• Title, Summary, Keyword: polytopes

### POLYTOPES OF MINIMAL NULL DESIGNS

• Cho, Soo-Jin
• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.143-153
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• 2002
• Null designs form a vector space and there are only finite number of minimal null designs(up to scalar multiple), hence it is natural to look at the convex polytopes of minimal null designs. For example, when t = 0, k = 1, the convex polytope of minimal null designs is the polytope of roofs of type An. In this article, we look at the convex polytopes of minimal null designs and find many general properties on the vertices, edges, dimension, and some structural properties that might help to understand the structure of polytopes for big n, t through the structure of smaller n, t.

### FACIAL STRUCTURES OF A GENERALIZED WEDGE OF POLYTOPES

• Kim, Sangwook
• Honam Mathematical Journal
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• v.38 no.2
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• pp.425-433
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• 2016
• In this paper, we study face and flag information of a generalized wedge of two polytopes over a face. In particular, we provide the explicit formula for the cd-index of the wedge of a polytope and a one-dimensional simplex over an arbitrary face.

### CONVEX POLYTOPES OF GENERALIZED DOUBLY STOCHASTIC MATRICES

• Cho, Soo-Jin;Nam, Yun-Sun
• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.679-690
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• 2001
• Doubly stochastic matrices are n$\times$n nonnegative ma-trices whose row and column sums are all 1. Convex polytope $\Omega$$_{n} of doubly stochastic matrices and more generally (R,S), so called transportation polytopes, are important since they form the domains for the transportation problems. A theorem by Birkhoff classifies the extremal matrices of , \Omega$$_{n}$ and extremal matrices of transporta-tion polytopes (R,S) were all classified combinatorially. In this article, we consider signed version of $\Omega$$_{n}$ and (R.S), obtain signed Birkhoff theorem; we define a new class of convex polytopes (R,S), calculate their dimensions, and classify their extremal matrices, Moreover, we suggest an algorithm to express a matrix in (R,S) as a convex combination of txtremal matrices. We also give an example that a polytope of signed matrices is used as a domain for a decision problem. In this context of finite reflection(Coxeter) group theory, our generalization may also be considered as a generalization from type $A_{*}$ n/ to type B$_{n}$ D$_{n}$. n/.

### ON THE VOLUME OF GRAPH POLYTOPES

• JU, HYEONG-KWAN;KIM, SANGWOOK;SEO, SOO-JEONG
• Honam Mathematical Journal
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• v.37 no.3
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• pp.361-376
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• 2015
• We have calculated the volume of the graph polytopes associated with several types of graphs.

### The change of mathematical representations and behavioral characteristics in the class using manipulative materials - Focused on teaching regular polytopes - (교구를 활용한 수업에서의 수학적 표현과 행동 특성의 변화 - 정다면체 지도를 중심으로 -)

• Choi, Jeong-Seon;Park, Hye-Sook
• The Mathematical Education
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• v.48 no.3
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• pp.303-328
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• 2009
• In this study, we developed the teaching methods using manipulative materials to teach regular polytopes, and applied these to first-year student of middle school who is attending the extra math class. In that class, we focused on the change of the mathematical representations -especially verval, visual and symbolic representations- and mathematical behavioral. By analyzing characterstics the students' work sheets, we obtained affirmative results as follows. First, manipulative materials played an important role on drawing a development figure of regular polyhtopes describing the verval representation definition of regular polytopes. Second, classes utilizing manipulative materials changed students verbalism level of representations the definition of regular polytopes. For example, in the first class about 60% of students are in the $0{\sim}2$ vervalism level, but in the third class, about 65% of students are in the $6{\sim}7$ level. Third, classes utilizing manipulative materials improved visual representation about development figure. After experiences making several development figures about regular octahedron directly, and discussion, students found out key points to be considered for draws development figure and this helped to draw development figures about other regular polytopes. Fourth, students were unaccustomed to make symbolic representations of regular polytopes. But, they obtained same improvement in symbolic representations, so in fifth the class some students try to make symbol about something in common of whole regular polytopes. Fifth, after the classes, we have significant differences in the students, especially behavioral characteristics in II items such as mind that want to study own fitness, interest, attachment, spirit of inquiry, continuously mathematics posthumously. This means that classes using manipulative materials. Specially, 'mind that want to study mathematics continuously' showed the biggest difference, and it may give positive influence to inculcates mathematics studying volition while suitable practical use of manipulative materials. To conclude, classes using manipulative materials may help students enhance the verbal, visual representation, and gestates symbol representation. Also, the class using manipulative materials may give positive influence in some part of mathematical behavioral characteristic. Therefore, if we use manipulative materials properly in the class, we have more positive effects on the students cognitive perspect and behavioral cteristics.

### MATROID BASE POLYTOPES FOR SERIES-PARALLEL EXTENSIONS

• Kim, Sangwook
• Honam Mathematical Journal
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• v.38 no.2
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• pp.393-401
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• 2016
• In this article, we study the matroid base polytope for a matroid obtained from another matroid by a series or parallel extension of an element. We express this polytope as a wedge of a polytope. In particular, we provide the facial structure of the matroid base polytope corresponding to a series-parallel matroid.

### A study on the manipulability measures of robot manipulators (로봇의 조작도 지수에 관한 연구)

• Lee, Yeong-Il;Lee, Ji-Hong
• Journal of Institute of Control, Robotics and Systems
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• v.4 no.1
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• pp.105-112
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• 1998
• Regarding the measure of dexterity of robot manipulators, two geometric tools, manipulability ellipsoids and manipulability polytopes, are examined and compared with each other. Even though the manipulability ellipsoid approach is the most widely used technique, it is shown that the manipulability ellipsoid transforms the inexact joint velocity constraints into task space and so it may fail to give an exact measure of dexterity and optimal direction of motion in task space. After showing that the polytope approach can handle such problems, we propose a practical polytope method which can be applied to 3-dimensional task space in general. The relation between manipulability ellipsoids and manipulability polytopes are also explored for a redundant case and a non-redundant one.

### DIFFERENT VOLUME COMPUTATIONAL METHODS OF GRAPH POLYTOPES

• Ju, Hyeong-Kwan;Kim, Sangwook;Lee, Daeseok
• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1405-1417
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• 2018
• The aim of this work is to introduce several different volume computational methods of graph polytopes associated with various types of finite simple graphs. Among them, we obtained the recursive volume formula (RVF) that is fundamental and most useful to compute the volume of the graph polytope for an arbitrary finite simple graph.

### Impact minimization by impact ellipsoids (임팩트 타원을 이용한 임팩트의 최소화)

• Lee, Ji-Hong;Lee, Young-Il;Yoo, Jun
• 제어로봇시스템학회:학술대회논문집
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• pp.726-729
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• 1996
• A weighted impact ellipsoid normalized by maximum allowable angular velocity changes is defined and compared with conventional impact ellipsoids and impact polytopes. The results shows that the conventional impact ellipsoid may give false solution as far as the optimal direction of motion is concerned.

### VOLUME OF GRAPH POLYTOPES FOR THE PATH-STAR TYPE GRAPHS

• Ju, Hyeong-Kwan;Seo, Soo-Jeong
• Honam Mathematical Journal
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• v.38 no.1
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• pp.71-84
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• 2016
• The aim of this work is to compute the volume of the graph polytope associated with various type of finite simple graphs composed of paths and stars. Recurrence relations are obtained using the recursive volume formula (RVF) which was introduced in Lee and Ju (). We also discussed the relationship between the volume of the graph polytopes and the number of linear extensions of the associated posets for given bipartite graphs.