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澤　正憲　（神戸大学）

統計的最適点配置問題について（前半）. 澤　正憲　（神戸大学）. 大阪組合せ論セミナ， 2014 年 9 月 20 日. The goal of this talk. Optimal designs. What’s optimal design?. Linear Regression. Gauss- markov model. Optimal design. Theorem (S., to appear in Sugaku ). Cubature and Hilbert identity. What’s cubature?.

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澤　正憲　（神戸大学）

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  1. 統計的最適点配置問題について（前半） 澤　正憲　（神戸大学） 大阪組合せ論セミナ，2014年9月20日

  2. The goal of this talk

  3. Optimal designs

  4. What’s optimal design?

  5. Linear Regression

  6. Gauss-markov model

  7. Optimal design

  8. Theorem (S., to appear in Sugaku)

  9. CubatureandHilbert identity

  10. What’s cubature? Definition ((Spherical) Cubature)

  11. Just for readers’ information… Theorem (Gauss, 1814) Christoffel Number

  12. Example

  13. What’s hilbert identity? Definition (Hilbert Identity)

  14. Waring’s problem Theorem (Liouville, 1859) ∎

  15. Hilbert’s theorem Theorem (Hilbert, 1909)

  16. Fundamental Problem

  17. Reznick’S Theorem Theorem (Reznick, 1992)

  18. Theorem (Stroud (1971), Delsarte et al.(1977)) ∎

  19. How good is this bound?

  20. My originals

  21. Combinatorial design Definition (Combinatorial t-design)

  22. Example (2-design)

  23. Cubature on the simplex Proposition 1 (S.-Xu, 2012)

  24. Proposition 2 (S.-Xu, 2012)

  25. Theorem (Nozaki-S., 2012)

  26. translating CF into identities Bajnok Theorem (Identity Version) ∎ Bajnok Theorem

  27. coming back to optimal designs…

  28. Generalizing Mean-value Thm Theorem (S., 2014+)

  29. Theorem (S., 2014+)

  30. Time!! 前半、 お疲れ様でした。

  31. What’s cubature?

  32. Invariant Cubature Definition (Invariant Spherical Cubature)

  33. Bajnok theorem (detailed version) Theorem (Bajnok, 2007)

  34. Theorem (Nozaki-S., 2011/2012)

  35. Key ingredients of the proof Theorem (Sobolev, 1962) Theorem (Molien-Poincaré Series)

  36. Generalizing Victoir’s idea Definition (t-wise balanced design)

  37. Proof of Reznick’s Theorem ∎

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