ゼミ発表 (2019年8月)

Seminar Presentation (August, 2019)

Transcript

1. θϛൃද ۚઘେհ (ؙ໺ݚڀࣨ, ਫ਼౓อূ෇͖਺஋ܭࢉάϧʔϓ OB) 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ, ਺ֶԠ༻਺ཧઐ߈ ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 1 / 27
2. ຊൃදͷྲྀΕ 1 લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ ཭ࢄՄੵ෼ܥͱ࠷దԽ໰୊ͷؔ܎ q-ྨࣅͱҰҙੑ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ q-ྨࣅͰఆٛҬ͕ڱ·Δྫ ղੳ઀ଓͰ͖Δྫ ಛҟ఺ɾଟՁੑճආʢॲཧʣςΫχοΫҰཡ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ 2 ޙ൒ɿօ͞Μʹ͓ฉ͖͠·͢ େҬ࠷దԽ Derivative-free optimization (DFO) ૊Έ߹Θͤ࠷దԽ 3 ͓·͚ɿJSIAM ೥ձ 2019 ஫໨ߨԋ (ಠஅ) 4 ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 2 / 27
3. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ ཭ࢄՄੵ෼ܥͱ࠷దԽ໰୊ͷؔ܎ ཭ࢄՄੵ෼ܥͱ࠷దԽ໰୊ͷؔ܎ ཭ࢄՄੵ෼ܥͱ࠷దԽ໰୊ʹؔ܎͋Γ·͔͢ ? ௐ΂ͯΈͨΒͲ͏΍Β͋ΔΑ͏Ͱ͢ 1,2. ۙ೥ͷՄੵ෼ܥݚڀͷಈ޲ͷதͰಛච͢΂͖͸Ԡ༻਺ֶ/Ԡ༻਺ཧతଆ ըͷਐలͰ͋Ζ͏. ͱΓΘ͚, ࠷దԽΞϧΰϦζϜ, ݻ༗஋ܭࢉ๏, Ճ଎ ๏ͳͲͷ਺஋ܭࢉ๏ͱՄੵ෼ܥͱͷີ઀ͳ͔͔ΘΓͷೝࣝΛ଍ֻ͔Γ ͱͯ͠৽͍͠ݚڀྖҬ͕ܗ੒͞Εͭͭ͋Δ. 1தଜՂਖ਼ (1997), ཭ࢄ࣌ؒՄੵ෼ܥͱ਺஋ܭࢉ๏, ਺ཧղੳݚڀॴߨڀ࿥ 1005 ר, 132-148. 2V. Jurdjevic (2016), Optimal Control and Geometry: Integrable Systems, Cambridge University Press. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 3 / 27
4. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͱҰҙੑ લճͷ࣭໰ 1 Fact q-ྨࣅͷݩ͸ҰҙͰ͋Δ. (∵) ҰҙͰͳ͍৔߹, ࿈ଓۃݶͷҰҙੑ (ඍੵͰطश) ʹ൓͢Δ. Fact q-ྨࣅ͸ҰҙͰ͸ͳ͍ → ͜Ε͕ q ͷੈքΛ๛͔ʹ͍ͯ͠Δ !! ൓ྫ 1 q-Bessel ͸গͳ͘ͱ΋ 3 ͭ͋Δ (Jackson q-Bessel × 2, Hahn-Exton q-Bessel). ൓ྫ 2 Painleve ํఔࣜ͸ 6 ͚͕ͭͩͩ q-Painleve ํఔࣜ͸ͦΕҎ্͋Δ. → ୔ࢁ͋Δ q-Painleve ํఔࣜ (& ପԁ Painleve) Λ౷Ұతʹѻ͏ͷ͕ Sakai ཧ࿦. ྫ͑͹ https://arxiv.org/abs/1804.10341 Figure 1, Figure 2 Λࢀর. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 4 / 27
5. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ લճͷ࣭໰ 2 q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ Definition (ղੳ઀ଓ) Riemann ٿ໘ C ∪ {∞} ্ͷྖҬͰఆٛͨ͠༗ཧܕ (meromorphic, ۃΛআ͍ͯ holomorphic) ؔ਺ʹର͢ΔఆٛҬͷ֦ு. (cf: ௚઀઀ଓ) ਆอಓ෉. (2003). ෳૉؔ਺ೖ໳. ؠ೾ॻళ. Ablowitz, M. J., Fokas, A. S. (2003). Complex variables: introduction and applications. Cambridge University Press. q-ྨࣅ͸ಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓ͢Δ͜ͱΛओ໨తͱͨ͠ ΋ͷͰ͸͋Γ·ͤΜ͕, ·ΕʹͰ͖Δ͜ͱ͕͋Γ·͢. ·ͨ, q-ྨࣅͯ͠ఆٛҬ͕ ڱ·Δ͜ͱ΋͋Γ·͢. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 5 / 27
6. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ q-ྨࣅͰఆٛҬ͕ڱ·Δྫ q-ࢦ਺ؔ਺ eq (z) := ∞ ∑ n=0 zn (q; q)n , |z| < 1. Jackson ͷୈ 1 छ q-Bessel ؔ਺ J(1) ν (x; q) := (qν+1; q)∞ (q; q)∞ (x/2)ν 2 ϕ1 (0, 0; qν+1; q, −x2/4), |x| < 2. Gasper, G., Rahman, M., (2004), Basic Hypergeometric Series, Cambridge University Press. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 6 / 27
7. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ ղੳ઀ଓͰ͖Δྫ r Fr−1 , r ϕr−1 ͷऩଋ൒ܘ͸ |z| < 1 ͕ͩ, ପԁ௒زԿؔ਺ 3: r Er−1 (a1 , · · · , ar ; b1 , · · · , br−1 ; p, q; z) := ∞ ∑ n=0 r ∏ i=1 (ai ; p, q)n zn r−1 ∏ j=1 (bj ; p, q)n (q; p, q)n . (z; p, q)n :=      ∏ n−1 k=0 θ(zqk; p) (n = 1, 2, · · · ) 1/ ∏ −n−1 k=0 θ(zqn+k; p) (n = −1, −2, · · · ) 1 (n = 0) . θ(z; q) := (z; q)∞ (qz−1; q)∞ , p, q ∈ (0, 1). ͷऩଋ൒ܘ͸ແݶͰ͋Δ (cf: Gasper-Rahman, chapter 11). (z; p, q)n ͸ p → 0 ͷͱ͖ (z; q)n ʹ໭ΔͷͰ, p → 0 ͱͨ͠ͱ͖, r Er−1 → r ϕr−1 ͱͳΔ. r Er−1 ͸ެൺ͕ପԁؔ਺ (༗ཧܕೋॏपظؔ਺) ʹͳΔ. 3Gasper, G., Rahman, M., (2004), Basic Hypergeometric Series, Cambridge University Press. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 7 / 27
8. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ ࢀߟ: ਺஋ղੳͰ࢖ΘΕΔಛҟ఺ɾଟՁੑճආʢॲཧʣςΫχοΫҰཡ Duffy ม׵ 4 (ႈܕಛҟ఺ΛؚΉ਺஋ੵ෼) → আڈՄೳಛҟ఺ (ྫ: f(x) = sin(x)/x ʹ͓͚Δ x = 0) ΋Ͱ͖Δ͔΋ ? generalized Duffy ม׵ 5 (4 ࣍ݩҎ্ͷ Duffy ม׵) →4 ࣍ݩҎ্ͷ਺஋ੵ෼͸ͲΜͳ࣌ʹඞཁ ? (cf: Davis-Rabinowitz) →PDE ͷ਺஋ղ๏ (FEM ౳) Ͱ 4 ࣍ݩҎ্ͷ਺஋ੵ෼͸ඞཁ ? ෦෼ੵ෼ → ႈܕಛҟ఺ΛؚΉ਺஋ੵ෼, ઴ۙల։ (cf: Ablowitz-Fokas) Schwartz ௒വ਺ → தඌཧ࿦Ͱඞཁ (ؔ਺ۭؒͷઃఆ 6) ෳૉ࣌ؒ →ODE/PDE Ͱരൃ࣌ࠁΛճආ (ߴ҆ઌੜͳͲ) 4Duffy, M. G. (1982). Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM journal on Numerical Analysis, 19(6), 1260-1262. 5Mousavi, S. E., & Sukumar, N. (2010). Generalized Duffy transformation for integrating vertex singularities. Computational Mechanics, 45(2-3), 127. 6http://mathsoc.jp/office/prize/haruaki/nakao2012aki.html ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 8 / 27
9. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ ࢀߟ: ਺஋ղੳͰ࢖ΘΕΔಛҟ఺ɾଟՁੑճආʢॲཧʣςΫχοΫҰཡ (ෳૉؔ਺࿦) ओ஋, Riemann ໘ 7,8→ log z (∀z ∈ C\(R≤0 )) ΛఆΊΔͷʹඞཁ (਺஋ܭࢉͱͷؔ࿈͕ݚڀ͞Ε͍ͯΔ 9) Riemann ٿ໘ (C ∪ ∞, ࣮ࣹӨ௚ઢͷ 2 ࣍ݩ൛) → ͜Ε΋਺஋ܭࢉʹ໾ཱͭͱ৴͡ΒΕ͍ͯΔ 10. ͜͏͍͏෼໺Λ Applied and Computational Complex Analysis ͱ͍͏. 7ਆอಓ෉. (2003). ෳૉؔ਺ೖ໳. ؠ೾ॻళ. 8Ablowitz, M. J., Fokas, A. S. (2003). Complex variables: introduction and applications. Cambridge University Press. 9Bobenko, A. I. (2011). Computational approach to Riemann surfaces. Springer Science & Business Media. 10https://tech.speee.jp/entry/2017/11/14/105047 ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 9 / 27
10. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ ࢀߟ: ਺஋ղੳͰ࢖ΘΕΔಛҟ఺ɾଟՁੑճආʢॲཧʣςΫχοΫҰཡ (ෳૉؔ਺࿦) hyperfunction11 (ࠤ౻௒വ਺, ਖ਼ଇؔ਺ಉ࢜ͷڥք্Ͱͷࠩ) f(x) = F (x + i0) − F (x − i0) (Fourier ڃ਺ʹ͓͚Δෆ࿈ଓ఺΋͜Μͳײ͡Ͱॲཧ͠·͢.) ୆ܗެࣜͱ૊Έ߹ΘͤΔ͜ͱͰ਺஋ੵ෼ʹԠ༻ 12,13 (ࠓ౓ͷ JSIAM ೥ձͰൃද͋Γ) Gauss-Legendre ౳͕ಋग़Ͱ͖Δ 14 ” ิؒ, ਺஋ඍ෼, ਺஋ੵ෼, Fourier ղੳͷΑ͏ͳ਺஋ղੳͷجૅతͳ ໰୊Λ௒വ਺ (hyperfunction) ͷཱ৔͔ΒோΊͯΈΔͱ, ౷ҰతͳऔΓ ѻ͍͕ՄೳʹͳΔ্ʹ, ޡࠩධՁͳͲʹ͓͍࣮ͯ༻্༗ޮͳํ๏ΛಘΔ ͜ͱ͕Ͱ͖Δ.” (by ৿ਖ਼෢ઌੜ) ” ࠤ౻௒വ਺͸ແݶ۠ؒੵ෼, Hadamard ༗ݶ෦෼ੵ෼ʹ΋Ԡ༻Ͱ͖Δ. ࠓޙ͸ Fourier ม׵, ੵ෼ํఔࣜͳͲͰ΁ͷԠ༻Λߟ͍͖͍͑ͯͨ.” 11ࠓҪޭ, Ԡ༻௒ؔ਺࿦ I, II. 12http://www.uec-ogata-lab.jp/research/research1/ 13ॹํलڭ, ฏࢁ߂, ਺஋ੵ෼ʹର͢Δ௒വ਺๏, ೔ຊԠ༻਺ཧֶձ࿦จࢽ, Vol.26 (2016) 33-43. 14৿ਖ਼෢, ਺஋ղੳͱ௒വ਺࿦, ژ౎େֶ਺ཧղੳݚڀॴߨڀ࿥, 145 (1972) 1-11. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 10 / 27
11. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ ࢀߟ: ͦͷଞͷಛҟ఺ɾଟՁੑճආʢॲཧʣςΫχοΫҰཡ (୅਺زԿ) ୅਺زԿʹ͓͚Δಛҟ఺ղফ (resolution of singularities, cf: Wikipedia ӳޠ൛) blow up (೔ຊޠʹ௚༁͢Δͱʮരൃʯ͕ͩ͜ͷݴ͍ํ͸͋·Γͳ͍ͱࢥ͏) தࠃޠͩͱʮ፮։ʯ Մੵ෼ܥݚڀऀͨͪͷେ޷෺ രൃղ (blow up solution) ͱͷؔ܎ ? Newton, Riemann, Noether, Zariski, ޿த, · · · → ͜ΕΒͷٕज़͕਺஋ղੳʹ࢖͑Δͱ৴͍ͯ͡Δઌੜͨͪ΋͍Δ cf: ਺஋୅਺زԿ (numerical algebraic geometry) ܭࢉػͰ୅਺ଟ༷ମΛݚڀ͢Δֶ໰ ୅਺زԿ͸࣮ݧՊֶ (by ᑸઌੜ, ਺ֶՊύϯϑϨοτ) ਫ਼౓อূͰྗֶܥʹ͓͚Δ҆ఆଟ༷ମΛݚڀ͢Δͷͱࣅ͍ͯΔ͔΋͠Εͳ͍. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 11 / 27
12. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ྨࣅͰಛҟੑɾଟՁੑΛճආʢॲཧʣͨ͠Γղੳ઀ଓͰ͖Δ͔ʁ ࢀߟ: ͦͷଞͷಛҟ఺ɾଟՁੑճආʢॲཧʣςΫχοΫҰཡ আ๏ͷఆٛΛม͑ͯ zero আࢉΛՄೳʹ͢Δ wheel theory (ྠ)15 q-আࢉ 16 x / ⃝q y := ( x1−q − y1−q + 1 ) 1 1−q + , (A)+ := max{A, 0}. ແݶԕ఺ΛؚΊͨ਺ͷऔΓѻ͍ affine ֦େ࣮਺ [−∞, ∞] →extended interval arithmetic17 ௒࣮਺ 15Carlstrom, J. (2004), ”Wheels - On Division by Zero”, Mathematical Structures in Computer Science, Cambridge University Press, 14 (1): 143-184. 16Borges, E. P. (2004). A possible deformed algebra and calculus inspired in nonextensive thermostatistics. Physica A: Statistical Mechanics and its Applications, 340(1-3), 95-101. 17Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to interval analysis. SIAM. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 12 / 27
13. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ 1990 ೥୅ʹʮq-ಛघؔ਺Λඍ෼͠Α͏ʯͱ͍͏ݚڀ͕ྲྀߦͬͨ (Swarttouw18 ౳). ͔͠͠ 20 ೥ܦͬͨࠓͰ͸ͦͷΑ͏ͳ࿩Λશ͘ฉ͔ͳ͘ͳͬͨ. ͔ͭͯऔΓ ૊ΜͰ͍ͨઌੜํ΋શһఫୀͨ͠Α͏ͩ. ݚڀऀͨͪʹ௚઀ฉ͍ͨΘ͚Ͱ͸ͳ͍ͷͰਖ਼֬ͳ͜ͱ͸෼͔Βͳ͍͕, Կ͕ ೉͍͠ͷ͔Λࣗ෼ͳΓʹߟ͑ͯΈ·ͨ͠. 18௚ަଟ߲ࣜۀքͰఱ࠽ͱ΋ݺ͹ΕΔઌੜͰ͢. Koekoek, R., & Swarttouw, R. F. (1996). The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. arXiv preprint math/9602214. Koekoek, R., Lesky, P. A., & Swarttouw, R. F. (2010). Hypergeometric orthogonal polynomials and their q-analogues. Springer Science & Business Media. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 13 / 27
14. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ Theorem (ඍ෼ੵ෼ͷ෮श, ߲ผඍ෼) ∑ an zn ͕ऩଋ͢ΔͳΒ d dz ∑ an zn = ∑ an d dz zn r ϕs (α1 , · · · , αr ; β1 , · · · , βs ; q, z) := ∞ ∑ n=0 r ∏ i=1 (αi ; q)n [ (−1)nq(n 2 ) ]1+s−r zn s ∏ j=1 (βj ; q)n (q; q)n . ∴ d dz (r ϕs (q, z)) = ∞ ∑ n=0 r ∏ i=1 (αi ; q)n [ (−1)nq(n 2 ) ]1+s−r nzn−1 s ∏ j=1 (βj ; q)n (q; q)n . μϥϯϕʔϧͷ൑ఆ๏ΑΓऩଋ൒ܘ͸ແݶͰ͋Δ. ͔͠͠ඍ෼ͨ͜͠ͱͰެൺ͕ େ͖͘ͳΔͷͰਫ਼౓͕ѱԽ͠, n ͕ཅతʹݱΕΔͷͰ overflow ͕ΑΓى͖΍͘͢ ͳ͍ͬͯΔ. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 14 / 27
15. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ d dz (rϕs(q, z)) = ∞ ∑ n=0 r ∏ i=1 (αi; q)n [ (−1)nq(n 2 ) ] 1+s−r nzn−1 s ∏ j=1 (βj; q)n(q; q)n . overflow Λճආ͢ΔͨΊʹ͸ asymptotics (઴ۙల։) ͕ඞཁͳͷ͕ͩ, ͜ͷؔ਺ ͸΋͸΍௒زԿͰ΋ͳ͚Ε͹ q-௒زԿͰ΋ͳ͍ͷͰ௒زԿ, q-௒زԿͷಓ۩͕શ ͘࢖͑ͳ͍. ௒زԿؔ਺ := ެൺ͕ n ͷ༗ཧؔ਺ q-௒زԿؔ਺ := ެൺ͕ qn ͷ༗ཧؔ਺ Theorem (௒زԿ͸ඍ෼ͯ͠΋௒زԿͷ··) d dz exp(z) = exp(z), d dz sin(z) = cos(z), z d dz dJν (z) dz − νJν (z) = −zJν+1(z), d dz (2F1(a, b; c; z)) = ab c 2F1(a + 1, b + 1; c + 1; z). ࣌߂఩࣏, ޻ֶʹ͓͚Δಛघؔ਺, ڞཱग़൛, 2006. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 15 / 27
16. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ d dz (r ϕs (q, z)) = ∞ ∑ n=0 r ∏ i=1 (αi ; q)n [ (−1)nq(n 2 ) ]1+s−r nzn−1 s ∏ j=1 (βj ; q)n (q; q)n . overflow ͷճආʹ͸ asymptotics (઴ۙల։) ͕ඞཁ͕ͩ, ͜ͷؔ਺͸ੵ෼ද͕ࣔ ݱ࣌఺Ͱ෼͔Βͳ͍ͷͰ Watson ͷิ୊, Mellin ม׵, WKB ۙࣅͳͲ͕࢖͑ͳ͍. Reference Ablowitz, M. J., Fokas, A. S. (2003). Complex variables: introduction and applications. Cambridge University Press. Chapter 6: Asymptotic Evaluation of Integrals ͜͏ͳͬͯ͘Δͱ, ࢖͑Δಓ۩͕ඍੵ͙Β͍͔͠ͳ͍. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 16 / 27
17. લ൒ɿօ͞Μ͔Βͷ࣭໰ʹ౴͑·͢ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ q-ಛघؔ਺ͷඍ෼͸ͳͥ೉͍͠ͷ͔ʁ ߲ผඍ෼Մೳ͔Ͳ͏͔෼͔Βͳ͍έʔε΋͋Δ (Ͱ͖ͯ΋ܭࢉ͕໘౗). Theorem (q-Bessel ؔ਺ͷผදݱ) J(2) ν (x; q) = (x/2)ν (q; q)∞ 1 ϕ1 (−x2/4; 0; q, qν+1). J(3) ν (x; q) = xν (x2q; q)∞ (q; q)∞ 1 ϕ1 (0; x2q; q, qν+1). ڃ਺ͷத਎ʹ xn Λ࣋ͨͳ͍Α͏ͳผදݱͰ͋Δ. Koelink, H. (1993). Hansen-Lommel Orthogonality Relations for Jackson’s q-Bessel Functions. Journal of Mathematical Analysis and Applications, 175, 425-437. Daalhuis, A. (1994). Asymptotic Expansions for q-Gamma, q-Exponential, and q-Bessel functions. Journal of Mathematical Analysis and Applications, 186, 896-913. Theorem (ඍ෼ੵ෼ͷ෮श, ߲ผඍ෼) ∑ an zn ͕ऩଋ͢ΔͳΒ d dz ∑ an zn = ∑ an d dz zn ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 17 / 27
18. ޙ൒ɿօ͞Μʹ͓ฉ͖͠·͢ େҬ࠷దԽ ޙ൒ɿօ͞Μʹ͓ฉ͖͠·͢ ͜͜·Ͱ, લճʹօ͞Μ͔Β͍͍࣭ͨͩͨ໰ʹ౴͑ͨͷͰ͔͜͜Β͸ٯʹօ͞Μ ʹ͓ฉ͖͠·͢. େҬ࠷దԽ Derivative-free optimization (DFO) ૊Έ߹Θͤ࠷దԽ ࠷ۙ, ࠷దԽؔ࿈χϡʔε͕ଟ͍ؾ͕͢Δ͕, ࠷దԽ͸ੈքతʹྲྀߦ͍ͬͯΔͷ ͩΖ͏͔ ? (࠷దԽ໰୊Λਫ਼౓อূ͢Δधཁ͕ߴ·ͬͯΔ ?) ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 18 / 27
19. ޙ൒ɿօ͞Μʹ͓ฉ͖͠·͢ େҬ࠷దԽ େҬ࠷దԽχϡʔε ࠷ۙ͜Μͳχϡʔε͕͋Γ·ͨ͠ (2019 ೥ 7 ݄) 19. ϕτφϜΛ୅ද͢Δ਺ֶऀͰɺԠ༻਺ֶʹ͓͚ΔʮେҬత࠷దԽ (Global Optimization)ʯ෼໺ͷ෕ͱ͞ΕɺInstitutes of Development Studies (IDS) ͷ૑ઃऀͰ΋͋ΔϗΞϯɾτΡΠ (Hoang Tuy) ത͕࢜ 7 ݄ 14 ೔ɺϋϊΠࢢͰࢮڈͨ͠ɻ92 ࡀͩͬͨɻ ʢதུʣ ݚڀʹ·͍ਐ͠ɺ100 ຊҎ্ͷ࿦จ͕ࠃࡍతʹ༗໊ͳࡶࢽʹܝࡌ͞Ε ͨɻϥΠφʔɾϗʔετ (Reiner Horst) ത࢜ͱڞಉࣥචͨ͠େҬత࠷ద Խʹؔ͢Δॻ੶ 20 ͸ɺେҬత࠷దԽ෼໺ͷʮ੟ॻʯͱ͞Ε͍ͯΔɻ 1980ʙ1990 ೥ʹ͸਺ֶݚڀॴͷॴ௕Λ຿ΊɺIDS Λ૑ઃɻಉݚڀॴ͸ 1994 ೥ʹୈࡾੈքՊֶΞΧσϛʔ͔Βൃల్্ࠃͷ༏Εͨηϯλʔͱ ͯ͠ೝΊΒΕͨɻτΡΠത࢜͸Պֶ΁ͷଟେͳޭ࿑ʹΑΓɺ1996 ೥ʹ ϕτφϜ੓෎͔ΒՊֶٕज़෼໺ͷϗʔνϛϯ৆Λत༩͞Εͨɻ େҬ࠷దԽͷਫ਼౓อূͬͯͲΕ͙Β͍ਐΜͰΔΜͰ͔͢ʁ → େੴઌੜͷຊʹ͸ INTLAB Ͱղ͍ͨྫ͕Ұ͚ͭͩࡌ͍ͬͯΔ (304 ϖʔδ). 19https://www.viet-jo.com/news/social/190716165705.html 20Horst, R., & Tuy, H. (2013). Global optimization: Deterministic approaches. Springer Science & Business Media. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 19 / 27
20. ޙ൒ɿօ͞Μʹ͓ฉ͖͠·͢ Derivative-free optimization (DFO) DFO χϡʔε ͜Ε·Ͱઆ໌͍ͯ͠ΔΑ͏ʹ, ࢲ͸ derivative-free algorithm ͷݚڀΛࢤ͍ͯ͠Δ ͷ͕ͩ, ࠷దԽͷ෼໺Ͱ΋͜ͷΑ͏ͳಈ͖͕͋Δ͜ͱΛ࠷ۙ஌ͬͨ. ઌि, NAG (Numerical Algorithms Group) ͔Β͜Μͳൃද͕͋ͬͨ (2019 ೥ 7 ݄ 26 ೔) A new set of Derivative-free Optimization (DFO) solvers are now available in the latest NAG Library. The DFO solvers for general nonlinear objective with bound constraints and for least squares (data fitting, calibration) problems with bound constraints, are available with both direct and reverse communication interfaces. These solvers should show an improved convergence rate compared to the existing DFO solutions in the NAG Library. They also have features designed to specifically handle noisy or expensive problems. DFO ͷਫ਼౓อূͬͯͲΕ͙Β͍ਐΜͰΔΜͰ͔͢ʁ → େੴઌੜͷຊʹ͸هࡌͳ͠. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 20 / 27
21. ޙ൒ɿօ͞Μʹ͓ฉ͖͠·͢ ૊Έ߹Θͤ࠷దԽ ૊Έ߹Θͤ࠷దԽχϡʔε 2019 ೥ 7/30 ͷهࣄ 21 ࠓɺྔࢠίϯϐϡʔλͷҰछͰ͋ΔʮྔࢠΞχʔϦϯάϚγϯʯͰߴ଎ ʹղ͚Δͱ͞ΕΔʮ૊߹ͤ࠷దԽ໰୊ʯΛΑΓ଎͘ɾେن໛ʹղ͘΂ ͘ɺ֤͕ࣾ͠ͷ͗Λ࡟͍ͬͯΔɻ(தུ) ֤͕ࣾ૊߹ͤ࠷దԽܭࢉʹऔΓ૊Ήͷ͸ɺ͜ΕΛߴ଎ʹղ͚Δͱަ௨ ौ଺ͷղফ΍ۚ༥ϙʔτϑΥϦΦͷ࠷దԽͳͲɺࣾձ໰୊ͷղܾ΍Ϗ δωε΁Ԡ༻͕ݟࠐΊΔ͔Βͩɻ 8/2 ͷهࣄ (౦ࣳͷ૊Έ߹Θͤ࠷దԽ࠷଎ΞϧΰϦζϜ, Ϋϥ΢υͰҰൠެ։)22 ౦ࣳ͸͜ͷ΄Ͳɺ૊Έ߹Θͤ࠷దԽܭࢉʹಛԽͨ͠طଘͷྔࢠίϯ ϐϡʔλΑΓ΋ߴ଎ɾେن໛ʹ໰୊Λղ͚ΔʮγϛϡϨʔςου෼ذΞ ϧΰϦζϜʯΛ࣮૷ͨ͠ϚγϯΛΫϥ΢υ্ʹެ։ͨ͠ɻAmazon Web Services ্ͷԾ૝αʔόར༻ྉۚʢ1 ࣌ؒ໿ 3 υϧʣͷΈͰར༻Ͱ͖Δɻ ૊Έ߹Θͤ࠷దԽͷਫ਼౓อূͬͯͲΕ͙Β͍ਐΜͰΔΜͰ͔͢ʁ → େੴઌੜͷຊʹ͸هࡌͳ͠. 21https://www.itmedia.co.jp/news/articles/1907/30/news030.html 22https://www.itmedia.co.jp/news/articles/1908/02/news104.html ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 21 / 27
22. ͓·͚ɿJSIAM ೥ձ 2019 ஫໨ߨԋ (ಠஅ) ͓·͚ɿJSIAM ೥ձ 2019 ஫໨ߨԋ (ಠஅ) ߴ҆ઌੜ͕ (NLS ͷݚڀʹଓ͍ͯ) ·ͨͯ͠΋Մੵ෼ܥʹҾ͖ͣΓࠐ·Εͨʂ ʂ Gauss ͷ௒زԿඍ෼ํఔࣜͷϞϊυϩϛʔߦྻʹର͢Δਫ਼౓อূ෇͖਺஋ܭࢉ ʢߴ҆ઌੜ et. al., 9 ݄ 4 ೔ɿ09:00-10:20ɿC (K213), ܭࢉͷ඼࣭ (1)ʣ Gauss ͷ௒زԿඍ෼ํఔࣜʹରͯ͠, ղͷଟՁੑΛදݱ͢ΔϞϊυϩ ϛʔߦྻͷ஋Λಘͨ. ਫ਼౓อূ෇͖਺஋ܭࢉΛ༻͍ͯ͋Δಛҟ఺पΓͷ ดܦ࿏ʹԊͬͨղੳ઀ଓΛߦ͏͜ͱͰ, ಛҟ఺पΓͷϞϊυϩϛʔߦྻ Λݫີʹแؚ͢Δ͜ͱ͕Ͱ͖ͨ. Gauss ͷ௒زԿඍ෼ํఔࣜͷ৔߹͸Ϟ ϊυϩϛʔߦྻΛಘΔཅతͳެ͕ࣜ͋Γ, ͜ͷ਺஋݁Ռ͕ਅͷ஋ΛؚΉ ͜ͱΛ֬ೝͰ͖Δ. ϞϊυϩϛʔߦྻɿՄੵ෼ܥݚڀऀͨͪͷେ޷෺ → ͜Ε͕෼͔Ε͹ ODE ͷղ͕ղੳ઀ଓʹΑΓͲ͏มΘΔ͔Λ׬શʹ೺ѲͰ͖Δ. ൵ใ Ԡ༻Մੵ෼ܥηογϣϯʹ q ؔ࿈ͷߨԋͳ͠ ਆอಓ෉. (2003). ෳૉؔ਺ೖ໳. ؠ೾ॻళ. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 22 / 27
23. ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ͜Ε·ͰͷθϛͰ͍ΖΜͳ࿩Λ͍ͯ͠·͕͢, ௒زԿɾPainleve eq. ΛڭΘͬͨؾ ʹͳΒͳ͍ͰԼ͍͞. (௒زԿɾPainleve ํఔࣜͷୀԽ͸গ͚ͩ͠આ໌͠·ͨ͠.) Painleve eq. ΛڭΘΔ ⇐⇒ Okamoto ॳظ஋ۭؒΛཧղͰ͖Δ. Young ਤܗ, Dynkin ਤܗΛ࢖͍͜ͳͤΔ. ϞϊυϩϛʔอଘมܗΛ࢖͍͜ͳͤΔ. Fuchs ܕ ODE ͷҰൠ࿦Λ஌Δ. ௒زԿ eq. ΛڭΘΔ ⇐⇒ GKZ, Aomoto-Gelfand ΛཧղͰ͖Δ. Grassman ଟ༷ମΛ࢖͍͜ͳͤΔ. Ԭຊ࿨෉. (2009). ύϯϧϰΣํఔࣜ. ؠ೾ॻళ. ໺ւਖ਼ढ़. (2000). ύϯϧϰΣํఔࣜ-ରশੑ͔Βͷೖ໳. ே૔ॻళ. Aomoto, K., Kita, M., et. al. (2011). Theory of hypergeometric functions. Tokyo: Springer. ໦ଜ߂৴ (2007): ௒زԿؔ਺ೖ໳-ಛघؔ਺΁ͷ౷Ұతࢹ఺͔ΒͷΞϓϩʔν, αΠΤϯεࣾ. ݪԬتॏ. (2002). ௒زԿؔ਺. ே૔ॻళ. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 23 / 27
24. ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ͜Ε·ͰͷθϛͰ͍ΖΜͳ࿩Λͯ͠·͕͢, Մੵ෼ܥΛڭΘͬͨؾʹͳΒͳ͍Ͱ Լ͍͞. (཭ࢄՄੵ෼ܥͱ਺஋ܭࢉͷؔ܎ʹ͍ͭͯ͸গ͚ͩ͠આ໌͠·ͨ͠.) Մੵ෼ܥΛڭΘΔ ⇐⇒ ޿ాͷ௚઀๏, ٯࢄཚ๏, Backlund ม׵, Darboux ม׵ Ͱ KdV, mKdV, KP, NLS, DS ౳͕ղ͚ΔΑ͏ʹͳΔ. ӃϨϕϧͷෳૉؔ਺࿦, ପԁؔ਺࿦ΛϚελʔ͢Δ. ߦྻࣜͷ߃౳ࣜΛಋग़, ҉هͰ͖ΔΑ͏ʹ͢Δ. ௒཭ࢄԽ, ٯ௒཭ࢄԽ͕ख଍ͷΑ͏ʹ࢖͍͜ͳͤΔ. Lax ܗࣜ, τ ؔ਺͕ख଍ͷΑ͏ʹ࢖͍͜ͳͤΔ. soliton ղ, cusp ղ, breather ղͷҧ͍͕આ໌Ͱ͖Δ. ୅਺తΤϯτϩϐʔ͕ܭࢉͰ͖Δ. ޿ాྑޗ. (1992). ௚઀๏ʹΑΔιϦτϯͷ਺ཧ. ؠ೾ॻళ. Ablowitz, M. J., & Segur, H. (1981). Solitons and the inverse scattering transform. SIAM. കଜߒ. (2000). ପԁؔ਺࿦. ౦ژେֶग़൛ձ. Vein, R., & Dale, P. (2006). Determinants and their applications in mathematical physics. Springer Science & Business Media. ޿ాྑޗ, & ߴڮେี. (2003). ࠩ෼ͱ௒཭ࢄ. ڞཱग़൛. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 24 / 27
25. ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ Մੵ෼ܥ͸ݫີͳఆ͕ٛͳ͍ (Մੵ෼ܥΛఆٛ͢Δ͜ͱ͕໨ඪͱͳ͍ͬͯΔ)23. ू߹࿦͕ू߹Λఆٛ͠Α͏ͱ͍ͯ͠ΔͷͱҰॹͰ͢. ͦͷͨΊ, ਓʹΑͬͯղऍ ͕ҧ͏͜ͱʹཹҙ͢Δඞཁ͕͋Γ·͢. ղऍ͕ҟͳΔྫ ࢲͷղऍ: q-ྨࣅ, q-ղੳֶ ∈ Մੵ෼ܥ ∩ ੔਺࿦ Α͋͘Δղऍ: q-ྨࣅ, q-ղੳֶ ∈ ਺ֶ \ Մੵ෼ܥ ྫ͑͹౦େ਺ཧͩͱՄੵ෼ܥͷڭһ͕ 5 ਓ͍Δ (࣌߂, Willox, தా, ੢੒, ࠃ৔). Αֶͬͯੜͨͪ͸ڭһ͝ͱʹͲ͏ఆ͍ٛͯ͠Δ͔ݟۃΊΔඞཁ͕͋Δ. (RIMS Ͱ͸) Մੵ෼ܥͷཧ࿦ΛΩʔϫʔυʹ, ৗඍ෼ํఔࣜ࿦, ֬཰࿦, ૊Έ߹Θͤ࿦, ࣮ղੳ, Ԡ༻਺ཧ, ୅਺زԿֶ, ඍ෼زԿֶ, ਺࿦ͳͲ෯ ޿͍෼໺ͷݚڀऀͷ৘ใަ׵Λ໨తͱ͍ͯ͠·͢ 24. 23https://mathoverflow.net/questions/6379/what-is-an-integrable-system 24https://sites.google.com/math.kindai.ac.jp/rims2019integrablesystems/ ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 25 / 27
26. ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ҰํͰ, ࠓ೥ͷ೔தՄੵ෼ܥूձ͸͜ͷΑ͏ͳϥΠϯφοϓʹͳ͍ͬͯΔ. The main themes of this workshop are devoted to the reviews of the research topics by both the Chinese and Japanese communities in integrable systems, which include (but not limited to)25: Algebra and combinatorics of integrable systems Backlund and Darboux transformations Discrete and ultradiscrete integrable systems Geometry of integrable systems Hamiltonian systems Hirota bilinear method and τ-functions Numerical algorithms and numerical computations Painlev´ e systems Solitons and applications Special functions and orthogonal polynomials Symmetry and supersymmetry of integrable systems ͜ͷΑ͏ʹ, Մੵ෼ܥ͸ਓʹΑͬͯࢦ͢ൣғ͕ҟͳΔͷͰ͋Δ. 25http://www.f.waseda.jp/kmaruno/cjjwis2019.html ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 26 / 27
27. ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ࠷ޙʹ: ࢲ͔Βͷ͓ئ͍ ʮθϛʯͱ͍͏ݴ༿͸෼໺ʹΑͬͯҙຯ͕ҧ͏ͷͰ஫ҙ͕ඞཁͰ͢. ਺஋ղੳͳͲͷ޻ֶܥ: ݚڀଧͪ߹Θͤ Մੵ෼ܥ: Տ౦ઌੜํࣜ 26 (౎਺ํࣜ 27) Ͱ GTM, GSM28 ౳Λྠಡ͢Δ. → Տ౦ઌੜ͕ 1996 ೥ʹηϛφʔͷ΍ΓํΛఏএͯ͠Ҏདྷ, શࠃͷ਺ֶՊɾ ७ਮ਺ֶͷ͋ΒΏΔ෼໺ʹ޿·ͬͨ. Տ౦ହ೭ઌੜ (ˏ౦େ਺ཧ) ઐ໳͸࡞༻ૉ؀࿦ (தུ) , ͞Βʹ͜ΕΒͱଞͷ෼໺ ʢྔࢠ܈, ڞܗ৔ཧ࿦, Մղ ֨ࢠ໛ܕͳͲʣ ͱͷؔ࿈Ͱ͢. ؔ਺ղੳత, ૊߹ͤ࿦తଆ໘ͷ૒ํ͕޷͖. → Մੵ෼ܥʹ΋ͷ͍ۙ͘͢͝. https://www.ms.u-tokyo.ac.jp/˜yasuyuki/ 26https://www.ms.u-tokyo.ac.jp/˜yasuyuki/sem.htm 27https://twitter.com/tosuu set 28Graduate Texts in Mathematics (Springer), Graduate Studies in Mathematics (AMS). ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 8 ݄, ˏദ໦ݚڀࣨ 27 / 27