MATH-PH / Math-Ph

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Confirmation of Lagrange Hypothesis for Twisted Elastic Rod

Confirmation of Lagrange Hypothesis for Twisted Elastic Rod

: ๋ณธ ๋…ผ๋ฌธ์€ ๊ตฌ์กฐ ์ตœ์ ํ™” ๋ถ„์•ผ์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋Š” ๋ผ๊ทธ๋ž‘์ฃผ ๋ฌธ์ œ๋ฅผ ๋‹ค๋ฃจ๋ฉฐ, ํŠนํžˆ ๊ทธ๋ฆฐํž ๋ถ€๋Ÿฌ์ง ๋ฌธ์ œ์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ์ ‘๊ทผ๋ฒ•์„ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ์–‡์€ ํƒ„์„ฑ ๋ง‰๋Œ€์˜ ์ตœ์  ํ˜•ํƒœ๋ฅผ ์ฐพ๋Š”๋ฐ ์ดˆ์ ์„ ๋งž์ถ”๊ณ  ์žˆ์œผ๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ๋ผ๊ทธ๋ž‘์ฃผ ๊ฐ€์„ค์ด ์‹ค์ œ๋กœ ์œ ํšจํ•จ์„ ์ฆ๋ช…ํ•˜๊ณ ์ž ํ•ฉ๋‹ˆ๋‹ค. 1. ์„œ๋ก  ์„œ๋ก ์—์„œ๋Š” ๋ณธ ๋…ผ๋ฌธ์˜ ์ฃผ์š” ๋ชฉํ‘œ์™€ ์—ฐ๊ตฌ ๋ฐฐ๊ฒฝ์„ ์„ค๋ช…ํ•ฉ๋‹ˆ๋‹ค. ๊ตฌ์กฐ ์ตœ์ ํ™” ๊ณผํ•™์—์„œ ์ค‘์š”ํ•œ ๋ฌธ์ œ ์ค‘ ํ•˜๋‚˜์ธ ๊ทธ๋ฆฐํž ๋ถ€๋Ÿฌ์ง ๋ฌธ์ œ๋ฅผ ๋‹ค๋ฃจ๋ฉฐ, ์ด ๋ฌธ์ œ๋Š” ์–‡์€ ํƒ„์„ฑ ๋ง‰๋Œ€๊ฐ€ ํŠน์ • ๋ชจ๋ฉ˜ํŠธ ํ•˜์ค‘์— ๋Œ€ํ•ด ์–ด๋–ป๊ฒŒ ํœ˜์–ด์ง€๋Š”์ง€๋ฅผ ๋ถ„์„ํ•˜๋Š” ๊ฒƒ์ž…๋‹ˆ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ๋ผ๊ทธ๋ž‘์ฃผ

MATH-PH Mathematics Nonlinear Sciences
The quadratic WDVV solution $E_8(a_1)$

The quadratic WDVV solution $E_8(a_1)$

์ด ๋…ผ๋ฌธ์€ ๊ณ ์ฐจ์› ๋ฆฌ ๋Œ€์ˆ˜์™€ ํ”„๋กœ๋ฒค๋ฆฌ์šฐ์Šค ๋งŒ๋Œ(Frobenius Manifold) ์‚ฌ์ด์˜ ๊นŠ์€ ๊ด€๊ณ„๋ฅผ ํƒ๊ตฌํ•ฉ๋‹ˆ๋‹ค. ํŠนํžˆ, Eโ‚ˆ(aโ‚)๋ผ๋Š” ํŠน์ •ํ•œ ๊ฒฝ์šฐ์— ์ดˆ์ ์„ ๋งž์ถ”๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” Witten Dijkgraaf Verlinde Verlinde (WDVV) ๋ฐฉ์ •์‹์ด๋ผ๋Š” ์ค‘์š”ํ•œ ํŽธ๋ฏธ๋ถ„ ๋ฐฉ์ •์‹ ์‹œ์Šคํ…œ์˜ ํ•ด๋ฅผ ๊ตฌ์„ฑํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ํƒ๊ตฌํ•ฉ๋‹ˆ๋‹ค. ํ”„๋กœ๋ฒค๋ฆฌ์šฐ์Šค ๋งŒ๋Œ์€ ๋Œ€์ˆ˜์  ๊ตฌ์กฐ์™€ ๊ธฐํ•˜ํ•™์  ๊ตฌ์กฐ๊ฐ€ ๊ฒฐํ•ฉ๋œ ๋ณต์žกํ•œ ์ˆ˜ํ•™์  ๊ฐ์ฒด์ž…๋‹ˆ๋‹ค. ์ด ๋…ผ๋ฌธ์—์„œ๋Š” ์ด๋Ÿฌํ•œ ๊ตฌ์กฐ๊ฐ€ ์–ด๋–ป๊ฒŒ Eโ‚ˆ ๋ฆฌ ๋Œ€์ˆ˜์™€ ์—ฐ๊ฒฐ๋˜๋Š”์ง€, ๊ทธ๋ฆฌ๊ณ  ๊ทธ ์ž ์žฌ ํ•จ์ˆ˜๋Š” ์–ด๋–ค ํ˜•ํƒœ๋ฅผ ๊ฐ€์ง€๋Š”์ง€๋ฅผ ๋ถ„

MATH-PH Mathematics Nonlinear Sciences
Toda tau functions with quantum torus symmetries

Toda tau functions with quantum torus symmetries

: ๋ณธ ๋…ผ๋ฌธ์€ ์œ„์ƒ ๋ฌธ์ž์—ด์˜ ๋…น๋Š” ๊ฒฐ์ • ๋ชจ๋ธ๊ณผ 5์ฐจ์› ๊ฒŒ์ด์ง€ ์ด๋ก  ๊ฐ„์˜ ํ†ตํ•ฉ ๊ฐ€๋Šฅ์„ฑ์„ ํƒ๊ตฌํ•œ๋‹ค. ๋…น๋Š” ๊ฒฐ์ • ๋ชจ๋ธ์€ ๋ฌผ๋ฆฌํ•™์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋Š” ๋ชจ๋ธ๋กœ, ํŠนํžˆ ์œ„์ƒ ๋ฌธ์ž์—ด๊ณผ ๊ด€๋ จ๋œ ๋ฌธ์ œ๋ฅผ ๋‹ค๋ฃจ๋Š”๋ฐ ํ™œ์šฉ๋œ๋‹ค. ๋…ผ๋ฌธ์—์„œ๋Š” ์ด๋Ÿฌํ•œ ๋ชจ๋ธ์˜ ๋ถ„ํ•  ํ•จ์ˆ˜๊ฐ€ ํŠน์ • ์™ธ๋ถ€ ์ž ์žฌ๋ ฅ์— ์˜ํ•ด ๋ณ€ํ˜•๋  ๋•Œ Toda ๊ณ„์ธต์˜ ํƒ€์šฐ ํ•จ์ˆ˜์™€ ๋ณธ์งˆ์ ์œผ๋กœ ๋™์ผํ•˜๋‹ค๋Š” ๊ฒƒ์„ ๋ณด์—ฌ์ค€๋‹ค. Toda ๊ณ„์ธต์€ ํ†ต์ƒ์ ์ธ ํŽ˜๋ฅด๋ฏธ์˜จ ์‹œ์Šคํ…œ์—์„œ ํŒŒ์ƒ๋œ ์ค‘์š”ํ•œ ์ˆ˜ํ•™์  ๊ตฌ์กฐ๋กœ, ์ด๋ก  ๋ฌผ๋ฆฌํ•™๊ณผ ์ˆ˜ํ•™์—์„œ ๋„๋ฆฌ ์—ฐ๊ตฌ๋˜๊ณ  ์žˆ๋‹ค. ๋…ผ๋ฌธ์—์„œ๋Š” Toda ํƒ€์šฐ ํ•จ์ˆ˜์™€ ๋ณ€ํ˜•๋œ ๋ถ„ํ•  ํ•จ์ˆ˜ ๊ฐ„์˜ ๊ด€๊ณ„๋ฅผ ์–‘์ž ํ† 

HEP-TH Nonlinear Sciences Mathematics MATH-PH

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