Điều kiện tối ưu cấp cao cho cực tiểu địa phương chặt và cực tiểu pareto địa phương chặt

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM QUỲNH TRANG ĐIỀU KIỆN TỐI ƯU CẤP CAO CHO CỰC TIỂU ĐỊA PHƯƠNG CHẶT VÀ CỰC TIỂU PARETO ĐỊA PHƯƠNG CHẶT LUẬN VĂN THẠC SĨ TOÁN HỌC Thái Nguyên - Năm 2015 Số hóa Trung tâm Học liệu - ĐHTN http://www.lrc-tnu.edu.vn/ ✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tr✉♥❣ t❤ù❝ ✈➭ ❦❤➠♥❣ trï♥❣ ❧➷♣ ✈í✐ ❝➳❝ ➤Ị t➭✐ ❦❤➳❝✳ ❚➠✐ ❝ị♥❣ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ♠ä✐ sù ❣✐ó♣ ➤ì ❝❤♦ ✈✐Ư❝ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ➤➢ỵ❝ ❝➯♠ ➡♥ ✈➭ ❝➳❝ t❤➠♥❣ t✐♥ trÝ❝❤ ❞➱♥ tr♦♥❣ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢ỵ❝ ❝❤Ø râ ♥❣✉å♥ ❣è❝✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺ ◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥ P❤➵♠ ◗✉ú♥❤ ❚r❛♥❣ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ t❤ù❝ ❤✐Ö♥ ✈➭ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ọ ủ P ỗ t ❣✐➯ ①✐♥ ➤➢ỵ❝ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬② ❣✐➳♦✱ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❝đ❛ ♠×♥❤✱ P●❙✳ ỗ t tì tr♦♥❣ s✉èt q✉➳ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ t➳❝ ❣✐➯✳ ➜å♥❣ t❤ê✐ t➳❝ ❣✐➯ ❝ò♥❣ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝ t❤➬② ❝➠ tr♦♥❣ ❦❤♦❛ ❚♦➳♥✱ ❦❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠✱ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ➤➲ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ ➤Ĩ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚➳❝ ❣✐➯ ❝ị♥❣ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ ➤Õ♥ ❣✐❛ ➤×♥❤ ✈➭ ❝➳❝ ❜➵♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ❚♦➳♥ ❑✷✶❇✱ ➤➲ ➤é♥❣ ✈✐➟♥ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥✳ ▲✉❐♥ ✈➝♥ ❦❤➠♥❣ t❤Ĩ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✱ t➳❝ ❣✐➯ r✃t ♠♦♥❣ ợ ỉ t tì ủ t ❝➠ ✈➭ ❜➵♥ ❜❒ ➤å♥❣ ♥❣❤✐Ö♣✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺ ◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥ P❤➵♠ ◗✉ú♥❤ ❚r❛♥❣ ✐✐✐ ▼ô❝ ❧ô❝ ▲ê✐ ❝❛♠ ➤♦❛♥ ✐ ▲ê✐ ❝➯♠ ➡♥ ✐✐ ▼ơ❝ ❧ơ❝ ✐✐✐ ▼ë ➤➬✉ ✶ ✶ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❲❛r❞ ✸ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ➤Þ♥❤ ♥❣❤Ü❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✻ ✶✳✸ ❍➭♠ ✷ C 1,1 m ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✈➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❘❛❤♠♦✲❙t✉❞♥✐❛rs❦✐ ✷✷ ✷✳✶ ❈➳❝ ❦Õt q✉➯ ❜ỉ trỵ ✷✳✷ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸ ➜✐Ị✉ ❦✐Ư♥ ➤đ tè✐ ➢✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ ➜➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✶ ▼ë ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ❧✉❐♥ ✈➝♥ ▲ý t❤✉②Õt ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❧➭ ♠ét ❜é ♣❤❐♥ q✉❛♥ trä♥❣ ❝đ❛ ❧ý t❤✉②Õt tè✐ ➢✉ ❤ã❛✳ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ tố ột é t ị ợ t❐♣ ❝➳❝ ➤✐Ĩ♠ ❞õ♥❣✳ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ é t tì r ợ ệ tố ➢✉ tr♦♥❣ t❐♣ ❝➳❝ ➤✐Ĩ♠ ❞õ♥❣ ➤ã✳ ❑❤➳✐ ♥✐Ư♠ ❝ù❝ tể ị t m ợ ị ĩ ❈r♦♠♠❡ ❬✷❪✳ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ĩ✉ ❝❤➷t ❝✃♣ m ➤➢ỵ❝ t❤✐Õt ❧❐♣ ❜ë✐ ❆✉s❧❡♥❞❡r ❬✶❪✱ ❙t✉❞♥✐❛rs❦✐ ❬✶✷❪✱ ❉✳❱✳ ▲✉✉ ❬✶✵❪✱ ❲❛r❞ ❬✶✹❪✳ ❉✳❊✳ ❲❛r❞ ✭✶✾✾✹✮ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ự tể ị t ữ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦✳ ❊✳❉✳ ❘❛❤♠♦ ✲ ▼✳ ❙t✉❞♥✐❛rs❦✐ ✭✷✵✶✷✮ ➤➲ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ö♠ ➤➵♦ ❤➭♠ ❙t✉❞♥✐❛rs❦✐ ➤➢❛ r❛ ✶✾✽✻ ❝❤♦ ❤➭♠ ✈Ð❝t➡ ✈➭ ❞➱♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ơ❝ t✐➟✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉✳ ➜➞② ❧➭ ➤Ị t➭✐ ➤➢ỵ❝ ♥❤✐Ị✉ t➳❝ ❣✐➯ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ❈❤Ý♥❤ ✈× ✈❐② ❡♠ ❝❤ä♥ ➤Ị t➭✐✿ ✧➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t✳✧ ✷✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❙➢✉ t➬♠ ✈➭ ➤ä❝ t➭✐ ❧✐Ö✉ tõ ❝➳❝ s➳❝❤✱ t➵♣ ❝❤Ý t♦➳♥ ❤ä❝ tr♦♥❣ ♥➢í❝ ✈➭ q✉è❝ tÕ ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦✳ ◗✉❛ ➤ã✱ t×♠ ❤✐Ĩ✉ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ✈✃♥ ➤Ị ♥➭②✳ ✸✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ t×♠ ❤✐Ĩ✉ ✈Ị ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ ✷ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t✳ ❈ơ t❤Ĩ✱ ❝❤ó♥❣ t➠✐ ➤ä❝ ❤✐Ĩ✉ ✈➭ tr×♥❤ ❜➭② ❧➵✐ ♠ét ❝➳❝❤ t➢ê♥❣ ♠✐♥❤ ❤❛✐ ❜➭✐ ❜➳♦ s❛✉✿ ✶✳ ❉✳❊✳ ❲❛r❞✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ str✐❝t ❧♦❝❛❧ ♠✐♥✐♠❛ ❛♥❞ ♥❡❝❡ss❛r② ❝♦♥❞✐✲ t✐♦♥s ❢♦r ✇❡❛❦ s❤❛r♣ ♠✐♥✐♠❛✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✽✵✭✶✾✾✹✮✱ ✺✺✶✲✺✼✶✳ ✷✳ ❊✳❉✳ ❘❛❤♠♦✱ ▼✳ ❙t✉❞♥✐❛rs❦✐✱ ❍✐❣❤❡r ♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r str✐❝t ❧♦❝❛❧ P❛r❡t♦ ♠✐♥✐♠❛ ✐♥ t❡r♠s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✾✸✭✷✵✶✷✮✱ ✷✶✷✲✷✷✶✳ ✹✳ ◆é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ♣❤➬♥ ♠ë ➤➬✉✱ ✷ ❝❤➢➡♥❣✱ ❦Õt ❧✉❐♥ ✈➭ ❞❛♥❤ ♠ơ❝ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❈❤➢➡♥❣ ✶✿ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ự tể ị t ủ r rì ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❲❛r❞ ❬✶✸❪ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ❦❤➳❝ ♥❤❛✉✳ ❱í✐ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉②✱ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❝✃♣ ❝❛♦ trë t❤➭♥❤ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦✳ ❈❤➢➡♥❣ ✷✿ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐ ❚r×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝✃♣ m ❝❤♦ ❤➭♠ ✈❡❝t➡ ✈➭ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐ ✭❬✶✵❪✱ ✷✵✶✷✮✳ m ✸ ❈❤➢➡♥❣ ✶ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❲❛r❞ ❚r♦♥❣ ❝❤➢➡♥❣ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦ ❝đ❛ ❲❛r❞ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❝➳❝ ➤➵♦ ❤➭♠ ❝✃♣ ❝❛♦ t❤❡♦ ♣❤➢➡♥❣ ❦❤➳❝ ♥❤❛✉✳ ❱í✐ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ trë t❤➭♥❤ ➤✐Ị✉ ❦✐Ư♥ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ĩ✉ ❝❤➷t ❝✃♣ ❝❛♦✳ ❈➳❝ ❦Õt q✉➯ tr×♥❤ ❜➭② tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ❧➭ ❝ñ❛ ❲❛r❞ ❬✶✹❪✳ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ➤Þ♥❤ ♥❣❤Ü❛ ❳Ðt ❜➭✐ t♦➳♥ tè✐ ➢✉ s❛✉✿ ✭✶✳✶✮ {f (x) |x ∈ S } , tr♦♥❣ ➤ã f ✿ Rn → R ∪ {+∞} ✈➭ S ột t rỗ tr Rn ị ♥❣❤Ü❛ ✶✳✶✳✶ ❈❤♦ · ❧➭ ❝❤✉➮♥ ➙❝❧✐t tr♦♥❣ Rn ✳ ❱í✐ ε > 0✱ ➤➷t B (x, ε) := {y ∈ Rn | y − x ≤ ε} ✭❛✮ ❚❛ ♥ã✐ r➺♥❣ tå♥ t➵✐ x¯ ∈ S ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❜➭✐ t♦➳♥ ε > s❛♦ ❝❤♦ f (x) > f (¯ x) (∀x ∈ S ∩ B (¯ x, ε) \ {¯ x}) (1.1) ♥Õ✉ ✹ ✭❜✮ ❈❤♦ m ≥ ❧➭ ♠ét sè ♥❣✉②➟♥✳ ❚❛ ♥ã✐ r➺♥❣ x¯ ∈ S ❝❤➷t ❝✃♣ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ m ❝đ❛ (1.1) ♥Õ✉ tå♥ t➵✐ ε > 0✱ β > s❛♦ ❝❤♦ m f (x) − f (¯ x) ≥ β x − x¯ ✭✶✳✷✮ (∀x ∈ S ∩ B (¯ x, ε)) ◆❤❐♥ ①Ðt ✶✳✶✳✶ ✭❛✮ ◆❤❐♥ t❤✃② r➺♥❣✱ ♥Õ✉ x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ j ✈í✐ ♠ä✐ m✱ t❤× ♥ã ❝ị♥❣ j > m✳ ✭❜✮ ❘â r➭♥❣ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❜✃t ❦ú ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t✳ ❚✉② ♥❤✐➟♥✱ ❦❤➠♥❣ ỗ ự tể ị t ột ự t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ✈í✐ m ♥➭♦ ➤ã✳ ❈❤➻♥❣ ❤➵♥✱ ❝❤♦ ❤➭♠ f : [0, +∞) → R f (x) = x1/x , ✈í✐ x > 0, f (0) = 0, ✈➭ S := [0, +∞) ❑❤✐ ➤ã✱ x = ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ♠➭ ❦❤➠♥❣ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ✈í✐ m ❜✃t ❦ú✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷ ✭❛✮ ❈❤♦ S⊂ Rp ◆ã♥ ❧ï✐ ①❛ ❝đ❛ S ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ 0+ S := {y ∈ Rp |s + ty ∈ S, ∀s ∈ S, t ≥ 0} ✭❜✮ ◆ã♥ t✐Õ♣ t✉②Õ♥ ❧➭ ♠ét ➳♥❤ ①➵ ✈➭ p p A : 2R × Rp → 2R s❛♦ ỗ S Rp x Rp , A (S, x) ột ó ó tể rỗ ỗ S Rp x S, t ❝ã 0+ S ⊂ 0+ A (S, x) ❈➳❝ ♥ã♥ t✐Õ♣ t✉②Õ♥ q✉❛♥ trä♥❣ ë ➤➞② ❧➭ ♥ã♥ t✐Õ♣ ❧✐➟♥✱ ♥ã♥ t✐Õ♣ t✉②Õ♥ ❯rs❡s❝✉ ✈➭ ❝➳❝ ♥ã♥ ♣❤➬♥ tr♦♥❣ t ứ ó tế ợ ị ĩ K (S, x) := y ∃ (tn , yn ) → 0+ , y s❛♦ ❝❤♦ x + tn yn ∈ S, n ; ó tế tế rss ợ ị ĩ ❜ë✐ k (S, x) := y ∀ (tn ) → 0+ , ∃ (yn ) → y ✈í✐ x + tn yn ∈ S, ∀n ; ✺ ✈➭ ❝➳❝ ♥ã♥ ♣❤➬♥ tr♦♥❣ t➢➡♥❣ ø♥❣ ❝đ❛ ❝❤ó♥❣ ❧➭ IK (S, x) := y ∃ (tn ) → 0+ s❛♦ ❝❤♦ ∀ (yn ) → y, x + tn yn ∈ S, ∀n ➤đ ❧í♥ ✈➭ Ik (S, x) := y ∀ (tn , yn ) → 0+ , y , x + tn yn ∈ S, ∀n ➤đ ❧í♥ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✸ : Rn → R ∪ {+∞} ❤÷✉ ❤➵♥ t➵✐ x ∈ Rn ❑Ý ●✐➯ sö A ❧➭ ♠ét ♥ã♥ t✐Õ♣ t✉②Õ♥✱ ✈➭ f ❤✐Ư✉ tr➟♥ ➤å t❤Þ ❝ñ❛ y ❧➭ ❡♣✐f f A✲ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ủ f t x t ợ ị ĩ f A (x; y) := inf {r |(y, r) ∈ A ( ❡♣✐f, (x, f (x)))} ❱í✐ ❝➳❝ ♥ã♥ tế tế ợ ị ĩ tr ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ t➢➡♥❣ ø♥❣ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ ♥❤➢ ❝➳❝ ❣✐í✐ ❤➵♥ s❛✉ ✭①❡♠ ❬✶✺❪✮ f K (x; y) = lim inf (f (x + tv) − f (x)) /t, + (t,v)→(0 ,y) f k (x; y) = lim sup inf (f (x + tv) − f (x)) /t v→y t→0+ (f (x + tv) − f (x)) /t, := sup lim sup inf ε>0 t→0+ v∈B(y,ε) f IK (x; y) = lim inf sup (f (x + tv) − f (x)) /t + t→0 v→y sup (f (x + tv) − f (x)) /t, := inf lim inf + ε>0 t→0 v∈B(y,ε) f Ik (x; y) = lim sup (f (x + tv) − f (x)) /t (t,v)→(0+ ,y) ◆Õ✉ f ❧➭ ❦❤➯ ✈✐ ❋rÐ❝❤❡t t➵✐ ♣❤➢➡♥❣ tr➟♥ ❜➺♥❣ tr♦♥❣ x ✈í✐ ➤➵♦ ❤➭♠ ∇f (x) , t❤× ❝➯ ❜è♥ ➤➵♦ ❤➭♠ t❤❡♦ ∇f (x) , y , tr♦♥❣ ➤ã ❧➭ ❦Ý ❤✐Ư✉ tÝ❝❤ ✈➠ ❤➢í♥❣ tr♦♥❣ Rn ế f st ị t x, tì f K (x; ·) = f IK (x; ·) ✈➭ f k (x; ·) = f Ik (x; ·) ✭①❡♠ ❬✶✻❪✮✳ ♠➭ ·, · f K (x; ·) > −∞ ✈➭ ▼ét ❧í♣ ❤➭♠ ➤➳♥❣ ❝❤ó ý ❜❛♦ ❣å♠ ❝➳❝ ❤➭♠ f K (x; ·) = f k (x; ·) tr➟♥ ❦❤➯ ✈✐ ❡♣✐❞✐❢❢❡r❡♥t✐❛❜❧❡ t➵✐ x✳ f ❈➳❝ ❤➭♠ sè ♥❤➢ t❤Õ ➤➢ỵ❝ ❣ä✐ ❧➭ ✻ ❈ị♥❣ ♥❤➢ tr♦♥❣ ❬✶✷❪✱ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ➤➵♦ ❤➭♠ ❝✃♣ ❝❛♦ dm f K (x; y) = lim inf (f (x + tv) − f (x)) /tm + (t,v)→(0 ,y) ✈➭ ➤Þ♥❤ ♥❣❤Ü❛ t➢➡♥❣ tù ❝❤♦ dm f k (x; y) , dm f IK ✶✳✷ (x; y) ✈➭ dm f Ik (x; y) ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ●✐➯ sö x¯ ∈ S m ✈➭ ❦ý ❤✐Ö✉ K (¯ x) := K (S, x¯) ∩ y f K (x; y) ≤ ; is ❧➭ ❤➭♠ ❝❤Ø ❝ñ❛ t❐♣ S:   0, ♥Õ✉ x ∈ S, is (x) =  +∞, ♥Õ✉ x ∈ / S ➜Þ♥❤ ❧ý ✶✳✷✳✶ ❈❤♦ ✭❛✮ [12] m > 1✱ ❝➳❝ ♣❤➳t ❜✐Ó✉ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❝đ❛ ❜➭✐ t♦➳♥ ✭✶✳✶✮❀ ✭❜✮ ❱í✐ ♠ä✐ y ∈ Rn \ {0} , dm (f + iS )K (¯ x; y) > 0; ✭❝✮ ❇✃t ➤➻♥❣ t❤ø❝ ✭✶✳✸✮ ➤ó♥❣ ✈í✐ ♠ä✐ ◆Õ✉ m = 1✱ ✭✶✳✸✮ y ∈ K (¯ x) \ {0} t❤× ✭❛✮✱ ✭❜✮✱ ✭❝✮ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ♥Õ✉ K (¯ x) tr♦♥❣ ✭❝✮✳ ◆❤❐♥ ①Ðt ✶✳✷✳✶ ➜✐Ị✉ ❦✐Ư♥ ủ ị ý ó tể ợ t ✭ˆ b✮ ❚å♥ t➵✐ β > s❛♦ ❝❤♦✱ ✈í✐ ♠ä✐ y ∈ Rn , dm (f + iS )K (¯ x; y) ≥ β y m ➤➢ỵ❝ t❤❛② ❜ë✐ K (S, x¯) ✷✼ ➤➢➡♥❣ ❧➭ t❤❡♦ ♥ã♥ ❞➢➡♥❣ Rp+ := [0, ∞)p ▼➷❝ ❞ï ë ➤➞② f ỉ trị t ữ í tết trì tr trớ ữ í tr ✈✐Ư❝ ①Ðt ♠ét sè ❧♦➵✐ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝đ❛ ❈❤ó♥❣ t❛ ❦Ý ❤✐Ư✉ S✳ N (x) ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ ❧➞♥ ❝❐♥ ❝đ❛ x ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✸ ●✐➯ sư ✈➭ ❤➭♠ ❝❤Ø ✈❡❝t➡ ❝đ❛ f ✭❬✺❪✮ m ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ ✈➭ x¯ ∈ S ✭❛✮ ❚❛ ♥ã✐ r➺♥❣ x ¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ t m ệ ữ ệ ị ❝❤➷t ❝✃♣ ♥Õ✉ tå♥ t➵✐ m✮ ❝ñ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ ❦Ý ❤✐Ö✉ α > ✈➭ U ∈ N (¯ x) s❛♦ ❝❤♦ m (f (x) + Rp+ ) ∩ B (f (¯ x) , α x − x¯ ✭❜✮ ❚❛ ♥ã✐ r➺♥❣ m ) = ∅, ∀x ∈ S ∩ U \ {¯ x} ✭✷✳✶✾✮ x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ s✐➟✉ ❝❤➷t ❝✃♣ m❀ s✉♣❡r✲str✐❝t ❧♦❝❛❧ P❛r❡t♦ ♠✐♥✐♠✐③❡r ♦❢ ♦r❞❡r ❝✃♣ x¯ ∈ StrL (m, f, S) m ệ ữ ệ ị s ❝❤➷t ❝đ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ ❦Ý ❤✐Ư✉ x¯ ∈ SStrL (m, f, S) ♥Õ✉ tå♥ t➵✐ α > 0✱ U ∈ N (¯ x) , ✈➭ t➵✐ ♥❤✐Ò✉ ♥❤✃t p ♥ã♥ ♠ë Ai ✭ ❦❤➠♥❣ ❝❤ø❛ ✵✮ i ∈ I ⊂ I, s❛♦ ❝❤♦ {Vi := x¯ + Ai : i ∈ I } ❧➭ ♠ét ♣❤ñ ❝ñ❛ S ∩ U \ {¯ x} ✈➭ m fi (x) > fi (¯ x) + α x − x¯ , ∀x ∈ S ∩ U ∩ Vi ✭✷✳✷✵✮ ▼Ư♥❤ ➤Ị ✷✳✶✳✺ ✭❬✹❪✮ x¯ ∈ ❙tr▲(m, f, S) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐ η ∈ intRp+ f (x) tø❝ ❧➭✱ ✈í✐ ỗ f ( x) + x x m ✈➭ U ∈ N (¯ x) s❛♦ ❝❤♦ , ∀x ∈ S ∩ U \ {¯ x} , ✭✷✳✷✶✮ x ∈ S ∩ U \ {¯ x} , ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❦❤➠♥❣ ➤ó♥❣ fi (x) fi (¯ x) + ηi x − x¯ fj (x) < fj (¯ x) + ηj x − x¯ m , m , ∀i ∈ I, ✈í✐ j ♥➭♦ ➤ã ∈ I ▼ë ré♥❣ ➤Þ♥❤ ♥❣❤Ü❛ tõ ❬✶✷❪ ❝❤♦ ❝➳❝ ❤➭♠ ✈❡❝t➡✱ ❜➞② ❣✐ê ❝❤ó♥❣ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝✃♣ ❞ m m✿ f (¯ x + tv) − f (¯ x) , ,y) tm f (¯ x; y) := lim inf + (t,v)→(0 ✭✷✳✷✷✮ ✷✽ f (¯ x + tv) − f (¯ x) d¯m f (¯ x; y) := lim sup , tm (t,v)→(0+ ,y) ✭✷✳✷✸✮ tr ó ợ ị ĩ ị ♥❣❤Ü❛ ✷✳✶✳✷✳ ❈❤Ý♥❤ ①➳❝ ❤➡♥✱ t❛ ❝ã ❞ m f (¯ x; y) := sup inf t∈(0,δ) δ>0 f (¯ x + tv) − f (¯ x) , tm ✭✷✳✷✹✮ f (¯ x + tv) − f (¯ x) tm ✭✷✳✷✺✮ v∈B(y,δ) d¯m f (¯ x; y) := inf sup δ>0 t∈(0,δ) v∈B(y,δ) ➳♣ ❞ơ♥❣ ♠Ư♥❤ ➤Ị ✷✳✶✳✸✱ t❛ ❝ã f (¯ x; y) = (❞m f1 (¯ x; y) , , ❞m fp (¯ x; y)) , ✭✷✳✷✻✮ x; y) , , d¯m fp (¯ x; y) , d¯m f (x; y) = d¯m f1 (¯ ✭✷✳✷✼✮ ❞ m tr♦♥❣ ➤ã ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝đ❛ ✈Õ ♣❤➯✐ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ ♥❤➢ tr♦♥❣ ❬✶✶❪✳ ❈❤ó♥❣ ❝ã t❤Ĩ ❝ã ❣✐➳ trÞ ✈➠ ❤➵♥✱ ✈➭ ❞♦ ➤ã ❝➳❝ ➤➵♦ ❤➭♠ ✭✷✳✷✷✮✲✭✷✳✷✸✮ ♥ã✐ ❝❤✉♥❣ t❤✉é❝ ❱í✐ ¯ p✳ R ¯ (¯ m = 1, t❛ ❦ý ❦✐Ö✉ ❞f (¯ x; y) ✈➭ df x; y) t❤❛② ❝❤♦ ❞1 f (¯ x; y) ✈➭ d¯1 f (¯ x; y) ❚❛ ❦ý ❤✐Ö✉ dm f (¯ x; y) := lim+ (t,v)→(0 ❦❤✐ ❣✐í✐ ❤➵♥ ♥➭② tå♥ t➵✐ ❤÷✉ ❤➵♥ tr♦♥❣ t➢➡♥❣ tù ❝ã t❤Ĩ sư ❞ơ♥❣ ❝❤♦ ❤➭♠ ❑ý ❤✐Ư✉ g f (¯ x + tv) − f (¯ x) , ,y) tm ✭✷✳✷✽✮ ¯ p ✭tø❝ ❧➭✱ dm f (¯ R x; y) ∈ Rp ✮✳ ❈➳❝ ❦ý ❤✐Ö✉ ♥Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ t➢➡♥❣ ø♥❣ tr♦♥❣ K (S, x¯) ❧➭ ♥ã♥ t✐Õ♣ ❧✐➟♥ ❝ñ❛ S t➵✐ Y x¯ : K (S, x¯) := y ∈ X : ∃ (tn , yn ) → 0+ , y s❛♦ ❝❤♦ x + tn yn ∈ S, ∀n ✭✷✳✷✾✮ ➜➷t x; y) Kf (S, x¯) := K (S, x¯) ∩ {y ∈ X : ❞f (¯ 0} p = K (S, x¯) ∩ {y ∈ X : ❞fi (¯ x; y) 0} ✭✷✳✸✵✮ i=1 ❱× K (S, x¯) ❧➭ ♠ét ♥ã♥ ➤ã♥❣ ✈➭ ỗ số fi ( x; Ã) t t ❞➢➡♥❣ ✈➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✱ ♥➟♥ t❐♣ ❤ỵ♣ ✭✷✳✸✵✮ ❝ị♥❣ ❧➭ ♠ét ♥ã♥ ➤ã♥❣ ✭❝❤ø❛ ✵✮✳ ❚❛ sÏ sư ❞ơ♥❣ ❤➭♠ ❝❤Ø ✈Ð❝t➡ ❝đ❛ t❐♣ S✿ ¯p ∆ (x |S ) := (δ (x |S ) , , δ (x |S )) ∈ R ✈í✐ x ∈ X, ✭✷✳✸✶✮ ✷✾ tr♦♥❣ ➤ã   ♥Õ✉ x ∈ S, δ (x |S ) :=  ∞ ♥Õ✉ x ∈ / S ❚❛ ①➳❝ ➤Þ♥❤ ❤➭♠ ¯p fS : X → R f S = f1S , , fpS , ✷✳✷ f S := f + ∆ (· |S ) ❑❤✐ ➤ã✱ ♥❤➢ s❛✉✿ S tr♦♥❣ ➤ã fi ✭✷✳✸✷✮ := fi + δ (· |S ) , i ∈ I ✭✷✳✸✸✮ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉ ➜Ĩ ♣❤➳t ❜✐Ĩ✉ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉✱ t❛ sÏ ❝➬♥ ➤Õ♥ ♥ã♥ ❞➢➡♥❣ ♠ë ré♥❣ ¯ p+ := R [0, ∞]p ➜Þ♥❤ ❧ý ✷✳✷✳✶ ●✐➯ sö x¯ ∈ ❙tr▲(m, f, S) ✭❛✮ ●✐➯ sö D = ∅ ✈➭ dg (¯ x; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y ∈ X ❑❤✐ ➤ã tå♥ t➵✐ β > s❛♦ ❝❤♦ m d¯m f (¯ x; y) ∈ / B (0, β y ¯ p+ , )−R ∀y ∈ K (C, x¯) ∩ {u ∈ X : dg (¯ x; u) ∈ −intD} ✭❜✮ ●✐➯ sö Y = Rq ✈➭ ✭✷✳✸✹✮ D = Rq+ ❑❤✐ ➤ã tå♥ t➵✐ β > s❛♦ ❝❤♦ m d¯m f (¯ x; y) ∈ / B (0, β y ¯ p+ , )−R ¯ (¯ ∀y ∈ K (C, x¯) ∩ u ∈ X : dg x; u) < ✭✷✳✸✺✮ ❈❤ø♥❣ ♠✐♥❤ ✭❛✮ ✭❙ö ❞ơ♥❣ ý t➢ë♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ✸✳✶✭❛✮⇒ ✭❜✮ tr♦♥❣ ❬✼❪✮✳ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ tå♥ t➵✐ α > ✈➭ U ∈ N (¯ x) s❛♦ ❝❤♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✾✮ ➤ó♥❣✳ ➜✐Ị✉ ➤ã t➢➡♥❣ ➤➢➡♥❣ ✈í✐ f (x) − f (¯ x) ∈ / B 0, α x − x¯ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ ✭✷✳✸✹✮ ➤ó♥❣ ✈í✐ ❑❤✐ ➤ã tå♥ t➵✐ m ¯ p+ , ∀x ∈ S ∩ U \ {x} −R β = α/2m y ∈ K (C, x¯) , u ∈ B (0, β y ✭✷✳✸✻✮ ●✐➯ sư r➺♥❣ ➤✐Ị✉ ➤ã ❦❤➠♥❣ ➤ó♥❣✳ m ¯ p+ ) ✈➭ z ∈ R dg (¯ x; y) ∈ −intD s❛♦ ❝❤♦ ✭✷✳✸✼✮ ✸✵ ✈➭ d¯m f (¯ x; y) = u − z ✭❈❤ó ý r➺♥❣ d¯m f (¯ x; y) ❝ã t❤Ó ❝ã t❤➭♥❤ ♣❤➬♥ ❜➺♥❣ −∞✮✳ ▲✃② ε > s❛♦ ❝❤♦ u + εe ∈ B (0, β y tr♦♥❣ ➤ã ✭✷✳✸✽✮ u m ✭✷✳✸✾✮ ), e := (1, , 1)T ∈ Rp ❑❤✐ ➤ã ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✽✮ ❦Ð♦ t❤❡♦ d¯m f (¯ x; y) < u + εe ❧➭ ♠ét ♥ã♥ ✈➭ ✭✷✳✹✵✮ ❱× D ➤ã y = ✭❝➬♥ ❧➢✉ ý r➺♥❣ dg (¯ x; 0) = ❞♦ ➤Þ♥❤ ♥❣❤Ü❛ ✭✷✳✷✽✮ ❜ë✐ ✈× v = y ❧➭ ♠ét ❝➳❝❤ ❝❤ä♥ ❝đ❛ v ✮✳ D=Y ♥➟♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✼✮ ❦Ð♦ t❤❡♦ dg (¯ x; y) = 0, ✈➭ ❞♦ ❉♦ ➤ã✱ t❤❡♦ ♠Ư♥❤ ➤Ị ✷✳✶✳✹ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✼✮✱ t❛ t×♠ ➤➢ỵ❝ δ > s❛♦ ❝❤♦ δ ✭✷✳✹✶✮ y /2, ✭✷✳✹✷✮ x¯ + (0, δ) B (y, δ) ⊂ U, f (¯ x + tv) − f (¯ x) < u + εe, ∀t ∈ (0, δ) ✈➭ ∀v ∈ B (y, δ) tm g (¯ x + tv) − g (¯ x) ∈ −D, ∀t ∈ (0, δ) ✈➭ ∀v ∈ B (y, δ) t ❱× y ∈ K (C, x ¯) ♥➟♥ g (¯ x + tv) ∈ g (¯ x) − D ⊂ −D ✈í✐ ♠ä✐ D t❛ s✉② r❛ t ∈ (0, δ) ❚õ ✭✷✳✹✶✮ ✲ ✭✷✳✹✹✮ ✈➭ ✭✷✳✹✻✮ t❛ s✉② r❛ tå♥ t➵✐ ✭✷✳✹✹✮ ✭✷✳✹✺✮ (¯ x + (0, δ) B (y, δ)) ∩ C = ∅ ❍➡♥ ♥÷❛✱ tõ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✹✮ ✈➭ tÝ♥❤ ❧å✐ ❝ñ❛ ✭✷✳✹✸✮ ✈➭ v ∈ B (y, δ) ✭✷✳✹✻✮ λ ∈ (0, δ) ✈➭ ω ∈ B (y, δ) s❛♦ ❝❤♦ x¯ + λω ∈ C ∩ U \ {x} , ✭✷✳✹✼✮ f (¯ x + λω) − f (¯ x) < u + εe, λm ✭✷✳✹✽✮ ✸✶ ✭✷✳✹✾✮ g (¯ x + λω) ∈ −D ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✾✮ ✈➭ ✭✷✳✹✽✮ ❦Ð♦ t❤❡♦ f (¯ x + λω) − f (¯ x) ∈ B (0, β y λm m ) − RP+ , ❤♦➷❝ t➢➡♥❣ ➤➢➡♥❣ f (¯ x + λω) − f (¯ x) ∈ B (0, β λy m ) − RP+ ✭✷✳✺✵✮ ✭❈❤ó ý r➺♥❣ f (¯ x + λω) − f (¯ x) ❦❤➠♥❣ t❤Ĩ ♥❤❐♥ ❣✐➳ trÞ ✈➠ ❤➵♥ ❜ë✐ ✈× f ❇➞② ❣✐ê✱ ❧✃② x := x¯ + λω, tõ ✭✷✳✹✼✮✱ ✭✷✳✹✾✮ ✈➭ ✭✷✳✺✵✮ t❛ ❝ã x ∈ C ∩ U \ {¯ x} , g (x) ∈ −D, f (x) − f (¯ x) ∈ B (0, β λy m ❤÷✉ ❤➵♥✮✳ ) − RP+ ✭✷✳✺✶✮ ❚õ ✭✷✳✹✶✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ω−y x4 , ✈➭ ❞♦ ➤ã (u1 , u2 ) = u21 + u22 > x2 xm ❚õ ➤ã s✉② r❛ x3 , x2 + R2+ ∩ B ((0, 0) , αxm ) = ∅ ❚✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❝ã ✈í✐ dg (0; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y∈R ✈➭ dg (0; y) = −y < y > ❍➡♥ ♥÷❛✱ K (C, x¯) = R ❙ư ❞ơ♥❣ ✭✷✳✷✼✮✱ ✈í✐ y > 0, t❛ ❝ã     0, y , ♥Õ✉ m = 2, d¯m f (0; y) = y , ∞ , ♥Õ✉ m = 3,    (∞, ) , ế m > ó ỗ β ∈ (0, 1) , ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✺✮ ➤ó♥❣✱ tø❝ ❧➭ ¯ 2+ d¯m f (0; y) ∈ / B (0, βy m ) − R ✈í✐ ♠ä✐ ✈í✐ m ✈➭ y > ▼è✐ q✉❛♥ ❤Ư ♥➭② ➤ó♥❣ ✈í✐ m = ❜ë✐ ✈× β < 1, ✈➭ ✈× d¯m f (0; y) ❝❤ø❛ ❝➳❝ t❤➭♥❤ ♣❤➬♥ , ò t ủ m ỗ t tr♦♥❣ ¯ 2+ B (0, βy m ) − R ❤♦➷❝ ❤÷✉ ❤➵♥ ❤♦➷❝ ❜➺♥❣ −∞ ✸✸ ❱Ý ❞ơ ✷✳✷✳✷ ❈❤♦ f = (f1 , f2 ) : R2 → R2 ✈➭ g : R2 → R ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ f (x1 , x2 ) := x21 + x2 , x21 − x2 , g (x1 , x2 ) := − |x2 | ❳Ðt ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ tr♦♥❣ ➤ã S := (x1 , x2 ) ∈ R2 : −g (x1 , x2 ) = |x2 | ∈ R+ = R2 ë ➤➞② D ✭ := R+ C := R2 ✮✳ ✈➭ ➜✐Ó♠ x¯ = (0, 0) ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐✳ ➜Ĩ t❤✃② ➤✐Ị✉ ♥➭②✱ t❛ ❧✃② ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ α = 1/2 ✈➭ U := B (¯ x, 1) ❑❤✐ ➤ã✱ S ∩ U \ {¯ x} = R2 ∩ U \ {¯ x} = (x1 , x2 ) : < x21 + x22 < ❇➞② ❣✐ê✱ ❧✃② (x1 , x2 ) ∈ S ∩ U \ {¯ x} ✈➭ (h1 , h2 ) ∈ R2+ ; t❛ ①Ðt trờ ợ ế x2 < tì x2 f (x1 , x2 ) + (h1 , h2 ) > ✭✐✐✮ ◆Õ✉ x22 ❉♦ ➤ã✱ x1 + x22 = α (x1 , x2 ) − x¯ ; −1 < x2 < t❤× −x2 f (x1 , x2 ) + (h1 , h2 ) > x21 + x22 |f1 (x1 , x2 ) + h1 | = x21 + x2 + h1 ✭✷✳✺✹✮ x22 ❉♦ ➤ã✱ x21 + x22 |f2 (x1 , x2 ) + h2 | = x21 − x2 + h2 x1 + x22 = α (x1 , x2 ) − x¯ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✹✮ ✈➭ ✭✷✳✺✺✮ ❦Ð♦ t❤❡♦ ✭✷✳✶✾✮ ➤ó♥❣ ✈í✐ ❈❤ó♥❣ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✺✮✳ ❚❛ ❝ã dg (¯ x; (y1 , y2 )) = − |y2 | < ✈í✐ y2 = ✭✷✳✺✺✮ m = K (C, x¯) = R2 ▲✃② ✈❡❝t➡ ❜✃t ❦ú ✈➭ y = (y1 , y2 ) y2 = ❑❤✐ ➤ã✱ f1 (tv1 , tv2 ) t2 v12 + tv2 ¯ d f1 (¯ x; (y1 , y2 )) = lim sup = lim sup t2 t2 (t,v)→(0+ ,y) (t,v)→(0+ ,y)   ∞, ♥Õ✉ y > 0, v2 2 = lim sup v1 + =  −∞, ♥Õ✉ y2 < 0, t (t,v)→(0+ ,y) ✈í✐ ✸✹ ✈➭ f2 (tv1 , tv2 ) t2 v12 − tv2 ¯ d f2 (¯ x; (y1 , y2 )) = lim sup = lim sup t2 t2 (t,v)→(0+ ,y) (t,v)→(0+ ,y)   −∞, ♥Õ✉ y > 0, v2 2 = lim sup v1 − =  ∞, ♥Õ✉ y2 < t (t,v)→(0+ ,y) ❉♦ ➤ã✱   (∞, −∞) , ♥Õ✉ y > 0, 2 ¯ d f (¯ x; (y1 , y2 )) =  (−∞, ∞) , ♥Õ✉ y2 < 0, tø❝ ❧➭ ✭✷✳✸✺✮ ➤ó♥❣ ❜ë✐ ✈× ❦❤➠♥❣ ❝ã ✈❡❝t➡ ♥➭♦ tr♦♥❣ ❝ã t❤➭♥❤ ♣❤➬♥ ❜➺♥❣ ✷✳✸ B (0, β y m ¯ p+ )−R ❝ã t❤Ĩ ∞ ➜✐Ị✉ ❦✐Ư♥ ➤đ tè✐ ➢✉ ➜Þ♥❤ ❧ý ➤➬✉ t✐➟♥ ✈Ị ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤➢ỵ❝ ♣❤➳t ❜✐Ĩ✉ ❝❤♦ t❐♣ ❤ỵ♣ r➭♥❣ ❜✉é❝ S tï② ý ✭❝❤ó♥❣ t❛ ❣✐➯ sư ❦❤➠♥❣ ❝ã ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✽✮✮✳ ➜Þ♥❤ ❧ý ✷✳✸✳✶ ●✐➯ sö ❞✐♠X ✭❛✮ ◆Õ✉ < ∞✱ ✈➭ x¯ ❧➭ ♠ét ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✳ m > ✈➭ ❞ t❤× ¯ p+ , ∀y ∈ Kf (S, x¯) \ {0} f (¯ x; y) ∈ / −R m S ✭✷✳✺✻✮ x¯ ∈ ❙❙tr▲(m, f, S) ✭❜✮ ◆Õ✉ ❞f t❤× S ¯ p+ , ∀y ∈ K (S, x¯) \ {0} (¯ x; y) ∈ / −R ✭✷✳✺✼✮ x¯ ∈ ❙❙tr▲(1, f, S) ❈❤ø♥❣ ♠✐♥❤ ✭❛✮ ❚õ ✭✷✳✷✻✮ ✈➭ ✭✷✳✸✸✮✱ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✻✮ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ max ❞m fiS (¯ x; y) > 0, ∀y ∈ Kf (S, x¯) \ {0} i∈I ✭✷✳✺✽✮ ✸✺ ❚õ s✉② ❧✉❐♥ ✭❝✮⇒ ✭❛✮ tr♦♥❣ ❬✻✱ ➤Þ♥❤ ❧Ý ✸✳✶❪ t❛ s✉② r❛ x¯ ∈ SStrL (m, f, S) ✷ ❈❤ø♥❣ ♠✐♥❤ ♣❤➬♥ ✭❜✮ t➢➡♥❣ tù✳ ❚r♦♥❣ ➤Þ♥❤ ❧Ý t✐Õ♣ t❤❡♦ t❛ sư ❞ơ♥❣ ❦Ý ❤✐Ư✉ s❛✉ ➤➞② ❝❤♦ ❜❛♦ ➤ã♥❣ ❝ñ❛ ♥ã♥ s✐♥❤ ❜ë✐ D + g (¯ x)✿ ✭✷✳✺✾✮ Dg(¯x) := cl cone (D + g (¯ x)) ❱× D ❧➭ t❐♣ ❧å✐ ♥➟♥ Dg(¯x) ❧➭ ♠ét ♥ã♥ ❧å✐ ➤ã♥❣✳ ➜Þ♥❤ ❧ý ✷✳✸✳✷ ●✐➯ sư ❞✐♠X < ∞, ✭✷✳✶✽✮✳ ●✐➯ sö dg (¯ x; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y ∈ X ◆Õ✉ ❞ m ✈➭ x¯ ❧➭ ♠ét ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮ ✲ ¯ p+ , ∀y ∈ K (C, x¯) ∩ u ∈ X : dg (¯ f (¯ x; y) ∈ / −R x; u) ∈ −Dg(¯x) \ {0} ✭✷✳✻✵✮ t❤× x¯ ∈ ❙tr▲(m, f, S) ❈❤ø♥❣ ♠✐♥❤ ✭❙ư ❞ơ♥❣ ý t➢ë♥❣ ❝đ❛ ➤Þ♥❤ ❧Ý ✹✳✶ tr♦♥❣ ❬✶✵❪✮✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ x ¯ ∈ / StrL (m, f, S) ❑❤✐ ➤ã✱ tõ ➤Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✸✭❛✮ s✉② r ỗ số xn S B (¯ x, 1/n) \ {¯ x} ✈➭ dn = (dn,1 , , dn,p ) ∈ Rp+ f (xn ) − f (¯ x) + dn ∈ B 0, xn − x¯ n n, tå♥ t➵✐ s❛♦ ❝❤♦ m ➜✐Ị✉ ♥➭② t➢➡♥❣ ➤➢➡♥❣ ✈í✐ dn f (xn ) − f (¯ x) m + xn − x¯ xn − x¯ m ∈ B 0, ❑❤➠♥❣ ♠✃t tÝ♥❤ ❝❤✃t tỉ♥❣ q✉➳t t❛ ❝ã t❤Ĩ ❣✐➯ sư tơ ➤Õ♥ ♠ét ✈❡❝t➡ xn = x¯ + tn ❇ë✐ ✈× v ✈í✐ ✈í✐ ♠ä✐ v n = ❉♦ ❑Ý ❤✐Ö✉ tn xn ∈ S ⊂ C = n ✭✷✳✻✶✮ := (xn − x¯) / xn − x¯ xn − x¯ ♥➟♥ t❛ ❝ã t❤× tn v ∈ K (C, x¯) → 0+ ❤é✐ ✈➭ ❞♦ ✭✷✳✷✾✮✳ dg (¯ x; v) tå♥ t➵✐✱ ❝❤♦ ♥➟♥ g (¯ x + tn ) − g (¯ x) n→∞ tn dg (¯ x; v) := lim ✭✷✳✻✷✮ ✸✻ ❍➡♥ ♥÷❛✱ g (¯ x + tn ) = g (xn ) ∈ −D ❉♦ ➤ã✱ g (¯ x + tn ) − g (¯ x) ∈ cone (−D − g (¯ x)) ⊂ −Dg(¯x) , ∀n tn ❚õ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✻✷✮ ✈➭ ✭✷✳✻✸✮✱ Dg(¯x) v ∈ K (C, x¯) , v = 1, ✭✷✳✻✸✮ ✈➭ tÝ♥❤ ➤ã♥❣ ❝ñ❛ t❛ ❝ã v ∈ K (C, x¯) ∩ u ∈ X : dg (¯ x; u) ∈ −Dg(¯x) \ {0} ✭✷✳✻✹✮ ❚õ ✭✷✳✻✶✮ s✉② r❛ fi (xn ) − fi (¯ x) dn,i + m xn − x¯ xn − x¯ lim n→∞ = 0, ∀i ∈ I m ❇✐Ó✉ t❤ø❝ tr➟♥ ❝ã t❤Ó ✈✐Õt ❧➵✐ ♥❤➢ s❛✉ lim n→∞ fi (¯ x + tn ) − fi (¯ x) dn,i + m tm tn n ❚õ s r ỗ m fi (¯ x; v) = 0, ∀i ∈ I ✭✷✳✻✺✮ i, t❛ ❝ã lim inf n→∞ fi (¯ x + tn ) − fi (¯ x) tm n fi (¯ x + tn ) − fi (¯ x) dn,i dn,i + + lim inf − n→∞ n→∞ tm tm tm n n n dn,i = lim inf − m n→∞ tn p ➜✐Ị✉ ❦✐Ư♥ dn ∈ R+ ❦Ð♦ t❤❡♦ −dn,i /tm ❚õ ✭✷✳✻✻✮ s✉② r❛ n = lim ❞ m fi (¯ x; v) ✭✷✳✻✻✮ (∀i) ❉♦ ➤ã✱ ❞ m ¯ p+ f (¯ x; v) ∈ −R ✷ ❉♦ ✭✷✳✻✹✮✱ ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ✭✷✳✻✵✮✳ ❱Ý ❞ơ ✷✳✸✳✶ ❳Ðt ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ tr♦♥❣ ➤ã f = (f1 , f2 ) : R → R2   x sin , x − sin x x f (x) :=  (0, 0) , , ợ ị ế ế x = 0, x = 0, ✸✼ ✈➭ t❐♣ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ➤➢ỵ❝ ❝❤♦ ❜ë✐✿ S := {x ∈ R : −g (x) = |x| ∈ R+ , x ∈ R+ } = R+ ë ➤➞② D := R+, C ✭ sử := R+ g:RR ợ ị g (x) := − |x|✳✮ x¯ = ❱í✐ ♠ä✐ y ∈ R, t❛ ❝ã dg (0; y) = − |y| ✈➭ dg (0; y) ∈ −Dg(0) = −R+ ❘â r➭♥❣ K (R+ , 0) = R+ ❉♦ ➤ã✱ K (R+ , 0) ∩ u ∈ R : dg (¯ x; u) ∈ −Dg(0) = R+ ❱í✐ ♠ä✐ y ∈ R+ , t❛ ❝ã ¯ (0; y) = y, ❞f1 (0; y) = −y, df ❞ f2 ¯ (0; y) = 3y (0; y) = y, df ❉♦ ➤ã✱ ❞f ¯ 2+ , ∀y ∈ R+ \ {0} (0; y) = (−y, y) ∈ / −R ❱× ✈❐② ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✻✵✮ t❤á❛ ♠➲♥✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ t❛ ❧✃② f (x) |f2 (x)| = x − sin ❉♦ ➤ã ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✾✮ ➤ó♥❣ ✈í✐ α = 1/2 t❤× > αx, ∀x ∈ S\ {0} = R+ \ {0} x m = ✈➭ U = R ➜✐Ò✉ ➤ã ❝ã ♥❣❤Ü❛ ❧➭ x¯ = ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ❝❤➷t ✭t♦➭♥ ❝ơ❝✮ ❝✃♣ ♠ét✳ ▲➢✉ ý r➺♥❣ ✈Ý ❞ơ ♥➭② ❦❤➠♥❣ t❤Ĩ ❣✐➯✐ q✉②Õt ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➤Þ♥❤ ❧Ý ✹✳✶ tr♦♥❣ ❬✶✵❪ ❜ë✐ ✈× ❞f ✈í✐ ♠ä✐ ✷✳✹ ¯ (0; y) (0; y) = df y = ❉♦ ➤ã df (0; ·) ❦❤➠♥❣ tå♥ t➵✐✳ ➜➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ Prt ị t ú t trì ột tí ❝❤✃t ➤➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❝❤♦ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ô❝ t✐➟✉ ✭✷✳✶✼✮ ♠➭ ❦❤➠♥❣ ❝ã ❜✃t ❦× ❤➵♥ ❝❤Õ ❣× ✈Ị t❐♣ r➭♥❣ ❜✉é❝ S ♥❤➢ tr♦♥❣ ✭✷✳✶✽✮✳ ❑Õt q✉➯ ♥➭② t➢➡♥❣ tù ✈í✐ ❬✶✷✱ ➤Þ♥❤ ❧Ý ✸✳✶❪ ♥❤➢♥❣ ❜❛♦ ❣å♠ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ tï② ý✳ ➜Þ♥❤ ❧ý ✷✳✹✳✶ ●✐➯ sö ❞✐♠X < ∞, x¯ ∈ S, dm f (¯ x; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y ∈ X ❑❤✐ ➤ã ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❞➢í✐ ➤➞② ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✸✽ ✭❛✮ x¯ ∈ ❙tr▲(m, f, S) ✭❜✮ ❚å♥ t➵✐ β > s❛♦ ❝❤♦ dm f (¯ x; y) ∈ / B (0, β y ✭❝✮ m ¯ p+ , ∀y ∈ K (S, x¯) \ {0} )−R ✭✷✳✻✼✮ ¯ p+ , ∀y ∈ K (S, x¯) \ {0} dm f (¯ x; y) ∈ / −R ❈❤ø♥❣ ♠✐♥❤ ➜Ó ❝❤ø♥❣ ♠✐♥❤✭❛✮ C =S ⇒ ✈➭ ❦❤➠♥❣ ❝ã ✭❜✮✱ ❝❤ó♥❣ t❛ ❧➷♣ ❧➵✐ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➤Þ♥❤ ❧Ý ✷✳✷✳✶✭❛✮ ✈í✐ g ❈❤ó ý r➺♥❣ t❛ ♣❤➯✐ t❤➟♠ ❤➵♥ ❝❤Õ ➜✐Ị✉ ♥➭② ❝ã ➤➢ỵ❝ ❞♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✼✮✳ ❚➢➡♥❣ tù ✭❝✮ ✷✳✸✳✷✳ ❈ß♥ s✉② ❧✉❐♥ ✭❜✮ ⇒ ✭❝✮ ❧➭ ❤✐Ó♥ ♥❤✐➟♥✳ ⇒ y =0 tr♦♥❣ ✭✷✳✻✼✮✳ ó ợ ị í ết ▲✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❉✳❊✳ ❲❛r❞ ✭✶✾✾✹✮ ✈➭ ❊✳❉✳ ❘❛❤♠♦ ✲ ▼✳ ❙t✉❞♥✐❛rs❦✐ ✭✷✵✶✷✮✳ ❈➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❜❛♦ ❣å♠✿ • ❑❤➳✐ ♥✐Ư♠ A ✲ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝đ❛ ❲❛r❞❀ • ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ị t ủ r ã ệ t❤❡♦ ♣❤➢➡♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝✃♣ ❝❛♦ ❝❤♦ ❤➭♠ ✈❡❝t➡ ❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐❀ • ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦ ❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐✳ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ị t ề t ợ ♥❤✐Ị✉ t➳❝ ❣✐➯ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ✹✵ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❆✳ ❆✉s❧❡♥❞❡r ✭✶✾✽✹✮✱ ❙t❛❜✐❧✐t② ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ ♥♦♥❞✐❢✲ ❢❡r❡♥t✐❛❜❧❡ ❞❛t❛✱ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♥tr♦❧ ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✷✷✱ ♣♣✳ ✷✸✾✲✷✺✹✳ ❬✷❪ ▲✳ ❈r♦♠♠❡ ✭✶✾✼✽✮✱ ❙tr♦♥❣ ✉♥✐q✉❡♥❡s✿ ❆ ❢❛r r❡❛❝❤✐♥❣ ❝r✐t❡r✐♦♥ ❢♦r t❤❡ ❝♦♥✲ ✈❡r❣❡♥❝❡ ♦❢ ✐t❡r❛t✐✈❡ ♣r♦❝❡❞✉r❡s✱ ◆✉♠❡r✐s❝❤❡ ▼❛t❤❡♠❛t✐❦✱ ✈♦❧✳ ✷✾✱ ♣♣✳ ✶✼✾✲ ✶✾✸✳ ❬✸❪ ❆✳❱✳ ❋✐❛❝❝♦✱ ●✳P✳ ▼❝❈♦r♠✐❝❦ ✭✶✾✻✽✮✱ ◆♦♥❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✿ ❙❡q✉❡♥t✐❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♠✐♥✐♠✐③❛t✐♦♥ t❡❝❤♥✐q✉❡s✱ ❏♦❤♥ ❲✐❧❡②✱ ◆❡✇ ❨♦r❦✳ ❬✹❪ ❆✳ ●✉♣t❛✱ ❉✳ ❇❤❛t✐❛✱ ❆✳ ▼❡❤r❛ ✭✷✵✵✼✮✱ ❍✐❣❤❡r ♦r❞❡r ❡❢❢✐❝✐❡♥❝②✱ s❛❞❞❧❡ ♣♦✐♥t ♦♣t✐♠❛❧✐t②✱ ❛♥❞ ❞✉❛❧✐t② ❢♦r ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳ ✷✽ ✭✸✲✹✮✱ ✸✸✾✲✸✺✷✳ ❬✺❪ ❇✳ ❏✐♠Ð♥❡③ ✭✷✵✵✷✮✱ ❙tr✐❝t ❡❢❢✐❝✐❡♥❝② ✐♥ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✻✺✱ ✷✻✹✲✷✽✹✳ ❬✻❪ ❇✳ ❏✐♠Ð♥❡③ ✭✷✵✵✸✮✱ ❙tr✐❝t ♠✐♥✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♥♦♥❞✐❢❢❡r❡♥t✐❛❜❧❡ ♠✉❧t✐✲ ♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✶✻✱ ✾✾✲✶✶✻✳ ❬✼❪ ❇✳ ❏✐♠Ð♥❡③ ✭✷✵✵✾✮✱ ❱✳◆♦✈♦✱ ▼✳ ❙❛♠❛✱ ❙❝❛❧❛r✐③❛t✐♦♥ ❛♥❞ ♦♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s ❢♦r str✐❝t ♠✐♥✐♠✐③❡rs ✐♥ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ✈✐❛ ❝♦♥t✐♥❣❡♥t ❡♣✐❞❡r✐✈❛t✐✈❡s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✺✷ ✼✽✽✲✼✾✽✳ ❬✽❪ ❉✳ ❑❧❛tt❡✱ ❑✳ ❚❛♠♠❡r ✭✶✾✽✽✮✱ ❖♥ s❡❝♦♥❞✲♦r❞❡r s✉❢❢✐❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐✲ t✐♦♥s ❢♦r C 1,1 ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✱ ❖♣t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✶✾✱ ♣♣✳✶✻✾✲✶✼✾✳ ✹✶ ❬✾❪ ❏✳ ❑②♣❛r✐s✐s ✭✶✾✽✺✮✱ ❖♥ ✉♥✐q✉❡♥❡ss ♦❢ ❑✉❤♥✲❚✉❝❦❡r ♠✉❧t✐♣❧✐❡rs ✐♥ ♥♦♥❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✱ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠♠✐♥❣✱ ✈♦❧✳ ✸✷✱ ♣♣✳ ✷✹✷✲✷✹✻✳ ❬✶✵❪ ❉✳❱✳ ▲✉✉ ✭✷✵✵✽✮✱ ❍✐❣❤❡r✲♦r❞❡r ♥❡❝❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r str✐❝t P❛r❡t♦ ♠✐♥✐♠❛ ✐♥ t❡r♠s ♦❢ ❙t✉❞♥✐❛rs❦✐✬s ❞❡r✐✈❛t✐✈❡s✱ ❖♣t✐♠✐③❛t✐♦♥ ✺✼ ✭✹✮✱ ✺✾✸✲✻✵✺✳ ❬✶✶❪ ❊✳❉✳ ❘❛❤♠♦✱ ▼✳ ❙t✉❞♥✐❛rs❦✐ ✭✷✵✶✷✮✱ ❍✐❣❤❡r✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r str✐❝t ❧♦❝❛❧ P❛r❡t♦ ♠✐♥✐♠❛ ✐♥ t❡r♠s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛✲ t✐✈❡s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✾✸✱ ✷✶✷✲✷✷✶✳ ❬✶✷❪ ▼✳ ❙t✉❞♥✐❛rs❦✐ ✭✶✾✽✻✮✱ ◆❡❝❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r ✐s♦❧❛t❡❞ ❧♦❝❛❧ ♠✐♥✐♠❛ ♦❢ ♥♦♥s♠♦♦t❤ ❢✉♥❝t✐♦♥s✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✷✹✱ ✶✵✹✹✲✶✵✹✾✳ ❬✶✸❪ ❉✳❊✳ ❲❛r❞✱ ❏✳▼✳ ❇♦r✇❡✐♥ ✭✶✾✽✼✮✱ ◆♦♥s♠♦♦t❤ ❝❛❧❝✉❧✉s ✐♥ ❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♥tr♦❧ ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✷✺✱ ♣♣✳ ✶✸✶✷✲✶✸✹✵✳ ❬✶✹❪ ❉✳❊✳ ❲❛r❞ ✭✶✾✾✹✮✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ str✐❝t ❧♦❝❛❧ ♠✐♥✐♠❛ ❛♥❞ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ✇❡❛❦ s❤❛r♣ ♠✐♥✐♠❛✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✽✵✱ ✺✺✶✲✺✼✶✳ ❬✶✺❪ ❉✳❊✳ ❲❛r❞ ✭✶✾✽✼✮✱ ■s♦t♦♥❡ t❛♥❣❡♥t ❝♦♥❡s ❛♥❞ ♥♦♥s♠♦♦t❤ ♦♣t✐♠✐③❛t✐♦♥✱ ❖♣✲ t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✶✽✱ ♣♣✳ ✼✻✾✲✼✽✸✳ ❬✶✻❪ ❉✳❊✳ ❲❛r❞ ✭✶✾✽✽✮✱ ❚❤❡ q✉❛♥t✐❢✐❝❛t✐♦♥❛❧ t❛♥❣❡♥t ❝♦♥❡s✱ ❈❛♥❛❞✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✈♦❧✳ ✹✵✱ ♣♣✳ ✻✻✻✲✻✾✹✳