Mathematics / Math.RA

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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
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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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