• The Mathematical Education
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• v.10 no.2
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• pp.4-8
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• 1972

We consider the class (equation omitted) of all k-degenerate graphs, for k a non-negative integer. The class (equation omitted) and (equation omitted) are exactly the classes of totally disconnected graphs and of forests, respectively; the classes (equation omitted) and (equation omitted) properly contain all outerplanar and planar graphs respectively. The advantage of this view point is that many of the known results for chromatic number and point arboricity have natural extensions, for all larger values of k. The purpose of this note is to show that a graph G is (P$^3$)-realizable if G is planar and 3-degenerate.

• Proceedings of the IEEK Conference
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• 2003.07c
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• pp.2725-2728
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• 2003

The manipulability of robot provides useful Information for the design and path planning of robots. This paper shows an influence of joint velocities to acceleration of robot end-effector using a dynamic manipulability polytope. The main idea of this paper is that the dynamic manipulability polytope of robot can be divided to three intermediate polytope, the torque-dependant polytope, velocity-dependent polytope, and gravity-dependant polytope. The velocity-dependant polytope is made from the limits of robot joint velocities while the torque-dependant polytope is made from the limits of the joint torques. Combining of these two intermediate polytopes and considering the gravity-dependant polytope, the overall dynamic manipulability polytope of robot is obtained. This investigation will be useful on the field of space robot and high-speed application.

• Journal of Institute of Control, Robotics and Systems
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• v.5 no.3
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• pp.249-256
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• 1999

The conventional robot manipulability is decomposed into linear manipulability and angular manipulability so that they may be analysed and visualized in easy way even in the case of 3 dimensional task space with 6 variables. After the Jacobian matrix is decomposed into linear part and angular part, constraint on joint velocities is transformed into linear task velocity and angular task velocity through the decomposed Jacobian matrices. Under the assumption of redundant robot manipulators, several optimization problems which utilize the redundancy are formulated to be solved by linear programming technique or sequential quadratic programming technique. After deriving the solutions of the optimization problems, we give graphical interpretations for the solutions.

• 제어로봇시스템학회:학술대회논문집
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• 1996.10b
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• pp.739-742
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• 1996

Several figures representing velocity transmission from joint space to task space are analyzed and compared with each other. The figures include velocity ellipsoid derived from Jacobian matrix, scaled velocity ellipsoid derived from normalized joint velocities, polytope derived by numerical scaling, and polytopes derived by linear combinations of Jacobian column vectors. The results show that the optimal directions given by the measures are not the same and the conventional velocity ellipsoid is not good choice as optimization measure as far as the moving direction is concerned. Simulation examples for 3 d.o.f. redundant robot manipulators in 2-dimensional task space are given for comparison study.

• Journal of the Korean Institute of Telematics and Electronics C
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• v.34C no.7
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• pp.51-60
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• 1997

Prototype based methods are commonly used in cluster analysis and the results may be highly dependent on the prototype used. In this paper, we propose a fuzzy clustering method that involves adaptively expanding convex polytopes. Thus, the dependency on the use of prototypes can be eliminated. The proposed method makes it possible to effectively represent an arbitrarily distributed data set without a priori knowledge of the number of clusters in the data set. Specifically, nonlinear membership functions are utilized to determine whether a new cluster is created or which vertex of the cluster should be expanded. For this, the membership function of a new vertex is assigned according to not only a distance measure between an incoming pattern vector and a current vertex, but also the amount how much the current vertex has been modified. Therefore, cluster expansion can be only allowed for one cluster per incoming pattern. Several experimental results are given to show the validity of our mehtod.

• Proceedings of the KIEE Conference
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• 1997.07b
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• pp.431-433
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• 1997

In this paper, we consider the robust pole assignment inside the circular region for linear systems with time-varing uncertainty. The considered uncertainties are the norm bounded uncertainty and the matrix polytopes type uncertainty. Since the considered uncertainties are time-varying, it should remind that the closed loop system be time-varing system. Therefore, we should consider not only the pole assignment but also the stability. We present new conditions that guarantee the robust pole assignment and the robust stability. Finally, we show the usefulness of our results by an example.

• Journal of Institute of Control, Robotics and Systems
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• v.12 no.5
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• pp.497-504
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• 2006

This paper presents a new technique that derives the dynamic acceleration bounds of multiple cooperating robot systems from given individual torque limits of robots. A set of linear algebraic homogeneous equation is derived from the dynamic equations of multiple robots with friction contacts. The mobility of the robot system is analyzed by the decomposition of the null space of the linear algebraic equation. The acceleration bounds of multiple robot systems are obtained from the joint torque constraints of robots by the medium of the decomposed null space. As the joint constraints of the robots are given in the infinite norm sense, the resultant acceleration bounds of the systems are described as polytopes. Several case studies are presented to validate the proposed method in this paper.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.371-384
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• 2022

Fix an integer n ≥ 1. Then the simplex Π_n, Birkhoff polytope Ω_n, and Latin square polytope Λ_n each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of Ω_n are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.671-683
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• 2022

The aim of this paper is to deal with the realization problem of a given Lagrangian submanifold of a symplectic manifold as the fixed point set of an anti-symplectic involution. To be more precise, let (X, ω, µ) be a toric Hamiltonian T-space, and let ∆ = µ(X) denote the moment polytope. Let τ be an anti-symplectic involution of X such that τ maps the fibers of µ to (possibly different) fibers of µ, and let p₀ be a point in the interior of ∆. If the toric fiber µ^-1(p₀) is real Lagrangian with respect to τ, then we show that p₀ should be the origin and, furthermore, ∆ should be centrally symmetric.

• Journal of Educational Research in Mathematics
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• v.13 no.4
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• pp.495-506
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• 2003

In this study, 1 investigated some properties on the special hyper-dimensional figures made by the principle of the performance of equivalent forms representation. I supposed 2 definitions on the making n-dimensional figure : a cone type(hypercube) and a pillar type(simplex). We can explain that there exists only 6 4-dimensional regular polytopes as there exists only 5 regular polygons. And there are many hyper-dimensional figures, they all have sufficient condition to show the general Euler' Characteristics. And especially, we could certificate that the simplest cone type and pillar types are fitted to Pascal's Triangle and Hasse's Diagram, each other.