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研究生:陳瑟云
研究生(外文):Se-Yun Chen
論文名稱:關於集合系對的研究
論文名稱(外文):On pairs of set systems
指導教授:黃大原
指導教授(外文):Tayuan Huang
學位類別:碩士
校院名稱:國立交通大學
系所名稱:應用數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:47
中文關鍵詞:集合系對群試組合設計
外文關鍵詞:set systemnon-adaptive pooling designd-disjunct
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一個 (0,1) -矩陣的行向量及列向量可以分別被看成一個集合系對(set system) 的特徵向量。在這篇論文中,我們透過這個模型探討一組集合系對的disjunct性質及其從相交個數的觀點探討一些其他的組合性質。同時,本文也包含了我們針對Hirasaka所提出利用群的乘積建構一個disjunct矩陣的方法,所做的一些修正。

The row vectors and the column vectors of a (0,1)-matrix can be treated as characteristic vectors of a pair of set systems respectively. With this model, we study in this thesis pairs of set systems with disjunctness up to some degrees and with some other combinatorial properties in terms of the intersection sizes among them. Some modifications of Hirasaka’s construction of disjunct matrices in terms of products of groups are included.

Contents
1 Introduction
2 Set Families with Specified Intersecting Properties
2.1 Codes, Anticodes and Erd¨os-Ko-Rado Theorem
2.2 Association Schemes and Some Combinatorial Designs
2.3 d-disjunct Matrices and Non-adaptive Pooling Designs
3 Disjunctness of Matrices and Association Schemes
3.1 Johnson Association Schemes, Grassmanian Association Schemes and Atomic Ranked Semilattices
3.2 Hwang’s Conjecture
3.3 Random Designs form Deterministic Designs
4 Some Remarks on Disjunct Matrices
4.1 An Interpretation of Wu’s Matrix
4.2 Some Remarks on Hirasaka’s Construction
4.2.1 The Case d=10
4.2.2 The Case d=129
5 Disjunctness of Some (0,1)-matrices
5.1 Adjacency Matrices of Some Cages
5.2 Adjacency Matrices of Some Graphs
6 Reference
7 Appendix

[1. ] T. Beth, D. Jungnickel and H. Lenz, Design theory, 2nd edition Cambridge University 1996.
[2. ] Erd¨os. P., Frankel. P. and Furedi. Z., Families of finite set in which no set is covered by the union of r others, Israel J. Math. 51 (1985), no. 1-2.
[3. ] C. J. Colbourn, J. H. Dinitz and D. R. Stinson, Applications of combinatorial designs to communications, cryptograph and networking, Nov. 2000 preprint.
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[7. ] S. Huang and F. K. Hwang, When is individual testing optimal ?, SIAM Discrete Math 2002.
[8. ] F. K. Hwang and V. T. Sos, Non-adaptive hypergeometric group testing, Studia Scientiarum Mathematicarum Hungarica 22(1987), 257-263.
[9. ] F. K. Hwang and Ding-Zhu Du, Combinatorial group testing and its applications, World Scientific 2000. 42
[10. ] H. Q. Ngo and Ding-Zhu Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Series in Discrete Mathematics and Theoretical Science, Vol 55, 2000, p171-182.
[11. ] H. Q. Ngo and D. Z. Du, New constructions of non-adaptive and error-tolerance pooling design, Discrete Mathematics 243 (2002) 161-170.
[12. ] D. R. Stinson, Combinatorial designs with selected applications, preprint Dec. 1996.
[13. ] Chih-weng Weng, Construct non-adaptive pooling designs form a ranked atomic semi-lattice, Feb. 2002, preprint.

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