish fortuitously, the field consistent solution coincides with the best-fit strain solution.

• Journal of the Korean Mathematical Society
• /
• v.33 no.1
• /
• pp.169-181
• /
• 1996

Let $x : M^n \longrightarrow E^m$ be an isometric immersion of an n-dimensional connected Riemannian manifold into the m-dimensional Euclidean space. Then the metric tensor on $M^n$ is naturally induced from that of $E^m$.

• Structural Engineering and Mechanics
• /
• v.38 no.3
• /
• pp.261-281
• /
• 2011

A new method of formulation of a class of elements that are immune to mesh distortion effects is proposed here. The simple three-noded bar element with an offset of the internal node from the element center is employed here to demonstrate the method and the principles on which it is founded upon. Using the function space approach, the modified formulation is shown here to be superior to the conventional isoparametric version of the element since it satisfies the completeness requirement as the metric formulation, and yet it is in agreement with the best-fit paradigm in both the metric and the parametric domains. Furthermore, the element error is limited to only those that are permissible by the classical projection theorem of strains and stresses. Unlike its conventional counterpart, the modified element is thus not prone to any errors from mesh distortion. The element formulation is symmetric and thus satisfies the requirement of the conservative nature of problems associated with all self-adjoint differential operators. The present paper indicates that a proper mapping set for distortion immune elements constitutes geometric and displacement interpolations through parametric and metric shape functions respectively, with the metric components in the displacement/strain replaced by the equivalent geometric interpolation in parametric co-ordinates.

• Journal of the Korea Academia-Industrial cooperation Society
• /
• v.11 no.4
• /
• pp.1171-1177
• /
• 2010

This paper presents a robust design procedure for minimizing warpage in plastic injection-molded products, where the Pick-the-Winner rule based on Taguchi's Orthogonal Array experiments and the Design Space Reduction Method are integrated for optimization. Two-step optimization approach is applied to reduce warpage in the part design stage and additionally to minimize the warpage in the process conditions design stage. Taguchi's S/N ratio is introduced as a design metric to evaluate robustness against process variations. The effectiveness of proposed optimization process is shown with an example of warpage minimization problem.

• 제어로봇시스템학회:학술대회논문집
• /
• 1994.10a
• /
• pp.628-633
• /
• 1994

A new Riemannian geometric model for the controlled plant is proposed by imbedding the control vector space in the state space, so as to reduce the dimension of the model. This geometric model is derived by replacing the orthogonal straight coordinate axes on the state space of a linear system with the curvilinear coordinate axes. Therefore the integral manifold of the geometric model becomes homeomorphic to that of fictitious linear system. For the lower dimensional Riemannian geometric model, a nonlinear optimal regulator with a quadratic form performance index which contains the Riemannian metric tensor is designed. Since the integral manifold of the nonlinear regulator is determined to be homeomorphic to that of the linear regulator, it is expected that the basic properties of the linear regulator such as feedback structure, stability and robustness are to be reflected in those of the nonlinear regulator. To apply the above regulator theory to a real nonlinear plant, it is discussed how to distort the curvilinear coordinate axes on which a nonlinear plant behaves as a linear system. Consequently, a partial differential equation with respect to the homeomorphism is derived. Finally, the computational algorithm for the nonlinear optimal regulator is discussed and a numerical example is shown.

• The Transactions of the Korea Information Processing Society
• /
• v.3 no.1
• /
• pp.43-54
• /
• 1996

This paper presents new heuristic search algorithms for searching rectilinear r(L1), link metric, and combined shortest paths in the presence of orthogonal obstacles. The GMD(GuidedMinimum Detour) algorithm combines the best features of maze-running algorithms and line-search algorithms. The SGMD(Line-by-Line GuidedMinimum Detour)algorithm is a modiffication of the GMD algorithm that improves efficiency using line-by-line extensions. Our GMD and LGMD algorithms always find a rectilinear shortest path using the guided A search method without constructing a connection graph that contains a shortest path. The GMD and the LGMD algorithms can be implemented in O(m＋eloge＋NlogN) and O(eloge＋NlogN) time, respectively, and O(e＋N) space, where m is the total number of searched nodes, is the number of boundary sides of obstacles, and N is the total number of searched line segment. Based on the LGMD algorithm, we consider not only the problems of finding a link metric shortest path in terms of the number of bends, but also the combined L1 metric and Link Metric shortest path in terms of the length and the number of bands.