Về iđêan đường dẫn của đồ thị đường

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ▲➊ ❚❍➚ ❍➬◆● ❆◆❍ ❱➋ ■✣➊❆◆ ✣×❮◆● ❉❼◆ ❈Õ❆ ✣➬ ❚❍➚ ✣×❮◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ▲➊ ❚❍➚ ❍➬◆● ❆◆❍ ❱➋ ■✣➊❆◆ ✣×❮◆● ❉❼◆ ế ì ị ▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✹ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ▲❯❾◆ ữớ ữợ ❚❙✳ ❚❍■➋❯ ✣➐◆❍ P❍❖◆● ◆❣❤➺ ❆♥ ✲ ✷✵✶✽ ✶ ▼Ö❈ ▲Ư❈ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ỗ t ữớ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ P❤ù❝ ✤ì♥ ❤➻♥❤✱ ✐✤➯❛♥ ❙t❛♥❧❡②✲❘❡✐s♥❡r ✈➔ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✳ ✷ ✺ ✺ ✼ ❈❤÷ì♥❣ ✷✳ ❈❤✐➲✉ ①↕ ↔♥❤ ✈➔ ❝❤➾ sè q ữớ ỗ t ữớ ✶✸ ✷✳✶✳ ▼ët sè ❦➳t q✉↔ ✈➲ ❝❤✐➲✉ ①↕ ↔♥❤ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷✳ ❈❤✐➲✉ ①↕ ↔♥❤ ✈➔ số q ữớ ỗ t❤à ✤÷í♥❣ ✶✼ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✾ é ữớ ỗ t t ữủ ợ t ◆❡❣r✐ tr♦♥❣ ❬✸❪✳ ❱ỵ✐ ♠ët sè ♥❣✉②➯♥ m ≥ 2✱ sỷ r ởt ỗ t õ ữợ ợ t V = {x1, , xn}✱ tù❝ ❧➔✱ ♠é✐ ❝↕♥❤ ✤➣ ✤÷đ❝ ❝❤➾ ✤à♥❤ ởt ữợ ởt ỗ m xi , , xi ❧➔ ♠ët ✤÷í♥❣ ❞➝♥ ❝â ✤ë ❞➔✐ m ♥➳✉ ❝â m − ❝→❝ ❝↕♥❤ ❦❤→❝ ♥❤❛✉ e1 , , em−1 s❛♦ ❝❤♦ ej = (xi , xi ) ❧➔ ♠ët ❝↕♥❤ ❝â ữợ tứ xi xi ỗ t ❝→❝ ✤➾♥❤ ✈ỵ✐ ❝→❝ ❜✐➳♥ tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ R = K[x1, , xn] tr➯♥ ♠ët tr÷í♥❣ K ✱ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ ❝õ❛ Γ ❝â ✤ë ❞➔✐ m ❧➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝ m j j j+1 j+1 Jm (Γ) = ({xi1 · · · xim | xi1 , , xim ❧➔ ♠ët ✤÷í♥❣ m }) ữ ỵ r m = 2✱ t❤➻ J2(Γ) ✤ì♥ ❣✐↔♥ ❧➔ ✐✤➯❛♥ ❝↕♥❤ ❝õ❛ Γ✱ ✤✐➲✉ ♥➔② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❱✐❧❧❛rr❡❛❧ tr♦♥❣ ❬✶✵❪✳ ■✤➯❛♥ ✤÷í♥❣ ❞➝♥ ✤➣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✱ ✤➦❝ ❜✐➺t ❧➔ ✈✐➺❝ t➻♠ ❝→❝ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝→❝ t➼♥❤ ❝❤➜t ✤↕✐ sè ❝õ❛ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t tê ❤đ♣ ❝õ❛ ỗ t tữỡ ự t q ữủ r ởt ữợ t❤➻ ✤↕✐ sè ❘❡❡s R(Jm(Γ)) ❝õ❛ ♥â ❧➔ ❝❤✉➞♥ t➢❝ ữớ ỗ t ♣❤➛♥ ✤➛② ✤õ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❧➔ ❝❤✉➞♥ t➢❝❀ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ ❝õ❛ ♠ët ❝❤✉ tr➻♥❤ ✤÷đ❝ ❝❤➾ r❛ ❧➔ ❝â ❦✐➸✉ t✉②➳♥ t➼♥❤ ✭①❡♠ ❬✷❪✮✳ ❚r♦♥❣ ❬✻❪✱ ❍❡ ✈➔ ❚✉②❧ ♥❣❤✐➯♥ ❝ù✉ Jm(Γ) tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t Γ ỗ t ữớ Ln r õ ỗ t ữớ Ln ởt ỗ t ợ t V = {x1, , xn} ✈➔ ❝→❝ ❝↕♥❤ ❝â ữợ ej = (xj , xj+1) ợ j = 1, , n õ ỗ t❤à ✤÷í♥❣ Ln ❝â ❞↕♥❣ ✸ ❍❡ ✈➔ ❚✉②❧ ❝ơ♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ R/Jm(Ln) ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✈➔ ✤÷❛ r❛ ♠ët ❝ỉ♥❣ t❤ù❝ ❝❤➼♥❤ ①→❝ ❝❤♦ ❝❤✐➲✉ ①↕ ↔♥❤ ❝õ❛ Jm(Ln) t❤❡♦ m ✈➔ n✳ ❚r♦♥❣ ❬✶❪✱ sû ❞ư♥❣ ❤♦➔♥ t♦➔♥ ❝→❝ ❧➟♣ ❧✉➟♥ tê ❤đ♣✱ ❆❧✐❧♦♦❡❡ ✈➔ ❋❛r✐❞✐ ❝ơ♥❣ ✤÷❛ r❛ ❝ỉ♥❣ t❤ù❝ tr➯♥ ❝❤♦ ❝❤✐➲✉ ①↕ ↔♥❤ ❝õ❛ Jm(Ln)✳ ❍ì♥ ♥ú❛✱ ❤❛✐ ♥❤➔ ❦❤♦❛ ❤å❝ ❝ơ♥❣ ✤➣ ✤÷❛ r❛ ♠ët ❝ỉ♥❣ t❤ù❝ t÷í♥❣ ♠✐♥❤ ❝❤♦ ❝❤➾ sè ❝❤➼♥❤ q✉② ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ ❝õ❛ Jm(Ln) t❤❡♦ m ✈➔ n✳ ❚r♦♥❣ ❬✶✷❪✱ ●✉❛♥❣❥✉♥ ❩❤✉ ✤➣ ❦❤→✐ q✉→t ❦❤→✐ ♥✐➺♠ ✐✤➯❛♥ ữớ ữ s ỗ t õ ữợ ợ t V = {x1, , xn}✱ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ ❝õ❛ Γ ❝â ✤ë ❞➔✐ m ❧➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝ Im,k (Γ) = (u1 , , uk ), ✈ỵ✐ u1, , uk ❧➔ ♠ët sè ✤÷í♥❣ ❞➝♥ ✤ë ❞➔✐ m tr♦♥❣ Γ ❑❤✐ u1, , uk ❧➔ t➜t ❝↔ ❝→❝ ✤÷í♥❣ ❞➝♥ ✤ë ❞➔✐ m tr♦♥❣ Γ✱ t❛ ❝â Im,k (Γ) = Jm(Γ)✳ ❱ỵ✐ ❦❤→✐ ♥✐➺♠ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ tê♥❣ q✉→t ♥➔②✱ ♠ët ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ ❝❤✐➲✉ ①↕ ↔♥❤ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ❝❤ó♥❣ ❝â ❝á♥ t➼♥❤ ✤÷đ❝ t÷í♥❣ ♠✐♥❤ ❤❛② ❦❤ỉ♥❣✳ ❚r➯♥ ❝ì sð ✤â✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✧❱➲ ữớ ỗ t ữớ tr ❧↕✐ ❦➳t q✉↔ ❝õ❛ ●✉❛♥❣❥✉♥ ❩❤✉ tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❬✶✷❪ ✈➲ ❝❤✐➲✉ ①↕ ↔♥❤ ✈➔ ❝❤➾ sè q ữớ tờ qt ỗ t❤à ✤÷í♥❣✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ❝õ❛ ✣↕✐ sè ❣✐❛♦ ❤♦→♥ ✈➔ ✣↕✐ sè ❣✐❛♦ ❤♦→♥ ❚ê ❤đ♣ ♥❤➡♠ ❝❤✉➞♥ ❜à ♥ë✐ ❞✉♥❣ ❝❤♦ ❝❤÷ì♥❣ s ữỡ ỗ t t ỡ ỗ t ữớ ỗ t ự ỡ ✐✤➯❛♥ ❙t❛♥❧❡②✲❘❡✐s♥❡r ✈➔ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❈❤÷ì♥❣ ✷✳ ❈❤✐➲✉ ①↕ ↔♥❤ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ ỗ t ữớ ữỡ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❝❤✐➲✉ ①↕ ↔♥❤✱ ❝❤➾ sè ❝❤➼♥❤ q✉②✱ ✤ë s➙✉ ❝õ❛ ♠ët ♠ỉ✤✉♥✳ ❚ø ✤â ❧➔♠ ❝ì sð tr➻♥❤ ❜➔② ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝õ❛ ♠ët sè ❦➳t q✉↔ tr♦♥❣ t➔✐ ✹ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❬✶✷❪ ✈➲ ❝❤✐➲✉ ①↕ ↔♥❤ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ✐✤➯❛♥ ữớ ỗ t ữớ ữủ t ữợ sỹ ữợ t t ❚❙✳ ❚❤✐➲✉ ✣➻♥❤ P❤♦♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧í✐ ❝→♠ ì♥ s➙✉ s➢❝ ♥❤➜t ✤➳♥ ❚❙✳ ❚❤✐➲✉ ✣➻♥❤ P❤♦♥❣✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tợ t ổ tở ữ ỹ ♥❤✐➯♥✱ P❤á♥❣ ✤➔♦ t↕♦ ❙❛✉ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ♥❤✐➺t t➻♥❤ ❞↕② ❞é✱ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ✤➣ ♥❤➟♥ ✤÷đ❝ sü õ♥❣ ❤ë✱ ✤ë♥❣ ✈✐➯♥ t✐♥❤ t❤➛♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➔ ❣✐ó♣ ✤ï tø ❣✐❛ ✤➻♥❤✱ ỳ ữớ t ũ ợ ❜➧ t❤➙♥ ❤ú✉ ❣➛♥ ①❛✳ ❚→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ♥❤➜t tỵ✐ ♥❤ú♥❣ t➻♥❤ ❝↔♠✱ sü ú ù qỵ õ ũ õ ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ❝→❝ ❣â♣ þ✱ sû❛ ❝❤ú❛✱ ❜ê s✉♥❣ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦✱ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✈➔ ♥❣÷í✐ ✤å❝✳ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ✺ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t K ❧➔ ♠ët tr÷í♥❣✱ R = K[x1, , xn] ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ ♣❤➙♥ ❜➟❝ ❝❤✉➞♥ n ❜✐➳♥ tr➯♥ K ✈➔ M ❧➔ ♠ët R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr tự ỡ ỗ t ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❧➔♠ ❝ì sð ❝❤♦ ❝→❝ ♥ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ s❛✉✳ ❈→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t✱ ❦➳t q✉↔ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❧↕✐ tø ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ỗ t ữớ ởt ỗ t G = (V, E) ỗ ởt t ởt t ❝↕♥❤ E ♥è✐ ❝→❝ ❝➦♣ ✤➾♥❤ ♥➔♦ ✤â ✈ỵ✐ ♥❤❛✉✳ ởt v V ỗ t G ữủ ❣å✐ ❧➔ ✤➾♥❤ ❝æ ❧➟♣ ♥➳✉ ♥â ❦❤æ♥❣ t❤✉ë❝ ❜➜t ❝ù ❝↕♥❤ ♥➔♦ ❝õ❛ G✱ ❤❛② v ❦❤ỉ♥❣ ♥è✐ ✈ỵ✐ ❜➜t ❦➻ ✤➾♥❤ ♥➔♦ ❦❤→❝✳ V ✶✳✶✳✷ ❱➼ ❞ö✳ ❈❤♦ ỗ t G ữ tr õ d ❧➔ ✤➾♥❤ ❝æ ❧➟♣ ❝õ❛ G ✈➻ ♥â ❦❤æ♥❣ ố ợ t ỗ t G ỗ t ữớ ỵ Ln ởt ỗ t ợ t V = {x1 , , xn } ✈➔ õ ữợ ej = (xj , xj+1 ) ✈ỵ✐ j = 1, , n − ỗ t ữớ Ln õ ữ s ✈➔♥❤ ✤❛ t❤ù❝ n ❜✐➳♥ R = K[x1, , xn] õ t ỗ t x1 , , xn ❝õ❛ R ✈ỵ✐ ❝→❝ ✤➾♥❤ t÷ì♥❣ ù♥❣ tr♦♥❣ t➟♣ ✤➾♥❤ V = {x1 , , xn } ỗ t G õ ỗ t G s tữỡ ự ợ ♠ët ✐✤➯❛♥ ✤ì♥ t❤ù❝ ❞✉② ♥❤➜t ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ G = (V, E) ❧➔ ♠ët ỗ t ợ t V = {x1, , xn}✳ ■✤➯❛♥ ❝↕♥❤ ❝õ❛ G ❧➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ①→❝ ✤à♥❤ ❜ð✐ I(G) = (xi xj : {xi , xj } ∈ E) ⊆ R = K[x1 , , xn ] ✶✳✶✳✺ ❱➼ ❞ö✳ ❈❤♦ ỗ t G ữ õ G ❝â ✐✤➯❛♥ ❝↕♥❤ ❧➔ I(G) = (x1 x3 , x1 x4 , x2 x3 , x3 x4 ) ⊆ R = K[x1 , , xn ] ỗ t G ởt ❝→❝ ✤➾♥❤ ♣❤➙♥ ❜✐➺t xi , , xi ✤÷đ❝ ❣å✐ ❧➔ ♠ët t ❝â ✤ë ❞➔✐ t õ t õ ữợ ❜✐➺t e1, , et−1✱ tr♦♥❣ ✤â ej ởt õ ữợ ố tứ xi ✤➾♥❤ xi ✳ ■✤➯❛♥ ✤÷í♥❣ ❞➝♥ ❝õ❛ G t❤❡♦ ✤ë ❞➔✐ t ❧➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝ It (G) tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ R = K[x1, , xn] tr➯♥ tr÷í♥❣ K ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✤÷í♥❣ ❞➝♥ j It (G) = xi1 · · · xit : xi1 , , xit j+1 ❧➔ ♠ët ✤÷í♥❣ ❞➝♥ ✤ë ❞➔✐ t ❝õ❛ G ✼ ✶✳✶✳✼ ❱➼ G ởt ỗ t ữ ữợ õ ữớ õ ✸ ❧➔✿ (x1 , x4 , x7 ); (x1 , x3 , x6 ); (x1 , x2 , x5 ) ✈➔ (x2, x5, x8) ■✤➯❛♥ ✤÷í♥❣ ❞➝♥ ❝â ✤ë ❞➔✐ ✸ ❧➔ I3 (G) = x1 x4 x7 , x1 x3 x6 , x1 x2 x5 , x2 x5 x8 ⊆ R = K[x1 , , x8 ] ỗ t G Pự ỡ ✐✤➯❛♥ ❙t❛♥❧❡②✲❘❡✐s♥❡r ✈➔ t➼♥❤ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❞➣② ◆ë✐ ❞✉♥❣ ❝õ❛ ♣❤➛♥ ♥➔② ♥❤➡♠ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤ù❝ ✤ì♥ ❤➻♥❤✱ ✐✤➯❛♥ ❙t❛♥❧❡②✲❘❡✐s♥❡r✱ ❤é♥ t↕♣ ✈➔ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❝❤ó♥❣✳ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ▼ët ♣❤ù❝ ✤ì♥ ❤➻♥❤ ∆ tr➯♥ t➟♣ ✤➾♥❤ V ❧➔ ♠ët ❤å ❝→❝ t➟♣ ❝♦♥ ❝õ❛ V t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t✿ ♥➳✉ F ∈ ∆ t❤➻ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ F t❤✉ë❝ ∆✳ ▼ët ♣❤➛♥ tû ❝õ❛ ∆ ✤÷đ❝ ởt t t F ỵ ❤✐➺✉ dim(F )✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ |F | − ỵ dim() tữợ ợ t õ t õ ởt t ∆✳ ❈→❝ ♠➦t tè✐ ✤↕✐ t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠ tr♦♥❣ ∆ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♠➦t ❝ü❝ ✤↕✐✱ ✈➔ t ủ t ỹ ữủ ỵ ❤✐➺✉ ❜ð✐ ❋❛❝❡ts(∆)✳ P❤ù❝ ✤ì♥ ❤➻♥❤ ∆ ✤÷đ❝ ❣å✐ ❧➔ t❤✉➛♥ tó② ♥➳✉ t➜t ❝↔ ❝→❝ ♠➦t ❝ü❝ ✤↕✐ ❝õ❛ õ õ ũ tữợ ổ ữủ ❣å✐ ❧➔ ❦❤ỉ♥❣ t❤✉➛♥ tó②✳ ◆➳✉ F acets(∆) = {F1, , Fq } t❤➻ t❛ ✈✐➳t ∆ = F1, , Fq ✳ ✶✳✷✳✷ ❱➼ ự ỡ ữ ữợ ✽ ❍➻♥❤ ✶✳✹✿ P❤ù❝ ✤ì♥ ❤➻♥❤ ∆ ❑❤✐ ✤â✱ t➟♣ ❝→❝ ♠➦t ❝õ❛ ∆ ❧➔ ∅, {x1 }, {x2 }, {x3 }, {x4 },{x1 , x2 }, {x1 , x3 }, {x1 , x4 }, {x2 , x3 }, {x2 , x4 }, {x3 , x4 }, {x1 , x2 , x4 }, {x2 , x3 , x4 } ❚❛ ❝â ❋❛❝❡ts(∆) = {{x1, x2, x4}, {x2, x3, x4}, {x1, x3}}, dim(∆) = ✈➔ ∆ ❧➔ ♣❤ù❝ ✤ì♥ ❤➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ tó② ✈➻ dim({x1 , x3 }) = = dim({x1 , x2 , x4 }) = ✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ ♣❤ù❝ ✤ì♥ ❤➻♥❤ ∆✳ ■✤➯❛♥ ❙t❛♥❧❡② ✕ ❘❡✐s♥❡r ❝õ❛ ∆ ❧➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ①→❝ ✤à♥❤ ❜ð✐ I∆ = ( x:F ∈ / ∆) ⊆ R xF t sr ỵ K[∆]✱ ❧➔ ✈➔♥❤ t❤÷ì♥❣ K[∆] = R/I∆✳ ✶✳✷✳✹ ❱➼ ❞ư✳ ❈❤♦ ♣❤ù❝ ✤ì♥ ❤➻♥❤ ∆ ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✶✳✷✳✷✳ ❚❛ ❝â I∆ = (x1 x2 x3 , x1 x3 x4 ); K[∆] = R/I∆ = K[x1 , , x4 ]/(x1 x2 x3 , x1 x3 x4 ) ✶✳✷✳✺ ✣à♥❤ ♥❣❤➽❛✳ ▼ët ❤é♥ t↕♣ C tr➯♥ t➟♣ ✤➾♥❤ V ❧➔ ♠ët ❤å ❝→❝ t➟♣ ❝♦♥ ❝õ❛ V✱ ❣å✐ ❧➔ ❝→❝ ❝↕♥❤✱ s❛♦ ❝❤♦ ❦❤æ♥❣ ❝â ❝↕♥❤ ♥➔♦ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❝→❝ ❝↕♥❤ ❝á♥ ❧↕✐✳ ❚➟♣ t➜t ❝↔ C tữỡ ự ỵ ❤✐➺✉ ❜ð✐ VC ✈➔ EC ✳ ✶✼ ❙✉② r❛ ♣❞ (I + J) = = = = ♣❞ (R/I + J) − ♣❞ (R1/I) + ♣❞ (R2/J) − (♣❞ (I) + 1) + (♣❞ (J) + 1) − ♣❞ (I) + ♣❞ (J) + 1, ✣è✐ ✈ỵ✐ ❦❤➥♥❣ ✤à♥❤ t❤ù ❤❛✐ ✈➔ t❤ù ❜❛✱ ❜➡♥❣ ❇ê ✤➲ ✸✳✷ ❝õ❛ ❬✼❪✱ t❛ ❝â r❡❣ (R/I + J) = r❡❣ (R1/I) + r❡❣ (R2/J) ✈➔ r❡❣ (R/IJ) = r❡❣ (R1/I) + r❡❣ (R2/J) + ❱➻ t❤➳✱ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ r❡❣ (I + J) = = = = r❡❣ (R/I + J) + r❡❣ (R1/I) + r❡❣ (R2/J) + (r❡❣ (I) − 1) + (r❡❣ (J) − 1) + r❡❣ (I) + r❡❣ (J) − 1, ✈➔ r❡❣ (IJ) = = = = r❡❣ (R/IJ) + r❡❣ (R1/I) + r❡❣ (R2/J) + (r❡❣ (I) − 1) + (r❡❣ (J) − 1) + r❡❣ (I) + r❡❣ (J) ✷✳✷ ❈❤✐➲✉ ①↕ ↔♥❤ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ ỗ t ữớ ♥❤➡♠ tr➻♥❤ ❜➔② ♠ët sè ❝æ♥❣ t❤ù❝ ❝❤♦ ❝❤✐➲✉ ①↕ ↔♥❤ ♣❞ (Im,l,k ) ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣ (Im,l,k ) ❝õ❛ ✐✤➯❛♥ ✤÷í♥❣ ❞➝♥ Im,l,k tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ♠ët sè ổ ữủ t tr tr ỵ ❤✐➺✉ G(I) ✤➸ ❜✐➸✉ t❤à t➟♣ s✐♥❤ tè✐ t❤✐➸✉ ❞✉② ♥❤➜t ❝õ❛ ♠ët ✐✤➯❛♥ ✤ì♥ t❤ù❝ I ✳ ✷✳✷✳✶ ✣à♥❤ ỵ k, l, m, n số t❤ä❛ ♠➣♥ n = k(m − l) + l ✈ỵ✐ k ≥ 1✱ m ≥ ✈➔ l < m ✱ tr♦♥❣ ✤â sû Im,l,k = (u1 , , uk ) ✈ỵ✐ ui = m ❜✐➸✉ t❤à sè ♥❣✉②➯♥ ♥❤ä ♥❤➜t ≥ m m 2✳ ●✐↔ x(i−1)(m−l)+j ✈ỵ✐ ♠å✐ ≤ i ≤ k ✳ ❑❤✐ ✤â j=1 ♣❞ (Im,l,k ) = k − 1, r❡❣ (Im,l,k ) = (k − 1)(m − l − 1) + m ✣➛✉ t✐➯♥ t❛ ❝â ♥❤➟♥ ✤à♥❤ r➡♥❣ m − 2l − ≥ 0✳ ❚❤➟t ✈➟②✱ ♥➳✉ m = 2s + 1✱ t❤➻ m2 = s + 1✳ ❚❤❡♦ ❣✐↔ t❤✉②➳t✱ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ 2l + ≤ 2( ▼➦t ❦❤→❝✱ ♥➳✉ m = 2s✱ t❤➻ m 2l + ≤ 2( m − 1) + = 2s + = m = s✳ ◆❤÷ ✈➟② m − 1) + = 2s − < m ✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ ♥❤➟♥ ✤à♥❤ ð tr➯♥✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ k✳ ❘ã r➔♥❣ ❧➔ ♠➺♥❤ ✤➲ ✤ó♥❣ ✈ỵ✐ k = 1✳ ◆➳✉ k = 2✱ t❛ ❣✐↔ sû J1 = Im,l,1 ✈➔ K1 = (u2 ) ❝❤ù❛ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû s✐♥❤ ❝õ❛ Im,l,2 ❝❤✐❛ ❤➳t ❝❤♦ ❜✐➳♥ x2m−l ✳ ❇ð✐ ✈➻ K1 ❝â ❣✐↔✐ tü ❞♦ t✉②➳♥ t➼♥❤✱ Im,l,2 = J1 + K1 ❝â ♠ët ♣❤➨♣ t tt t ml ỵ J1 K1 = K1( xj ) ú ỵ r J1, K1 ✈➔ J1 ∩ K1 ❧➔ ♥❤ú♥❣ j=1 ✐✤➯❛♥ ❝❤➼♥❤✱ s✉② r❛ ♣❞ (J1) = ♣❞ (K1) = ♣❞ (J1 ∩ K1) = ❉♦ ✤â✱ ❜ð✐ ❍➺ q✉↔ ✷✳✶✳✼✱ t❛ ❝â ❝æ♥❣ t❤ù❝ s❛✉ ♣❞ (Im,l,2) = max{♣❞ (J1), ♣❞ (K1), ♣❞ (J1 ∩ K1) + 1} = m−l ❇ð✐ ✈➻ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ K1 ✈➔ ( xj ) j=1 ❧➔ ❦❤→❝ ♥❤❛✉✱ t❛ ❝â r❡❣ (J1 ∩ K1) = r❡❣ (J1) + r❡❣ (K1) = m + (m − l) ✶✾ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✽✳ ❉♦ ✤â✱ ❜ð✐ ❍➺ q✉↔ ✷✳✶✳✼✱ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ r❡❣ (Im,l,2) = max{r❡❣ (J1), r❡❣ (K1), r❡❣ (J1 ∩ K1) − 1} = max{m, m, m + (m − l) − 1} = m + (m − l − 1) ✣✐➲✉ ♥➔② ❣✐↔✐ q✉②➳t tr÷í♥❣ ❤đ♣ k = 2✳ ●✐↔ sû r➡♥❣ k ≥ ✈➔ ♠➺♥❤ ✤➲ ✤ó♥❣ ❝❤♦ t➜t ❝↔ Im,l,t ✈ỵ✐ t < k✳ ❚❛ ①➨t ❝→❝ m−l ✐✤➯❛♥ L0 = Im,l,k ✈➔ Li = Im,l,k−i−1 + ( x(k−i−1)(m−l)+j ) ✈ỵ✐ ≤ i ≤ k j=1 ml ỵ Ji = Im,l,ki ❝❤♦ ≤ i ≤ k − 1✱ K1 = (uk )✱ Ki = ( x(k−i)(m−l)+j ) j=1 ✈ỵ✐ ≤ i ≤ k − 1✳ ❚÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤đ♣ k = 2✱ t❛ ❝â ✈ỵ✐ ≤ i ≤ k − 2✱ Li = Ji+1 + Ki+1 ❝â ởt t tt ú ỵ r Ji Ki = Ki Li ✱ ✈ỵ✐ ♠å✐ m−l ≤ i ≤ k − 2✱ Jk−1 ∩ Kk−1 = Kk−1 ( xj ) ✈➔ t❤ü❝ t➳ ❧➔ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ j=1 tr♦♥❣ Ki ✈➔ Li ❧➔ ❦❤→❝ ♥❤❛✉ ✈➔ ❦❤æ♥❣ ❝â ❜✐➳♥ ♥➔♦ ❝❤✐❛ ❤➳t Kk−1 ❝❤✐❛ ❤➳t ❜➜t m−l ❦ý ♣❤➛♥ tû s✐♥❤ ❝â ❞↕♥❣ xj ✱ t❛ t❤✉ ữủ ợ i k j=1 ♣❞ (Ji ∩ Ki) = ♣❞ (Li) = max{♣❞ (Ji+1), ♣❞ (Ki+1), ♣❞ (Ji+1 ∩ Ki+1) + 1}, r❡❣ (Li) = max{r❡❣ (Ji+1), r❡❣ (Ki+1), r❡❣ (Ji+1 ∩ Ki+1) − 1}, r❡❣ (Ji ∩ Ki) = r❡❣ (KiLi) = r❡❣ (Ki) + r❡❣ (Li) ≥ r❡❣ (Ki) + 1, (1) m−l r❡❣ (Jk−1 ∩ Kk−1) = r❡❣ (Kk−1) + r❡❣ (( xj )) = 2(m − l) j=1 ❉♦ Jk−1 ∩ Kk−1 ✈➔ Ki ❧➔ ♥❤ú♥❣ ✐✤➯❛♥ ❝❤➼♥❤✱ t❛ ❝â ♣❞ (Jk−1 ∩ Kk−1) = ♣❞ (Ki) = ✈ỵ✐ ≤ i ≤ k − ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❧➦♣ ✤✐ ❧➦♣ ❧↕✐ ✤➥♥❣ t❤ù❝ ✭✶✮ ✈➔ ❣✐↔ t❤✐➳t q✉② ♥↕♣ ♣❞ (Ji) = k − i − 1, r❡❣ (Ji) = (k − i − 1)(m − l − 1) + m ✈➔ m − 2l − ≥ 0, t❛ t❤✉ ✤÷đ❝ ♣❞ (J1 ∩ K1) = ♣❞ (L1) = k − ✈➔ r❡❣ (J1 ∩ K1) = (k − 1)(m − l − 1) + m + ✷✵ ❙✉② r❛ ♣❞ (L0) = max{♣❞ (J1), ♣❞ (K1), ♣❞ (J1 ∩ K1) + 1} = max{k − 2, 0, k − 1} = k − 1, r❡❣ (L0) = max{r❡❣ (J1), r❡❣ (K1), r❡❣ (J1 ∩ K1) − 1} = max{(k − 2)(m − l − 1) + m, m, (k − 1)(m − l − 1) + m + − 1} = (k − 1)(m − l − 1) + m ữ ởt q ỵ tr t ❝â✿ ✷✳✷✳✷ ❍➺ q✉↔✳ ❈❤♦ k, l, m, n ✈➔ Im,l,k ữ tr ỵ õ t (Im,l,k ) = n − k + ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ❆✉s❧❛♥❞❡r✲❇✉❝❤s❜❛✉♠✱ s✉② r❛ ❞❡♣t❤ (Im,l,k ) = n − ♣❞ (Im,l,k ) = n − k + ỵ s tờ qt ỵ q ỵ ❈❤♦ k, l, m, n ❧➔ sè ♥❣✉②➯♥ s❛♦ ❝❤♦ n = k(m−l)+lm tr♦♥❣ ✤â k ≥ 1✱ m ≥ ✈➔ m ≤ l < m✳ ✣➦t Im,l,k = (u1 , , uk ) ✈ỵ✐ ui = x(i−1)(m−l)+j j=1 ❜➜t ❦ý ≤ i ≤ k ✳ ◆➳✉ m ≡ (♠♦❞ (m − l)) ✈➔ n = p(2m − l) + d ✈ỵ✐ ≤ d < 2m − l✱ t❤➻ ✭✶✮ ♣❞ (Im,l,k ) = 2p − ♥➳✉ d = m; 2p ♥➳✉ d = m ✭✷✮ r❡❣ (Im,l,k ) = p(2m − l − 2) + ♥➳✉ d = m; p(2m − l − 2) + m ♥➳✉ d = m ✣➦t t = 2m−l m−l ✱ t❤➻ t > 2✳ ❚r➯♥ t❤ü❝ t➳✱ ♥➳✉ t = 2✱ t❤➻ l = 0✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t r➡♥❣ l ≥ m2 ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♥❤ú♥❣ ❦❤➥♥❣ ✤à♥❤ ♥➔② ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ k✳ ❈→❝ tr÷í♥❣ ❤đ♣ k = 1, ữủ s r tứ ỵ sỷ r➡♥❣ k ≥ ✈➔ ♠➺♥❤ ✤➲ ✤ó♥❣ ✈ỵ✐ t➜t ❝↔ Im,l,s ✈ỵ✐ s < k✳ ◆➳✉ ≤ k ≤ t✱ t❤➻ n = (2m − l) + d ❈❤ù♥❣ ♠✐♥❤✳ ✷✶ ✈ỵ✐ d = (k − 2)(m − l) < m✳ ✣➦t J1 = Im,l,k−1 ✈➔ K1 m−l x(k−2)(m−l)+j )✳ ❉♦ ✤â J ∩ K1 = K1 ( = (uk )✱ t❛ t❤✉ ✤÷đ❝ j=1 ♣❞ (J1 ∩ K1) = ✈➔ r❡❣ (J1 ∩ K1) = m + (m − l) = 2m − l ❱➻ sè ❧÷đ♥❣ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J1 ❧➔ (2m − l) + d − (m − l)✱ sû ❞ö♥❣ ❣✐↔ t❤✉②➳t q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1) = ✈➔ r❡❣ (J1) = 2m − l − 1✳ ❉♦ ✤â ♣❞ (Im,l,k ) = max{♣❞ (J1), ♣❞ (K1), ♣❞ (J1 ∩ K1) + 1} = 1, ✈➔ r❡❣ (Im,l,k ) = max{r❡❣ (J1), r❡❣ (K1), r❡❣ (J1 ∩ K1) − 1} = max{2m − l − 1, m, 2m − l − 1} = 2m − l − ✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ✤ó♥❣ ❝❤♦ ≤ k ≤ t✳ ◆➳✉ k ≥ qt + ✈ỵ✐ q ≥ 1✳ ✣➦t J1 = Im,l,k−1 ✈➔ K1 = (uk )✳ ❇➡♥❣ ❝→❝ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ỵ t s r Im,l,k = J1 + K1 ❝â ♠ët ♣❤➨♣ t→❝❤ ❇❡tt✐ ✈➔ m−l J1 ∩ K1 = K1 (Im,l,k−t + ( x(k−2)(m−l)+j )) j=1 m−l ữ ỵ r t tr K1, Im,l,kt ✈➔ ( x(k−2)(m−l)+j ) ❧➔ ❦❤→❝ j=1 ♥❤❛✉✱ s✉② r❛ ♣❞ (J1 ∩ K1) = ♣❞ (Im,l,k−t + ( m−l x(k−2)(m−l)+j )) j=1 m−l = = ♣❞ (Im,l,k−t) + ♣❞ ( ♣❞ (Im,l,k−t) + x(k−2)(m−l)+j ) + j=1 tr♦♥❣ ✤â ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ ✤÷đ❝ s✉② r❛ tø ❇ê ✤➲ ✷✳✶✳✽ ✭✶✮✳ ❚❛ ❝❤✐❛ ❜❛ tr÷í♥❣ ❤đ♣ s❛✉✿ ✷✷ ✭✶✮ ◆➳✉ k − = qt ✈ỵ✐ q ≥ 1✱ t❤➻ sè ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J1 ✈➔ Im,l,k−t t÷ì♥❣ ù♥❣ ❧➔ p(2m − l) + l ✈➔ (p − 1)(2m − l) + m✳ ❇ð✐ ❣✐↔ t❤✉②➳t q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1) = 2p − 1, ♣❞ (Im,l,k−t) = 2(p − 1), r❡❣ (J1) = p(2m − l − 2) + r❡❣ (Im,l,k−t) = (p − 1)(2m − l − 2) + m ❉♦ ✤â ♣❞ (J1 ∩ K1) = ♣❞ (Im,l,k−t) + = 2p − 1, ♣❞ (Im,l,k ) = max{♣❞ (J1), ♣❞ (K1), ♣❞ (J1 ∩ K1) + 1} = max{2p − 1, 0, (2p − 1) + 1} = 2p; r❡❣ (J1 ∩ K1) = r❡❣ (K1) + r❡❣ (Im,l,k−t + ( m−l x(k−2)(m−l)+j )) j=1 m−l = r❡❣ (K1 ) + r❡❣ (Im,l,k−t ) + r❡❣ ( x(k−2)(m−l)+j ) − j=1 = m + [(p − 1)(2m − l − 2) + m] + (m − l) − = p(2m − l − 2) + m + 1, r❡❣ (Im,l,k ) = max{r❡❣ (J1), r❡❣ (K1), r❡❣ (J1 ∩ K1) − 1} = max{p(2m − l − 2) + 1, m, p(2m − l − 2) + m} = p(2m − l − 2) + m ✭✷✮ ◆➳✉ k − = qt + ✈ỵ✐ q ≥ 1✱ t❤➻ sè ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J1 ✈➔ Im,l,k−t t÷ì♥❣ ù♥❣ ❧➔ (p − 1)(2m − l) + m ✈➔ (p − 1)(2m − l)✳ ❉♦ ✤â✱ ❜ð✐ q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1) = 2(p − 1), ♣❞ (Im,l,k−t) = 2(p − 1) − 1, r❡❣ (J1) = (p − 1)(2m − l − 2) + m ✈➔ r❡❣ (Im,l,k−t) = (p − 1)(2m − l − 2) + ❉♦ ✤â✱ t÷ì♥❣ tü ♥❤÷ s✉② ❧✉➟♥ tr➯♥✱ t❛ s✉② r❛ ♣❞ (Im,l,k ) = 2p − ✈➔ r❡❣ (Im,l,k ) = p(2m − l − 2) + ✷✸ ✭✸✮ ◆➳✉ k − = qt + c ✈ỵ✐ q ≥ ✈➔ ≤ c < t✱ t❤➻ sè ❧÷đ♥❣ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J1 ✈➔ Im,l,k−t t÷ì♥❣ ù♥❣ ❧➔ p(2m − l) + (c − 2)(m − l) ✈➔ (p − 1)(2m − l) + (c − 1)(m − l)✳ ❉♦ ✤â✱ ❜ð✐ q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1) = 2p − 1, ♣❞ (Im,l,k−t) = 2(p − 1) − 1, r❡❣ (J1) = p(2m − l − 2) + ✈➔ r❡❣ (Im,l,k−t) = (p − 1)(2m − l − 2) + ▼ët ❝→❝ t÷ì♥❣ tü✱ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ ♣❞ (Im,l,k ) = 2p − ✈➔ r❡❣ (Im,l,k ) = p(2m − l − 2) + ❱➔ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ✷✳✷✳✹ ❍➺ q✉↔✳ ❈❤♦ k, l, m, n ✈➔ Im,l,k ♥❤÷ tr♦♥❣ ỵ õ depth(Im,l,k ) = n + − n + (m − l) n + (m − l) − 2m − l 2m − l ✣➦t k − = qt + c✱ tr♦♥❣ ✤â q ≥ ✈➔ ≤ c < t✳ ❚ø ự ỵ t õ c = 0✱ t❤➻ d = m✱ ♥➳✉ ❦❤æ♥❣ t❤➻✱ d = (c − 1)(m − l)✳ ❉♦ ✤â✱ ❜➡♥❣ ♠ët sè ♣❤➨♣ t➼♥❤ ✤ì♥ ❣✐↔♥✱ ❝❤ó♥❣ t❛ ❝â ♥➳✉ c = 0✱ t❤➻ ❈❤ù♥❣ ♠✐♥❤✳ n + (m − l) n + (m − l) = = p + 1, 2m − l 2m − l ♥➳✉ ❦❤æ♥❣ t❤➻✱ n + (m − l) =p+1 2m − l ✈➔ n + (m − l) = p 2m − l ⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❆✉s❧❛♥❞❡r✲❇✉❝❤s❜❛✉♠✱ t❛ s✉② r❛ ❞❡♣t❤ (Im,l,k ) = n − ♣❞ (Im,l,k ), ❦➳t ❧✉➟♥ ♠♦♥❣ ♠✉è♥ s ỵ k, l, m, n ❧➔ ❝→❝ sè ♥❣✉②➯♥ t❤ä❛ ♠➣♥ n = k(m − l) + l tr♦♥❣ ✤â k ≥ 1✱ m ≥ ✈➔ m ui = ≤ l < m✳ ✣➦t Im,l,k = (u1 , , uk ) ✈ỵ✐ m x(i−1)(m−l)+j ✤è✐ ✈ỵ✐ ❜➜t ❦ý ≤ i ≤ k ✳ ◆➳✉ m ≡ s (♠♦❞ (m − l)) ✈ỵ✐ j=1 ≤ s < m − l ✈➔ ❝❤ó♥❣ t❛ ❝â t❤➸ ✈✐➳t n ❧➔ n = p(2m − l − s) + d tr♦♥❣ ✤â ≤ d < 2m − l − s✱ t❤➻ ♣❞ (Im,l,k ) = 2p − d = m; 2p d = m ✣➦t t = 2m−l−s m−l ✱ ❦❤✐ ✤â t > ❜ð✐ ❝→❝ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tỹ ữ tr ỵ ự ✤➲ ♥➔② ❜➡♥❣ ❝→❝❤ q✉② ♥↕♣ t❤❡♦ k✳ ❈→❝ tr÷í♥❣ ủ k = 1, tứ ỵ ●✐↔ sû k ≥ ✈➔ ♠➺♥❤ ✤➲ ✤ó♥❣ ❝❤♦ t➜t ❝↔ Im,l,s ✈ỵ✐ s < k✳ ◆➳✉ ≤ k ≤ t✱ t❤➻ n = (2m − l − s) + d ✈ỵ✐ s + (m − l) ≤ d < m✳ ✣➦t J1 = Im,l,k−1 ✈➔ K1 = (uk )✱ t❛ s✉② r❛ J1 ∩ K1 = m−l x(k−2)(m−l)+j )✳ ❉♦ ✤â ♣❞ (J1 ∩ K1 ) = 0✳ ❱➻ sè ❧÷đ♥❣ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ K1 ( j=1 tr♦♥❣ J1 ❧➔ (2m − l − s) + d − (m − l)✱ sû ❞ö♥❣ ❣✐↔ t❤✉②➳t q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1) = 1✳ ❙✉② r❛ ❈❤ù♥❣ ♠✐♥❤✳ ♣❞ (Im,l,k ) = max{♣❞ (J1), ♣❞ (K1), ♣❞ (J1 ∩ K1) + 1} = ✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ❝❤♦ tr÷í♥❣ ❤đ♣ ≤ k ≤ t✳ ◆➳✉ k ≥ qt + ✈ỵ✐ q ≥ 1✳ ❚❛ ①➨t ❝→❝ ✐✤➯❛♥ s❛✉ L0 = Im,l,k , J1 = Im,l,k−1 , K1 = (uk ), m−l L1 = Im,l,k−t + ( x(k−2)(m−l)+j ), J2q = Im,l,k−qt , j=1 m−l K2q = ( x[k−(q−1)t−2](m−l)+j ), j=1 ✈➔ ✈ỵ✐ ≤ i ≤ q − 1✱ ✷✺ J2i = Im,l,k−it (Γ), J2i+1 = Im,l,k−it−1 (Γ), m−l K2i = ( x[k−(i−1)t−2](m−l)+j ), j=1 (t−1)(m−l) K2i+1 = ( x(k−it−1)(m−l)+j ), j=1 (t−1)(m−l) L2i = Im,l,k−it−1 (Γ) + ( x(k−it−1)(m−l)+j ), j=1 m−l x(k−it−2)(m−l)+j ) L2i+1 = Im,l,k−(i+1)t (Γ) + ( j=1 tữỡ tỹ ữ tr ỵ ✷✳✷✳✶✱ t❛ s✉② r❛ ✈ỵ✐ ❜➜t ❦ý ≤ i ≤ 2q ✱ t❛ ❝â Li = Ji+1 + Ki+1 ❝â ♠ët ♣❤➨♣ t→❝❤ ❇❡tt✐ ✈➔ Ji ∩ Ki = Ki Li ữ ỵ r t tr♦♥❣ Ki ✈➔ Li ❧➔ ❦❤→❝ ♥❤❛✉✱ t❛ s✉② r❛ r➡♥❣✱ ✈ỵ✐ ❜➜t ❦ý ≤ i ≤ 2q − 1✱ t❛ ❝â ♣❞ (Ji ∩ Ki) = ♣❞ (Li) = max{♣❞ (Ji+1), ♣❞ (Ki+1), ♣❞ (Ji+1 ∩ Ki+1) + 1} ❈â ❜❛ tr÷í♥❣ ❤đ♣ ❝➛♥ ①❡♠ ①➨t✿ ✭✶✮ ◆➳✉ k − = qt ❝❤♦ ♠ët sè q ≥ 1✱ t❤➻ n = k(m − l) + l = (qt + 1)(m − l) + l = qt(m − l) + m = q(2m − l − s) + m ❇➡♥❣ ✈✐➯❝ s♦ s→♥❤ ✤➥♥❣ t❤ù❝ ♥➔② ✈ỵ✐ ✤➥♥❣ t❤ù❝ n = p(2m − l − s) + d✱ t❛ ❝â q = p ✈➔ d = m✳ ❙è ❧÷đ♥❣ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J1 ✈➔ J2q t÷ì♥❣ ù♥❣ ❧➔ p(2m − l − s) + l ✈➔ m ởt tữỡ tỹ ố ợ t ý ≤ i ≤ q − 1✱ sè ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J2i+1 ✈➔ J2i t÷ì♥❣ ù♥❣ ❧➔ (p − i)(2m − l − s) + l ✈➔ (p − i)(2m − l − s) + m✳ ❉♦ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1) = 2p − 1, ♣❞ (J2q ) = 0, ♣❞ (J2i+1) = 2(p − i) − 1, ♣❞ (J2i) = 2(p − i) ợ i q ữ þ r➡♥❣ (t−1)(m−l) J2q ∩ K2q = K2q ( xj ) j=1 ✷✻ ✈➔ Ki ❧➔ ♥❤ú♥❣ ✐✤➯❛♥ ❝❤➼♥❤ ✈ỵ✐ ≤ i ≤ 2q✱ t❛ ❝â ♣❞ (J2q ∩ K2q ) = ♣❞ (Ki) = 0✳ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❧➦♣ ❧↕✐ ✤➥♥❣ t❤ù❝ ♣❞ (Ji ∩ Ki) = max{♣❞ (Ji+1), ♣❞ (Ki+1), ♣❞ (Ji+1 ∩ Ki+1) + 1} ✈ỵ✐ i = 2q − 1, 2q − 2, , 1, t❛ t❤✉ ✤÷đ❝ ♣❞ (J1 ∩ K1) = 2p − 1✳ ❱➻ t❤➳ ♣❞ (Im,l,k ) = max{♣❞ (J1), ♣❞ (K1), ♣❞ (J1 ∩ K1) + 1} = max{2p − 1, 0, (2p − 1) + 1} = 2p ✣✐➲✉ ♥➔② ❣✐↔✐ q✉②➳t tr÷í♥❣ ❤đ♣ k − = qt ✈ỵ✐ q ≥ ♥➔♦ ✤â✳ ✭✷✮ ◆➳✉ k − = qt + ✈ỵ✐ q ≥ ♥➔♦ ✤â✱ t❤➻ t÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤ñ♣ ✭✶✮✱ t❛ ❝â q = p + ✈➔ d = s✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ sè ❧÷đ♥❣ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J1 ✈➔ J2q t÷ì♥❣ ù♥❣ ❧➔ (p − 1)(2m − l − s) + m ✈➔ · (2m − l − s) + s✳ ❚÷ì♥❣ tü✱ ✤è✐ ✈ỵ✐ ❜➜t ❦ý ≤ i ≤ q − 1✱ sè ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J2i+1 ✈➔ J2i t÷ì♥❣ ù♥❣ ❧➔ (p − i − 1)(2m − l − s) + m ✈➔ (p − i)(2m − l − s) + s✳ ❉♦ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1) = 2(p − 1)✱ ♣❞ (J2q ) = 1✱ ♣❞ (J2i+1) = 2(p − i − 1) ✈➔ ♣❞ (J2i) = 2(p − i) − ✤è✐ ✈ỵ✐ ≤ i ≤ q − 1✳ ✣➦t (t−1)(m−l) L2q = Im,l,k−qt−1 + ( x(k−qt−1)(m−l)+j ), j=1 J2q+1 = Im,l,k−qt−1 = Im,l,1 , (t−1)(m−l) K2q+1 = ( x(m−l)+j ), j=1 ❦❤✐ ✤â L2q = J2q+1 + K2q+1 ❝â ♠ët ♣❤➨♣ t→❝❤ ❇❡tt✐ ✈➔ J2q ∩ K2q = K2q L2q ml ữ ỵ r J2q+1 K2q+1 = K2q+1( xj ) ✈➔ Ki ❧➔ ♥❤ú♥❣ ✐✤➯❛♥ ❝❤➼♥❤ ✈ỵ✐ j=1 ≤ i ≤ 2q + 1✱ t❛ ❝â ♣❞ (J2q+1 ∩ K2q+1 ) = ♣❞ (Ki ) = 0✳ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❧➦♣ ❧↕✐ ✤➥♥❣ t❤ù❝ ♣❞ (Ji ∩ Ki) = max{♣❞ (Ji+1), ♣❞ (Ki+1), ♣❞ (Ji+1 ∩ Ki+1) + 1} ✈ỵ✐ i = 2q, 2q − 1, , 1, t❛ s✉② r❛ ♣❞ (J1 ∩ K1) = 2(p − 1)✳ ❱➻ t❤➳ ♣❞ (Im,l,k ) = max{♣❞ (J1), ♣❞ (K1), ♣❞ (J1 ∩ K1) + 1} = max{2(p − 1), 0, 2(p − 1) + 1} = 2p − ✷✼ ✣✐➲✉ ♥➔② ❣✐↔✐ q✉②➳t tr÷í♥❣ ❤đ♣ k − = qt + ✤è✐ ✈ỵ✐ q ≥ ♥➔♦ ✤â✳ ✭✸✮ ◆➳✉ k − = qt + c ✈ỵ✐ q ≥ ♥➔♦ ✤â ✈➔ ≤ c < t✱ t❤➻✱ t÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤đ♣ ✭✶✮✱ ❝❤ó♥❣ t❛ ❝â p = q + ✈➔ d = s + (c − 1)(m − l)✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ d = m✳ ❚❤➟t ✈➟②✱ ♥➳✉ d = m✱ t❤➻ c − = m−s m−l = t − 1✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ c = t✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t r➡♥❣ c < t✳ ✣✐➲✉ ♥➔② s✉② r❛ s + (m − l) ≤ d < m − l + (c − 1)(m − l) < t(m − l) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ sè ❧÷đ♥❣ ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J1 ✈➔ J2q t÷ì♥❣ ù♥❣ ❧➔ p(2m − l − s) + s + (c − 2)(m − l) ✈➔ · (2m − l − s) + s + (c − 1)(m − l) ❚÷ì♥❣ tü✱ ✤è✐ ✈ỵ✐ ❜➜t ❦ý ≤ i ≤ q − 1✱ sè ❝→❝ ❜✐➳♥ ①✉➜t ❤✐➺♥ tr♦♥❣ J2i+1 ✈➔ J2i t÷ì♥❣ ù♥❣ ❧➔ (p−i)(2m−l−s)+s+(c−2)(m−l) ✈➔ (p−i)(2m−l−s)+s+(c− 1)(m − l)✳ ❉♦ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ t❛ ❝â ♣❞ (J1 ) = 2p − 1✱ ♣❞ (J2q ) = 1✱ ♣❞ (J2i+1) = 2(p − i) − ✈➔ ♣❞ (J2i) = 2(p − i) − ✈ỵ✐ ≤ i ≤ q − 1✳ ✣➦t (t−1)(m−l) L2q = Im,l,k−qt−1 + ( x(k−qt−1)(m−l)+j ), j=1 J2q+1 = Im,l,k−qt−1 = Im,l,c , (t−1)(m−l) K2q+1 = ( xc(m−l)+j ), j=1 ❦❤✐ ✤â L2q = J2q+1 + K2q+1 ❝â ♠ët ♣❤➨♣ t→❝❤ ❇❡tt✐✱ J2q ∩ K2q = K2q L2q ✈➔ m−l J2q+1 ∩ K2q+1 = K2q+1 ( xj ) j=1 ❚÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤đ♣ ✭✷✮ ð tr➯♥✱ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ♣❞ (J2q+1 ∩ K2q+1) = ♣❞ (Ki) = ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❧➦♣ ❧↕✐ ✤➥♥❣ t❤ù❝ ♣❞ (Ji ∩ Ki) = max{♣❞ (Ji+1), ♣❞ (Ki+1), ♣❞ (Ji+1 ∩ Ki+1) + 1} ✷✽ ✈ỵ✐ i = 2q, 2q − 1, , 1, ❝❤ó♥❣ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ ♣❞ (J1 ∩ K1) = 2(p − 1)✳ ❱➻ t❤➳ ♣❞ (Im,l,k ) = max{♣❞ (J1 ), ♣❞ (K1 ), ♣❞ (J1 ∩ K1 ) + 1} = max{2p − 1, 0, 2(p − 1) + 1} = 2p ởt q trỹ t ỵ tr➯♥ ❧➔ ♥❤÷ s❛✉✿ ✷✳✷✳✻ ❍➺ q✉↔✳ ❈❤♦ k, l, m, n, s Im,l,k ữ tr ỵ ❑❤✐ ✤â ❞❡♣t❤ (Im,l,k ) = n + − n+m−l−s n+m−l−s − 2m − l − s 2m − l − s ✣➦t k − = qt + c✱ tr♦♥❣ ✤â q ≥ ✈➔ ≤ c < t ứ ự ỵ t❛ s✉② r❛ r➡♥❣ ♥➳✉ c = 0✱ t❤➻ d = m✱ ♥➳✉ ❦❤æ♥❣ t❤➻✱ d = s + (c − 1)(m − l)✳ ❉♦ ✤â✱ ❜➡♥❣ t➼♥❤ t♦→♥ ✤ì♥ ❣✐↔♥✱ t❛ ❝â ♥➳✉ c = 0✱ t❤➻ ❈❤ù♥❣ ♠✐♥❤✳ n + (m − l − s) n + (m − l − s) = = p + 1, 2m − l − s 2m − l − s ♥➳✉ ❦❤æ♥❣ t❤➻ ✈➔ n + (m − l − s) =p+1 2m − l − s n + (m − l − s) = p 2m − l − s ⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❆✉s❧❛♥❞❡r✲❇✉❝❤s❜❛✉♠✱ t❛ s✉② r❛ ❞❡♣t❤ (Im,l,k ) = n − ♣❞ (Im,l,k ) ✷✾ ❑➌❚ ▲❯❾◆ ❚r➯♥ ❝ì sð t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❬✶✷❪✱ ❧✉➟♥ ✈➠♥ ✤➣ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❝→❝ ♥ë✐ ❞✉♥❣ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ỡ số q st r ỗ t ữớ ỗ t ữớ ♥✐➺♠ tê ❤đ♣ ❦❤→❝ ❝ị♥❣ ✈ỵ✐ ♠ët sè ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ✷✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ữớ ỗ t ✤÷í♥❣✳ ✸✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ❝❤➾ số q ữớ ỗ t ✤÷í♥❣✳ ✸✵ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❉❛♥❤ ♠ư❝ t➔✐ ❧✐➺✉ t ỗ t s tr♦♥❣ ✤â t➔✐ ❧✐➺✉ ❝❤➼♥❤ ❧➔ ❬✶✷❪✳ ❬✶❪ ❆✳ ❆❧✐❧♦♦❡❡ ❛♥❞ ❙✳ ❋❛r✐❞✐ ✭✷✵✶✺✮✱ ●r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs ♦❢ ♣❛t❤ ✐❞❡❛❧s ♦❢ ❝②❝❧❡s ❛♥❞ ❧✐♥❡s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✹✸✱ ✺✹✶✸✲✺✹✸✸✳ ❬✷❪ P✳ ❇r✉♠❛tt✐ ❛♥❞ ❆✳ ❋✳ ❞❛ ❙✐❧✈❛ ✭✷✵✵✶✮✱ ❖♥ t❤❡ s②♠♠❡tr✐❝ ❛♥❞ ❘❡❡s ❛❧❣❡✲ ❜r❛s ♦❢ (n, k)✲❝②❝❧✐❝ ✐❞❡❛❧s✱ ✶✻t❤ ❙❝❤♦♦❧ ♦❢ ❆❧❣❡❜r❛✱ P❛rt ■■ ✭P♦rt✉❣✉❡s❡✮ ✭❇r❛s✐❧✐❛✱ ✷✵✵✵✮✳ ▼❛t✳ ❈♦♥t❡♠♣✳ ✷✶✱ ✷✼✲✹✷✳ ❬✸❪ ❆✳ ❈♦♥❝❛ ❛♥❞ ❊✳ ❉❡ ◆❡❣r✐ ✭✶✾✾✾✮✱ ▼✲❙❡q✉❡♥❝❡s✱ ❣r❛♣❤ ✐❞❡❛❧s ❛♥❞ ❧❛❞❞❡r ✐❞❡❛❧s ♦❢ ❧✐♥❡❛r t②♣❡✱ ❏✳ ❆❧❣❡❜r❛✱ ✷✶✶✱ ✺✾✾✲✻✷✹✳ ❬✹❪ ❈✳ ❆✳ ❋r❛♥❝✐s❝♦✱ ❍✳ ❚✳ ❍➔ ❛♥❞ ❆✳ ❱❛♥ ❚✉②❧ ✭✷✵✵✾✮✱ ❙♣❧✐tt✐♥❣s ♦❢ ♠♦♥♦♠✐❛❧ ✐❞❡❛❧s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✸✼ ✭✶✵✮✱ ✸✷✼✶✲✸✷✽✷✳ ❬✺❪ ❍✳ ❚✳ ❍➔ ❛♥❞ ❆✳ ❱❛♥ ❚✉②❧ ✭✷✵✵✽✮✱ ▼♦♥♦♠✐❛❧ ✐❞❡❛❧s✱ ❡❞❣❡ ✐❞❡❛❧s ♦❢ ❤②♣❡r✲ ❣r❛♣❤s✱ ❛♥❞ t❤❡✐r ❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✱ ❏✳ ❆❧❣❡❜r❛✐❝ ❈♦♠❜✐♥✳✱ ✷✼ ✭✷✮✱ ✷✶✺✲✷✹✺✳ ❬✻❪ ❏✐♥❣ ❍❡ ❛♥❞ ❆✳ ❱❛♥ ❚✉②❧ ✭✷✵✶✵✮✱ ❆❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♣❛t❤ ✐❞❡❛❧ ♦❢ ❛ tr❡❡✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✸✽ ✭✺✮✱ ✶✼✷✺✲✶✼✹✷✳ ❬✼❪ ▲✳ ❚✳ ❍♦❛ ❛♥❞ ◆✳ ❉✳ ❚❛♠ ✭✷✵✶✵✮✱ ❖♥ s♦♠❡ ✐♥✈❛r✐❛♥ts ♦❢ ❛ ♠✐①❡❞ ♣r♦❞✉❝t ♦❢ ✐❞❡❛❧s✳ ❆r❝❤✳ ▼❛t❤✳✱ ✾✹ ✭✹✮✱ ✸✷✼✲✸✸✼✳ ❬✽❪ ❆✳ ❱✳ ❚✉②❧ ❛♥❞ ❘✳ ❍✳ ❱✐❧❧❛rr❡❛❧ ✭✷✵✵✽✮✱ ❙❤❡❧❧❛❜❧❡ ❣r❛♣❤s ❛♥❞ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❜✐♣❛rt✐t❡ ❣r❛♣❤s✱ ❏♦✉r♥❛❧ ♦❢ ❈♦♠❜✐♥❛t♦r✐❛❧ ❚❤❡♦r②✱ ✶✶✺ ✭✺✮✱ ✼✾✾✲✽✶✹✳ ✸✶ ❬✾❪ ❘✳ ❍✳ ❱✐❧❧❛rr❡❛❧ ✭✷✵✵✶✮✱ ▼♦♥♦♠✐❛❧ ❛❧❣❡❜r❛s✱ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✱ ◆❨✳ ❬✶✵❪ ❘✳ ❍✳ ❱✐❧❧❛rr❡❛❧ ✭✶✾✾✵✮✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❣r❛♣❤s✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✳✱ ✻✻✱ ✷✼✼✲✷✾✸✳ ❬✶✶❪ ●✉❛♥❣❥✉♥ ❩❤✉ ✭✷✵✶✻✮✱ ❙❤❡❧❧❛❜✐❧✐t② ♦❢ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s ❛♥❞ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s ✇✐t❤ t❤❡ ❢r❡❡ ✈❡rt❡① ♣r♦♣❡rt②✱ ❚✉r❦✐s❤ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✹✵ ✭✶✮✱ ✶✽✶✲✶✾✵✳ ❬✶✷❪ ●✉❛♥❣❥✉♥ ❩❤✉ ✭✷✵✶✼✮✱ Pr♦❥❡❝t✐✈❡ ❞✐♠❡♥s✐♦♥ ❛♥❞ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♣❛t❤ ✐❞❡❛❧ ♦❢ t❤❡ ❧✐♥❡ ❣r❛♣❤✱ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❖♥❧✐♥❡ ❛✈❛✐❧❜❧❡✳