TAILIEUCHUNG - Báo cáo toán học: "Omittable Plane"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Omittable Planes. | Omittable Planes Branko Griinbaum Jonathan Lenchner t Submitted Jul 16 2010 Accepted Aug 26 2011 Published Sep 2 2011 Mathematics Subject Classification 52C35 Abstract In analogy to omittable lines in the plane we initiate the study of omittable planes in 3-space. Given a collection of n planes in real projective 3-space a plane n is said to be omittable if n is free of ordinary lines of intersection - in other words if all the lines of intersection of n with other planes from the collection come at the intersection of three or more planes. We provide two infinite families of planes yielding omittable planes in either a pencil or near-pencil together with examples having between three and seven omittable planes examples that we call sporadic which do not fit into either of the two infinite families. 1 Introduction A finite family L of straight lines in the plane determines several different objects. One of these is the aggregate associated with L defined as the family of points P in which two or more of the lines intersect together with the lines L themselves. To avoid trivialities and exceptions it is customary to stipulate that P consists of more than a single point. A point of P that is on precisely two lines is called ordinary. A line L of L that is incident with no ordinary point is called omittable since the deletion of L from L does not change P. Figure 1 shows an example of a family L of 15 lines for this L each of the 6 heavily drawn lines is omittable. In fact all 4 of these lines that pass through the center are simultaneously omittable. A famous problem still unsolved after many decades is to determine the best lower bound for the number s n of ordinary points in aggregates of n lines. The conjecture is that s n Ln 2j for all n with a higher lower bound in case n 3 is a multiple of 4. The best available result is s n 6n 13 see Csima and Sawyer 2 . There are many accounts of the history and variants of this problem the most recent one is that of .

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• báo cáo của tạp chí Department of Mathematic
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