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Register a new userSpecifying the Kernels of the Starshaped Orthogonal Polygons which are Secondly Connected when the Component of the Complement is a Rectangle
تعيين نواة المضلعات المتعامدة النجمية ثنائية الترابط عندما تكون المركبة المحدودة للمتممة مستطيلاً
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Tishreen University ورقة بحثية
Publication date
2014
fields
Mathematics
and research's language is
العربية
Authors
نجود حسن( باحث )
Created by
Shamra Editor
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تهتم الرؤية الدرجية بدراسة المضلعات المتعامدة، ومن أهم المواضيع التي دُرست هو تعيين نواة المجموعة النجمية درجياً، فقد توصل الباحث تورانزوس إلى نتيجة هامة جداً في تعيين نواة المجموعة النجمية في حالة الرؤية العادية. ثم أثبتت الباحثة مارلين برين صحة نظريّة مرادفة لهذه النتيجة في حالة الرؤية الدرجية، كما تمكنت من إيجاد طريقة لتعين نواة المضلع المتعامد النجمي درجياً عندما يكون المضلع المتعامد بسيط الترابط. يهدف البحث إلى تعميم الطريقة السابقة عندما يكون المضلع المتعامد ثنائي الترابط والمركبة المحدودة للمتممة مستطيلاً حيث سنثبت صحة النتيجة الآتية: لتكن ، مضلعاً متعامداً مغلقاً ثنائي الترابط، حيث إنّ المجموعة نجمية درجياً، عندئذ إذا كانت جبهة المركبة المحدودة B للمجموعة مستطيلاً فإن نواة تتألف من مركبة أو مركبتين أو أربع مركبات.
The staircase visibility concerns with the study of orthogonal polygon, one of the most important subjects which are studied is the Specification kernel of the orthogonal starshaped set. Toranzos represent a very important result in Specifying the kernel of the starshaped set in the usual notion of visibility via segments, after that Breen presented an analogue to this result of the staircase visibility. She also could find a way for Specifying the kernel of starshaped orthogonal polygon when this orthogonal polygon is a simply connected. The aim of this paper is generalizing the previous way when the orthogonal polygon is secondly connected and the bounded component for the complement is a rectangular; we will prove the following result: Let , be secondly connected closed orthogonal polygon, and staircase starshaped set. If the boundary of the bounded component for the complement is a rectangle ,so the kernel of is either one component or two or four ones.
References used
TORANZOS,F.A. Radial functions of convex and star-shaped bodies. Am.Math.Monthly , Vol. 74, 1967, 278–280
BREEN,M. Staircase kernels in orthogonal polygons. Arch. Math, Vol.59, 1992,588-594
MOTWANI,R.؛RAGHUNATHAN,A.؛SARAN,H. Covering orthogonal polygons with star polygons: The Perfect Graph Approach. J.Comput.System Sci,Vol.40,1990,19-48
VALENTINE,F.A.Convex sets. McGraw. Hill, New York, 1964
BREEN,M. Generating the kernel of a staircase starshaped set from certain staircase convex subsets. Periodica Math. Hungarica,Vol.64,N1, 2012,29-37
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Many mathematicians were interested in specifying the kernel of the starshaped set. In staircase visibility the researcher Rajeev Motwani proved that the points of separating regions with dents cannot see each other, and then he proved that these po
ints are seen from other points of an orthogonal polygon. After that Breen could find a way for specifying the kernel of starshaped orthogonal polygon when this orthogonal polygon is simply connected.
The aim of this paper is to generalize the previous way when the closed orthogonal polygon is secondly connected and the bounded component for the complement contains one staircase path or two staircase paths, every path consists of more than two edges. We will prove that the kernel is either one component or two or three ones.
One of the most important subjects that the starshaped sets theory concerned withis
specifying the kernel of the starshaped set and vision the points and regions for each other.
So in staircase visibilitytheresearcher Rajeev Motwani proved that the
points of separating
regions with dents cannot see each other. After that Breen could find a way for specifying
the kernel of starshaped orthogonal polygon when this orthogonal polygon is simply
connected.
In this paper we will generalize the previous way when the closed orthogonal
polygon is secondly connected and the bounded component for the complement is union of
three staircase paths, every path consists of more than two edges. We will prove that the
kernel is only one component.
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