Mathematics / Math.LO

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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
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
SPM Bulletin 32

SPM Bulletin 32

[Catchy Title KO]: ์นœ๊ตฌ Ljubiลกa Koฤinac์˜ ์ƒ์ผ์„ ๊ธฐ๋…ํ•˜๋Š” ํ† ํฌ๋กœ์ง€ ์›Œํฌ์ˆ [Abstract KO]: ์ด ๋ฌธ์„œ๋Š” ์นœ๊ตฌ์ธ Ljubiลกa Koฤinac ์”จ์˜ 65์„ธ ์ƒ์ผ์„ ๊ธฐ๋…ํ•˜์—ฌ ์กฐ์ง๋œ IV ์›Œํฌ์ˆ: ํ† ํฌํ•™์—์„œ์˜ ๋ฎ๊ฐœ, ์„ ํƒ ๋ฐ ๊ฒŒ์ž„ ์— ๋Œ€ํ•œ ์•ˆ๋‚ด์ž…๋‹ˆ๋‹ค. ์ด ์›Œํฌ์ˆ์€ ์„ธ์ฝ˜๋‹ค ์œ ๋‹ˆ๋ฒ ๋ฅด์‹œํƒ€ ๋”” ๋‚˜ํด๋ฆฌ ์ˆ˜ํ•™๋ถ€์—์„œ 3์›” 2์ผ(๊ธˆ)์— ๊ฐœ์ตœ๋  ์˜ˆ์ •์ด๋ฉฐ, ๊ฐ ๋ฐœํ‘œ๋Š” ์•ฝ 30๋ถ„๊ฐ„ ์ง„ํ–‰๋ฉ๋‹ˆ๋‹ค. ๋˜ํ•œ, ์žฌ์ • ์ƒํ™ฉ์œผ๋กœ ์ธํ•ด ์ฐธ๊ฐ€ ์ง€์› ๋ฐ ๋“ฑ๋ก๋น„ ์ •๋ณด๋Š” ์•„์ง ํ™•์ •๋˜์ง€ ์•Š์•˜์Šต๋‹ˆ๋‹ค. ์‹ฌ๋„ ๋ถ„์„ (Deep Analysis KO): ์ด

Mathematics
mizar-items: Exploring fine-grained dependencies in the Mizar Mathematical Library

mizar-items: Exploring fine-grained dependencies in the Mizar Mathematical Library

๋ณธ ๋…ผ๋ฌธ์€ MML(Mathematical Meta Language) ๋‚ด์˜ ๋ฏธ์„ธํ•œ ์˜์กด์„ฑ์„ ๋ถ„์„ํ•˜๋Š” ์‹œ์Šคํ…œ์„ ๊ฐœ๋ฐœํ•˜๊ณ  ๊ทธ ํ™œ์šฉ ๋ฐฉ๋ฒ•์— ๋Œ€ํ•ด ์„ค๋ช…ํ•œ๋‹ค. MML์€ ์ˆ˜ํ•™์  ์ฆ๋ช…๊ณผ ์ •๋ฆฌ๊ฐ€ ๊ณต์‹ํ™”๋œ ๊ฐ€์žฅ ํฐ ๋ฐ์ดํ„ฐ๋ฒ ์ด์Šค ์ค‘ ํ•˜๋‚˜๋กœ, ์•ฝ 1,100๊ฐœ์˜ ๊ธฐ์‚ฌ์™€ 250๋งŒ ์ค„ ์ด์ƒ์˜ ํ…์ŠคํŠธ๋ฅผ ํฌํ•จํ•˜๋ฉฐ, ์ด๋Š” 5๋งŒ ๊ฐœ ์ด์ƒ์˜ ์ •๋ฆฌ์™€ 1๋งŒ ๊ฐœ ์ด์ƒ์˜ ์ •์˜๋ฅผ ์ˆ˜ํ•™์  ๊ธฐํ˜ธ๋กœ ํ‘œํ˜„ํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ๋ฐฉ๋Œ€ํ•œ ๋ฐ์ดํ„ฐ๋Š” ๋‹ค์–‘ํ•œ ์‹คํ—˜์„ ์ˆ˜ํ–‰ํ•˜๊ธฐ ์œ„ํ•œ ์ฝ”ํผ์Šค๋กœ์„œ ๊ฐ€์น˜๊ฐ€ ํฌ๋‹ค. ๋…ผ๋ฌธ์—์„œ ์ œ์‹œ๋œ Mizar items ์‹œ์Šคํ…œ์€ MML ๋‚ด์˜ ๋ฏธ์„ธ ์˜์กด์„ฑ์„ ๊ณ„์‚ฐํ•˜๋Š” ๋ฐ ์ค‘์ ์„ ๋‘๊ณ 

Digital Libraries Mathematics Computer Science
$sigma$-homogeneity of Borel sets

$sigma$-homogeneity of Borel sets

๋ณธ ๋…ผ๋ฌธ์€ ๋ณด๋  ์ง‘ํ•ฉ์˜ ๋™์งˆ์„ฑ์— ๋Œ€ํ•œ ๊นŠ์ด ์žˆ๋Š” ์ดํ•ด๋ฅผ ์ œ๊ณตํ•˜๋ฉฐ, ํŠนํžˆ ์นธํ† ์–ด ์ง‘ํ•ฉ C ๋‚ด์—์„œ ๋ณด๋  ์ง‘ํ•ฉ์˜ ๊ตฌ์กฐ์™€ ๊ทธ ์„ฑ์งˆ์„ ํƒ๊ตฌํ•œ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” h ๋™์งˆ ๊ณต๊ฐ„์ด๋ผ๋Š” ๊ฐœ๋…์„ ์ค‘์‹ฌ์œผ๋กœ ์ง„ํ–‰๋˜๋ฉฐ, ์ด๋ฅผ ํ†ตํ•ด ๋ณด๋  ์ง‘ํ•ฉ์˜ ๋™์งˆ์„ฑ์„ ๋ถ„์„ํ•˜๊ณ  ์žˆ๋‹ค. 1. ๊ธฐ๋ณธ ๊ฐœ๋… ๋ฐ ์ •์˜ ๋…ผ๋ฌธ์—์„œ๋Š” ๋จผ์ € ๋ช‡ ๊ฐ€์ง€ ์ค‘์š”ํ•œ ์šฉ์–ด๋ฅผ ์ •์˜ํ•œ๋‹ค: ์นธํ† ์–ด ์ง‘ํ•ฉ (C) : ์ด์‚ฐ์  ์ˆ˜์™€ ํ•ฉ๋ฆฌ์  ์ˆ˜๋ฅผ ๊ฐ๊ฐ P, Q๋กœ ํ‘œ๊ธฐํ•˜๋ฉฐ, ์‹ค์ˆ˜ R์€ P์™€ Q์˜ ํ•ฉ์ง‘ํ•ฉ์œผ๋กœ ํ‘œํ˜„๋œ๋‹ค. h ๋™์งˆ ๊ณต๊ฐ„ : 0์ฐจ์› ํ† ํด๋กœ์ง€ ๊ณต๊ฐ„ X๊ฐ€ ๋ชจ๋“  ๋น„๊ณตํ—ˆ ํด๋กœํ”„ ์—ด๋ถ„ U์— ๋Œ€ํ•ด U์™€ X๊ฐ€ ํ™ˆ๋ชจ๋ฅดํ”ฝ(h

Mathematics

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