Mathematics

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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
No Image

Smarandache Curves According to Bishop Frame in Euclidean 3-Space

: ๋ณธ ๋…ผ๋ฌธ์€ ๋น„์ˆ ํ”„๋ ˆ์ž„(Bishop Frame)์„ ์ด์šฉํ•˜์—ฌ ์œ ํด๋ฆฌ๋“œ 3์ฐจ์› ๊ณต๊ฐ„์—์„œ ํŠน์ • ์Šค๋งˆ๋ž€๋‹ค์ฒด ๊ณก์„ (Special Smarandache Curves)์— ๋Œ€ํ•œ ์‹ฌ๋„ ์žˆ๋Š” ๋ถ„์„์„ ์ œ๊ณตํ•ฉ๋‹ˆ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ๊ธฐ์กด์˜ ํ”„๋ ˆ๋„คํŠธ ์„ธ๋ฅด๋ ›(Frenet Serret) ํ”„๋ ˆ์ž„์ด ์ œํ•œ๋˜๋Š” ๊ฒฝ์šฐ์—๋„ ๋น„์ˆ ํ”„๋ ˆ์ž„์€ ํšจ๊ณผ์ ์œผ๋กœ ์ ์šฉ๋  ์ˆ˜ ์žˆ๋‹ค๋Š” ์ ์—์„œ ์ค‘์š”ํ•œ ์˜๋ฏธ๋ฅผ ๊ฐ€์ง‘๋‹ˆ๋‹ค. 1. ์Šค๋งˆ๋ž€๋‹ค์ฒด ๊ณก์„ ์˜ ์ •์˜์™€ ์ค‘์š”์„ฑ ์Šค๋งˆ๋ž€๋‹ค์ฒด ๊ณก์„ ์€ ๋ฏธ๋ถ„ ๊ธฐํ•˜ํ•™์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋Š” ํŠน์ˆ˜ํ•œ ๊ณก์„ ์ž…๋‹ˆ๋‹ค. ์ด ๋…ผ๋ฌธ์—์„œ๋Š” T N 1, T N 2, N 1 N 2, ๊ทธ๋ฆฌ๊ณ  T

Mathematics
A Brief Review of SIAM Review

A Brief Review of SIAM Review

๋ณธ ๋…ผ๋ฌธ์€ SIAM ๋ฆฌ๋ทฐ ์ €๋„์˜ ๋ณ€ํ™”์™€ ๊ทธ ์˜ํ–ฅ๋ ฅ์„ ๋ถ„์„ํ•˜๊ณ , ํ–ฅํ›„ ๊ฐœ์„  ๋ฐฉ์•ˆ์„ ์ œ์‹œํ•œ๋‹ค. 1999๋…„ ์žฌ์กฐ์ง ์ดํ›„, SIAM ๋ฆฌ๋ทฐ ๋Š” ์—ฌ๋Ÿฌ ์„น์…˜์œผ๋กœ ๊ตฌ๋ถ„๋˜๋ฉฐ, ์ด๋กœ ์ธํ•ด ์ €๋„์˜ ๊ตฌ์„ฑ์ด ํฌ๊ฒŒ ๋ฐ”๋€Œ์—ˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ด๋Ÿฌํ•œ ๋ณ€ํ™”๊ฐ€ ์ €๋„์˜ ์˜ํ–ฅ๋ ฅ์„ ํ–ฅ์ƒ์‹œํ‚ค์ง€ ๋ชปํ•œ ๊ฒƒ์œผ๋กœ ๋‚˜ํƒ€๋‚ฌ๋‹ค. 1. SIAM ๋ฆฌ๋ทฐ์˜ ๊ฐœ์š”์™€ ๋ณ€ํ™” SIAM ๋ฆฌ๋ทฐ๋Š” SIAM์—์„œ ๋ฐœํ–‰ํ•˜๋Š” 12๊ฐœ ์ด์ƒ์˜ ํ•™์ˆ  ์ €๋„ ์ค‘ ํ•˜๋‚˜๋กœ, ๋ชจ๋“  ํšŒ์›์—๊ฒŒ ๋ฐฐํฌ๋˜๋Š” ์„ ๋„์ ์ธ ์ €๋„์ด๋‹ค. 1999๋…„์— ์ด๋ฃจ์–ด์ง„ ์ฃผ์š” ๋ณ€๊ฒฝ ์‚ฌํ•ญ์€ ์ €๋„์˜ ํ”„๋กœํ•„์„ ๋†’์ด๊ธฐ ์œ„ํ•œ ๊ฒƒ์ด์—ˆ๋‹ค. ์ด ๋ณ€ํ™”๋Š” ์ปฌ๋Ÿฌ ์ธ์‡„ ๋„์ž…๊ณผ ํ•จ๊ป˜

Mathematics
Computation of copulas by Fourier methods

Computation of copulas by Fourier methods

์ด ๋…ผ๋ฌธ์€ ๋‹ค์ฐจ์› ๋žœ๋ค ๋ณ€์ˆ˜์˜ ์•”์‹œ์  ์ปคํ”Œ๋ผ ํ‘œํ˜„์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ์ ‘๊ทผ๋ฒ•์„ ์ œ์•ˆํ•˜๋ฉฐ, ํŠนํžˆ ํ‘ธ๋ฆฌ์— ๋ฐฉ๋ฒ•์„ ํ™œ์šฉํ•œ ๊ณ„์‚ฐ ๊ธฐ๋ฒ•์„ ์†Œ๊ฐœํ•œ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ๋ฌด์ž‘์œ„ ๋ณ€์ˆ˜ ๊ฐ„์˜ ์˜์กด์„ฑ ๊ตฌ์กฐ๋ฅผ ํšจ๊ณผ์ ์œผ๋กœ ๋ฌ˜์‚ฌํ•˜๋Š” ๋ฐ ์ค‘์ ์„ ๋‘๊ณ  ์žˆ์œผ๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ๋™์  ๊ณผ์ •์—์„œ ๋ฐœ์ƒํ•˜๋Š” ๋ณต์žกํ•œ ์ƒํ˜ธ ์ž‘์šฉ์„ ์ดํ•ดํ•˜๊ณ  ๋ถ„์„ํ•  ์ˆ˜ ์žˆ๋Š” ๋„๊ตฌ๋ฅผ ์ œ๊ณตํ•œ๋‹ค. 1. ์ปคํ”Œ๋ผ์™€ ์˜์กด์„ฑ ๊ตฌ์กฐ ์ปคํ”Œ๋ผ๋Š” ๋ฌด์ž‘์œ„ ๋ณ€์ˆ˜ ๊ฐ„์˜ ์˜์กด์„ฑ์„ ์™„๋ฒฝํ•˜๊ฒŒ ๋ฌ˜์‚ฌํ•˜๋ฉฐ, ์Šคํด๋ผ๋ฅด์˜ ์ •๋ฆฌ๋ฅผ ํ†ตํ•ด ๊ณต๋™ ๋ถ„ํฌ์™€ ์ด๋ถ„ ๋ถ„ํฌ ์‚ฌ์ด์˜ ๊ด€๊ณ„๋ฅผ ์šฐ์•„ํ•˜๊ฒŒ ์—ฐ๊ฒฐํ•œ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ปคํ”Œ๋ผ๋Š” ๋™์  ๊ณผ์ •๊ณผ ์ž˜ ์–ด์šธ๋ฆฌ์ง€ ์•Š์œผ๋ฉฐ, ์˜ˆ๋ฅผ

Mathematics Quantitative Finance
A Majorization Order on Monomials and Termination of a Successive   Difference Substitution Algorithm

A Majorization Order on Monomials and Termination of a Successive Difference Substitution Algorithm

๋ณธ ๋…ผ๋ฌธ์€ ์„ฑ๊ณต์ ์ธ ์ฐจ๋ถ„ ๋Œ€์ฒด ์•Œ๊ณ ๋ฆฌ์ฆ˜(KSDS)์˜ ์ข…๊ฒฐ์„ฑ์— ๋Œ€ํ•œ ์‹ฌ๋„ ์žˆ๋Š” ์—ฐ๊ตฌ๋ฅผ ์ œ๊ณตํ•˜๋ฉฐ, ํŠนํžˆ ์ฃผ์š”ํ™” ์ˆœ์„œ๋ฅผ ์ด์šฉํ•˜์—ฌ ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด ์–ธ์ œ ์ข…๋ฃŒ๋˜๋Š”์ง€ ๋ถ„์„ํ•œ๋‹ค. KSDS๋Š” ๋‹คํ•ญ์‹ ํ•จ์ˆ˜์—์„œ ํŠน์ • ์กฐ๊ฑด์„ ๋งŒ์กฑํ•˜๋Š” ๊ฒฝ์šฐ ์–‘์ ์œผ๋กœ ์ข…๋ฃŒ๋˜๋ฉฐ, ์ด๋Ÿฌํ•œ ์กฐ๊ฑด์€ ์ฃผ๋กœ ๋‹จํ•ญ์‹์˜ ๊ณ„์ˆ˜์™€ ๊ทธ๋“ค์˜ ๊ด€๊ณ„์— ๊ธฐ๋ฐ˜ํ•œ๋‹ค. 1. KSDS์˜ ์ •์˜ ๋ฐ ๋ฐฐ๊ฒฝ KSDS๋Š” ์ž…๋ ฅ ํ•จ์ˆ˜ f๋ฅผ ๋‹คํ•ญ์‹ ํ˜•ํƒœ๋กœ ํ‘œํ˜„ํ•˜๊ณ , ์ด ๋‹คํ•ญ์‹์—์„œ ํŠน์ •ํ•œ ๋Œ€์ฒด ๊ทœ์น™์„ ์ ์šฉํ•˜์—ฌ ์ƒˆ๋กœ์šด ๋‹คํ•ญ์‹์„ ์ƒ์„ฑํ•˜๋Š” ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด๋‹ค. ๋…ผ๋ฌธ์—์„œ๋Š” ๋‹จํ•ญ์‹์ด ์–‘์ˆ˜ ๋˜๋Š” ์Œ์ˆ˜ ๊ณ„์ˆ˜๋ฅผ ๊ฐ€์งˆ ๋•Œ์˜ ํŠน์„ฑ์„ ์ •์˜ํ•œ๋‹ค. ํŠน

Symbolic Computation Mathematics Computer Science
Symmetries of the Black-Scholes equation

Symmetries of the Black-Scholes equation

: ๋ณธ ๋…ผ๋ฌธ์€ ๋ธ”๋ž™ ์‡ผ์Šค ๋ฐฉ์ •์‹์˜ ๋Œ€์ˆ˜๊ธฐํ•˜ํ•™์  ๊ตฌ์กฐ๋ฅผ ํƒ๊ตฌํ•˜๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ๊ธˆ์œต์ˆ˜ํ•™์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋Š” ๋ฏธ๋ถ„๋ฐฉ์ •์‹์˜ ๋Œ€์นญ์„ฑ์„ ๋ถ„์„ํ•˜๊ณ ์ž ํ•ฉ๋‹ˆ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ๊ณ ์ „ ๋ฌผ๋ฆฌํ•™๊ณผ ์œ ํด๋ฆฌ๋“œ ์–‘์ž์—ญํ•™์—์„œ ์ด๋ฏธ ์ž…์ฆ๋œ ๋ฐฉ๋ฒ•๋ก ์„ ๊ธˆ์œต์ˆ˜ํ•™์— ์ ์šฉํ•˜๋ ค๋Š” ์‹œ๋„๋กœ, ๋ธ”๋ž™ ์‡ผ์Šค ๋ฐฉ์ •์‹์˜ ๊ตฌ์กฐ๋ฅผ ๋” ๊นŠ์ด ์ดํ•ดํ•˜๊ณ ์ž ํ•ฉ๋‹ˆ๋‹ค. ๋…ผ๋ฌธ์€ ๋จผ์ € ๋ธ”๋ž™ ์‡ผ์Šค ๋ฐฉ์ •์‹์˜ ์ผ๋ฐ˜์ ์ธ ํ‹€์„ ์„ค์ •ํ•œ ํ›„, ์ด๋ถ„์ˆ˜๋ฅผ ๊ฒฐ์ •ํ•˜๊ธฐ ์œ„ํ•ด ์—ด๋ฐฉ์ •์‹๊ณผ ์ž ์žฌํ•ญ ํ•ญ๋ชฉ์— ๋Œ€ํ•œ ์—ญ์—ด๋ฐฉ์ •์‹์˜ ๋ฐฉ๋ฒ•๋ก ์„ ์ ์šฉํ•ฉ๋‹ˆ๋‹ค. ์ด๋ฅผ ํ†ตํ•ด ๋ธ”๋ž™ ์‡ผ์Šค ๋ฐฉ์ •์‹์˜ ์›๋ž˜ ํ•ด๊ฒฐ ๋ฐฉ๋ฒ•์ด ์ž์—ฐ์Šค๋Ÿฝ๊ฒŒ ๋‚˜ํƒ€๋‚˜๋ฉฐ, ํŠนํžˆ r ฯƒยฒโ‚‚์™€

Quantitative Finance Mathematics
Nordhaus-Gaddum for Treewidth

Nordhaus-Gaddum for Treewidth

: ๋ณธ ์—ฐ๊ตฌ์—์„œ๋Š” ๋…ธ๋“œ ๊ฐ€๋“œ๋‹ด ์œ ํ˜• ์ •๋ฆฌ๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ํŠธ๋ฆฌ ๋„ˆ๋น„(treewidth)์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ๊ด€์ ์„ ์ œ์‹œํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ์ด๋Š” ํŠนํžˆ ๊ทธ๋ž˜ํ”„ ์ด๋ก ์—์„œ ์ค‘์š”ํ•œ ๋งค๊ฐœ๋ณ€์ˆ˜๋กœ, ๊ทธ๋ž˜ํ”„์˜ ๋ณต์žก์„ฑ์„ ์ธก์ •ํ•˜๋Š” ๋ฐ ํ™œ์šฉ๋ฉ๋‹ˆ๋‹ค. ํŠธ๋ฆฌ ๋„ˆ๋น„์™€ ๊ทธ ์˜๋ฏธ: ํŠธ๋ฆฌ ๋„ˆ๋น„๋Š” ๊ทธ๋ž˜ํ”„ G๋ฅผ ๋ถ€๋ถ„ ๊ทธ๋ž˜ํ”„๋กœ ํฌํ•จํ•  ์ˆ˜ ์žˆ๋Š” k ํŠธ๋ฆฌ(k tree)์˜ ์ตœ์†Œ ์ •์ˆ˜ k๋ฅผ ๋‚˜ํƒ€๋ƒ…๋‹ˆ๋‹ค. ์ด๋Š” ํŠธ๋ฆฌ ๋ถ„ํ•ด(tree decomposition)๋ฅผ ํ†ตํ•ด ์ •์˜๋˜๋ฉฐ, ๊ทธ๋ž˜ํ”„์˜ ๋ณต์žก์„ฑ์„ ์ธก์ •ํ•˜๋Š” ๋ฐ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•ฉ๋‹ˆ๋‹ค. ๋…ธ๋“œ ๊ฐ€๋“œ๋‹ด ์œ ํ˜• ์ •๋ฆฌ์™€ ํŠธ๋ฆฌ ๋„ˆ๋น„: ๋ณธ ๋…ผ๋ฌธ์—์„œ ์ œ์‹œ๋œ ๋…ธ๋“œ ๊ฐ€๋“œ๋‹ด

Mathematics Computer Science Discrete Mathematics
No-go theorems for functorial localic spectra of noncommutative rings

No-go theorems for functorial localic spectra of noncommutative rings

๋ณธ ๋…ผ๋ฌธ์€ ๋น„์ปค๋ฎค๋‹ˆํ‹ฐ๋ธŒ C ๋Œ€์ˆ˜์— ๋Œ€ํ•œ ๊ฒ”ํŒ๋“œ ์ŠคํŽ™ํŠธ๋Ÿผ์˜ ์ผ๋ฐ˜ํ™”๋ฅผ ์‹œ๋„ํ•˜๋Š” ๋ฐ ์žˆ์–ด ์ค‘์š”ํ•œ ์ œํ•œ์  ๊ฒฐ๊ณผ๋ฅผ ์ œ๊ณตํ•œ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ํŠนํžˆ ํ–‰๋ ฌ ๋Œ€์ˆ˜ M n(C) (n โ‰ฅ 3)์—์„œ ์ž๋ช…ํ•œ ๋กœ์ปฌ์„ ์ƒ์„ฑํ•œ๋‹ค๋Š” ์ ์—์„œ, ์ „ํ†ต์ ์ธ ํ† ํฌ์Šค ๊ฐœ๋…์ด ๋น„์ปค๋ฎค๋‹ˆํ‹ฐ๋ธŒ C ๋Œ€์ˆ˜์— ๋Œ€ํ•œ ์ข‹์€ ์ŠคํŽ™ํŠธ๋Ÿผ ๊ฐœ๋…์„ ํฌ์šฉํ•˜๋Š” ๋ฐ ๋ถ€์ ์ ˆํ•จ์„ ๊ฐ•๋ ฅํ•˜๊ฒŒ ๋ณด์—ฌ์ค€๋‹ค. 1. ๋ ˆ์˜ˆ์Šค ์ •๋ฆฌ์˜ ํ™•์žฅ ๋ ˆ์˜ˆ์Šค์˜ ๋…ผ๋ฌธ์€ ๋ชจ๋“  ์ปค๋ฎคํ‹ฐ๋ธŒ C ๋Œ€์ˆ˜์— ๋Œ€ํ•ด ๊ฒ”ํŒ๋“œ ์ŠคํŽ™ํŠธ๋Ÿผ์„ ํ• ๋‹นํ•˜๋Š” ํŽ‘ํ„ฐ๊ฐ€ ํ–‰๋ ฌ ๋Œ€์ˆ˜ M n(C) (n โ‰ฅ 3)์—์„œ ์ž๋ช…ํ•œ ๊ฒฐ๊ณผ๋ฅผ ์–ป๋Š”๋‹ค๋Š” ๊ฒƒ์„ ๋ณด์—ฌ์ค€๋‹ค. ๋ณธ ๋…ผ๋ฌธ์€ ์ด ์ •๋ฆฌ๋ฅผ ํ™•์žฅํ•˜

Quantum Physics Mathematics
A well-balanced finite volume scheme for 1D hemodynamic simulations

A well-balanced finite volume scheme for 1D hemodynamic simulations

๋ณธ ๋…ผ๋ฌธ์€ ํ˜ˆ๋ฅ˜ ํ๋ฆ„์˜ 1์ฐจ์› ๋ชจ๋ธ ๋ฐ ์ˆ˜์น˜ ์‹œ๋ฎฌ๋ ˆ์ด์…˜์— ์ดˆ์ ์„ ๋งž์ถ”๊ณ  ์žˆ๋‹ค. ํŠนํžˆ, ๋น„์ผ์ • ํƒ„์„ฑ์„ฑ์„ ๊ฐ€์ง„ ๋™๋งฅ์—์„œ์˜ ํ˜ˆ๋ฅ˜๋ฅผ ์ •ํ™•ํ•˜๊ฒŒ ๋ชจ๋ธ๋งํ•˜๊ณ  ์‹œ๋ฎฌ๋ ˆ์ด์…˜ํ•˜๊ธฐ ์œ„ํ•œ ๊ท ํ˜• ์žกํžŒ ์œ ํ•œ ์ฒด์  ์Šคํ‚ด์„ ์ œ์•ˆํ•œ๋‹ค. 1. ๋ณด์กดํ˜• ์งˆ๋Ÿ‰ ๋ฐ ์šด๋™๋Ÿ‰ ๋ฐฉ์ •์‹ ๋…ผ๋ฌธ์€ ๋น„์ผ์ • ํƒ„์„ฑ์„ฑ์„ ๊ฐ€์ง„ ๋™๋งฅ์—์„œ์˜ ํ˜ˆ๋ฅ˜ ํ๋ฆ„์„ ๋ชจ๋ธ๋งํ•˜๊ธฐ ์œ„ํ•ด ๋ณด์กดํ˜• ์งˆ๋Ÿ‰ ๋ฐ ์šด๋™๋Ÿ‰ ๋ฐฉ์ •์‹์„ ๊ณ ๋ คํ•˜๊ณ  ์žˆ๋‹ค. ์ด ๋ฐฉ์ •์‹๋“ค์€ ์ผ๋ฐ˜์ ์œผ๋กœ ์‚ฌ์šฉ๋˜๋Š” ํ˜•ํƒœ์™€ ๋‹ฌ๋ฆฌ ๋ณด์ˆ˜์ ์ธ ํ˜•ํƒœ๋กœ ์žฌํ‘œํ˜„๋˜์–ด ์žˆ์œผ๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ์ •์  ์ƒํƒœ๋ฅผ ๋ณด์กดํ•˜๋Š” ๊ท ํ˜• ์žกํžŒ ์Šคํ‚ด์„ ๋„์ž…ํ•  ์ˆ˜ ์žˆ๊ฒŒ ๋œ๋‹ค. 2. ๊ท ํ˜• ์žกํžŒ ์œ ํ•œ ์ฒด์ 

Mathematics Numerical Analysis Computer Science
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The Art of Space Filling in Penrose Tilings and Fractals

1. ์—์…” ์Šคํƒ€์ผ ํƒ€์ผ๋ง์˜ ๋„์ „๊ณผ ๊ฐ€๋Šฅ์„ฑ MC ์—์…”๋Š” ๊ทธ์˜ ๋…ํŠนํ•œ ํƒ€์ผ ์•„ํŠธ๋กœ ์„ธ๊ณ„์ ์ธ ๋ช…์„ฑ์„ ์–ป์—ˆ์Šต๋‹ˆ๋‹ค. ๊ทธ์˜ ์ž‘ํ’ˆ์€ ๋‹จ์ˆœํžˆ ์˜ˆ์ˆ ์  ๊ฐ€์น˜๋ฅผ ๋„˜์–ด์„œ, ์ˆ˜ํ•™์  ๊ฐœ๋…์„ ์‹œ๊ฐ์ ์œผ๋กœ ํ‘œํ˜„ํ•˜๋Š” ๋ฐ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ–ˆ์Šต๋‹ˆ๋‹ค. ํŠนํžˆ, ์—์…”์˜ ํƒ€์ผ๋ง ์ž‘์—…์€ ์ƒ๋ฌผ ํ˜•ํƒœ์™€ ๊ธฐํ•˜ํ•™์  ํŒจํ„ด์„ ๊ฒฐํ•ฉํ•˜์—ฌ ์ƒˆ๋กœ์šด ๋ฏธ์  ๊ฒฝํ—˜์„ ์ œ๊ณตํ–ˆ์Šต๋‹ˆ๋‹ค. ์—์…” ์Šคํƒ€์ผ ํƒ€์ผ๋ง์˜ ํ•ต์‹ฌ ๋„์ „์  ์ค‘ ํ•˜๋‚˜๋Š” ์ด๋ฏธ์ง€๊ฐ€ ํƒ€์ผ ๊ฒฝ๊ณ„๋ฅผ ๋„˜์–ด์„œ ์ผ๊ด€๋˜๊ฒŒ ์—ฐ๊ฒฐ๋˜๋Š” ๊ฒƒ์ž…๋‹ˆ๋‹ค. ์ด ๊ณผ์ •์—์„œ ์—์…”๋Š” ์–‘๋Œ€์นญ์„ ํ™œ์šฉํ•ด ๊ฐ ๊ฐ€์žฅ์ž๋ฆฌ๊ฐ€ ๋ณด์™„์ ์ธ ๋ถ€๋ถ„์œผ๋กœ ๊ตฌ์„ฑ๋˜์–ด์•ผ ํ•จ์„ ์ธ์‹ํ–ˆ์Šต๋‹ˆ๋‹ค. ์ด๋Ÿฌํ•œ ์ ‘๊ทผ๋ฒ•์€ ๋‹จ์ˆœํ•œ

Physics Mathematics
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An Amendment of Fast Subspace Tracking Methods

: ๋ณธ ๋…ผ๋ฌธ์€ ์‹ ํ˜ธ ๋ฐ ์žก์Œ ํ•˜์œ„ ๊ณต๊ฐ„ ์ถ”์  ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ์•ˆ์ •์„ฑ๊ณผ ์„ฑ๋Šฅ ํ–ฅ์ƒ์„ ์œ„ํ•œ ์—ฐ๊ตฌ๋ฅผ ๋‹ค๋ฃน๋‹ˆ๋‹ค. ํŠนํžˆ, DPM(ํ™•์‚ฐ ํ”„๋กœ์„ธ์Šค ๋ชจํ˜•) ํด๋ž˜์Šค์™€ Oja ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ค‘์‹ฌ์œผ๋กœ ์ƒˆ๋กœ์šด ํ•˜์œ„ ๊ณต๊ฐ„ ๊ธฐ์ €์˜ ์—…๋ฐ์ดํŠธ ๋ฐฉ์‹์„ ๋ถ„์„ํ•˜๊ณ , ์ด๋ฅผ ํ†ตํ•ด ์ŠคํŒŒํฌ ๋ฌธ์ œ๋ฅผ ํ•ด๊ฒฐํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์•ˆํ•ฉ๋‹ˆ๋‹ค. ์„œ๋ธŒ์ŠคํŽ˜์ด์Šค ์ถ”์ ์˜ ์ค‘์š”์„ฑ ์„œ๋ธŒ์ŠคํŽ˜์ด์Šค ์ถ”์ ์€ ์„ผ์„œ ๋ฐฐ์—ด์—์„œ ๊ด€์ฐฐ๋œ ๋ฌด์ž‘์œ„ ๋ฒกํ„ฐ ์‹œํ€€์Šค๋กœ๋ถ€ํ„ฐ ๊ณต๊ฐ„ ๋˜๋Š” ๊ทธ ๊ธฐ์ €๋ฅผ ์ถ”์ •ํ•˜๋Š” ๊ณผ์ •์ž…๋‹ˆ๋‹ค. ์ด๋Š” ํ†ต์‹ , ๋ ˆ์ด๋”, ์†Œ๋‚˜, ํ•ญ๋ฒ•๊ณผ ๊ฐ™์€ ์‹ ํ˜ธ ์ฒ˜๋ฆฌ ๋ถ„์•ผ์—์„œ ๊ฐ•๋ ฅํ•œ ๋„๊ตฌ๋กœ ํ™œ์šฉ๋˜๋ฉฐ ์ ์‘ ํ•„ํ„ฐ, ๋„์ƒ ์œ„์น˜ ์ถ”์ •(DOA ์ถ”์ •

Computer Science Mathematics Neural Computing
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Banach Contraction Principle for Cyclical Mappings on Partial Metric Spaces

: ์ด ๋…ผ๋ฌธ์€ ๋ถ€๋ถ„ ๋ฏธ์  ๊ณต๊ฐ„(Partial Metric Space, PMS)์— ๋Œ€ํ•œ ๋ฐ”๋‚˜ํ ์ˆ˜์ถ• ์›๋ฆฌ(Banach Contraction Principle)๋ฅผ ํ™•์žฅํ•˜๊ณ , ์ด๋ฅผ ์ˆœํ™˜ ๋งตํ•‘(cyclical mapping)์œผ๋กœ ์ผ๋ฐ˜ํ™”ํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. ์ด๋Š” ๊ธฐ์กด์˜ ๊ณ ์ •์  ์ •๋ฆฌ๋ฅผ ๋”์šฑ ๋„“์€ ๋ฒ”์œ„๋กœ ํ™•์žฅ์‹œํ‚ค๊ณ , ๋ถ€๋ถ„ ๋ฏธ์  ๊ณต๊ฐ„์—์„œ์˜ ์ˆ˜ํ•™์  ๊ตฌ์กฐ์™€ ๊ทธ ํ™œ์šฉ์„ฑ์„ ๋” ๊นŠ์ด ์ดํ•ดํ•  ์ˆ˜ ์žˆ๋Š” ์ค‘์š”ํ•œ ์—ฐ๊ตฌ์ด๋‹ค. 1. ๋ถ€๋ถ„ ๋ฏธ์  ๊ณต๊ฐ„(PMS)์˜ ๊ฐœ๋…๊ณผ ์„ฑ์งˆ ๋…ผ๋ฌธ์—์„œ๋Š” PMS์˜ ์ •์˜์™€ ๊ธฐ๋ณธ์ ์ธ ์„ฑ์งˆ์„ ์„ค๋ช…ํ•œ๋‹ค. ํŠนํžˆ ๋Œ€์นญ์„ฑ, ์ผ์น˜์„ฑ, ์ž‘์€ ์ž๊ธฐ ๊ฑฐ๋ฆฌ,

Mathematics
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The Fiedler Rose: On the extreme points of the Fiedler vector

๋งค๋ ฅ์ ์ธ ํ•œ๊ธ€ ์ œ๋ชฉ: ํ”ผ๋”๋Ÿฌ ์žฅ๋ฏธ: Fiedler ๋ฒกํ„ฐ์˜ ๊ทน์ ์— ๋Œ€ํ•œ ์—ฐ๊ตฌ ์ดˆ๋ก ์ „์ฒด ๋ฒˆ์—ญ ๋ฐ ์ •๋ฆฌ: ์ด ๋…ผ๋ฌธ์—์„œ๋Š” ๊ทธ๋ž˜ํ”„ ๋ผํ”Œ๋ผ์‹œ์•ˆ๊ณผ ๊ทธ ๊ณ ์œ ๋ฒกํ„ฐ ์ค‘ ํ•˜๋‚˜์ธ ํ”ผ๋”๋Ÿฌ ๋ฒกํ„ฐ(Fiedler vector)๋ฅผ ์ค‘์‹ฌ์œผ๋กœ, ํŠนํžˆ ์ด ๋ฒกํ„ฐ์˜ ๊ทน๊ฐ’๋“ค์ด ์–ด๋–ค ์˜๋ฏธ๋ฅผ ๊ฐ–๋Š”์ง€์— ๋Œ€ํ•ด ์—ฐ๊ตฌํ•œ๋‹ค. ํ”ผ๋”๋Ÿฌ ๋ฒกํ„ฐ๋Š” ๊ทธ๋ž˜ํ”„ ๋ถ„ํ•  ๋ฌธ์ œ์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋ฉฐ, ์ด ๋…ผ๋ฌธ์—์„œ๋Š” ์ด ๋ฒกํ„ฐ์™€ ์ด์‚ฐ ์—ด ๋ฐฉ์ •์‹ ์‚ฌ์ด์˜ ๊ด€๊ณ„๋ฅผ ํ†ตํ•ด ๊ทธ ํŠน์„ฑ์„ ์ดํ•ดํ•˜๋ ค๊ณ  ํ•œ๋‹ค. ํŠนํžˆ, ํ”ผ๋”๋Ÿฌ ๋ฒกํ„ฐ์˜ ๊ทน๊ฐ’์ด ๊ฐ€์žฅ ๋ฉ€๋ฆฌ ๋–จ์–ด์ง„ ๋‘ ์ •์ ์ผ ๊ฒƒ์ด๋ผ๋Š” ์ผ๋ฐ˜์ ์ธ ์ถ”์ธก์— ๋Œ€ํ•œ ๋ฐ˜๋ก€๋กœ 'ํ”ผ๋”๋Ÿฌ ์žฅ๋ฏธ'๋ผ๋Š”

Mathematics
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A Criticism on 'A Mathematicians Apology' by G. H. Hardy

: G.H. ํ•˜๋””์˜ ์ˆ˜ํ•™์ž์˜ ๋ณ€๋ช… ์€ 20์„ธ๊ธฐ ์ดˆ๋ฐ˜ ์ˆ˜ํ•™๊ณผ ๊ณผํ•™์— ๋Œ€ํ•œ ์ฒ ํ•™์  ์ ‘๊ทผ์„ ๋…ผํ•˜๋Š” ์ค‘์š”ํ•œ ๋ฌธํ—Œ์ด๋‹ค. ์ด ์ฑ…์—์„œ ํ•˜๋””๋Š” ์ˆ˜ํ•™์„ ์ˆœ์ˆ˜ํ•œ ์ง€์  ํ˜ธ๊ธฐ์‹ฌ์˜ ๊ฒฐ๊ณผ๋กœ ๋ณด๋ฉฐ, ์‘์šฉ ์ˆ˜ํ•™์— ๋Œ€ํ•ด ๋ถ€์ •์ ์ธ ํƒœ๋„๋ฅผ ์ทจํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ๊ด€์ ์€ ๊ทธ๊ฐ€ ์ˆ˜ํ•™์ž๋กœ์„œ์˜ ์ž๋ถ€์‹ฌ๊ณผ ํ•จ๊ป˜ ๊ณผํ•™์˜ ๋ฐœ์ „์— ๋Œ€ํ•œ ์ฒ ํ•™์  ๊ฒฌํ•ด๋ฅผ ๋ฐ˜์˜ํ•˜๊ณ  ์žˆ๋‹ค. ํ•˜๋””๋Š” ์ˆœ์ˆ˜ ์ˆ˜ํ•™์„ ๊ฐ€์žฅ ๋†’์ด ํ‰๊ฐ€ํ•˜๋ฉฐ, ์‘์šฉ ์ˆ˜ํ•™์„ ํŽธ๊ฒฌ์ ์œผ๋กœ ๋ฐ”๋ผ๋ณธ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์•„์ด๋Ÿฌ๋‹ˆํ•˜๊ฒŒ๋„, ๊ทธ ์ž์‹ ์˜ ์—ฐ๊ตฌ ์ค‘ ์ผ๋ถ€๋Š” ์ค‘์š”ํ•œ ์‘์šฉ ๊ฒฐ๊ณผ๋ฅผ ๋‚ณ์•˜๋‹ค. ์˜ˆ๋ฅผ ๋“ค์–ด, ํ•˜๋”” ์™€์ธ๋ฒ ๋ฅดํฌ ๋ฒ•์น™์€ ์œ ์ „ํ•™์—์„œ ํ•ต์‹ฌ์ ์ธ ์—ญํ• ์„ ํ•˜๋Š”๋ฐ,

Mathematics
Elementary trigonometry based on a first order differential equation

Elementary trigonometry based on a first order differential equation

: ๋ณธ ๋…ผ๋ฌธ์€ ๊ธฐ์กด์˜ ์‚ผ๊ฐํ•จ์ˆ˜ ์ •์˜๋ฅผ ๋„˜์–ด์„œ, 1์ฐจ ๋ฏธ๋ถ„ ๋ฐฉ์ •์‹ $f'(x) f(x + a)$๋ฅผ ํ†ตํ•ด ์‚ผ๊ฐํ•จ์ˆ˜์˜ ์„ฑ์งˆ์„ ์žฌํ•ด์„ํ•˜๊ณ ์ž ํ•œ๋‹ค. ์ด๋Š” ์‚ผ๊ฐํ•จ์ˆ˜๊ฐ€ ์ฃผ๊ธฐ์ ์ด๊ณ  ์—ฌ๋Ÿฌ ์‹์ฆ์„ ๋งŒ์กฑํ•˜๋Š” ํ•จ์ˆ˜๋ผ๋Š” ๊ธฐ์กด์˜ ์ดํ•ด๋ฅผ ํ™•์žฅ์‹œํ‚ค๋ฉฐ, ์ƒˆ๋กœ์šด ๊ด€์ ์—์„œ ์‚ผ๊ฐํ•จ์ˆ˜์˜ ๋ณธ์งˆ์„ ํƒ๊ตฌํ•œ๋‹ค. 1. ์„œ๋ก  ์„œ๋ก ์—์„œ๋Š” ์‚ฌ์ธ๊ณผ ์ฝ”์‚ฌ์ธ ํ•จ์ˆ˜๊ฐ€ 2์ฐจ ๋ฏธ๋ถ„ ๋ฐฉ์ •์‹ $f'' f$์˜ ํ•ด๋กœ์„œ ์ •์˜๋œ๋‹ค๋Š” ์ ์„ ๊ฐ•์กฐํ•œ๋‹ค. ์ด๋Š” ์ฃผ๊ธฐ์„ฑ, ์ œํ•œ์„ฑ, ๊ทธ๋ฆฌ๊ณ  ๋‹ค์–‘ํ•œ ์‚ผ๊ฐํ•จ์ˆ˜ ์‹์ฆ์„ ๋งŒ์กฑํ•˜๋Š” ํ•จ์ˆ˜๋ผ๋Š” ์˜๋ฏธ์ด๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ด๋Ÿฌํ•œ ์„ฑ์งˆ๋“ค์€ ๋‹ค๋ฅธ ์ •์˜๋“ค์—์„œ ๋„์ถœ๋  ์ˆ˜ ์žˆ์œผ๋ฉฐ, ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” 1

Mathematics
Zum 200. Geburtstag von Evariste Galois

Zum 200. Geburtstag von Evariste Galois

: ๊ฐˆ๋ฃจ์•„์˜ ์‚ถ๊ณผ ์‹œ๋Œ€์  ๋ฐฐ๊ฒฝ ์—๋ฐ”๋ฆฌ์Šค ๊ฐˆ๋ฃจ์•„๋Š” ํ”„๋ž‘์Šค ํ˜๋ช… ์ดํ›„์˜ ๋ถˆ์•ˆ์ •ํ•œ ์ •์น˜ ํ™˜๊ฒฝ ์†์—์„œ ํƒœ์–ด๋‚ฌ๋‹ค. ๊ทธ์˜ ์ƒ์• ๋Š” 1820๋…„๋Œ€์™€ 30๋…„๋Œ€, ์ฆ‰ ๋‘ ๋ฒˆ์งธ ํ˜๋ช…์ด ์ผ์–ด๋‚œ ์‹œ๊ธฐ์™€ ๋งž๋ฌผ๋ ค ์žˆ๋‹ค. ์ด๋Ÿฌํ•œ ์‹œ๋Œ€์  ๋ฐฐ๊ฒฝ์€ ๊ฐˆ๋ฃจ์•„์—๊ฒŒ ํฐ ์˜ํ–ฅ์„ ๋ฏธ์ณค์œผ๋ฉฐ, ๊ทธ๋Š” ์ •์น˜์ ์œผ๋กœ ๋งค์šฐ ํ™œ๋™์ ์ด์—ˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ด๋กœ ์ธํ•ด ํ•™๊ต์—์„œ ์ถ”๋ฐฉ๋‹นํ•˜๊ฑฐ๋‚˜ ์‚ฌํšŒ์  ์ œ์•ฝ์„ ๊ฒช๊ฒŒ ๋˜์—ˆ๋‹ค. ์ˆ˜ํ•™์  ์—…์  ๊ฐˆ๋ฃจ์•„์˜ ๊ฐ€์žฅ ์ค‘์š”ํ•œ ์ˆ˜ํ•™์  ์—…์  ์ค‘ ํ•˜๋‚˜๋Š” ๋ฐฉ์ •์‹์˜ ํ•ด๋ฒ•๊ณผ ๊ตฐ๋ก  ์‚ฌ์ด์˜ ๊นŠ์€ ์—ฐ๊ฒฐ์„ ํƒ๊ตฌํ•œ ๊ฒƒ์ด๋‹ค. ๊ทธ๋Š” ๋‹ค์„ฏ ์ฐจ์ˆ˜ ์ด์ƒ์˜ ๋ฐฉ์ •์‹์— ๋Œ€ํ•œ ์ผ๋ฐ˜์ ์ธ ํ•ด๋ฒ•์ด ์กด์žฌํ•˜์ง€ ์•Š๋Š”๋‹ค๋Š” ๊ฒƒ

Mathematics
Transfinite Sequences of Continuous and Baire Class 1 Functions

Transfinite Sequences of Continuous and Baire Class 1 Functions

๋ณธ ๋…ผ๋ฌธ์€ ์—ฐ์† ํ•จ์ˆ˜์™€ ๋ฐ”์ด์–ด ํด๋ž˜์Šค 1 ํ•จ์ˆ˜์˜ ์ˆœ์„œ ์œ ํ˜•์— ๋Œ€ํ•œ ์‹ฌ๋„ ์žˆ๋Š” ๋ถ„์„์„ ์ œ๊ณตํ•˜๋ฉฐ, ํŠนํžˆ ์ด๋Ÿฌํ•œ ํ•จ์ˆ˜๋“ค์˜ ์ฆ๊ฐ€ ๋˜๋Š” ๊ฐ์†Œํ•˜๋Š” ์ž˜ ์ •๋ ฌ๋œ ์‹œํ€€์Šค์˜ ๊ธธ์ด๋ฅผ ์กฐ์‚ฌํ•œ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ๊ธฐ์กด์˜ ๊ณ ์ „์ ์ธ ๊ฒฐ๊ณผ๋ฅผ ํ™•์žฅํ•˜๊ณ , ๋ฉ”ํŠธ๋ฆญ ๊ณต๊ฐ„์—์„œ ๋ฐ”์ด์–ด ํด๋ž˜์Šค 1 ํ•จ์ˆ˜์˜ ์‚ฌ์Šฌ์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ๊ด€์ ์„ ์ œ์‹œํ•œ๋‹ค. ์—ฐ์† ํ•จ์ˆ˜์™€ ๋ฐ”์ด์–ด ํด๋ž˜์Šค 1 ํ•จ์ˆ˜์˜ ์ˆœ์„œ ์œ ํ˜• ๋…ผ๋ฌธ์€ ๋จผ์ € ํด๋ž€๋“œ ๊ณต๊ฐ„์—์„œ ์—ฐ์† ํ•จ์ˆ˜์™€ ๋ฐ”์ด์–ด ํด๋ž˜์Šค 1 ํ•จ์ˆ˜์˜ ์‹œํ€€์Šค๋ฅผ ๋ถ„์„ํ•œ๋‹ค. ์ด๋Š” ๊ณ ์ „์ ์ธ ๊ฒฐ๊ณผ์ธ Kuratowski์˜ ์ •๋ฆฌ์— ๊ธฐ๋ฐ˜ํ•˜๋ฉฐ, ๋ฐ”์ด์–ด ํด๋ž˜์Šค 1 ํ•จ์ˆ˜์˜ ๋ชจ๋…ธํ†ค ์‹œํ€€์Šค์˜ ๊ธธ์ด

Mathematics
VC dimension of ellipsoids

VC dimension of ellipsoids

: ๋ณธ ๋…ผ๋ฌธ์€ ์—˜๋ฆฝ์†Œ์ด๋“œ์˜ ๋ณต์žก์„ฑ์„ ์ธก์ •ํ•˜๋Š” VC ์ฐจ์›์— ๋Œ€ํ•œ ์‹ฌ์ธต์ ์ธ ์—ฐ๊ตฌ๋ฅผ ์ˆ˜ํ–‰ํ•œ๋‹ค. ์ด๋Š” ํ•™์Šต ์ด๋ก ์—์„œ ์ค‘์š”ํ•œ ๊ฐœ๋…์œผ๋กœ, ๋ฐ์ดํ„ฐ ๋ถ„์„ ๋ฐ ๋ชจ๋ธ ์„ ํƒ์— ํ™œ์šฉ๋œ๋‹ค. 1. VC ์ฐจ์›์˜ ์ •์˜์™€ ์ค‘์š”์„ฑ VC ์ฐจ์›์€ ์ง‘ํ•ฉ์ด ์–ผ๋งˆ๋‚˜ ๋ณต์žกํ•œ ํ˜•ํƒœ๋ฅผ ๊ฐ€์งˆ ์ˆ˜ ์žˆ๋Š”์ง€๋ฅผ ์ธก์ •ํ•˜๋Š” ์ง€ํ‘œ๋กœ, ๊ฒฝํ—˜์  ๊ณผ์ • ์ด๋ก , ํ†ต๊ณ„ ๋ฐ ๊ณ„์‚ฐ ํ•™์Šต ์ด๋ก , ๊ทธ๋ฆฌ๊ณ  ์ด์‚ฐ ๊ธฐํ•˜ํ•™ ๋“ฑ ๋‹ค์–‘ํ•œ ๋ถ„์•ผ์—์„œ ํ™œ์šฉ๋œ๋‹ค. ํŠนํžˆ, VC ์ฐจ์›์€ ๋ฐ์ดํ„ฐ์˜ ๋ณต์žก์„ฑ์„ ์ธก์ •ํ•˜๊ณ  ๋ชจ๋ธ ์„ ํƒ์— ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•œ๋‹ค. 2. ์—˜๋ฆฝ์†Œ์ด๋“œ์™€ ๊ฐ€์šฐ์‹œ์•ˆ ํ˜ผํ•ฉ ๋ชจ๋ธ ์—˜๋ฆฝ์†Œ์ด๋“œ๋Š” d์ฐจ์› ๊ณต๊ฐ„์—์„œ ์ •์˜๋˜๋Š” ์ง‘ํ•ฉ์œผ๋กœ,

Mathematics Statistics Computer Science Machine Learning
A Note on the Grothendieck Group of an Additive Category

A Note on the Grothendieck Group of an Additive Category

๋งค๋ ฅ์ ์ธ ํ•œ๊ธ€ ์ œ๋ชฉ: ๊ฐ€์‚ฐ ๋ฒ”์ฃผ์™€ ๊ทธ๋กœํ…๋””ํฌ ๊ทธ๋ฃน์˜ ๋™ํ˜•์„ฑ ์ดˆ๋ก ์ „์ฒด ๋ฒˆ์—ญ ๋ฐ ์ •๋ฆฌ: ๋ณธ ๋…ผ๋ฌธ์€ ๊ฐ€์‚ฐ ๋ฒ”์ฃผ์˜ ์Šคํ”Œ๋ฆฟ ๊ทธ๋กœํ…๋””ํฌ ๊ทธ๋ฃน๊ณผ ํ•ด๋‹น ๋ฒ”์ฃผ์˜ ์œ ํ•œ ๋ณต์žก์ฒด์˜ ํ˜ธ๋ชจํ† ํ”ผ ๋ฒ”์ฃผ์—์„œ์˜ ์‚ผ๊ฐํ™”๋œ ๊ทธ๋กœํ…๋””ํฌ ๊ทธ๋ฃน์ด ๋™ํ˜•์ธ์ง€์— ๋Œ€ํ•œ ์งˆ๋ฌธ์„ ๋‹ค๋ฃฌ๋‹ค. ํŠนํžˆ, ๊ฐ€์‚ฐ ๋ฒ”์ฃผ A์™€ ๊ทธ์˜ ์œ ํ•œ ๋ณต์žก์ฒด์˜ ํ˜ธ๋ชจํ† ํ”ผ ๋ฒ”์ฃผ Kb(A)๋ฅผ ๊ณ ๋ คํ•  ๋•Œ, A์˜ ์Šคํ”Œ๋ฆฟ ๊ทธ๋กœํ…๋””ํฌ ๊ทธ๋ฃน KโŠ•(A)๊ฐ€ Kb(A)์˜ ์‚ผ๊ฐํ™”๋œ ๊ทธ๋กœํ…๋””ํฌ ๊ทธ๋ฃน Kโ–ณ(Kb(A))์™€ ๋™ํ˜•์ธ ๊ฒƒ์„ ์ฆ๋ช…ํ•œ๋‹ค. ์ด๋Š” ํ˜ธ๋ชจํ† ํ”ผ ๋ฒ”์ฃผ์—์„œ์˜ ์—์šธ๋Ÿฌ ํŠน์„ฑ์ด ์›๋ž˜ ๊ตฌ์กฐ๋ฅผ ๋ณต์›ํ•˜๋Š” ๋ฐ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•œ๋‹ค๋Š” ์ ์—์„œ

Mathematics
Complexity of Bondage and Reinforcement

Complexity of Bondage and Reinforcement

๋ณธ ๋…ผ๋ฌธ์€ ์ง€๋ฐฐ ์ง‘ํ•ฉ๊ณผ ๊ด€๋ จ๋œ ์—ฌ๋Ÿฌ ๋งค๊ฐœ๋ณ€์ˆ˜, ํŠนํžˆ ๊ฒฐํ•ฉ ์ˆซ์ž์™€ ๊ฐ•ํ™” ์ˆซ์ž์˜ ๋ณต์žก์„ฑ์„ ๋ถ„์„ํ•˜๊ณ  ์žˆ๋‹ค. ์ด๋Ÿฌํ•œ ๊ฐœ๋…๋“ค์€ ๊ทธ๋ž˜ํ”„ ์ด๋ก ์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋ฉฐ, ํŠนํžˆ NP ์™„์ „์„ฑ ๋ฌธ์ œ์— ๋Œ€ํ•œ ์ดํ•ด๋ฅผ ๊นŠ๊ฒŒ ํ•œ๋‹ค. 1. ์ง€๋ฐฐ ์ง‘ํ•ฉ๊ณผ ๊ด€๋ จ ๋งค๊ฐœ๋ณ€์ˆ˜ ์ง€๋ฐฐ ์ง‘ํ•ฉ์€ ๊ทธ๋ž˜ํ”„ G (V, E)์˜ ๋ชจ๋“  ๊ผญ์ง“์ ์ด ํ•ด๋‹น ์ง‘ํ•ฉ ๋‚ด์˜ ์ธ์ ‘ํ•œ ๊ผญ์ง“์ ์— ํฌํ•จ๋˜๋Š” ์ตœ์†Œ ์ง‘ํ•ฉ์ด๋‹ค. ์ด๋Š” ๊ทธ๋ž˜ํ”„์—์„œ ์ค‘์š”ํ•œ ์ •๋ณด๋ฅผ ํšจ์œจ์ ์œผ๋กœ ์ „๋‹ฌํ•˜๊ฑฐ๋‚˜ ์ปค๋ฒ„ํ•˜๋Š” ๋ฐ ์‚ฌ์šฉ๋œ๋‹ค. ์ง€๋ฐฐ ์ˆซ์ž ฮณ(G)๋Š” ์ด๋Ÿฌํ•œ ์ตœ์†Œ ์ง‘ํ•ฉ์˜ ํฌ๊ธฐ๋ฅผ ๋‚˜ํƒ€๋‚ด๋ฉฐ, ์ด๋ฅผ ๊ณ„์‚ฐํ•˜๋Š” ๋ฌธ์ œ๋Š” NP ์™„์ „์ž„์ด ์•Œ๋ ค์ ธ ์žˆ๋‹ค. ๊ฒฐํ•ฉ

Mathematics Computational Complexity Computer Science
No Image

The Igusa-Todorov function for comodules

๋ณธ ๋…ผ๋ฌธ์€ ์ด๊ตฌ์‚ฌ ํ† ๋„ํ”„(IT) ํ•จ์ˆ˜๋ฅผ ์ด์šฉํ•˜์—ฌ ์ฝ”๋ชจ๋“ˆ๊ณผ ๊ด€๋ จ๋œ ์—ฌ๋Ÿฌ ์„ฑ์งˆ์„ ๋ถ„์„ํ•˜๊ณ  ์žˆ๋‹ค. ์ฃผ์š” ๋‚ด์šฉ์€ ๋‹ค์Œ๊ณผ ๊ฐ™๋‹ค: 1. IT ํ•จ์ˆ˜์™€ quasi Frobenius ์„ฑ์งˆ: IT ํ•จ์ˆ˜๋Š” ์ฃผ์–ด์ง„ ์ฝ”์•Œ๊ฒŒ๋ผ์—์„œ ๊ฐ ์œ ํ•œ ์ƒ์„ฑ ์šฐ(์ขŒ) ๋ชจ๋“ˆ์— ๋Œ€ํ•œ ์ž๋ช…ํ•˜์ง€ ์•Š์€ ํ•ต์„ ์ •์˜ํ•˜๋Š” ์ƒˆ๋กœ์šด ํ˜ธ๋ชฐ๋กœ์ง€ ๋„๊ตฌ์ด๋‹ค. ์ด ํ•จ์ˆ˜๋ฅผ ํ†ตํ•ด ์šฐ(์ขŒ) ์•„ํ‹ด ๋ฐ˜ํ™˜ ๋ง์˜ selfinjectivity๋ฅผ ํ‘œํ˜„ํ•  ์ˆ˜ ์žˆ์œผ๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด quasi Frobenius ์„ฑ์งˆ๊ณผ IT ํ•จ์ˆ˜ ์‚ฌ์ด์˜ ๊ด€๊ณ„๋ฅผ ๋ถ„์„ํ•œ๋‹ค. 2. ์ขŒ(์šฐ) qcF ์ฝ”์•Œ๊ฒŒ๋ผ์˜ ์ •์˜์™€ IT ํ•จ์ˆ˜: ์ขŒ(์šฐ) qua

Mathematics
On the Intersection of All Critical Sets of a Unicyclic Graph

On the Intersection of All Critical Sets of a Unicyclic Graph

: ๋ณธ ๋…ผ๋ฌธ์€ ์œ ์‚ฌ์ดํด์  ๊ทธ๋ž˜ํ”„, ์ฆ‰ ๋‹จ์ผ ์‚ฌ์ดํด์„ ๊ฐ–๋Š” ์—ฐ๊ฒฐ ๊ทธ๋ž˜ํ”„์—์„œ ํ•ต์‹ฌ(core), ์ฝ”๋กœ๋‚˜(corona), ๊ทธ๋ฆฌ๊ณ  ker(G) ์ง‘ํ•ฉ๋“ค ๊ฐ„์˜ ๊ด€๊ณ„์— ๋Œ€ํ•ด ๊นŠ์ด ์žˆ๋Š” ๋ถ„์„์„ ์ œ๊ณตํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ๊ทธ๋ž˜ํ”„๋“ค์€ ์ด๋ก ์ ์œผ๋กœ ์ค‘์š”ํ•œ ์œ„์น˜๋ฅผ ์ฐจ์ง€ํ•˜๋ฉฐ, ํŠนํžˆ ๊ทธ๋“ค์˜ ๋…๋ฆฝ ์ง‘ํ•ฉ๊ณผ ๋งค์นญ์— ๋Œ€ํ•œ ์„ฑ์งˆ์€ ๊ทธ๋ž˜ํ”„ ์ด๋ก ์—์„œ ํ•ต์‹ฌ์ ์ธ ์—ญํ• ์„ ํ•œ๋‹ค. ๋…ผ๋ฌธ์—์„œ๋Š” ๋จผ์ € ํ•ต์‹ฌ(core)๊ณผ ์ฝ”๋กœ๋‚˜(corona)์˜ ์ •์˜๋ฅผ ์ œ์‹œํ•œ๋‹ค. ํ•ต์‹ฌ์€ ๋ชจ๋“  ์ตœ๋Œ€ ๋…๋ฆฝ ์ง‘ํ•ฉ๋“ค์˜ ๊ต์ง‘ํ•ฉ์ด๊ณ , ์ฝ”๋กœ๋‚˜๋Š” ์ด๋Ÿฌํ•œ ์ง‘ํ•ฉ๋“ค์˜ ํ•ฉ์ง‘ํ•ฉ์ด๋‹ค. ker(G)๋Š” G์˜ ์ค‘์š”ํ•œ ๋…๋ฆฝ ์ง‘ํ•ฉ๋“ค์˜ ๊ต์ง‘ํ•ฉ์œผ๋กœ ์ •

Mathematics Computer Science Discrete Mathematics
Brouwers fixed point theorem with sequentially at most one fixed point

Brouwers fixed point theorem with sequentially at most one fixed point

์ด ๋…ผ๋ฌธ์€ ๋ธŒ๋ฃจ์–ด ๊ณ ์ •์  ์ •๋ฆฌ์™€ ๊ด€๋ จ๋œ ์ƒˆ๋กœ์šด ์ ‘๊ทผ๋ฒ•์„ ์†Œ๊ฐœํ•˜๋ฉฐ, ํŠนํžˆ ๊ทผ์‚ฌ ๊ณ ์ •์ ๊ณผ ์ˆœ์ฐจ์ ์œผ๋กœ ๊ฐ€์žฅ ํฐ ๊ฐ’์ด ํ•˜๋‚˜์ธ ์กฐ๊ฑด์— ์ดˆ์ ์„ ๋งž์ถ˜๋‹ค. ์ด๋Ÿฌํ•œ ์ ‘๊ทผ๋ฒ•์€ ๊ธฐ์กด์˜ ๋ธŒ๋ฃจ์–ด ํŒฌ ์ •๋ฆฌ๋ฅผ ํ™•์žฅํ•˜๊ณ , ์ด๋ฅผ ํ†ตํ•ด ์ œ๋กœ์„ฌ ๊ฒŒ์ž„์—์„œ ์ตœ์  ์ „๋žต์„ ์ฐพ๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. 1. ๋ธŒ๋ฃจ์–ด ๊ณ ์ •์  ์ •๋ฆฌ์™€ ๊ทผ์‚ฌ ๋ฒ„์ „ ๋ธŒ๋ฃจ์–ด ๊ณ ์ •์  ์ •๋ฆฌ๋Š” ๋น„์Šˆํ”ผ์ฒ˜ ์ˆ˜ํ•™์—์„œ ๊ฑด์„ค์ ์œผ๋กœ ์ฆ๋ช…๋  ์ˆ˜ ์—†๋‹ค๋Š” ๊ฒƒ์ด ์ž˜ ์•Œ๋ ค์ ธ ์žˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์Šคํผ๋„ˆ์˜ ํ•จ์ˆ˜๋ฅผ ์ด์šฉํ•˜์—ฌ ๊ทผ์‚ฌ ๋ฒ„์ „์„ ์ œ์‹œํ•  ์ˆ˜ ์žˆ๋‹ค๋Š” ์ ์ด ์ค‘์š”ํ•˜๋‹ค. ์ด ๋…ผ๋ฌธ์—์„œ๋Š” ์ด๋Ÿฌํ•œ ๊ทผ์‚ฌ ๋ฒ„์ „์„ ํ†ตํ•ด ๋ธŒ๋ฃจ์–ด ๊ณ ์ •์  ์ •๋ฆฌ๋ฅผ ๊ฑด์„ค์ ์œผ๋กœ ์ฆ

Mathematics Game Theory Computer Science
Recursion Relations and Functional Equations for the Riemann Zeta   Function

Recursion Relations and Functional Equations for the Riemann Zeta Function

: ๋ณธ ๋…ผ๋ฌธ์€ ๋ฆฌ๋งŒ ์ œํƒ€ ํ•จ์ˆ˜(ฮถ(s))์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ์žฌ๊ท€ ๊ด€๊ณ„๋ฅผ ํƒ๊ตฌํ•˜๊ณ  ์ด๋ฅผ ํ†ตํ•ด ๋ณต์†Œ ํ‰๋ฉด ์ƒ์˜ ๋‹ค์–‘ํ•œ ์ง€์ ์—์„œ ์ œํƒ€ ํ•จ์ˆ˜์˜ ๊ฐ’์„ ๊ณ„์‚ฐํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ๊ธฐ์กด์˜ ์ œํ•œ์ ์ธ ์žฌ๊ท€ ๊ด€๊ณ„๋ฅผ ํ™•์žฅํ•˜์—ฌ, ๋ณต์žกํ•œ ํ‰๋ฉด ์ƒ์—์„œ ์ œํƒ€ ํ•จ์ˆ˜์˜ ํ–‰๋™์„ ๋” ์ž˜ ์ดํ•ดํ•˜๊ณ  ํ™œ์šฉํ•  ์ˆ˜ ์žˆ๋Š” ์ƒˆ๋กœ์šด ๋„๊ตฌ๋ฅผ ์ œ๊ณตํ•œ๋‹ค. 1. ๋ฆฌ๋งŒ ์ œํƒ€ ํ•จ์ˆ˜์™€ ๊ธฐ๋Šฅ ๋ฐฉ์ •์‹ ๋ฆฌ๋งŒ ์ œํƒ€ ํ•จ์ˆ˜๋Š” ๋ณต์†Œ์ˆ˜ s์— ๋Œ€ํ•ด ์ •์˜๋˜๋ฉฐ, ๊ทธ ๊ธฐ๋Šฅ ๋ฐฉ์ •์‹์€ ฮถ(s) 2^s ฯ€^(s 1) sin(ฯ€s/2) ฮ“(1 s)๋กœ ์ฃผ์–ด์ง„๋‹ค. ์ด ๋ฐฉ์ •์‹์€ ๋ณ€์ˆ˜ s๋ฅผ 1 s๋กœ ๋ฐ”๊พธ์–ด๋„ ๋Œ€์นญ์„ฑ์„ ์œ 

Mathematics
Decomposition of Cellular Balleans

Decomposition of Cellular Balleans

: ์ด ๋…ผ๋ฌธ์€ ์„ธํฌ ๊ตฌ ๊ตฌ์กฐ์— ๋Œ€ํ•œ ์‹ฌ๋„ ์žˆ๋Š” ์—ฐ๊ตฌ๋ฅผ ์ง„ํ–‰ํ•˜๋ฉฐ, ํŠนํžˆ ์ด๋Ÿฌํ•œ ๊ตฌ ๊ตฌ์กฐ์˜ ๋ถ„ํ•ด ๊ฐ€๋Šฅ์„ฑ๊ณผ ๋ฉ”ํŠธ๋ฆญํ™” ๊ฐ€๋Šฅ์„ฑ์„ ํƒ๊ตฌํ•œ๋‹ค. ๊ตฌ ๊ตฌ์กฐ๋Š” ์„ธ ๊ฐ€์ง€ ์š”์†Œ B ( X , P , B )๋กœ ๊ตฌ์„ฑ๋˜๋ฉฐ, ์—ฌ๊ธฐ์„œ X ์™€ P ๋Š” ๋น„๊ณตํ—ˆ ์ง‘ํ•ฉ์ด๊ณ , ๋ชจ๋“  x โˆˆ X ์™€ ฮฑ โˆˆ P ์— ๋Œ€ํ•ด B(x, ฮฑ) ๋Š” ๋ฐ˜์ง€๋ฆ„ ฮฑ ์˜ ๊ตฌ๋ฅผ ๋‚˜ํƒ€๋‚ด๋Š” X ์˜ ๋ถ€๋ถ„์ง‘ํ•ฉ์ด๋‹ค. ๋…ผ๋ฌธ์€ ์ด๋Ÿฌํ•œ ๊ตฌ ๊ตฌ์กฐ๊ฐ€ ๋ฉ”ํŠธ๋ฆญํ™” ๊ฐ€๋Šฅํ•˜๊ฑฐ๋‚˜ ์„ธํฌ ๊ตฌ ๊ตฌ์กฐ๋กœ ๋ถ„ํ•ด๋  ์ˆ˜ ์žˆ๋Š” ์กฐ๊ฑด์„ ์ •๋ฆฌํ•˜๊ณ  ์ฆ๋ช…ํ•œ๋‹ค. 1. ๊ตฌ ๊ตฌ์กฐ์˜ ๊ธฐ๋ณธ ๊ฐœ๋… ๊ตฌ ๊ตฌ์กฐ๋Š” ์„ธ ๊ฐ€์ง€ ์š”์†Œ B ( X , P , B )๋กœ ๊ตฌ์„ฑ๋˜๋ฉฐ

Mathematics
A Class of Special Solutions for the Ultradiscrete Painleve II   Equation

A Class of Special Solutions for the Ultradiscrete Painleve II Equation

: ๋ณธ ๋…ผ๋ฌธ์€ ํŒŒ์ธ๋ ˆ๋ธŒ II ๋ฐฉ์ •์‹์˜ ์ดˆ๊ณ ๊ธ‰ ์ด์‚ฐํ™”๋œ ํ˜•ํƒœ๋ฅผ ์—ฐ๊ตฌํ•˜๊ณ , ์ด๋ฅผ ํ†ตํ•ด ์–ป์–ด์ง€๋Š” ํŠน๋ณ„ํ•œ ํ•ด์— ๋Œ€ํ•ด ์‹ฌ๋„ ์žˆ๊ฒŒ ๋ถ„์„ํ•œ๋‹ค. ํŠนํžˆ, q ์ฐจ๋ถ„ ์œ ์‚ฌํ˜• ์—์–ด๋ฆฌ ๋ฐฉ์ •์‹์„ ๊ธฐ๋ฐ˜์œผ๋กœ udPII (์ดˆ๊ณ ๊ธ‰ ์ด์‚ฐํ™” ํŒŒ์ธ๋ ˆ๋ธŒ II) ๋ฐฉ์ •์‹์˜ ํŠน์ˆ˜ํ•ด๋ฅผ ๊ตฌ์ถ•ํ•˜๊ณ  ๊ทธ ์„ฑ์งˆ์„ ํƒ๊ตฌํ•œ๋‹ค. ์ดˆ๊ณผ ์ด์‚ฐํ™”์™€ p ์ดˆ๊ณผ ์ด์‚ฐํ™” ์ดˆ๊ณผ ์ด์‚ฐํ™”๋Š” ์ฃผ์–ด์ง„ ์ฐจ๋ถ„ ๋ฐฉ์ •์‹์„ ์…€ ์˜คํ† ๋งˆํ†ค์œผ๋กœ ๋ณ€ํ™˜ํ•˜๋Š” ๊ณผ์ •์ด๋‹ค. ์ด ๊ณผ์ •์—์„œ ์ข…์† ๋ณ€์ˆ˜ x<sub>n</sub> ์€ ์ด์‚ฐ ๊ฐ’์„ ๊ฐ€์ง€๊ฒŒ ๋˜๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ์›๋ž˜์˜ ๋ฏธ๋ถ„ ๋ฐฉ์ •์‹์ด ์กฐ๊ฐ ์„ ํ˜• ๋ฐฉ์ •์‹์œผ๋กœ ๊ทผ์‚ฌ๋œ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ '์Œ์˜ ๋ฌธ์ œ'๋กœ

Mathematics Nonlinear Sciences
Integration of Constraint Equations in Problems of a Disc and a Ball   Rolling on a Horizontal Plane

Integration of Constraint Equations in Problems of a Disc and a Ball Rolling on a Horizontal Plane

์ด ๋…ผ๋ฌธ์€ ์›ํ˜•๊ณผ ๊ตฌ๊ฐ€ ์ˆ˜ํ‰๋ฉด์—์„œ ๋ฏธ๋„๋Ÿฌ์ง ์—†์ด ๊ตด๋Ÿฌ๊ฐ€๋Š” ๋™์—ญํ•™์  ๋ฌธ์ œ๋ฅผ ์ฒด๊ณ„์ ์œผ๋กœ ๋ถ„์„ํ•˜๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ๊ณ ์ „ ์—ญํ•™์˜ ๋ณต์žกํ•œ ๋ฌธ์ œ๋ฅผ ๋‹จ์ˆœํ™”ํ•˜๊ณ  ํ•ด๊ฒฐํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค. ํŠนํžˆ, ์ž์—ฐ ๋ฐฉ์ •์‹์„ ํ™œ์šฉํ•˜์—ฌ ์ ‘์ด‰ ๊ถค์ ์˜ ๊ณก๋ฅ  ์˜์กด์„ฑ์„ ๋ช…ํ™•ํžˆ ํ‘œํ˜„ํ•จ์œผ๋กœ์จ, ๊ตด๋Ÿฌ๊ฐ€๋Š” ๋™์ž‘์— ๋Œ€ํ•œ ๊นŠ์€ ์ดํ•ด์™€ ์ •๋Ÿ‰์ ์ธ ๋ถ„์„์ด ๊ฐ€๋Šฅํ•ด์ง‘๋‹ˆ๋‹ค. 1. ์›ํ˜•๊ณผ ๊ตฌ์˜ ๊ตด๋Ÿฌ๊ฐ€๋Š” ์šด๋™ ์›ํ˜•๊ณผ ๊ตฌ๊ฐ€ ์ˆ˜ํ‰๋ฉด์—์„œ ๊ตด๋Ÿฌ๊ฐˆ ๋•Œ, ์ด๋“ค์˜ ์›€์ง์ž„์€ ๋น„ํ™€๋ก ์  ์ œ์•ฝ์ด๋ผ๋Š” ๋ณต์žกํ•œ ๋™์—ญํ•™์  ์กฐ๊ฑด์— ์˜ํ•ด ๊ฒฐ์ •๋ฉ๋‹ˆ๋‹ค. ์ด๋Ÿฌํ•œ ์ œ์•ฝ์€ ์ ‘์ด‰์  P์˜ ์œ„์น˜์™€ ์›ํ˜• ๋˜๋Š” ๊ตฌ์˜ ํšŒ์ „๊ฐ ฯ•, ์ „๋‹จ๊ฐ ฯˆ, ๊ทธ

Mathematics Nonlinear Sciences
Convex hyperspaces of probability measures and extensors in the   asymptotic category

Convex hyperspaces of probability measures and extensors in the asymptotic category

๋งค๋ ฅ์ ์ธ ํ•œ๊ธ€ ์ œ๋ชฉ: ํ™•๋ฅ  ์ธก๋„ ๊ณต๊ฐ„์—์„œ ์••์ถ• ๊ณต์ง‘์˜ ์ ˆ๋Œ€ ํ™•์žฅ์„ฑ ์—ฐ๊ตฌ ์ดˆ๋ก ์ „์ฒด ๋ฒˆ์—ญ ๋ฐ ์ •๋ฆฌ: ๋ณธ ๋…ผ๋ฌธ์€ ๋ฉ”ํŠธ๋ฆญ ๊ณต๊ฐ„ ์œ„์— ์ •์˜๋œ ํ™•๋ฅ  ์ธก๋„ ๊ณต๊ฐ„๊ณผ ๊ทธ ํ•˜์ดํผ์ŠคํŽ˜์ด์Šค์ธ ์••์ถ• ๊ณต์ง‘์˜ ์ ˆ๋Œ€ ํ™•์žฅ์ž ์„ฑ์งˆ์„ ํƒ๊ตฌํ•œ๋‹ค. ํŠนํžˆ, ๊ทผ์‚ฌ ๋ฒ”์ฃผ์—์„œ ์ด๋Ÿฌํ•œ ๊ณต๊ฐ„๋“ค์ด ์ ˆ๋Œ€ ํ™•์žฅ์ž๊ฐ€ ๋˜๋Š”์ง€ ์—ฌ๋ถ€๋ฅผ ๋ถ„์„ํ•œ๋‹ค. ๋…ผ๋ฌธ์€ ๋“œ๋ผ๋‹ˆ์‹œ๋‹ˆ์ฝ”ํ”„๊ฐ€ ์ œ๊ธฐํ•œ ๋ฌธ์ œ์— ๋Œ€ํ•œ ๋ถ€์ •์ ์ธ ๋‹ต๋ณ€์„ ์ œ๊ณตํ•˜๋ฉฐ, ์ด๋Š” ๋ฉ”ํŠธ๋ฆญ ๊ณต๊ฐ„์˜ ํ™•๋ฅ  ์ธก๋„ ๊ณต๊ฐ„์ด ์ผ๋ฐ˜์ ์œผ๋กœ ๊ทผ์‚ฌ ๋ฒ”์ฃผ์˜ ์ ˆ๋Œ€ ํ™•์žฅ์ž๊ฐ€ ์•„๋‹ˆ๋ผ๋Š” ๊ฒƒ์„ ๋ณด์—ฌ์ค€๋‹ค. ๋˜ํ•œ, ๋ณธ ์—ฐ๊ตฌ์—์„œ๋Š” ์••์ถ• ๊ณต์ง‘์˜ ํŠน์„ฑ์„ ๋ถ„์„ํ•˜๊ณ , ์ด๋ฅผ ํ†ตํ•ด ํ™•๋ฅ  ์ธก๋„ ๊ณต

Mathematics
Integral Value Transformations: A Class of Discrete Dynamical Systems

Integral Value Transformations: A Class of Discrete Dynamical Systems

์ด ๋…ผ๋ฌธ์€ Integral Value Transformations (IVTs)๋ฅผ ์ค‘์‹ฌ์œผ๋กœ ์ด์งˆ์  ๋™์—ญํ•™ ์‹œ์Šคํ…œ๊ณผ ๊ทธ ์•ˆ์ •์„ฑ์— ๋Œ€ํ•œ ๊นŠ์ด ์žˆ๋Š” ๋ถ„์„์„ ์ œ๊ณตํ•ฉ๋‹ˆ๋‹ค. IVT๋Š” Sk. S. Hassan ์™ธ ์—ฐ๊ตฌ์ž๋“ค์— ์˜ํ•ด ์†Œ๊ฐœ๋˜์—ˆ์œผ๋ฉฐ, p adic ์‹œ์Šคํ…œ์—์„œ Collatz ์œ ์‚ฌ ํ•จ์ˆ˜์™€ ๊ด€๋ จ์ด ์žˆ์Šต๋‹ˆ๋‹ค. ์ด๋Ÿฌํ•œ ๋ณ€ํ™˜์€ ์…€๋ฃฐ๋Ÿฌ ์˜คํ† ๋งˆํƒ€์™€ ์œ ์‚ฌํ•œ ๋ฐฉ์‹์œผ๋กœ ์—ฐ๊ตฌ๋˜์–ด ์™”์Šต๋‹ˆ๋‹ค. ๋™์—ญํ•™ ์‹œ์Šคํ…œ์˜ ์ •์˜ ๋ฐ ๋ถ„์„ ๋…ผ๋ฌธ์—์„œ๋Š” IVTs๊ฐ€ ๋ฐ˜๋ณต์ ์œผ๋กœ ์ ์šฉ๋  ๋•Œ ํ˜•์„ฑ๋˜๋Š” ์ด์งˆ์  ๋™์—ญํ•™ ์‹œ์Šคํ…œ์„ ํƒ๊ตฌํ•ฉ๋‹ˆ๋‹ค. ์ด๋Ÿฌํ•œ ํ•จ์ˆ˜๋“ค์ด ์‹œ๊ฐ„์— ๋”ฐ๋ผ ์–ด๋–ป๊ฒŒ ์ง„ํ™”ํ•˜๊ณ  ํ˜ผ๋ž€์Šค๋Ÿฌ์šด

System Mathematics Computer Science Discrete Mathematics
The Unlucky Door

The Unlucky Door

๋ณธ ๋…ผ๋ฌธ์€ ๋ชฌํ‹ฐ ํ™€ ๋ฌธ์ œ๋ฅผ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•œ ๊ฒŒ์ž„ ์ด๋ก ์  ์ ‘๊ทผ๋ฒ•๊ณผ ๋‹ค์–‘ํ•œ ๋ณ€ํ˜•์— ๋Œ€ํ•ด ์‹ฌ๋„ ์žˆ๊ฒŒ ๋ถ„์„ํ•˜๊ณ  ์žˆ๋‹ค. ํŠนํžˆ, ์„ธ ๋ฌธ ๊ฒŒ์ž„์—์„œ ์ฝ˜์ด์™€ ๋ชฌํ…Œ์˜ ์ƒํ˜ธ์ž‘์šฉ์„ ์กฐํ•ฉ์ ์œผ๋กœ ๋ถ„์„ํ•˜๋ฉฐ, ๋„ค ๊ฐœ ์ด์ƒ์˜ ๋ฌธ์„ ํฌํ•จํ•œ ํ™•์žฅ๋œ ๋ฒ„์ „์—์„œ๋Š” ํ˜‘๋ ฅ์ ์ธ ์ „๋žต์„ ์ œ์‹œํ•œ๋‹ค. 1. ์„ธ ๋ฌธ ๊ฒŒ์ž„ ๋ถ„์„ ์„ธ ๋ฌธ ๊ฒŒ์ž„์€ ํ€ด์ฆˆ ํŒ€์ด ํ•œ ๋ฌธ ๋’ค์— ์ƒํ’ˆ์„ ์ˆจ๊ธฐ๊ณ , ์ฝ˜์ด๊ฐ€ ์ฒซ ๋ฒˆ์งธ ์„ ํƒ์œผ๋กœ ๋ฌธ ํ•˜๋‚˜๋ฅผ ๊ณ ๋ฅธ๋‹ค. ๋ชฌํ…Œ๋Š” ์ƒํ’ˆ์ด ์•„๋‹Œ ๋‹ค๋ฅธ ๋ฌธ์„ ๊ณต๊ฐœํ•˜๊ณ , ์ฝ˜์ด๋Š” ์ž์‹ ์˜ ์„ ํƒ์„ ์œ ์ง€ํ•˜๊ฑฐ๋‚˜ ๋ณ€๊ฒฝํ•  ์ˆ˜ ์žˆ๋‹ค. ์ด ๊ฒŒ์ž„์—์„œ ์ฝ˜์ด์˜ ์ „๋žต์€ ์ƒํ™ฉ์— ๋”ฐ๋ผ ๋‹ฌ๋ผ์ง€๋ฉฐ, ์˜ˆ๋ฅผ ๋“ค์–ด '1ss'์™€ ๊ฐ™

Game Theory Mathematics Computer Science
No Image

An introduction to ML(n)BiCGStab

1. ์„œ๋ก  ๋ฐ ๋ฐฐ๊ฒฝ ML(n)BiCGStab๋Š” BiCGStab ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ์ž์—ฐ์Šค๋Ÿฌ์šด ์ผ๋ฐ˜ํ™”๋กœ, Yeung๊ณผ Chan์— ์˜ํ•ด 1999๋…„ ์†Œ๊ฐœ๋˜์—ˆ๋‹ค. ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ ์—ฌ๋Ÿฌ ์‹œ์ž‘ ๋žœํฌ๋กœ์Šค ๊ณผ์ •์„ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•˜๋ฉฐ, van der Vorst์˜ BiCGStab์—์„œ ํŒŒ์ƒ๋˜์—ˆ์ง€๋งŒ ๋” ์•ˆ์ •์ ์ด๊ณ  ํšจ์œจ์ ์ธ ์„ฑ๋Šฅ์„ ์ œ๊ณตํ•œ๋‹ค. Sonneveld์™€ van der Vorst๊ฐ€ CGS์™€ BiCGStab๋ฅผ ๊ตฌ์„ฑํ•˜๋Š” ๋ฐ ์‚ฌ์šฉํ•œ ๊ธฐ๋ฒ•์ด ML(n)BiCGStab์—๋„ ์ ์šฉ๋˜์—ˆ๋‹ค. 2. ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ์œ ๋„ ๋ฐ ๊ตฌ์กฐ ML(n)BiCGStab๋Š” ์—ฌ๋Ÿฌ ์‹œ์ž‘ ๋ฒกํ„ฐ๋ฅผ ์ด์šฉํ•˜์—ฌ Kryl

Computer Science Mathematics Numerical Analysis
Centre for Mathematical Sciences India (CMS): Professor A.M. Mathais   75th Birthday

Centre for Mathematical Sciences India (CMS): Professor A.M. Mathais 75th Birthday

CMS๋Š” ์ธ๋„ ์ผ€๋ž„๋ผ์ฃผ์—์„œ ์ค‘์š”ํ•œ ์—ฐ๊ตฌ ๋ฐ ๊ต์œก ์„ผํ„ฐ๋กœ, ๋‹ค์–‘ํ•œ ๋ถ„์•ผ์—์„œ ํ™œ๋ฐœํ•œ ํ™œ๋™์„ ํŽผ์น˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ์ด ์„ผํ„ฐ๋Š” 1977๋…„ ์„ค๋ฆฝ๋˜์–ด ํ˜„์žฌ๊นŒ์ง€ ์ˆ˜์‹ญ ๋…„ ๋™์•ˆ ์ง€์†์ ์œผ๋กœ ๋ฐœ์ „ํ•ด ์™”์Šต๋‹ˆ๋‹ค. CMS์˜ ์ฃผ์š” ํŠน์ง•๊ณผ ์—ญํ• ์€ ๋‹ค์Œ๊ณผ ๊ฐ™์Šต๋‹ˆ๋‹ค. 1. ์—ฐ๊ตฌ ๋ฐ ๊ต์œก ํ”„๋กœ๊ทธ๋žจ CMS๋Š” ๋‹ค์–‘ํ•œ ๋ถ„์•ผ์—์„œ ์—ฐ๊ตฌ๋ฅผ ์ˆ˜ํ–‰ํ•˜๊ณ , ์ด๋ฅผ ํ†ตํ•ด ํ•™๊ณ„์™€ ์‚ฐ์—…๊ณ„์— ๊ธฐ์—ฌํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ํŠนํžˆ ์ˆ˜ํ•™๊ณผ ํ†ต๊ณ„ํ•™ ๋ถ„์•ผ์—์„œ ๋งŽ์€ ์„ฑ๊ณผ๋ฅผ ๊ฑฐ๋‘๊ณ  ์žˆ์œผ๋ฉฐ, ์ด๋Š” CMS์˜ ์ถœํŒ๋ฌผ๊ณผ ๊ฐ•์—ฐ ์‹œ๋ฆฌ์ฆˆ๋ฅผ ํ†ตํ•ด ํ™•์ธํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. CMS๋Š” ๋งค๋…„ 5์ฃผ๊ฐ„ ์ง„ํ–‰๋˜๋Š” SERC ํ•™๊ต๋ผ๋Š” ์—ฐ๊ตฌ ๋ฐฉํ–ฅ์„ฑ ๊ณผ

Mathematics
Preservation of the Borel class under open-$LC$ functions

Preservation of the Borel class under open-$LC$ functions

๋ณธ ๋…ผ๋ฌธ์€ ๋ณด๋  ์ง‘ํ•ฉ๊ณผ ๊ทธ ๋ณด๋  ํด๋ž˜์Šค์— ๋Œ€ํ•œ ์ค‘์š”ํ•œ ์ด๋ก ์  ๊ฒฐ๊ณผ๋ฅผ ์ œ๊ณตํ•œ๋‹ค. ํŠนํžˆ, ๊ฐœ๋ฐฉํ˜• ํด๋กœ์ฆˆ๋“œ(clopen) ํ•จ์ˆ˜๋ผ๋Š” ์ƒˆ๋กœ์šด ๊ฐœ๋…์„ ๋„์ž…ํ•˜์—ฌ, ์ด๋Ÿฌํ•œ ํ•จ์ˆ˜๊ฐ€ ํŠน์ • ์กฐ๊ฑด ํ•˜์—์„œ ๋ณด๋  ์ง‘ํ•ฉ์˜ ๋ณด๋  ํด๋ž˜์Šค๋ฅผ ์œ ์ง€ํ•˜๋Š” ๊ฒƒ์„ ์ฆ๋ช…ํ•˜๊ณ  ์žˆ๋‹ค. 1. ์„œ๋ก  ์„œ๋ก ์—์„œ๋Š” ๋ณด๋  ์ง‘ํ•ฉ๊ณผ ๊ทธ ๋ณด๋  ํด๋ž˜์Šค์— ๋Œ€ํ•œ ๊ธฐ์กด ์—ฐ๊ตฌ๋ฅผ ๊ฐ„๋žตํžˆ ์†Œ๊ฐœํ•œ๋‹ค. ํŠนํžˆ, ๋ณด๋  ์ง‘ํ•ฉ C์˜ ๋ถ€๋ถ„ ์ง‘ํ•ฉ X์™€ ์—ฐ์† ํ•จ์ˆ˜ f: X โ†’ Y๊ฐ€ ์ฃผ์–ด์กŒ์„ ๋•Œ, ๋งŒ์•ฝ ๋ชจ๋“  ํด๋กญ์—” ์„ธํŠธ U์˜ ์ด๋ฏธ์ง€ f(U)๊ฐ€ ์—ด๋ฆฐ ์ง‘ํ•ฉ์ด๊ฑฐ๋‚˜ ๋‹ซํžŒ ์ง‘ํ•ฉ์ด๋ผ๋ฉด, Y๋Š” ๋™์ผํ•œ ๋ณด๋  ํด๋ž˜์Šค๋ฅผ ๊ฐ€์ง„๋‹ค๋Š” ๊ฒฐ๊ณผ๋ฅผ ์–ธ๊ธ‰

Mathematics
Another approach to parametric Bing and Krasinkiewicz maps

Another approach to parametric Bing and Krasinkiewicz maps

๋ณธ ๋…ผ๋ฌธ์€ ๋น„๋‹(Bing) ๋งต๊ณผ ํฌ๋ผ์‹ ํ‚ค์—๋น„์น˜(Krasinkiewicz) ๋งต์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ์ ‘๊ทผ๋ฒ•์„ ์ œ์‹œํ•˜๋ฉฐ, ํŠนํžˆ ํŒŒ๋ผ์ฝคํŒฉํŠธ ๊ณต๊ฐ„์—์„œ ์ด๋Ÿฌํ•œ ๋งต๋“ค์ด ์–ด๋–ป๊ฒŒ ํ•จ์ˆ˜ ๊ณต๊ฐ„ ๋‚ด์—์„œ ๋ฐ€์ง‘ ๋ถ€๋ถ„ ์ง‘ํ•ฉ์„ ํ˜•์„ฑํ•˜๋Š”์ง€ ๋ถ„์„ํ•œ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” Pasynkov์˜ ๊ธฐ๋ฒ•์„ ํ™œ์šฉํ•˜์—ฌ, ๋น„๋‹ ๋ฐ ํฌ๋ผ์‹ ํ‚ค์—๋น„์น˜ ๋งต์ด ์–ด๋–ค ์กฐ๊ฑด ํ•˜์—์„œ ํŠน์ • ์„ฑ์งˆ์„ ๊ฐ–๊ฒŒ ๋˜๋Š”์ง€๋ฅผ ํƒ๊ตฌํ•œ๋‹ค. 1. ๋„์ž… ๋ฐ ๋ฐฐ๊ฒฝ ๋…ผ๋ฌธ์€ ๋น„๋‹ ๋งต๊ณผ ํฌ๋ผ์‹ ํ‚ค์—๋น„์น˜ ๋งต์˜ ์ •์˜์™€ ๊ด€๋ จ๋œ ์ด๋ก ์  ๋ฐฐ๊ฒฝ์„ ์„ค๋ช…ํ•œ๋‹ค. ๋น„๋‹ ๋งต์€ ์ปดํŒฉํŠธ ๊ณต๊ฐ„ ์‚ฌ์ด์˜ ๋งต ์ค‘, ๋ชจ๋“  ์„ฌ์œ ๊ฐ€ ๋น„๋‹ ๊ณต๊ฐ„์ธ ๊ฒฝ์šฐ๋ฅผ ๋งํ•˜๋ฉฐ, ์—ฌ๊ธฐ์„œ ๋น„๋‹ ๊ณต

Mathematics
Determination of Different Biological Factors on the Base of Dried Blood   Spot Technology

Determination of Different Biological Factors on the Base of Dried Blood Spot Technology

๋ณธ ๋…ผ๋ฌธ์€ ๊ฑด์กฐํ˜ˆ์•ก๋ฐฉ์šธ(DBS) ๊ธฐ์ˆ ์„ ํ™œ์šฉํ•œ ํ˜ˆ๋Ÿ‰ ๊ณ„์‚ฐ ๋ฐฉ๋ฒ•์— ๋Œ€ํ•œ ์‹ฌ๋„ ์žˆ๋Š” ๋ถ„์„์„ ์ œ๊ณตํ•ฉ๋‹ˆ๋‹ค. DBS๋Š” ํ™˜์ž๊ฐ€ ์ง์ ‘ ํ˜ˆ์•ก ์ƒ˜ํ”Œ์„ ์ฑ„์ทจํ•˜๊ณ  ์ด๋ฅผ ์—ฌ๊ณผ์ง€๋‚˜ ์„ธํฌ์งˆ ์•„์„ธํ…Œ์ดํŠธ ๋ง‰ ๋“ฑ์— ๋ฐฉ์šธ ํ˜•ํƒœ๋กœ ๊ฑด์กฐํ•˜์—ฌ ์‹คํ—˜์‹ค๋กœ ๋ณด๋‚ด๋Š” ๊ธฐ์ˆ ์ž…๋‹ˆ๋‹ค. ์ด ๋ฐฉ๋ฒ•์€ ๋Œ€์‚ฌ๋ฌผ์งˆ, ํ˜ธ๋ฅด๋ชฌ, ํ˜ˆ๋‹น์ˆ˜์น˜, ๋ฉด์—ญ ์ฒด๊ณ„ ์ง€ํ‘œ ๋“ฑ์˜ ๋‹ค์–‘ํ•œ ์ƒ๋ฌผํ•™์  ํŠน์„ฑ์„ ๋ถ„์„ํ•  ์ˆ˜ ์žˆ๋Š” ๊ธฐํšŒ๋ฅผ ์ œ๊ณตํ•˜๋ฉฐ, ํŠนํžˆ DNA ๋ฐ RNA ๋ถ„์„ ๊ฐ€๋Šฅ์„ฑ์œผ๋กœ ์—์ด์ฆˆ๋‚˜ ๊ฐ„์—ผ๊ณผ ๊ฐ™์€ ๊ฐ์—ผ๋ณ‘์˜ ๋Œ€๋Ÿ‰ ์กฐ์‚ฌ๋ฅผ ๊ฐ€๋Šฅํ•˜๊ฒŒ ํ•ฉ๋‹ˆ๋‹ค. ๋ณธ ์—ฐ๊ตฌ๋Š” DBS ๋ฐฉ์šธ์˜ ์ฒด์ ์„ ๋„์ง• ์žฅ์น˜ ์—†์ด ๊ณ„์‚ฐํ•˜๋Š” ๋ณดํŽธ์ ์ธ ๋ฐฉ๋ฒ•์„ ๊ฐœ๋ฐœํ•˜๊ณ ์ž

Statistics Physics Mathematics
O Algoritmo usado no programa de criptografia PASME

O Algoritmo usado no programa de criptografia PASME

์ด ๋…ผ๋ฌธ์€ PASME ๋„๊ตฌ์˜ ํ•ต์‹ฌ ์•”ํ˜ธํ™” ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ƒ์„ธํžˆ ์„ค๋ช…ํ•˜๊ณ  ์žˆ๋‹ค. ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ ํฐ ์ˆ˜์˜ ์†Œ์ธ์ˆ˜ ๋ถ„ํ•ด๋ผ๋Š” ๊ณ„์‚ฐ์ ์œผ๋กœ ์–ด๋ ค์šด ๋ฌธ์ œ๋ฅผ ํ™œ์šฉํ•˜์—ฌ ๋ฐ์ดํ„ฐ๋ฅผ ์•ˆ์ „ํ•˜๊ฒŒ ์•”ํ˜ธํ™”ํ•œ๋‹ค. ์ด๋ฅผ ํ†ตํ•ด ๊ธฐ์กด ์•”ํ˜ธํ™” ๋ฐฉ๋ฒ•๋“ค์˜ ํ•œ๊ณ„๋ฅผ ๊ทน๋ณตํ•˜๋ ค๋Š” ์‹œ๋„๊ฐ€ ์ด๋ฃจ์–ด์ง€๊ณ  ์žˆ์œผ๋ฉฐ, ํŠนํžˆ ๋นˆ๋„ ๋ถ„์„๊ณผ ๊ฐ™์€ ๊ณต๊ฒฉ์— ์ทจ์•ฝํ•œ ๊ณ ์ „์ ์ธ ์•”ํ˜ธํ™” ๋ฐฉ๋ฒ•๋“ค๋ณด๋‹ค ๋ณด์•ˆ์„ฑ์ด ํ–ฅ์ƒ๋˜์—ˆ๋‹ค. I. ์„œ๋ก  ์„œ๋ก ์—์„œ๋Š” ์•”ํ˜ธํ™” ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ํ•„์š”์„ฑ๊ณผ ๋‹ค์–‘ํ•œ ์•”ํ˜ธํ™” ๊ธฐ๋ฒ•๋“ค์˜ ํ•œ๊ณ„๋ฅผ ์„ค๋ช…ํ•œ๋‹ค. ํŠนํžˆ, ๋ฌธ์ž ์ง‘ํ•ฉ ์ˆœ์„œ ๋ณ€๊ฒฝ์ด๋‚˜ ์ด์ง„ ๋ฉ”์‹œ์ง€์—์„œ ๋น„ํŠธ ์—ญ์ „ ๋“ฑ์˜ ๋ฐฉ๋ฒ•๋“ค์€ ๋นˆ๋„ ๋ถ„์„ ๊ณต๊ฒฉ์— ์ทจ์•ฝํ•˜๋‹ค๋Š” ์ ์„

Mathematics Computer Science Cryptography and Security
Series Prediction based on Algebraic Approximants

Series Prediction based on Algebraic Approximants

์ด ๋…ผ๋ฌธ์€ ๋ณต์žกํ•œ ํ•จ์ˆ˜์˜ ์‹œํ€€์Šค ์˜ˆ์ธก์„ ์œ„ํ•œ ์ƒˆ๋กœ์šด ์ ‘๊ทผ๋ฒ•์„ ์ œ์‹œํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ํŠนํžˆ, ์ด ์—ฐ๊ตฌ๋Š” ๋Œ€์ˆ˜์  ๊ทผ์‚ฌ์™€ ํ—ˆ๋ฏธํŠธ ํŒŒ๋ฐ ๋‹คํ•ญ์‹(HPP)์— ์ค‘์ ์„ ๋‘๊ณ  ์žˆ์œผ๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ๋ณต์žกํ•œ ์‹œํ€€์Šค๋ฅผ ํšจ๊ณผ์ ์œผ๋กœ ์˜ˆ์ธกํ•  ์ˆ˜ ์žˆ๋Š” ๋ฐฉ๋ฒ•๋ก ์„ ๊ฐœ๋ฐœํ•˜์˜€์Šต๋‹ˆ๋‹ค. 1. ๋Œ€์ˆ˜์  ๊ทผ์‚ฌ์˜ ๊ฐœ๋… ๋…ผ๋ฌธ์€ ๋Œ€์ˆ˜์  ๊ทผ์‚ฌ๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ๋ณต์žกํ•œ ํ•จ์ˆ˜์˜ ์ง€์ˆ˜ ๊ธ‰์ˆ˜๋ฅผ ๊ทผ์‚ฌํ•˜๋Š” ๋ฐฉ๋ฒ•์— ๋Œ€ํ•ด ์„ค๋ช…ํ•ฉ๋‹ˆ๋‹ค. ์ด๋Š” ํ—ˆ๋ฏธํŠธ ํŒŒ๋ฐ ๋‹คํ•ญ์‹(HPP)์„ ์ด์šฉํ•ด ์ˆ˜ํ–‰๋˜๋ฉฐ, HPP๋Š” ์ฃผ์–ด์ง„ ํ•จ์ˆ˜ f ๊ฐ€ ์ง€์ˆ˜ ๊ธ‰์ˆ˜ ํ˜•ํƒœ๋กœ ํ‘œํ˜„๋  ๋•Œ, ์ด๋ฅผ ๊ทผ์‚ฌํ•˜๋Š” ๋‹คํ•ญ์‹ ์ง‘ํ•ฉ์ž…๋‹ˆ๋‹ค. 2. ํ—ˆ๋ฏธํŠธ ํŒŒ๋ฐ ๋‹คํ•ญ์‹์˜ ๊ตฌ์„ฑ

Mathematics Physics
Another elementary proof of $: sum_{n ge 1}{1/{n^2}} = pi^2/6,$ and   a recurrence formula for $,zeta{(2k)}$

Another elementary proof of $: sum_{n ge 1}{1/{n^2}} = pi^2/6,$ and a recurrence formula for $,zeta{(2k)}$

: ๋ณธ ๋…ผ๋ฌธ์€ ๋ฆฌ๋งŒ ์ œํƒ€ ํ•จ์ˆ˜ ฮถ(s)์˜ ํŠน๋ณ„ํ•œ ๊ฒฝ์šฐ์ธ ฮถ(2k)์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ์ ‘๊ทผ ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•˜๋ฉฐ, ํŠนํžˆ ฮถ(2) ฯ€ยฒ/6์ด๋ผ๋Š” ์ค‘์š”ํ•œ ๊ฒฐ๊ณผ๋ฅผ ๊ฐ„๋‹จํ•˜๊ฒŒ ์ฆ๋ช…ํ•˜๊ณ  ์žฌ๊ท€ ๊ณต์‹์„ ๋„์ถœํ•ฉ๋‹ˆ๋‹ค. ์ด ๋…ผ๋ฌธ์€ Dancs์™€ He (2006)์˜ ์—ฐ๊ตฌ์—์„œ ์‹œ์ž‘ํ•˜์—ฌ, sin(nฯ€) ๋Œ€์‹  cos(nฯ€)๋ฅผ ์‚ฌ์šฉํ•œ ๊ธ‰์ˆ˜ ์ „๊ฐœ ๋ฐฉ๋ฒ•์„ ํ†ตํ•ด ฮถ(2k)์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ์ฆ๋ช…๊ณผ ์žฌ๊ท€ ๊ณต์‹์„ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค. 1. ์‹ฌํ”Œํ•œ ์ฆ๋ช…๊ณผ ์žฌ๊ท€ ๊ณต์‹ ๋…ผ๋ฌธ์€ ๋จผ์ € s 1์ผ ๋•Œ ํ•ด๋ฐ€ํ„ด ๊ธ‰์ˆ˜๊ฐ€ ๋ฐœ์‚ฐํ•จ์„ ์–ธ๊ธ‰ํ•˜๊ณ , ์ œ๊ณฑ ะ‘ะตั€ะฝัƒะปะปะธ ์ˆ˜ Bk๋ฅผ z/e^z 1์˜ ํƒ€์ผ๋Ÿฌ ๊ธ‰์ˆ˜ ์ „๊ฐœ์—์„œ z

Mathematics
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Recovery of a Sparse Integer Solution to an Underdetermined System of Linear Equations

๋ณธ ๋…ผ๋ฌธ์€ ์Šค๋งˆํŠธ ๊ทธ๋ฆฌ๋“œ์—์„œ ๋ฐœ์ƒํ•˜๋Š” ์ด์ง„ ํ•ด ๋ณต์› ๋ฌธ์ œ๋ฅผ ๋‹ค๋ฃจ๋ฉฐ, ํŠนํžˆ m < n ์ธ ๊ฒฝ์šฐ ๋ฌดํ•œํ•œ ์‹ค์ˆ˜ ํ•ด์™€ ์—ฌ๋Ÿฌ ์ด์ง„ ํ•ด๊ฐ€ ์กด์žฌํ•  ์ˆ˜ ์žˆ๋Š” ์ƒํ™ฉ์„ ๊ณ ๋ คํ•ฉ๋‹ˆ๋‹ค. ์ด๋Ÿฌํ•œ ๋ฌธ์ œ๋Š” NP ํ•˜๋“œ๋กœ ์•Œ๋ ค์ ธ ์žˆ์œผ๋ฉฐ, ์ด๋ฅผ ํ•ด๊ฒฐํ•˜๊ธฐ ์œ„ํ•ด Mangasarian ๋“ฑ์ด ์ œ์•ˆํ•œ ๋ฐฉ๋ฒ•๋ก ์„ ๊ธฐ๋ฐ˜์œผ๋กœ ์—ฐ๊ตฌ๋ฅผ ์ง„ํ–‰ํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. 1. ๋ฌธ์ œ์˜ ๋ฐฐ๊ฒฝ๊ณผ ์ค‘์š”์„ฑ ๋…ผ๋ฌธ์€ ์Šค๋งˆํŠธ ๊ทธ๋ฆฌ๋“œ์—์„œ ๋ฐœ์ƒํ•˜๋Š” ์ด์ง„ ํ•ด ๋ณต์› ๋ฌธ์ œ์— ๋Œ€ํ•ด ๋‹ค๋ฃน๋‹ˆ๋‹ค. ๊ฐ ๊ณ ๊ฐ ๊ฐ€๊ตฌ๋Š” ์ €์ „์•• ๋ณ€์••๊ธฐ์˜ ์„ธ ๊ฐ€์ง€ ์ „์œ„ ์ค‘ ํ•˜๋‚˜์— ์—ฐ๊ฒฐ๋˜์–ด ์žˆ์œผ๋ฉฐ, ์ด๋ฅผ ์ถ”์ถœํ•˜๊ธฐ ์œ„ํ•ด A ํ–‰๋ ฌ์„ ์‚ฌ์šฉํ•ฉ๋‹ˆ๋‹ค. ์—ฌ๊ธฐ์„œ A ์˜ ์—ด

Information Theory Machine Learning Computer Science System Discrete Mathematics Mathematics
A 1-dimensional Peano continuum which is not an IFS attractor

A 1-dimensional Peano continuum which is not an IFS attractor

: ๋ณธ ๋…ผ๋ฌธ์€ ๋ณต์žกํ•œ ์œ„์ƒ ๊ณต๊ฐ„ ์ด๋ก ๊ณผ ๊ด€๋ จ๋œ ์ค‘์š”ํ•œ ๋ฌธ์ œ๋ฅผ ๋‹ค๋ฃจ๊ณ  ์žˆ๋‹ค. ํŠนํžˆ, ๋ฌดํ•œ ์ฐจ์›์˜ ํ‰๋ฉด Peano ์—ฐ์†์ฒด๊ฐ€ IFS attractor์™€ ๋™ํ˜•์ด ๋  ์ˆ˜ ์žˆ๋Š”์ง€์— ๋Œ€ํ•œ ์งˆ๋ฌธ์„ ์ œ๊ธฐํ•˜๊ณ  ์ด๋ฅผ ๋ถ€์ •์ ์œผ๋กœ ํ•ด๊ฒฐํ•œ๋‹ค. IFS attractor๋Š” ๋ฐ˜๋ณต ํ•จ์ˆ˜ ์‹œ์Šคํ…œ(Iterated Function System)์„ ํ†ตํ•ด ์ƒ์„ฑ๋˜๋Š” ์ง‘ํ•ฉ์œผ๋กœ, ์••์ถ•๋œ ๋ฉ”ํŠธ๋ฆญ ๊ณต๊ฐ„์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•œ๋‹ค. ์ด ๋…ผ๋ฌธ์—์„œ๋Š” IFS attractor์˜ ์œ„์ƒํ•™์  ์„ฑ์งˆ์— ๋Œ€ํ•ด ๊นŠ์ด ์žˆ๊ฒŒ ๋ถ„์„ํ•œ๋‹ค. ํŠนํžˆ, ์—ฐ๊ฒฐ๋œ IFS attractor๋Š” ๊ตญ์†Œ์ ์œผ๋กœ ์—ฐ๊ฒฐ๋˜์–ด ์žˆ๊ณ , ์†์„ฑ

Mathematics
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A Useful Property of the Finite Nonabelian Groups

: ์ด ๋…ผ๋ฌธ์€ ์œ ํ•œ ๋น„์•„๋ฒจ ๊ตฐ์˜ ๊ตฌ์กฐ์— ๋Œ€ํ•ด ๊นŠ๊ฒŒ ํƒ๊ตฌํ•˜๋ฉฐ, ํŠนํžˆ ๊ทธ ์ค‘์‹ฌ๊ณผ ์ตœ๋Œ€ ์•„๋ฒจ ๋ถ€๋ถ„๊ตฐ ์‚ฌ์ด์˜ ๊ด€๊ณ„๋ฅผ ๋ถ„์„ํ•œ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ๊ตฐ๋ก ์—์„œ ์ค‘์š”ํ•œ ๊ฐœ๋…์ธ Z ์˜์กด์„ฑ์„ ํ†ตํ•ด ์ด๋Ÿฌํ•œ ๊ตฐ๋“ค์˜ ํŠน์„ฑ์„ ์ดํ•ดํ•˜๋Š”๋ฐ ์ค‘์ ์„ ๋‘”๋‹ค. 1. ์ •์˜์™€ ๊ธฐ๋ณธ ๊ฐœ๋… ๋…ผ๋ฌธ์€ ์œ ํ•œ ๋น„์•„๋ฒจ ๊ตฐ G์— ๋Œ€ํ•ด ์ค‘์‹ฌ Z์™€ ์ตœ๋Œ€ ์•„๋ฒจ ๋ถ€๋ถ„๊ตฐ H i (i 1,2,...,r)๋ฅผ ์ •์˜ํ•œ๋‹ค. ์—ฌ๊ธฐ์„œ ๊ฐ H i๋Š” ์„œ๋กœ ๋‹ค๋ฅธ ์ตœ๋Œ€ ์•„๋ฒจ ๋ถ€๋ถ„๊ตฐ์ด๋ฉฐ, ๋ชจ๋“  i, j์— ๋Œ€ํ•ด i j์ผ ๋•Œ๋งŒ H i H j๊ฐ€ ์„ฑ๋ฆฝํ•œ๋‹ค. ๋˜ํ•œ G๊ฐ€ Z ์˜์กด์ ์ด๋ผ๋Š” ๊ฐœ๋…์„ ๋„์ž…ํ•˜๋Š”๋ฐ, ์ด๋Š” ๋‘ ๋ถ€๋ถ„๊ตฐ H i์™€

Mathematics
Reproductive and non-reproductive solutions of the matrix equation AXB=C

Reproductive and non-reproductive solutions of the matrix equation AXB=C

Catchy Title KO: ์žฌ์ƒ์„ฑ๊ณผ ๋น„์žฌ์ƒ์„ฑ ํ•ด๋ฅผ ํ†ตํ•œ ํ–‰๋ ฌ ๋ฐฉ์ •์‹ AXB C์˜ ํ•ด๊ฒฐ Abstract KO: ๋ณธ ๋…ผ๋ฌธ์€ S. B. Preลกiฤ‡๊ฐ€ ๋„์ž…ํ•œ ์žฌ์ƒ์‹ ๋ฐฉ์ •์‹์˜ ๊ฐœ๋…์„ ๋ฐ”ํƒ•์œผ๋กœ, ํ–‰๋ ฌ ๋ฐฉ์ •์‹ AXB C์— ๋Œ€ํ•œ ํ•ด๋ฅผ ๋ถ„์„ํ•œ๋‹ค. ํŠนํžˆ, ์ด ๋…ผ๋ฌธ์—์„œ๋Š” ์žฌ์ƒ์„ฑ์  ํ•ด์™€ ๋น„์žฌ์ƒ์„ฑ์  ํ•ด์˜ ๊ตฌ๋ถ„๊ณผ ๊ทธ ํ•ด์˜ ์ผ๋ฐ˜์ ์ธ ํ˜•ํƒœ๋ฅผ ๋‹ค๋ฃฌ๋‹ค. R. Penrose์˜ ์ •๋ฆฌ์— ๋”ฐ๋ผ, ์ผ๊ด€๋œ ํ–‰๋ ฌ ๋ฐฉ์ •์‹ AXB C์˜ ์ผ๋ฐ˜ ํ•ด๋Š” ํŠน์ • ์กฐ๊ฑด ํ•˜์—์„œ {1} ์—ญํ–‰๋ ฌ์„ ์‚ฌ์šฉํ•˜์—ฌ ํ‘œํ˜„๋  ์ˆ˜ ์žˆ๋‹ค. ๋˜ํ•œ, Preลกiฤ‡์˜ ๊ฒฐ๊ณผ์™€ Haveriฤ‡์˜ ์—ฐ๊ตฌ๋ฅผ ํ†ตํ•ด ์žฌ์ƒ์„ฑ์  ํ•ด

Mathematics
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Monochromatic Progressions in Random Colorings

: 1. ๋ฐ˜๋”์™€๋ฅด๋ด ์ •๋ฆฌ์™€ W(k) ๋ฐ˜๋”์™€๋ฅด๋ด ์ •๋ฆฌ๋Š” ๋žจ์ง€ ์ด๋ก ์—์„œ ์ค‘์š”ํ•œ ์œ„์น˜๋ฅผ ์ฐจ์ง€ํ•˜๋ฉฐ, ๋ชจ๋“  ์–‘์˜ ์ •์ˆ˜ k์— ๋Œ€ํ•ด W(k)๋ผ๋Š” ์ƒํ•œ์„ ์„ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค. ์ด๋Š” {1, 2, ..., W(k)}์˜ 2์ƒ‰ ๋ถ„๋ฅ˜์—์„œ kํ•ญ ์•„๋ ˆํŠธ๋ฆญ ์ง„ํ–‰์ด ๋ฐ˜๋“œ์‹œ ๋ชจ๋…ธํฌ๋กฌ์œผ๋กœ ๋‚˜ํƒ€๋‚œ๋‹ค๋Š” ๊ฒƒ์„ ์˜๋ฏธํ•ฉ๋‹ˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ W(k)์˜ ์ •ํ™•ํ•œ ๊ฐ’์€ k๊ฐ€ ์ž‘์„ ๋•Œ๋งŒ ์•Œ๋ ค์ ธ ์žˆ์œผ๋ฉฐ, k๊ฐ€ ์ปค์งˆ์ˆ˜๋ก ์ด ๊ฐ’์„ ๊ตฌํ•˜๋Š” ๊ฒƒ์ด ๋งค์šฐ ์–ด๋ ค์›Œ์ง‘๋‹ˆ๋‹ค. 2. ํ™•๋ฅ ์  ์ ‘๊ทผ: N+(k)์™€ N (k) N+(k)๋Š” {1, 2, ..., N+(k)}์˜ 2์ƒ‰ ๋ถ„๋ฅ˜์—์„œ kํ•ญ ์•„๋ ˆํŠธ๋ฆญ ์ง„ํ–‰์ด ํฌํ•จ๋  ํ™•๋ฅ ์ด

Computer Science Mathematics Discrete Mathematics
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Five Exponential Diophantine Equations and Mayhem Problem M429

: ์ด ๋…ผ๋ฌธ์€ ์ง€์ˆ˜ํ˜• ๋””์˜คํŒํ† ์Šค ๋ฐฉ์ •์‹์— ๋Œ€ํ•œ ๊นŠ์ด ์žˆ๋Š” ์—ฐ๊ตฌ๋ฅผ ์ œ๊ณตํ•˜๋ฉฐ, ํŠนํžˆ ์„ธ ๋ณ€์ˆ˜ ๋””์˜คํŒํ† ์Šค ๋ฐฉ์ •์‹์˜ ํ•ด ์ง‘ํ•ฉ์„ ๋ถ„์„ํ•˜๋Š” ๋ฐ ์ค‘์ ์„ ๋‘๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ์ด๋Š” Crux Mathematicorum ์ €๋„์—์„œ ์ œ์‹œ๋œ ํ˜ผ๋ž€ ๋ฌธ์ œ M429๋กœ๋ถ€ํ„ฐ ์‹œ์ž‘๋˜๋ฉฐ, ๊ทธ ํ•ด๊ฒฐ์ฑ…์€ 2010๋…„ 12์›”์— ๊ฒŒ์žฌ๋˜์—ˆ์Šต๋‹ˆ๋‹ค. ๋…ผ๋ฌธ์˜ ํ•ต์‹ฌ ๋‚ด์šฉ์€ ์„ธ ๋ณ€์ˆ˜ ๋””์˜คํŒํ† ์Šค ๋ฐฉ์ •์‹ a(bc) (ab)c๋ฅผ ๋‹ค๋ฃจ๋Š” ๊ฒƒ์ž…๋‹ˆ๋‹ค. ์ด ๋ฐฉ์ •์‹์„ ๋งŒ์กฑํ•˜๋Š” ์ •์ˆ˜ ํ•ด ์ง‘ํ•ฉ์€ ๋‹ค์Œ๊ณผ ๊ฐ™์ด ๋ถ„๋ฅ˜๋ฉ๋‹ˆ๋‹ค: (1, b, c) ํ˜•ํƒœ: ์—ฌ๊ธฐ์„œ b์™€ c๋Š” ์–‘์˜ ์ •์ˆ˜์ผ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. (a, b, 1) ํ˜•

Mathematics
No Image

How to Lose with Least Probability

: ์ด ๋…ผ๋ฌธ์€ ์•จ๋ฆฌ์Šค์™€ ๋ฐฅ์ด ์ฐธ์—ฌํ•˜๋Š” ๊ฒŒ์ž„์—์„œ ์•จ๋ฆฌ์Šค๊ฐ€ ์Šน๋ฆฌํ•  ํ™•๋ฅ  I(p | n, ฮฑ, ฮฒ)๋ฅผ ์ˆ˜ํ•™์ ์œผ๋กœ ๋ถ„์„ํ•˜๊ณ  ์ตœ์ ์˜ ๋™์ „ ํŽธํ–ฅ์„ ์ฐพ๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. ์ด ๊ฒŒ์ž„์€ ํ”Œ๋ ˆ์ด์–ด๋“ค์ด ๋ฒˆ๊ฐˆ์•„๊ฐ€๋ฉฐ ๋™์ „์„ ๋˜์ ธ ๊ผฌ๋ฆฌ์— ฮฑ ํฌ์ธํŠธ, ๋จธ๋ฆฌ์— ฮฑ + ฮฒ ํฌ์ธํŠธ๋ฅผ ํš๋“ํ•˜๋ฉฐ, ๋จผ์ € n ํฌ์ธํŠธ๋ฅผ ์–ป๋Š” ํ”Œ๋ ˆ์ด์–ด๊ฐ€ ์Šน๋ฆฌํ•˜๋Š” ๋ฐฉ์‹์œผ๋กœ ์ง„ํ–‰๋œ๋‹ค. ๋…ผ๋ฌธ์€ I(p | n, ฮฑ, ฮฒ)์˜ ์„ฑ์งˆ์„ ๋ถ„์„ํ•˜๊ณ  ์ด๋ฅผ ์ตœ์†Œํ™”ํ•˜๋Š” ๋™์ „ ํŽธํ–ฅ p n์— ๋Œ€ํ•ด ๊นŠ์ด ์žˆ๊ฒŒ ๋‹ค๋ฃฌ๋‹ค. ๊ฒŒ์ž„์˜ ๊ธฐ๋ณธ ์›์น™๊ณผ ์ˆ˜ํ•™์  ๋ชจ๋ธ๋ง ๊ฒŒ์ž„์—์„œ ์•จ๋ฆฌ์Šค์™€ ๋ฐฅ์€ ๋ฒˆ๊ฐˆ์•„๊ฐ€๋ฉฐ ๋™์ „์„ ๋˜์ง„๋‹ค. ๋™์ „์˜ ๋จธ๋ฆฌ๋‚˜

Computer Science Game Theory Mathematics
No Image

Dominance in the Monty Hall Problem

๋งค๋ ฅ์ ์ธ ํ•œ๊ธ€ ์ œ๋ชฉ: ๋ชฌํ‹ฐ ํ™€ ๋ฌธ์ œ์—์„œ ์ „๋žต์˜ ์ง€๋ฐฐ์„ฑ๊ณผ ์ตœ์ ์„ฑ ์ดˆ๋ก ์ „์ฒด ๋ฒˆ์—ญ ๋ฐ ์ •๋ฆฌ: ๋ชฌํ‹ฐ ํ™€ ๋ฌธ์ œ๋Š” ์„ธ ๊ฐœ์˜ ๋ฌธ ์ค‘ ํ•˜๋‚˜๊ฐ€ ์ƒ์„ ์ˆจ๊ธฐ๊ณ , ๋‚˜๋จธ์ง€ ๋‘ ๋ฌธ์€ ํ—ˆ์šธ๋ฟ์ธ ๋‹ต๋ณ€์„ ์ œ๊ณตํ•˜๋Š” ๊ณ ์ „์ ์ธ ํ™•๋ฅ  ๋ฌธ์ œ๊ฐ€๋ฉฐ, ํ”Œ๋ ˆ์ด์–ด๋Š” ํ•œ ๋ฌธ์„ ์„ ํƒํ•˜๊ณ  ์ง„ํ–‰์ž๋Š” ์„ ํƒํ•˜์ง€ ์•Š์€ ๋ฌธ ์ค‘ ํ•˜๋‚˜๋ฅผ ์—ด์–ด ์ƒ์ด ์—†๋Š” ๊ฒƒ์„ ๋“œ๋Ÿฌ๋‚ด๋ฉฐ, ์ดํ›„ ํ”Œ๋ ˆ์ด์–ด์—๊ฒŒ ์„ ํƒํ•œ ๋ฌธ์„ ๊ณ ์ˆ˜ํ• ์ง€ ๋‹ค๋ฅธ ๋ฌธ์œผ๋กœ ์ „ํ™˜ํ• ์ง€๋ฅผ ๊ฒฐ์ •ํ•˜๊ฒŒ ํ•ฉ๋‹ˆ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ์ด ๋ฌธ์ œ์— ๋‚ด์žฌ๋œ ์ง€๋ฐฐ์„ฑ ๊ฐœ๋…์„ ๋ถ„์„ํ•˜๊ณ , ํ•ญ์ƒ ์ „ํ™˜ ์ „๋žต์˜ ์ตœ์ ์„ฑ์„ ์ฆ๋ช…ํ•˜๋ฉฐ, ๋ฒ ์ด์ฆˆ์•ˆ ๊ด€์ ์—์„œ ์ตœ์ ์˜ ์ „๋žต์„ ํƒ๊ตฌํ•ฉ๋‹ˆ๋‹ค. ์‹ฌ๋„ ๋ถ„์„

Computer Science Mathematics Game Theory
Generalizations of Holders and some related integral inequalities on   fractal space

Generalizations of Holders and some related integral inequalities on fractal space

: ๋ณธ ์—ฐ๊ตฌ๋Š” ์ˆ˜ํ•™์  ์ด๋ก  ์ค‘ ํ•˜๋‚˜์ธ ํ˜ธ๋”(Hรถlder) ๋ถˆํ‰๋“ฑ์— ์ดˆ์ ์„ ๋งž์ถ”๊ณ , ์ด๋ฅผ ์ง€์—ญ ๋ถ„์ˆ˜ ๋ฏธ์ ๋ถ„ํ•™์ด๋ผ๋Š” ๋ณต์žกํ•œ ์ˆ˜ํ•™ ์˜์—ญ์œผ๋กœ ํ™•์žฅํ•˜๋Š” ๋ฐ ์ฃผ๋ ฅํ•˜๊ณ  ์žˆ๋‹ค. ํ˜ธ๋” ๋ถˆํ‰๋“ฑ์€ ํ•จ์ˆ˜ ๊ณต๊ฐ„์—์„œ์˜ ๋ฒกํ„ฐ ๋‚ด์ ๊ณผ ๊ด€๋ จ๋œ ์ค‘์š”ํ•œ ๊ฒฐ๊ณผ๋กœ, ๋‹ค์–‘ํ•œ ์‘์šฉ ๋ถ„์•ผ์—์„œ ํ™œ์šฉ๋˜๊ณ  ์žˆ๋‹ค. 1. ํ˜ธ๋” ๋ถˆํ‰๋“ฑ์˜ ์ผ๋ฐ˜ํ™” ๋…ผ๋ฌธ์—์„œ๋Š” ๋จผ์ € ๊ธฐ์กด์˜ ํ˜ธ๋” ๋ถˆํ‰๋“ฑ์„ ๋ณต์Šตํ•˜๊ณ  ์ด๋ฅผ ์ง€์—ญ ๋ถ„์ˆ˜ ๋ฏธ์ ๋ถ„ํ•™์— ์ ์šฉํ•˜๊ธฐ ์œ„ํ•œ ์ƒˆ๋กœ์šด ํ˜•ํƒœ์˜ ๋ถˆํ‰๋“ฑ์„ ์ œ์‹œํ•œ๋‹ค. ํŠนํžˆ, pโ‚๊ณผ pโ‚‚๊ฐ€ ์–‘์ˆ˜์ผ ๋•Œ์™€ 0 < pโ‚ < 1์ด๊ณ  pโ‚‚ < 0์ธ ๊ฒฝ์šฐ์— ๋Œ€ํ•ด ๊ฐ๊ฐ ๋‹ค๋ฅธ ์ผ๋ฐ˜ํ™”๋œ ๊ฒฐ๊ณผ๋ฅผ ๋„์ถœํ•œ๋‹ค.

Mathematics
On a property of the $n$-dimensional cube

On a property of the $n$-dimensional cube

๋ณธ ๋…ผ๋ฌธ์€ ๊ณ ์ฐจ์› ๊ทธ๋ž˜ํ”„ ์ด๋ก ์—์„œ ์ค‘์š”ํ•œ ๋ฌธ์ œ ์ค‘ ํ•˜๋‚˜์ธ n์ฐจ์› ํ๋ธŒ์˜ ๋ถ€๋ถ„ ์ง‘ํ•ฉ์— ๋Œ€ํ•œ ์—ฐ๊ตฌ๋ฅผ ์ˆ˜ํ–‰ํ•˜๊ณ  ์žˆ๋‹ค. ํŠนํžˆ, n โ‰ฅ 4์ผ ๋•Œ, |V'| โ‰ฅ 2^(n 1) + 1์ธ ๋ชจ๋“  ๋ถ€๋ถ„ ์ง‘ํ•ฉ V'์— ๋Œ€ํ•ด ํด๋ผ์šฐ ๋˜๋Š” ๋‹จ์ˆœ ์ˆœํ™˜์„ ์œ ๋„ํ•˜๋Š” ์ •์ ๋“ค์˜ ์กด์žฌ๋ฅผ ์ฆ๋ช…ํ•œ๋‹ค. ์„œ๋ก  ๋ฐ ์ •์˜ ๋…ผ๋ฌธ์€ ๊ทธ๋ž˜ํ”„ ์ด๋ก ์—์„œ ์ค‘์š”ํ•œ ๊ฐœ๋…๋“ค์„ ์†Œ๊ฐœํ•˜๊ณ , n์ฐจ์› ํ๋ธŒ Qn๊ณผ ํด๋ผ์šฐ(K1,3)์˜ ์ •์˜๋ฅผ ์ œ๊ณตํ•œ๋‹ค. ํŠนํžˆ, n์ฐจ์› ํ๋ธŒ๋Š” ๊ฐ ์ฐจ์›๋งˆ๋‹ค ๋‘ ๊ฐœ์˜ ๊ฐ’์„ ๊ฐ€์งˆ ์ˆ˜ ์žˆ๋Š” ์ •์  ์ง‘ํ•ฉ์œผ๋กœ ๊ตฌ์„ฑ๋˜๋ฉฐ, ํด๋ผ์šฐ๋Š” ์ค‘์‹ฌ ์ •์ ์„ ํฌํ•จํ•˜๋Š” ์™„์ „ ์ด๋ถ„ ๊ทธ๋ž˜ํ”„ K1,3์„ ์˜๋ฏธํ•œ

Mathematics Computer Science Discrete Mathematics
Fourier Cosine and Sine Transform on fractal space

Fourier Cosine and Sine Transform on fractal space

์ด ๋…ผ๋ฌธ์€ ์–‘ ํ‘ธ๋ฆฌ์— ๋ณ€ํ™˜๊ณผ ๊ทธ ์‘์šฉ์„ ์ค‘์‹ฌ์œผ๋กœ ํ”„๋ž™ํƒˆ ๊ณต๊ฐ„์—์„œ์˜ ํ•จ์ˆ˜ ํ•ด์„ ๋ฐฉ๋ฒ•๋ก ์„ ํƒ์ƒ‰ํ•ฉ๋‹ˆ๋‹ค. ํŠนํžˆ, ์ด ์—ฐ๊ตฌ๋Š” ๋ถˆ๊ทœ์น™ํ•œ ์—ฐ์† ํ”„๋ž™ํƒˆ ํ•จ์ˆ˜๋ฅผ ๋‹ค๋ฃจ๋Š”๋ฐ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋Š” ์ง€์—ญ ๋ถ„์‚ฐ ๋ฏธ์ ๋ถ„ํ•™์„ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•œ ์–‘ ํ‘ธ๋ฆฌ์— ๋ณ€ํ™˜์„ ์†Œ๊ฐœํ•˜๊ณ , ์ด๋ฅผ ํ†ตํ•ด ์ฝ”์‚ฌ์ธ๊ณผ ์‚ฌ์ธ ๋ณ€ํ™˜์˜ ์ƒˆ๋กœ์šด ํ˜•ํƒœ๋ฅผ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค. ์ด๋Ÿฌํ•œ ๋ณ€ํ™˜๋“ค์€ ๊ณตํ•™ ๋ฌธ์ œ ํ•ด๊ฒฐ์—์„œ ์œ ์šฉํ•˜๋ฉฐ, ํŠนํžˆ ํ”„๋ž™ํƒˆ ์ฐจ์› ฮฑ ์— ๋Œ€ํ•œ ํ•จ์ˆ˜๋“ค์˜ ๋ถ„์„์— ์ค‘์š”ํ•œ ๋„๊ตฌ๊ฐ€ ๋ฉ๋‹ˆ๋‹ค. 1. ์–‘ ํ‘ธ๋ฆฌ์— ๋ณ€ํ™˜์˜ ์ •์˜์™€ ์„ฑ์งˆ ์–‘ ํ‘ธ๋ฆฌ์— ๋ณ€ํ™˜์€ ์ง€์—ญ ๋ถ„์‚ฐ ๋ฏธ์ ๋ถ„ํ•™์„ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•˜๋ฉฐ, ์ด๋Š” ํ”„๋ž™ํƒˆ ๊ณต๊ฐ„์—์„œ์˜ ์—ฐ์† ํ•จ์ˆ˜๋ฅผ

Mathematics
A conjecture on independent sets and graph covers

A conjecture on independent sets and graph covers

๋ณธ ๋…ผ๋ฌธ์€ ๊ทธ๋ž˜ํ”„ ์ด๋ก ์˜ ํ•ต์‹ฌ ๊ฐœ๋…์ธ ๋…๋ฆฝ ์ง‘ํ•ฉ๊ณผ ์ปค๋ฒ„๋ฅผ ๋‹ค๋ฃจ๋ฉฐ, ํŠนํžˆ M ์ปค๋ฒ„์™€ ๋ฒ ํ…Œ ๊ทผ์‚ฌ(Bethe approximation) ์‚ฌ์ด์˜ ๊ด€๊ณ„๋ฅผ ํƒ๊ตฌํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ์—ฐ๊ตฌ๋Š” ํ†ต๊ณ„ ๋ฌผ๋ฆฌํ•™์—์„œ ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•˜๋Š” ๋ถ„ํ•  ํ•จ์ˆ˜(partition function)์™€ ์ž์œ  ์—๋„ˆ์ง€(free energy)์— ๋Œ€ํ•œ ์ดํ•ด๋ฅผ ํ™•์žฅํ•˜๋Š” ๋ฐ ๊ธฐ์—ฌํ•œ๋‹ค. 1. ๋‹ค๋ณ€์ˆ˜ ๋…๋ฆฝ ์ง‘ํ•ฉ ๋‹คํ•ญ์‹๊ณผ M ์ปค๋ฒ„ ๋…ผ๋ฌธ์€ ๊ทธ๋ž˜ํ”„ G์˜ ๋‹ค๋ณ€์ˆ˜ ๋…๋ฆฝ ์ง‘ํ•ฉ ๋‹คํ•ญ์‹์„ ์ •์˜ํ•˜๊ณ , ์ด๋ฅผ ํ†ตํ•ด ๋…๋ฆฝ ์ง‘ํ•ฉ์˜ ํฌ๊ธฐ์— ๋”ฐ๋ฅธ ๊ฐ€์ค‘์น˜๋ฅผ ํ‘œํ˜„ํ•œ๋‹ค. ์ด ๋‹คํ•ญ์‹์€ G์˜ ๋ชจ๋“  ๋…๋ฆฝ ์ง‘ํ•ฉ I์— ๋Œ€ํ•ด x^{|I|

Mathematics Computer Science Discrete Mathematics

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