(Luận án tiến sĩ) stability and robust stability of singular linear difference equations , tính ổn định và ổn định vững của phương trình sai phân tuyến tính suy biến 624601

❱■❊❚◆❆▼ ◆❆❚■❖◆❆▲ ❯◆■❱❊❘❙■❚❨✱ ❍❆◆❖■ ❱◆❯ ❯◆■❱❊❘❙■❚❨ ❖❋ ❙❈■❊◆❈❊ ◆❣ô ❚❤✐✳ ❚❤❛♥❤ ◆❣❛ ❙❚❆❇■▲■❚❨ ❆◆❉ ❘❖❇❯❙❚ ❙❚❆❇■▲■❚❨ ❖❋ ❙■◆●❯▲❆❘ ▲■◆❊❆❘ ❉■❋❋❊❘❊◆❈❊ ❊◗❯❆❚■❖◆❙ ❚❍❊❙■❙ ❋❖❘ ❚❍❊ ❉❊●❘❊❊ ❖❋ ❉❖❈❚❖❘ ❖❋ P❍■▲❖❙❖P❍❨ ■◆ ▼❆❚❍❊▼❆❚■❈❙ ❍❆◆❖■ ✕ ✷✵✶✽ ❱■❊❚◆❆▼ ◆❆❚■❖◆❆▲ ❯◆■❱❊❘❙■❚❨✱ ❍❆◆❖■ ❱◆❯ ❯◆■❱❊❘❙■❚❨ ❖❋ ❙❈■❊◆❈❊ ◆❣ô ❚❤✐✳ ❚❤❛♥❤ ◆❣❛ ❙❚❆❇■▲■❚❨ ❆◆❉ ❘❖❇❯❙❚ ❙❚❆❇■▲■❚❨ ❖❋ ❙■◆●❯▲❆❘ ▲■◆❊❆❘ ❉■❋❋❊❘❊◆❈❊ ❊◗❯❆❚■❖◆❙ ❙♣❡❝✐❛❧✐t②✿ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ❙♣❡❝✐❛❧✐t② ❈♦❞❡✿ ✻✷ ✹✻ ✵✶ ✵✸ ❚❍❊❙■❙ ❋❖❘ ❚❍❊ ❉❊●❘❊❊ ❖❋ ❉❖❈❚❖❘ ❖❋ P❍❨▲❖❙❖P❍❨ ■◆ ▼❆❚❍❊▼❆❚■❈❙ Supervisors: ASSOC PROF DR HABIL VŨ HOÀNG LINH and PROF DR NGUYỄN HỮU DƯ ❍❆◆❖■ ✕ ✷✵✶✽ ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN Ngô Thị Thanh Nga TÍNH ỔN ĐỊNH VÀ ỔN ĐỊNH VỮNG CỦA PHƯƠNG TRÌNH SAI PHÂN TUYẾN TÍNH SUY BIẾN Chun ngành: Phương trình Vi phân Tích phân Mã số: 62 46 01 03 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng dẫn khoa học: PGS.TSKH VŨ HOÀNG LINH GS.TS NGUYỄN HỮU DƯ HÀ NỘI – 2018 ❉❡❝❧❛r❛t✐♦♥ This work has been completed at the Faculty of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, under the supervision of Assoc.Prof.Dr.habil Vu Hoang Linh and Prof.Dr Nguyen Huu Du I hereby declare that the results presented in the thesis are new and have never been published fully or partially in any other thesis/work Author: Ngô Thị Thanh Nga ✶ ❆❝❦♥♦✇❧❡❞❣♠❡♥ts Firstly, I would like to thank my two supervisors Prof.Dr Nguyễn Hữu Dư and especially Assoc.Prof.Dr.habil Vũ Hoàng Linh for the continuous support of my PhD study and related research; for their patience, motivation and immense knowledge Without their help I could not have overcome the difficulties in research and study I would like to express sincere thanks to Assoc.Prof.Dr Lê Văn Hiện and Dr Nguyễn Trung Hiếu for their useful comments and suggestions that led to the improvement of the thesis I would also like to thank Dr Đỗ Đức Thuận for his collaboration in research My deepest appreciation goes to Prof Phạm Kỳ Anh and other members of "Seminar on Computational and Applied Mathematics", and also to the members of "Seminar on Differential Equations and Dynamical Systems" at the Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Hanoi, for their valuable comments and discussions I am grateful to my parents, brother, my beloved daughters, my husband and other members in my big family, who have provided me moral and emotional support throughout my life A very special gratitude goes to all Thang Long University, National Foundation for Science and Technology Development, the MOET project 911 for providing the funding for me in the period of my study Last but not least, I would like to thank my colleagues in Thang Long University, the staffs of Vietnam Institute for Advanced Study in Mathematics, my friends, and many other people beside me for their love, motivation and constant guidance Thanks all for your love and support! ❆❜str❛❝t ❚❤✐s ✇♦r❦ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ ❧✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✭▲❙❉❊s✮ ♦❢ ✜rst ♦r❞❡r ❛♥❞ s❡❝♦♥❞ ♦r❞❡r✳ ❋♦r ▲❙❉❊s ♦❢ ✜rst ♦r❞❡r✱ ❜② ✉s✐♥❣ t❤❡ ♣r♦❥❡❝t♦r✲ ❜❛s❡❞ ❛♣♣r♦❛❝❤ ✇❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ s②st❡♠ ✉♥❞❡r ♣❡rt✉r❜❛✲ t✐♦♥s ❛♥❞ ❡st❛❜❧✐s❤ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ s♦❧✉t✐♦♥s ♦❢ ♥♦♥✲ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠s ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧✴ ✉♥✐❢♦r♠ st❛❜✐❧✐t② ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✲ ✐♥❣ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✳ ❲❡ ❛❧s♦ ❡①t❡♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❇♦❤❧ ❡①♣♦♥❡♥t ❢r♦♠ r❡❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s t♦ ▲❙❉❊s ❛♥❞ ✐♥✈❡st✐❣❛t❡ ✐ts ♣r♦♣❡rt✐❡s✳ ❋♦r ▲❙❉❊s ♦❢ s❡❝♦♥❞✲♦r❞❡r✱ ✇❡ ✉s❡ t❤❡ str❛♥❣❡♥❡ss✲✐♥❞❡① ❛♣♣r♦❛❝❤✳ ❯♥✲ ❞❡r t❤❡ str❛♥❣❡♥❡ss✲❢r❡❡ ❛ss✉♠♣t✐♦♥ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ ■❱Ps✱ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ ❛♥❞ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s♦❧✉t✐♦♥ s❡ts ♦❢ t❤❡ s②st❡♠s ❛♥❞ t❤♦s❡ ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ r❡❞✉❝❡❞ r❡❣✉❧❛r s②st❡♠s✳ ❇② ❛ ❝♦♠♣❛r✐s♦♥ ♣r✐♥❝✐♣❧❡✱ s♦♠❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❝r✐t❡r✐❛ ❛r❡ ♦❜t❛✐♥❡❞✳ ❆ ❇♦❤❧✲ P❡rr♦♥✲t②♣❡ t❤❡♦r❡♠ ✐s ❛❧s♦ ❣✐✈❡♥ t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ✐♥♣✉t✲s♦❧✉t✐♦♥ r❡❧❛t✐♦♥ ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✳ ❋✐♥❛❧❧②✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ r♦❜✉st st❛❜✐❧✐t② ✉♥❞❡r r❡str✐❝t❡❞ str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s ✐s ✐♥✈❡st✐❣❛t❡❞✳ ❆❧s♦ ✉s✐♥❣ t❤❡ ❝♦♠♣❛r✐s♦♥ ♣r✐♥❝✐♣❧❡✱ ❛♥ ❡①♣❧✐❝✐t ❜♦✉♥❞ ❢♦r ♣❡rt✉r❜❛t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ t❤❡ s②st❡♠s ♣r❡s❡r✈❡ t❤❡✐r ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✐s ♦❜t❛✐♥❡❞✳ ✐ Tóm tắt Trong cơng trình chúng tơi nghiên cứu phương trình sai phân suy biến tuyến tính cấp cấp hai Đối với phương trình sai phân suy biến tuyến tính cấp 1, chúng tơi sử dụng cách tiếp cận phép chiếu đưa kết như: đặc trưng hóa tính ổn định hệ tác động nhiễu; thiết lập mối quan hệ tính ổn định mũ/ ổn định hệ tính chất nghiệm hệ khơng nhất; mở rộng khái niệm số mũ Bohl cho hệ sai phân suy biến số tính chất Đối với phương trình sai phân suy biến cấp hai, sử dụng cách tiếp cận dùng số lạ Dưới giả thiết số lạ không, chúng tơi nghiên cứu tính giải tốn giá trị ban đầu điều kiện đầu tương thích, mối quan hệ tập nghiệm hệ ban đầu tập nghiệm hệ đưa dạng quy Bằng cách sử dụng nguyên lý so sánh, tiêu chuẩn cho ổn định mũ thiết lập Một định lý dạng Bohl-Perron đưa nhằm đặc trưng mối quan hệ đầu vào-nghiệm hệ không Cuối cùng, tốn tính ổn định vững tác động nhiễu có cấu trúc Tiếp tục sử dụng nguyên lý so sánh lần nữa, đưa chặn cho nhiễu để hệ bị nhiễu bảo toàn số tính ổn định mũ ✐✐ ▲✐st ♦❢ ◆♦t❛t✐♦♥s t❤❡ s❡t ♦❢ r❡❛❧ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡rs (C) R N t❤❡ s❡t ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs N(n0 ) t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs t❤❛t ❛r❡ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ❛ ❣✐✈❡♥ ✐♥t❡❣❡r K R C R ♦r C d t❤❡ d− ❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❧❡① ✈❡❝t♦r s♣❛❝❡ d t❤❡ d− ❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ Cd,d t❤❡ s❡t ♦❢ d×d ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ✐♥ GL(K ) t❤❡ s❡t ♦❢ d×d ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ✐♥ ❦❡rE t❤❡ ❦❡r♥❡❧ s♣❛❝❡ ♦❢ E ✐♠E t❤❡ ✐♠❛❣❡ s♣❛❝❡ ♦❢ E rank E t❤❡ r❛♥❦ ♦❢ ♠❛tr✐① E d x ♥♦r♠ ♦❢ ✈❡❝t♦r ∆ ♥♦r♠ ♦❢ ♠❛tr✐① C K x ∆ B(0, 1) ✉♥✐t ❞✐s❦ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ det A t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ♠❛tr✐① A t❤❡ ♠❛tr✐① t✉♣❧❡ rK t❤❡ ✭str✉❝t✉r❡❞✮ st❛❜✐❧✐t② r❛❞✐✉s H n0 A (A, B, C) A t❤❡ ❝♦♥❥✉❣❛t❡ tr❛♥s♣♦s❡ ♦❢ ♠❛tr✐① lp (n0 ) t❤❡ s♣❛❝❡ ♦❢ s❡q✉❡♥❝❡s {qn }n n0 A ⊂ Kd s✉❝❤ t❤❛t qn p < ∞✱ p n n0 s♣❡❝tr❛❧ r❛❞✐✉s ♦❢ ♠❛tr✐① ρ(A) ❞✐❛❣(σ1 , · · · , σp ) t❤❡ ♠❛tr✐① ✐♥ C m,n A ✇❤♦s❡ u>0 ❡❛❝❤ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❝t♦r u>v ♠❡❛♥s t❤❛t ii u u−v >0 ✐✐✐ ❡♥tr② ✐s σi ✐s ♣♦s✐t✐✈❡ ❢♦r ❛♥② i = 1, , p ❛♥❞ t❤❡ ♦t❤❡rs ❛r❡ ③❡r♦ ❈♦♥t❡♥ts P❛❣❡ ❆❜str❛❝t ✐ Tóm tắt ii ▲✐st ♦❢ ◆♦t❛t✐♦♥s ✐✐✐ ■♥tr♦❞✉❝t✐♦♥ ✶ ❈❤❛♣t❡r ✶ Pr❡❧✐♠✐♥❛r✐❡s ✶✳✶ ✶✳✷ ▲✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② tr❛❝t❛❜✐❧✐t②✲✐♥❞❡① ❛♣♣r♦❛❝❤ ✶✶ ✶✳✶✳✶✳ ❉❡✜♥✐t✐♦♥ ♦❢ ✐♥❞❡①✲✶ s②st❡♠s ❛♥❞ t❤❡✐r ♣r♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✷✳ ❙♦❧✉t✐♦♥s ♦❢ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ▲✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② str❛♥❣❡♥❡ss✲✐♥❞❡① ❛♣♣r♦❛❝❤ ✶✺ ✶✳✷✳✶✳ ❉❡✜♥✐t✐♦♥ ♦❢ str❛♥❣❡♥❡ss ✐♥❞❡① ❛♥❞ ❇rü❧❧✬s r❡s✉❧ts ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✷✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ t②♣❡s ♦❢ ✐♥❞❡① ❞❡✜♥✐t✐♦♥s ✳ ✷✶ ✶✳✷✳✸✳ ▲✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♦❢ s❡❝✲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ♦♥❞ ♦r❞❡r ✶✳✸ ✶✶ ❋✉rt❤❡r ❛✉①✐❧✐❛r② r❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✈ ✷✺ ❈❤❛♣t❡r ✷ ❙✐♥❣✉❧❛r s②st❡♠s ♦❢ ✜rst✲♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✷✽ ✷✳✶ ❙t❛❜✐❧✐t② ♥♦t✐♦♥s ❢♦r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷ ❙t❛❜✐❧✐t② ♦❢ ♣❡rt✉r❜❡❞ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷✳✶✳ ❚❤❡ ❝❛s❡ ♦❢ ♦♥❡✲s✐❞❡❞ ♣❡rt✉r❜❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✷✳ ❚❤❡ ❝❛s❡ ♦❢ t✇♦✲s✐❞❡❞ ♣❡rt✉r❜❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸ ✷✳✹ ✷✳✺ ❇♦❤❧✲P❡rr♦♥✲t②♣❡ st❛❜✐❧✐t② t❤❡♦r❡♠s ✷✳✸✳✶✳ ❇♦✉♥❞❡❞♥❡ss ♦❢ s♦❧✉t✐♦♥s ♦❢ ♥♦♥❤♦♠♦❣❡♥♦✉s ❡q✉❛t✐♦♥s ✳ ✹✶ ✷✳✸✳✷✳ ❇♦❤❧✲P❡rr♦♥✲t②♣❡ t❤❡♦r❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ❇♦❤❧ ❡①♣♦♥❡♥ts ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✹✳✶✳ ❇♦❤❧ ❡①♣♦♥❡♥ts ❛♥❞ t❤❡✐r ❜❛s✐❝ ♣r♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✹✳✷✳ ❘♦❜✉st♥❡ss ♦❢ ❇♦❤❧ ❡①♣♦♥❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ❚❤❡ ❝❛s❡ ♦❢ ✉♥❜♦✉♥❞❡❞ ❝❛♥♦♥✐❝❛❧ ♣r♦❥❡❝t♦r ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✷✳✺✳✶✳ ❯♥✐❢♦r♠ st❛❜✐❧✐t② ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ ♣❡rt✉r❜❡❞ ❡q✉❛t✐♦♥s ✷✳✺✳✷✳ ✷✳✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ❇♦❧❤ ❡①♣♦♥❡♥t ♦❢ s♦❧✉t✐♦♥s ❛♥❞ ❇♦❤❧ ❡①♣♦♥❡♥t ♦❢ t❤❡ s②st❡♠ ✻✼ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ❈❤❛♣t❡r ✸ ❙✐♥❣✉❧❛r s②st❡♠s ♦❢ s❡❝♦♥❞✲♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✼✷ ✸✳✶ ■♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✷ ❊①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✸✳✷✳✶✳ ◆♦t✐♦♥ ♦❢ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✸✳✷✳✷✳ ❈r✐t❡r✐❛ ❢♦r ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✸✳✷✳✸✳ ❇♦❤❧✲P❡rr♦♥ t❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✸✳✸ ❘♦❜✉st st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✸✳✹ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾ ✈ ❈♦♥❝❧✉s✐♦♥ ■♥ t❤✐s t❤❡s✐s✱ t❤❡ st❛❜✐❧✐t② ❛♥❞ r♦❜✉st st❛❜✐❧✐t② ♦❢ ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♦❢ ✜rst ❛♥❞ s❡❝♦♥❞ ♦r❞❡r ❤❛✈❡ ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞✳ ■♥ t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ t❤❡s✐s✱ ✇❡ ❤❛✈❡ ❛♥❛❧②③❡❞ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ s②st❡♠s ♦❢ ✜rst ♦r❞❡r ✇❤❡r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ s✉❜❥❡❝t t♦ ♣❡rt✉r❜❛t✐♦♥s✳ ❲❡ ❤❛✈❡ ❛❧s♦ ❝❤❛r❛❝✲ t❡r✐③❡❞ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧✴✉♥✐❢♦r♠ st❛❜✐❧✐t② ♦❢ t❤❡ ❤♦♠♦❣❡✲ ♥❡♦✉s s②st❡♠ ❛♥❞ t❤❡ ❜♦✉♥❞❡❞♥❡ss ♣r♦♣❡rt② ♦❢ s♦❧✉t✐♦♥s ♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✳ ❚❤r❡❡ ❇♦❤❧✲P❡rr♦♥✲t②♣❡ st❛❜✐❧✐t② t❤❡♦r❡♠s ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ❙❉❊s ♦❢ ✐♥❞❡①✲✶✳ ❋♦r t❤❡ ♣r♦♦❢s✱ t❤❡ ❢r❛♠❡✇♦r❦s ✐♥ ❬✽✱ ✼✵❪ ❝❛♥ ✇♦r❦ ♦♥❧② ✇✐t❤ ❝❡rt❛✐♥ ♠♦❞✐✜❝❛t✐♦♥s ❜❡❝❛✉s❡ ✇❡ ❤❛✈❡ t♦ ♦✈❡r❝♦♠❡ t❤❡ ❞✐✣❝✉❧t✐❡s ❝❛✉s❡❞ ❜② t❤❡ s✐♥❣✉❧❛r✐t②✳ ❚❤❡ ♣r♦❥❡❝t♦r✲❜❛s❡❞ ❛♥❛❧②s✐s ✐s ❛♣♣❧✐❡❞ t♦ ❞❡❝♦✉♣❧❡ t❤❡ s✐♥❣✉✲ ❧❛r s②st❡♠s✳ ❚❤❡♥✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ❝♦♥str✉❝t t❤❡ ❛♣♣r♦♣r✐❛t❡ ❛❞♠✐ss✐❜❧❡ s♣❛❝❡s ❛♥❞ ❢♦r♠✉❧❛t❡ t❤❡ ❣❡♥❡r❛❧✐③❡❞ P❡rr♦♥✬s ♣r♦♣❡rt②✱ ✇❤✐❝❤ ✐s ♦❜✈✐♦✉s❧② r❡❞✉❝❡❞ t♦ t❤❡ ❝❧❛ss✐❝❛❧ ♦♥❡ ❢♦r t❤❡ tr✐✈✐❛❧ ❝❛s❡ En ≡ I ✳ ■♥ ♦✉r ♦♣✐♥✐♦♥✱ t❤❡s❡ st❛❜✐❧✐t② t❤❡♦r❡♠ ❛r❡ ♣r♦♠✐s✐♥❣❧② ✉s❡❢✉❧ ❢♦r r❡s❡❛r❝❤❡rs ✇♦r❦✐♥❣ ♦♥ t❤❡ ❝♦♥tr♦❧ t❤❡♦r② ♦❢ ❞✐s❝r❡t❡✲t✐♠❡ s✐♥❣✉❧❛r s②st❡♠s ❞✉❡ t♦ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❞❡s❝r✐♣t♦r s②st❡♠s✳ ❚❤❡ ♥♦t✐♦♥ ♦❢ ❇♦❤❧ ❡①♣♦♥❡♥t ❤❛s ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❛♥❞ ✐ts ♣r♦♣❡rt✐❡s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ t❤❡s✐s✱ ✇❡ ❤❛✈❡ ✐♥✈❡st✐❣❛t❡❞ ❛ ❝❧❛ss ♦❢ ❧✐♥❡❛r t✐♠❡✲ ✈❛r②✐♥❣ s✐♥❣✉❧❛r s②st❡♠s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② t❤❡ str❛♥❣❡♥❡ss✲ ✐♥❞❡① ❛♣♣r♦❛❝❤✳ ❚❤❡ s♦❧✈❛❜✐❧✐t②✱ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t②✱ ❛♥❞ r♦❜✉st st❛❜✐❧✐t② ❤❛✈❡ ❜❡❡♥ ❛♥❛❧②③❡❞✳ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ t❤❡ ■❱Ps ❛s ✇❡❧❧ ❛s s❤♦✇♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❝♦♥s✐st❡♥t ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s♦✲ ❧✉t✐♦♥ s❡ts ♦❢ t❤❡ s②st❡♠s ❛♥❞ t❤♦s❡ ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ r❡❞✉❝❡❞ r❡❣✉❧❛r s②st❡♠s ✐s s❤♦✇♥✳ ❚❤❡♥✱ ❝r✐t❡r✐❛ ❢♦r ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❛r❡ ❛❝❤✐❡✈❡❞✳ ❋✉rt❤❡r♠♦r❡✱ ❛ ❇♦❤❧✲P❡rr♦♥✲t②♣❡ t❤❡♦r❡♠ ✐s ❣✐✈❡♥✳ ❋✐♥❛❧❧②✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ r♦❜✉st st❛❜✐❧✲ ✐t② ✉♥❞❡r r❡str✐❝t❡❞ str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s ❤❛s ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞✳ ✶✶✶ ❇② t❤❡ ❝♦♠♣❛r✐s♦♥ ♣r✐♥❝✐♣❧❡✱ ❛♥ ❡①♣❧✐❝✐t ❜♦✉♥❞ ❢♦r t❤❡ ❛❧❧♦✇❛❜❧❡ ♣❡rt✉r❜❛t✐♦♥s✱ ✉♥✲ ❞❡r ✇❤✐❝❤ t❤❡ str❛♥❣❡♥❡ss✲❢r❡❡ ❢♦r♠ ❛s ✇❡❧❧ ❛s t❤❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ t❤❡ s②st❡♠ ❛r❡ ♣r❡s❡r✈❡❞✱ ❤❛s ❜❡❡♥ ♦❜t❛✐♥❡❞✳ ❋✉t✉r❡ ✇♦r❦s t♦✇❛r❞ t❤❡ ❝♦♠♣❧❡t❡ ❛♥❛❧②s✐s ♦❢ ❣❡♥❡r❛❧ s✐♥❣✉❧❛r ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s ♦❢ ❤✐❣❤ ♦r❞❡r ✐♥❝❧✉❞✐♥❣ ❛ t❤❡♦r② ♦❢ str❛♥❣❡♥❡ss ✐♥❞❡① ❛♥❞ ❛ r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ s❤♦✉❧❞ ❜❡ ❞♦♥❡✱ ✇✐t❤ ✇❤✐❝❤ ✇❡ ❝❛♥ tr❛♥s❢♦r♠ ❛♥ ❛r❜✐tr❛r② ❣❡♥❡r❛❧ s②st❡♠ ✐♥t♦ t❤❡ str❛♥❣❡♥❡ss✲❢r❡❡ ❢♦r♠ ✉♥❞❡r ❝❡rt❛✐♥ ❛ss✉♠♣t✐♦♥s✳ ❆ ❝♦♠♣❧❡t❡ t❤❡♦r② ♦❢ ▲②❛♣✉♥♦✈✱ ❇♦❤❧✱ ❛♥❞ ❙❛❝❦❡r✲❙❡❧❧ s♣❡❝tr❛ ❢♦r ❙❉❊s ❛♥❞ t❤❡✐r ♥✉♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✱ ❡①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠② ❛♥❞ s❤❛❞♦✇✐♥❣ ♣r♦♣❡rt② ♦❢ ❙❉❊s ✇♦✉❧❞ ❜❡ ✐♥t❡r❡st✐♥❣ ❢✉t✉r❡ ✇♦r❦s✱ ❛s ✇❡❧❧✳ ✶✶✷ ▲✐st ♦❢ ♣✉❜❧✐❝❛t✐♦♥s ✉s❡❞ ✐♥ t❤❡ t❤❡s✐s ❬✶❪ ◆❣✉②❡♥ ❍✉✉ ❉✉✱ ❱✉ ❍♦❛♥❣ ▲✐♥❤ ❛♥❞ ◆❣♦ ❚❤✐ ❚❤❛♥❤ ◆❣❛ ✭✷✵✶✻✮✱ ✧❖♥ st❛❜✐❧✐t② ❛♥❞ ❇♦❤❧ ❡①♣♦♥❡♥t ♦❢ ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥ts✧✱ ❏✳ ❉✐✛❡r✳ ❊q✉❛t✐♦♥s ❆♣♣❧✳✱ ✷✷✱ ♣♣✳ ✶✸✺✵✕✶✸✼✼✳ ✭❙❈■❊✮ ❬✷❪ ❱✉ ❍♦❛♥❣ ▲✐♥❤✱ ◆❣♦ ❚❤✐ ❚❤❛♥❤ ◆❣❛ ✭✷✵✶✽✮✱ ✧❇♦❤❧✕P❡rr♦♥ ❚②♣❡ ❙t❛❜✐❧✐t② ❚❤❡♦r❡♠s ❢♦r ▲✐♥❡❛r ❙✐♥❣✉❧❛r ❉✐✛❡r❡♥❝❡ ❊q✉❛t✐♦♥s✧✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳✱ ✹✻✱ ♣♣✳ ✹✸✼✲✹✺✶✳ ✭❊❙❈■✱ ❙❝♦♣✉s✮ ❬✸❪ ❱✉ ❍♦❛♥❣ ▲✐♥❤✱ ◆❣♦ ❚❤✐ ❚❤❛♥❤ ◆❣❛✱ ❉♦ ❉✉❝ ❚❤✉❛♥ ✭✷✵✶✽✮✱ ✧❊①♣♦✲ ♥❡♥t✐❛❧ st❛❜✐❧✐t② ❛♥❞ r♦❜✉st st❛❜✐❧✐t② ❢♦r ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ s✐♥❣✉❧❛r s②st❡♠s ♦❢ s❡❝♦♥❞✲♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ❙■❆▼ ❏✳ ▼❛tr✐① ❆♥❛❧✳ ❆♣♣❧✳✱ ✶✷✵✹✲✷✸✸✳✭❙❈■✮ ✶✶✸ ✸✾✲✶✱ ♣♣✳ ❆♣♣❡♥❞✐① ❆✳ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✳✶✳✶ Pr♦♦❢✳ ❲❡ ✇✐❧❧ ♣r♦✈❡ t❤✐s ❧❡♠♠❛ ❜② s❤♦✇✐♥❣ t❤❛t (iii) ⇒ (i)✱ (i) ⇒ (iii) ❛♥❞ (ii) ⇔ (iii)✳ (iii) ⇒ (i) : ❙✉♣♣♦s❡ t❤❛t Nn−1 ∩ Sn = {0}✱ ✇❡ ♥❡❡❞ ♣r♦✈❡ t❤❛t Gn = En − An Tn Qn ✐s ♥♦♥s✐♥❣✉❧❛r✱ ✐✳❡✳ ❦❡rGn = {0}✳ ■♥❞❡❡❞✱ ❧❡t x ❜❡ ❛♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥t ✐♥ ❦❡rGn ✱ ✇❡ ❤❛✈❡ = Gn x = En x − An Tn Qn x✳ ❚❤✉s✱ En x = An Tn Qn x✳ ■t ✐♠♣❧✐❡s t❤❛t Tn Qn x ∈ Sn ✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Tn |Nn ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Nn ❛♥❞ Nn−1 ✱ s♦ Tn Qn x ∈ Nn−1 ✳ ❋r♦♠ t❤✐s✱ ✇❡ ❤❛✈❡ Tn Qn x ∈ Nn−1 ∩Sn = {0}✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t Tn Qn x = ♦r Qn x = 0✳ ❍❡♥❝❡✱ En x = An Tn Qn x = 0✱ s♦ x ∈ ❦❡rEn = Nn ✳ ❚❤✉s✱ x = Qn x = (i) ⇒ (iii) : ❆ss✉♠❡ t❤❛t Gn = En − An Tn Qn ✐s ♥♦♥s✐♥❣✉❧❛r✱ ✇❡ ♥❡❡❞ ♣r♦✈❡ t❤❛t Nn−1 ∩ Sn = {0} ■♥❞❡❡❞✱ ❧❡t x ❜❡ ❛♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥t ♦❢ Nn−1 ∩ Sn ✳ d ❙✐♥❝❡ x ∈ Sn ✱ t❤❡r❡ ❡①✐sts z ∈ K s✉❝❤ t❤❛t An x = En z = En Pn z ✳ ▼♦r❡♦✈❡r✱ x ∈ Nn−1 t❤❡♥ Tn−1 x ∈ Nn = ❦❡rEn ♦r Tn−1 x = Qn Tn−1 x✳ ❍❡♥❝❡✱ ✇❡ ♦❜t❛✐♥ (En − An Tn Qn )Tn−1 x = En Tn−1 x − An Tn Qn Tn−1 x = En Qn Tn−1 x − An Tn Tn−1 x = − An x = −An x ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ −(En − An Tn Qn )Pn z = −En Pn z + An Tn Qn Pn z = −An x + = −An x (En − An Tn Qn )Tn−1 x = −(En − An Tn Qn )Pn z ✳ ▼♦r❡♦✈❡r✱ s✐♥❝❡ En − An Tn Qn ✐s ♥♦♥s✐♥❣✉❧❛r✱ t❤❡♥ ✇❡ ❤❛✈❡ Tn−1 x = −Pn z ✳ ■t ✐♠♣❧✐❡s t❤❛t Tn−1 x = Qn Tn−1 x = −Qn Pn z = ♦r x = 0✳ ❍❡♥❝❡✱ Nn−1 ∩ Sn = {0}✳ ❚❤✉s✱ ✶✶✹ (ii) ⇔ (iii) : ▲❡t x ❜❡ ❛♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥t ✐♥ Kd ✱ ✇❡ ❤❛✈❡ x = (I + −1 −1 Tn Qn G−1 n An )x − Tn Qn Gn An x✳ ■t ✐s ♦❜✈✐♦✉s❧② t♦ s❡❡ t❤❛t Tn Qn Gn An x ∈ Nn−1 ✳ ▼♦r❡♦✈❡r✱ −1 −1 An (I + Tn Qn G−1 n An )x = An x − (En − An Tn Qn )Gn An x + En Gn An x −1 = −An x + Gn G−1 n An x + En Gn An x = En G−1 n An x (I + Tn Qn G−1 n An )x ∈ Sn ✳ ❚❤✉s✱ x ∈ Nn−1 + Sn ♦r Nn−1 + Sn = d ■t ❝❧❡❛r❧② ✐♠♣❧✐❡s t❤❛t Nn−1 ⊕ Sn = K ✐❢ ❛♥❞ ♦♥❧② ✐❢ Nn−1 ∩ Sn = {0} ❚❤✐s s❤♦✇s t❤❛t Kd ✳ ❇✳ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✳✶✳✸ Pr♦♦❢✳ ❋✐rst❧②✱ ✇❡ ♥♦t❡ t❤❛t Tn Qn x ∈ = 0✳ ■t ✐♠♣❧✐❡s t❤❛t En Tn−1 Qn−1 Tn |Nn Nn ❛♥❞ Nn−1 ✳ En−1 Tn Qn = ❛♥❞ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Nn−1 ❛♥❞ Tn−1 Qn−1 x ∈ Nn ✐✳❡✳ ✭✐✮ ❲❡ ❤❛✈❡ GnPn = (En − AnTnQn)Pn = EnPn − AnTnQnPn = EnPn = En✳ ❚❤✉s✱ Pn = G−1 n En ✭✭✐✮ ✐s ♣r♦✈❡♥✮✳ ✭✐✐✮ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t Tn−1Qn−1x ∈ ❦❡rEn = Nn ❢♦r ❛❧❧ x ∈ Kd ✳ ❚❤✉s✱ Tn−1 Qn−1 = Qn Tn−1 Qn−1 ✳ • ❲❡ ❤❛✈❡ −1 −1 Pn G−1 n An Qn−1 = Pn Gn An Tn Tn Qn−1 −1 = Pn G−1 n An Tn Qn Tn Qn−1 −1 = Pn G−1 n [−(En − An Tn Qn ) + En ]Tn Qn−1 −1 = Pn G−1 n [−(En − An Tn Qn ) + En ]Qn Tn Qn−1 −1 −1 −1 = −Pn G−1 n Gn Qn Tn Qn−1 + Pn Gn En Qn Tn Qn−1 −1 = −Pn Qn Tn−1 Qn−1 + Pn G−1 n En Qn Tn Qn−1 = (s✐♥❝❡ Pn Qn = ❍❡♥❝❡✱ ❛♥❞ En Qn = 0) −1 −1 Pn G−1 n An = Pn Gn An (Pn−1 + Qn−1 ) = Pn Gn An Pn−1 ❡q✉❛❧✐t② ♦❢ ✭✐✐✮ ✐s ♣r♦✈❡♥✮✳ ✶✶✺ ✭t❤❡ ✜rst • ❈❛❧❝✉❧❛t✐♥❣ s✐♠✐❧❛r❧② ❛s ❛❜♦✈❡✱ ✇❡ ♦❜t❛✐♥ −1 −1 Qn G−1 n An Qn−1 = Qn Gn An Tn Tn Qn−1 −1 = Qn G−1 n An Tn Qn Tn Qn−1 = Qn G−1 − (En − An Tn Qn ) + En Tn−1 Qn−1 n −1 −1 −1 = −Qn G−1 n Gn Tn Qn−1 + Qn Gn En Qn Tn Qn−1 = −Tn−1 Qn−1 + Pn Pn Qn Tn−1 Qn−1 (G−1 n En = Pn ✱ = −Tn−1 Qn−1 + = −Tn−1 Qn−1 ✭s✐♥❝❡ ❛❝❝♦r❞✐♥❣ t♦ ✭✐✮✮ Pn Qn = 0) ❚❤✉s✱ −1 −1 −1 Qn G−1 n An = Qn Gn An (Pn−1 + Qn−1 ) = Qn Gn An Pn−1 − Tn Qn−1 ❚❤❡ s❡❝♦♥❞ ❡q✉❛❧✐t② ♦❢ ✭✐✐✮ ✐s ♣r♦✈❡♥✳ ✭✐✐✐✮ ❚♦ ♣r♦✈❡ t❤❛t Qn−1 Qn−1 ✱ • ✐♠Qn−1 Nn−1 ❛❧♦♥❣ Sn ❦❡rQn−1 = Sn ✳ ✐s ❛ ♣r♦❥❡❝t♦r ♦♥t♦ = ❦❡rEn−1 = Nn−1 ❋✐rst❧②✱ ✇❡ ♣r♦✈❡ t❤❛t ❛♥❞ Q2n−1 = Qn−1 ✳ ✇❡ ♥❡❡❞ Q2n−1 = ■♥❞❡❡❞✱ −1 Q2n−1 = − Tn Qn G−1 n An (−Tn Qn Gn An ) = Tn Qn G−1 − (En − An Tn Qn ) + En G−1 n n An −1 −1 −1 = −Tn Qn G−1 n Gn Gn An + Tn Qn Gn En Gn An −1 = −Tn Qn G−1 n An + Tn Qn Pn Gn An = −Tn Qn G−1 n An = Qn−1 • En−1 Qn−1 = −En−1 Tn Qn G−1 n An = ✭s✐♥❝❡ En−1 Tn Qn = 0✮✳ ❍❡♥❝❡✱ ✐♠Qn−1 ⊂ ❦❡rEn−1 ✳ ▲❡t x ❜❡ ❛♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥t ✐♥ ❦❡rEn−1 ✱ ❢r♦♠ t❤❡ ♣r♦♣❡rt② t❤❛t Qn−1 ✐s ❛ ♣r♦❥❡❝t♦r ♦♥t♦ ❦❡rEn−1 ✇❡ ✐♠♣❧② Qn−1 x = x✳ ❈♦♥s✐❞❡r ❲❡ ♦❜t❛✐♥ −1 −1 Qn−1 x = −Tn Qn G−1 n An x = −Tn Qn Gn An Pn−1 − Tn Qn−1 x = −Tn Qn G−1 n An Pn−1 Qn−1 x + Qn−1 x = + Qn−1 x = x Qn−1 x = x✱ ✇❡ ❦❡rEn−1 = ✐♠Qn−1 ✳ ❋r♦♠ t❤❡ ❢❛❝t t❤❛t✱ ✐♠Qn−1 ✳ ❚❤✉s✱ • ❯s✐♥❣ ▲❡♠♠❛ ✶✳✶✳✶✱ ✇❡ ❤❛✈❡ Qn−1 ❛❧♦♥❣ Sn ♦♥t♦ Nn−1 ✱ ❤❛✈❡ x ∈ Nn−1 ⊕ Sn = Kd ✳ ❦❡rEn−1 ⊂ ❚♦ ♣r♦✈❡ t❤❛t t❤❡ ♣r♦❥❡❝t♦r ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t ✐❢ ✶✶✻ ✐♠Qn−1 ✱ ✐✳❡✳ x ✐s ❛♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥t Sn ✱ t❤❡♥ Qn−1 x = 0✳ An x = En z ✳ ❚❤❡♥ ✐♥ ❙✐♥❝❡ x ∈ Sn ✱ t❤❡r❡ ❡①✐sts z ∈ Kd s✉❝❤ t❤❛t −1 Qn−1 x = −Tn Qn G−1 n An x = −Tn Qn Gn En z = −Tn Qn Pn z = ❲❡ ❤❛✈❡ ♣r♦✈❡♥ ✭✐✐✐✮✳ ❈✳ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✳✶✳✹ Pr♦♦❢✳ Pn = I − Qn , Pn = I − Qn ✱ d r❡s♣❡❝t✐✈❡❧②✳ ▲❡t Tn , Tn ❜❡ ♦♣❡r❛t♦rs ✐♥ GL(K ) s✉❝❤ t❤❛t Tn |Nn , Tn |Nn ❛r❡ ✐s♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ Nn ❛♥❞ Nn−1 ❛♥❞ ❧❡t Gn ❜❡ ❞❡✜♥❡❞ s✐♠✐❧❛r❧② t♦ Gn ✳ ▲❡t Qn , Q n ❜❡ ♣r♦❥❡❝t♦rs ♦♥t♦ Nn ✱ ❛♥❞ s❡t ❚❤❡♥✱ ✇❡ ❤❛✈❡ −1 −1 −1 G−1 n Gn = Gn (En − An Tn Qn ) = Pn − Gn An Tn Tn Tn Qn = ❦❡rEn−1 ❛♥❞ ✐♠(Tn−1 Tn Qn ) = Pn Tn−1 Tn Qn = ❍❡♥❝❡✱ ◆♦t❡ t❤❛t✱ ✐♠(Tn Qn ) Qn Tn−1 Tn Qn ❛♥❞ ❦❡rEn ✱ s♦ Tn−1 Tn Qn = −1 −1 G−1 n Gn = Pn − Gn An Tn Qn Tn Tn Qn −1 = Pn + G−1 n (Gn − En )Tn Tn Qn = Pn + Tn−1 Tn Qn − Pn Tn−1 Tn Qn = Pn + Tn−1 Tn Qn ❚❤❡r❡❢♦r❡✱ −1 −1 −1 Pn G−1 n = Pn (Pn + Tn T n Q n )G n = Pn G n ❚♦ ♣r♦✈❡ t❤❛t −1 Tn−1 Tn Qn )G n −1 Tn Qn G−1 n = T nQ nG n ✳ ✱ ✇❡ ✉s❡ t❤❡ ❡q✉❛❧✐t② G−1 n = (Pn + ✳ ■t ✐♠♣❧✐❡s t❤❛t −1 −1 Tn Qn G−1 n = Tn Qn (Pn + Tn Tn Qn )G n −1 = Tn Qn Tn−1 Tn Qn G n ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Pn Tn−1 Tn Qn = 0✳ ❍❡♥❝❡✱ −1 −1 −1 Tn Qn G−1 n = Tn (Pn + Qn )Tn Tn Qn G n = T n Q n G n ❚❤❡ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡❞✳ ❉✳ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✳✸✳✷ ✶✶✼ Pr♦♦❢✳ ❲❡ ❤❛✈❡ (I + F N ) I − F (I + N F )−1 N = I + F N − F (I + N F )−1 N − F N F (I + N F )−1 N = I + (I + N F − I − N F )(I + N F )−1 N = I, ❛♥❞ I − F (I + N F )−1 N (I + F N ) = I + F N − F (I + N F )−1 N − F (I + N F )−1 N F N = I + F (I + N F )−1 (I + N F − I − N F )N = I ❚❤✉s✱ I − F (I + N F )−1 N ✐s t❤❡ ✐♥✈❡rs❡ ♠❛tr✐① ♦❢ ✶✶✽ I + F N✳ ❇✐❜❧✐♦❣r❛♣❤② ❉✐✛❡r❡♥❝❡ ❊q✉❛t✐♦♥s ❛♥❞ ■♥❡q✉❛❧✐t✐❡s✱ ❚❤❡♦r②✱ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧✳ ✷✷✽ ♦❢ ▼♦♥♦❣r❛♣❤s ❛♥❞ ❚❡①t❜♦♦❦s ✐♥ P✉r❡ ❬✶❪ ❘✳ P✳ ❆❣❛r✇❛❧ ✭✷✵✵✵✮✱ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✱ ◆❨✱ ❯❙❆✱ ✷♥❞ ❡❞✐✲ t✐♦♥✳ ❬✷❪ P✳❑✳ ❆♥❤✱ ◆✳ ❍✳ ❉✉✱ ❛♥❞ ▲✳ ❈✳ ▲♦✐ ✭✷✵✵✹✮✱ ✧❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❛♥❞ ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✧✱ ♥❛♠✳✱ ✷✾✱ ✷✸✕✸✾✳ ❆❝t❛ ▼❛t❤✳ ❱✐❡t✲ ❬✸❪ P✳ ❑✳ ❆♥❤✱ ◆✳ ❍✳ ❉✉✱ ❛♥❞ ▲✳ ❈✳ ▲♦✐ ✭✷✵✵✼✮✱ ✧❙✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✿ ❛♥ ♦✈❡r✈✐❡✇✧✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳✱ ✸✺ ✱ ✸✸✾✕✸✼✷✳ ❬✹❪ P✳ ❑✳ ❆♥❤ ❛♥❞ ❉✳ ❙✳ ❍♦❛♥❣ ✭✷✵✵✻✮✱ ✧❙t❛❜✐❧✐t② ♦❢ ❛ ❝❧❛ss ♦❢ s✐♥❣✉❧❛r ❞✐✛❡r✲ ❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ■♥t❡r✳ ❏✳ ❉✐✛❡r❡♥❝❡ ❊q✉✳✱ ✶✱ ✶✽✶✕✶✾✸ ✳ ❬✺❪ P✳❑✳ ❆♥❤ ❛♥❞ ❍✳❚✳◆✳ ❨❡♥ ✭✷✵✵✹✮✱ ✧❖♥ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ ✐♥✐t✐❛❧✲✈❛❧✉❡ ♣r♦❜✲ ❧❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ❆❞✈✳ ❉✐✛❡r❡♥❝❡ ❊q✉✳✱ ✸✱ ✶✾✺✕✷✵✵✳ ❬✻❪ P✳❑✳ ❆♥❤ ❛♥❞ ❍✳❚✳◆✳ ❨❡♥ ✭✷✵✵✻✮✱ ✧❋❧♦q✉❡t t❤❡♦r❡♠ ❢♦r ❧✐♥❡❛r ♥♦♥❛✉✲ t♦♥♦♠♦✉s ❞✐✛❡r❡♥❝❡ s②st❡♠s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✷✶✱ ♥♦✳ ✷✱ ✾✷✶✕✾✷✾✳ ❬✼❪ P✳❑✳ ❆♥❤ ❛♥❞ P✳❚✳ ▲✐♥❤ ✭✷✵✶✼✮✱ ✧❙t❛❜✐❧✐t② ♦❢ ♣❡r✐♦❞✐❝❛❧❧② s✇✐t❝❤❡❞ ❞✐s❝r❡t❡✲t✐♠❡ ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s✧✱ ❏✳ ❉✐✛❡r✳ ❊q✉❛t✐♦♥s ❆♣♣❧✳✱ ✷✸✱ ✶✻✽✵✲ ✶✻✾✸✳ ❬✽❪ ❇✳ ❆✉❧❜❛❝❤ ❛♥❞ ◆✳❱✳ ▼✐♥❤ ✭✶✾✾✻✮✱ ✧❚❤❡ ❝♦♥❝❡♣t ♦❢ s♣❡❝tr❛❧ ❞✐❝❤♦t♦♠② ❢♦r ❧✐♥❡❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ■■✧✱ ❏✳ ❉✐✛❡r❡♥❝❡✳ ❊q✳ ❆♣♣❧✳✱ ✷✱ ✷✺✶✕✷✻✷✳ ❬✾❪ ▲✳ ❇❡r❡③❛♥s❦②✱ ❊✳ ❇r❛✈❡r♠❛♥ ✭✷✵✵✺✮✱ ✧❖♥ ❡①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠②✱ ❇♦❤❧✲ P❡rr♦♥ t②♣❡ t❤❡♦r❡♠s ❛♥❞ st❛❜✐❧✐t② ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ❆♣♣❧✳✱ ✸✵✹✱ ✺✶✶✕✺✸✵✳ ✶✶✾ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❬✶✵❪ ❚✳ ❇❡r❣❡r ✭✷✵✶✷✮✱ ✧❇♦❤❧ ❡①♣♦♥❡♥t ❢♦r t✐♠❡✲✈❛r②✐♥❣ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧✲ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✧✱ ■♥t✳ ❏✳ ❈♦♥tr♦❧✱ ✽✺✱ ✶✹✸✸✕✶✹✺✶✳ ❬✶✶❪ ❚✳ ❇❡r❣❡r ✭✷✵✶✹✮✱ ✧❘♦❜✉st♥❡ss ♦❢ st❛❜✐❧✐t② ♦❢ t✐♠❡✲✈❛r②✐♥❣ ✐♥❞❡①✲✶ ❉❆❊s✱ ▼❛t❤✳ ❈♦♥tr♦❧ ❙✐❣♥❛❧s ❙②st❡♠s✧✱ ✷✻✱ ♥♦✳ ✸✱ ✹✵✸✕✹✸✸✳ ❬✶✷❪ ❊✳ ❇r❛✈❡r♠❛♥✱ ■✳▼✳ ❑❛r❛❜❛s❤ ✭✷✵✶✷✮✱ ✧❇♦❤❧✲P❡rr♦♥ t②♣❡ st❛❜✐❧✐t② t❤❡✲ ♦r❡♠s ❢♦r ❧✐♥❡❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✜♥✐t❡ ❞❡❧❛②✧✱ ❊q♥s✳ ❆♣♣❧✳✱ ✶✽✱ ✾✵✾✕✾✸✾✳ ❏✳ ❉✐✛❡r❡♥❝❡ ❬✶✸❪ ❚✳ ❇rü❧❧ ✭✷✵✵✾✮✱ ✧❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s ♦❢ ❧✐♥❡❛r ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥t ❞✐s❝r❡t❡✲t✐♠❡ ❞❡s❝r✐♣t♦r s②st❡♠s✧✱ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❆♣♣❧✳✱ ✹✸✶✱ ♥♦✳ ✶✲✷✱ ✷✹✼✕✷✻✺✳ ❬✶✹❪ ❚✳ ❇rü❧❧ ✭✷✵✵✾✮✱ ✧❊①♣❧✐❝✐t s♦❧✉t✐♦♥s ♦❢ r❡❣✉❧❛r ❧✐♥❡❛r ❞✐s❝r❡t❡✲t✐♠❡ ❞❡s❝r✐♣✲ t♦r s②st❡♠s ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts✱ ❜r❛✧✱ ❱♦❧✉♠❡ ✶✽✱ ♣♣✳ ✸✶✼✲✸✸✽ ❊❧❡❝tr♦♥✐❝ ❏♦✉r♥❛❧ ♦❢ ▲✐♥❡❛r ❆❧❣❡✲ ❬✶✺❪ P✳ ❇♦❤❧ ✭✶✾✶✸✮✱ ✧Ü❜❡r ❉✐✛❡r❡♥t✐❛❧❣❧❡✐❝❤✉♥❣❡♥✧✱ ✇❛♥❞t❡ ▼❛t❤❡♠❛t✐❦✱ ✶✹✹✱ ♣♣✳ ✷✽✹✕✸✶✸✳ ❏✳ ❢ür ❘❡✐♥❡ ✉♥❞ ❆♥❣❡✲ ❬✶✻❪ ❘✳ ❇②❡rs ❛♥❞ ◆✳❑✳ ◆✐❝❤♦❧s ✭✶✾✾✸✮✱ ✧❖♥ t❤❡ st❛❜✐❧✐t② r❛❞✐✉s ♦❢ ❛ ❣❡r♥❡r❛❧✐③❡❞ st❛t❡✲s♣❛❝❡ s②st❡♠s✧✱ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❆♣♣❧✳✱ ✶✽✽✲✶✽✾✱ ♣♣✳ ✶✶✸✕✶✶✹✳ ❬✶✼❪ ❙✳▲✳ ❈❛♠♣❜❡❧❧ ✭✶✾✽✵✮✱ ❙✐♥❣✉❧❛r ❙②st❡♠s ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ P✐t♠❛♥✱ ▲♦♥❞♦♥✳ ❬✶✽❪ ❙✳▲✳ ❈❛♠♣❜❡❧❧ ✭✶✾✽✷✮✱ ❙✐♥❣✉❧❛r ❙②st❡♠s ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ■■✱ P✐t✲ ♠❛♥✱ ▲♦♥❞♦♥✳ ❬✶✾❪ ❈✳ ❏✳ ❈❤②❛♥✱ ◆✳❍✳❉✉✱ ❛♥❞ ❱✳ ❍✳ ▲✐♥❤ ✭✷✵✵✽✮✱ ✧❖♥ ❞❛t❛✲❞❡♣❡♥❞❡♥❝❡ ♦❢ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❛♥❞ t❤❡ st❛❜✐❧✐t② r❛❞✐✐ ❢♦r ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ s②st❡♠s✧✱ ❏✳ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✷✹✺✱ ♣♣✳ ✷✵✼✽✕ ✷✶✵✷✳ ❬✷✵❪ ❆✳ ❈③♦r♥✐❦✱ ▼✳ ◆✐❡③❛❜✐t♦✇s❦✐ ✭✷✵✶✺✮✱ ✧❆❧t❡r♥❛t✐✈❡ ❢♦r♠✉❧❛❡ ❢♦r ❧♦✇❡r ❣❡♥✲ ❡r❛❧ ❡①♣♦♥❡♥t ♦❢ ❞✐s❝r❡t❡ ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ s②st❡♠s✧✱ ❏✳ ❋r❛♥❦❧✐♥ ■♥s✳✱✸✺✷✱ ♣♣✳ ✸✾✾✲✹✶✾✳ ❬✷✶❪ ▲✳ ❉❛✐ ✭✶✾✽✾✮✱ ❙✐♥❣✉❧❛r ❝♦♥tr♦❧ s②st❡♠s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♥tr♦❧ ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ❙❝✐❡♥❝❡✱ ❱♦❧✳ ✶✶✽✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✳ ✶✷✵ ❬✷✷❪ ❏✳ ▲✳ ❉❛❧❡❝❦✐✐ ❛♥❞ ▼✳ ●✳ ❑r❡✐♥ ✭✶✾✼✹✮✱ ❙t❛❜✐❧✐t② ♦❢ ❙♦❧✉t✐♦♥s ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✐♥ ❇❛♥❛❝❤ ❙♣❛❝❡s✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✳ ❬✷✸❪ ◆✳❍✳ ❉✉ ✭✷✵✵✽✮✱ ✧❙t❛❜✐❧✐t② r❛❞✐✐ ♦❢ ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s✧✱ ❙②st✳ ❈♦♥tr♦❧ ▲❡tt✳ ✺✼✱ ♣♣✳ ✺✹✻✕✺✺✸✳ ❬✷✹❪ ◆✳❍✳ ❉✉✱ ❱✳❍✳ ▲✐♥❤ ✭✷✵✵✺✮✱ ✧■♠♣❧✐❝✐t✲s②st❡♠ ❛♣♣r♦❛❝❤ t♦ t❤❡ r♦❜✉st st❛✲ ❜✐❧✐t② ❢♦r ❛ ❝❧❛ss ♦❢ s✐♥❣✉❧❛r❧② ♣❡rt✉r❜❡❞ ❧✐♥❡❛r s②st❡♠s✧✱ ❙②st✳ ❈♦♥tr♦❧ ▲❡tt✳ ✺✹✱ ♣♣✳ ✸✸✕✹✶✳ ❬✷✺❪ ◆✳ ❍✳ ❉✉ ❛♥❞ ❱✳ ❍✳ ▲✐♥❤ ✭✷✵✵✻✮✱ ✧❘♦❜✉st st❛❜✐❧✐t② ♦❢ ✐♠♣❧✐❝✐t ❧✐♥❡❛r s②s✲ t❡♠s ❝♦♥t❛✐♥✐♥❣ ❛ s♠❛❧❧ ♣❛r❛♠❡t❡r ✐♥ t❤❡ ❧❡❛❞✐♥❣ t❡r♠✧✱ ❈♦♥t✳ ■♥❢✳✱ ✷✸✱ ♣♣✳ ✻✼✕✽✹✳ ■▼❆ ❏✳ ▼❛t❤✳ ❬✷✻❪ ◆✳❍✳ ❉✉ ❛♥❞ ❱✳❍✳ ▲✐♥❤ ✭✷✵✵✻✮✱ ✧❙t❛❜✐❧✐t② r❛❞✐✐ ❢♦r ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ ❞②♥❛♠✐❝ ♣❡rt✉r❜❛t✐♦♥s✧✱ ❏✳ ❉✐✛❡r✳ ❊q✉✳ ✷✸✵✱ ♣♣✳ ✺✼✾✕✺✾✾✳ ❬✷✼❪ ◆✳❍✳ ❉✉✱ ❉✳❚✳ ▲✐❡♥✱ ❛♥❞ ❱✳❍✳ ▲✐♥❤ ✭✷✵✵✸✮✱ ✧❈♦♠♣❧❡① st❛❜✐❧✐t② r❛❞✐✐ ❢♦r ✐♠♣❧✐❝✐t ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s✧✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳ ✸✶✱ ♣♣✳ ✹✼✺✕✹✽✽✳ ❬✷✽❪ ◆✳❍✳ ❉✉✱ ❱✳ ❍✳ ▲✐♥❤✱ ❛♥❞ ❱✳ ▼❡❤r♠❛♥♥ ✭✷✵✶✸✮✱ ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✱ ✐♥✿ ❘♦❜✉st st❛❜✐❧✐t② ♦❢ ❙✉r✈❡②s ✐♥ ❉✐✛❡r❡♥t✐❛❧✲❆❧❣❡❜r❛✐❝ ❊q✉❛✲ t✐♦♥s ■✱ ❉❆❊✲❋ ✱ ✻✸✕✾✺✳ ❬✷✾❪ ◆✳❍✳ ❉✉✱ ❱✳❍✳ ▲✐♥❤✱ ❱✳ ▼❡❤r♠❛♥♥ ❛♥❞ ❉✳❉✳❚❤✉❛♥ ✭✷✵✶✸✮✱ ✧❙t❛❜✐❧✐t② ❛♥❞ r♦❜✉st st❛❜✐❧✐t② ♦❢ ❧✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t ❞❡❧❛② ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ ❡q✉❛✲ t✐♦♥s✧✱ ❙■❆▼ ❏✳ ▼❛tr✐① ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✹✱ ♣♣✳ ✶✻✸✶✕✶✻✺✹✳ ❬✸✵❪ ◆✳❍✳ ❉✉✱ ❱✳❍✳ ▲✐♥❤✱ ❛♥❞ ◆✳❚✳❚✳ ◆❣❛ ✭✷✵✶✻✮✱ ✧❖♥ st❛❜✐❧✐t② ❛♥❞ ❇♦❤❧ ❡①✲ ♣♦♥❡♥t ♦❢ ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦✲ ❡✣❝✐❡♥ts✧✱ ❏✳ ❉✐✛❡r✳ ❊q✉❛t✐♦♥s ❆♣♣❧✳✱ ✷✷✱ ♣♣✳ ✶✸✺✵✕✶✸✼✼✳ ❬✸✶❪ ◆✳❍✳ ❉✉✱ ❉✳❚✳ ▲✐❡♥✱ ❱✳❍✳ ▲✐♥❤ ✭✷✵✵✸✮✱ ✧❖♥ ❝♦♠♣❧❡① st❛❜✐❧✐t② r❛❞✐✐ ❢♦r ✐♠♣❧✐❝✐t ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s✧✱ ❬✸✷❪ ❙✳ ◆✳ ❊❧❛②❞✐ ✭✷✵✵✺✮✱ ❱✐❡t✳ ❏✳ ▼❛t❤✳✱ ✸✶✱ ♣♣✳ ✹✼✺✕✹✽✽✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❉✐✛❡r❡♥❝❡ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ▲♦♥❞♦♥✱ ❯❑✱ ✸r❞ ❡❞✐t✐♦♥✳ ❬✸✸❪ ❩✳ ❋❡♥❣✱ ❲✳ ▲✐ ❛♥❞ ❏✳ ▲❛♠ ✭✷✵✶✺✮✱ ✧◆❡✇ ❛❞♠✐ss✐❜✐❧✐t② ❛♥❛❧②s✐s ❢♦r ❞✐s✲ ❝r❡t❡ s✐♥❣✉❧❛r s②st❡♠s ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ❞❡❧❛②✧✱ ❈♦♠♣✉t❛t✐♦♥✱ ✷✻✺✱ ♣♣✳ ✶✵✺✽✲✶✵✻✻✳ ✶✷✶ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❬✸✹❪ ❩✳ ❋❡♥❣✱ ❲✳ ▲✐ ❛♥❞ ❏✳ ▲❛♠ ✭✷✵✶✻✮✱ ✧❉✐ss✐♣❛t✐✈✐t② ❛♥❛❧②s✐s ❢♦r ❞✐s❝r❡t❡ s✐♥❣✉❧❛r s②st❡♠s ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ❞❡❧❛②✧✱ ■❙❆ ❚r❛♥s❛❝t✐♦♥s✱ ✻✹✱ ♣♣✳ ✽✻✲ ✾✶✳ ❬✸✺❪ ❨✳ ❋❡♥❣✱ ❚✐♠❡ ❩✳▼✳ ◆♦♥❧✐♥❡❛r ▲✐ ❛♥❞ ❳✳❍✳ ❙✐♥❣✉❧❛r ❈❤❛♥❣ ❙②st❡♠s Pr♦❜❧❡♠s ✐♥ ❊♥❣✐♥❡❡r✐♥❣✱ ✈♦❧✳ ✭✷✵✶✼✮✱ ✇✐t❤ ✷✵✶✼✱ ✧❋✐❧t❡r✐♥❣ ■❉ ❉✐s❝r❡t❡✲ ▼❛t❤❡♠❛t✐❝❛❧ ◗✉❛♥t✐③❛t✐♦♥✧✱ ❆rt✐❝❧❡ ❢♦r ✾✺✹✽✹✵✼✱ ✾ ♣❛❣❡s✳ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✶✺✺✴✷✵✶✼✴✾✺✹✽✹✵✼✳ ❬✸✻❪ ❨✳ ●❡♥✐♥✱ ❘✳ ❙t❡❢❛♥✱ P✳ ❱❛♥ ❉♦♦r❡♥ ✭✷✵✵✷✮✱ ✧❘❡❛❧ ❛♥❞ ❝♦♠♣❧❡① st❛❜✐❧✐t② r❛❞✐✐ ♦❢ ♣♦❧②♥♦♠✐❛❧ ♠❛tr✐❝❡s✧✱ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❆♣♣❧✳✱ ✸✺✶✲✸✺✷✱ ♥♦✳ ✶✲✷✱ ♣♣✳ ✸✽✶✕✹✶✵✳ ❬✸✼❪ ◆✳❚✳ ❍❛✱ ◆✳❍✳ ❉✉ ❛♥❞ ❉✳❉✳ ❚❤✉❛♥ ✭✷✵✶✻✮✱ ✧❖♥ ❞❛t❛ ❞❡♣❡♥❞❡♥❝❡ ♦❢ st❛❜✐❧✐t② ❞♦♠❛✐♥s✱ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❛♥❞ st❛❜✐❧✐t② r❛❞✐✐ ❢♦r ✐♠♣❧✐❝✐t ❧✐♥❡❛r ❞②♥❛♠✐❝ ❡q✉❛t✐♦♥s✧✱ ▼❛t❤✳ ❈♦♥tr♦❧ ❙✐❣♥❛❧s ❙②st❡♠s✱ ✷✽✿ ✶✸✳ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴s✵✵✹✾✽✲✵✶✻✲✵✶✻✹✲✼ ❬✸✽❪ P✳ ❍❛ ❛♥❞ ❱✳ ▼❡❤r♠❛♥♥ ✭✷✵✶✻✮✱ ✧❆♥❛❧②s✐s ❛♥❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❧✐♥❡❛r ❞❡❧❛② ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✧✱ ❇■❚ ◆✉♠❡r✳ ▼❛t❤✳✱ ✺✻✱ ♣♣✳ ✻✸✸✕ ✻✺✼✳ ❬✸✾❪ ◆✳ ❍✐❣❤❛♠ ❛♥❞ ❋✳ ❚✐ss❡✉r ✭✷✵✵✷✮✱ ✧▼♦r❡ ♦♥ ♣s❡✉❞♦s♣❡❝tr❛ ❢♦r ♣♦❧②♥♦♠✐❛❧ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♥tr♦❧ t❤❡♦r②✧✱ ❆♣♣❧✳✱ ✸✺✶✲✸✺✷✱ ♣♣✳ ✹✸✺✕✹✺✸✳ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❬✹✵❪ ❉✳ ❍✐♥r✐❝❤s❡♥ ❛♥❞ ❆✳❏✳ Pr✐t❝❤❛r❞ ✭✶✾✽✻✮✱ ✧❙t❛❜✐❧✐t② r❛❞✐✐ ♦❢ ❧✐♥❡❛r s②s✲ t❡♠s✧✱ ❙②st✳ ❈♦♥tr♦❧ ▲❡tt✳ ✼✱ ♣♣✳ ✶✕✶✵✳ ❬✹✶❪ ❉✳ ❍✐♥r✐❝❤s❡♥ ❛♥❞ ❆✳❏✳ Pr✐t❝❤❛r❞ ✭✶✾✽✻✮✱ ✧❙t❛❜✐❧✐t② r❛❞✐✐ ❢♦r str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s ❛♥❞ t❤❡ ❛❧❣❡❜r❛✐❝ ❘✐❝❝❛t✐ ❡q✉❛t✐♦♥s✧✱ ❙②st✳ ❈♦♥tr♦❧ ▲❡tt✳ ✽✱ ♣♣✳ ✶✵✺✕✶✶✸✳ ❬✹✷❪ ❉✳ ❍✐♥r✐❝❤s❡♥✱ ◆✳❑✳ ❙♦♥ ❛♥❞ P✳❍✳❆✳ ◆❣♦❝ ✭✷✵✵✸✮✱ ✧❙t❛❜✐❧✐t② r❛❞✐✐ ♦❢ ♣♦s✲ ✐t✐✈❡ ❤✐❣❤❡r ♦r❞❡r ❞✐✛❡r❡♥❝❡ s②st❡♠s✧✱ ❙②st❡♠s ❈♦♥tr♦❧ ▲❡tt❡rs✱ ✹✾ ✱ ♣♣✳ ✸✼✼✕✸✽✽✳ ❬✹✸❪ ❇✳ ❏❛❝♦❜ ✭✶✾✾✽✮✱ ✧❆ ❢♦r♠✉❧❛ ❢♦r t❤❡ st❛❜✐❧✐t② r❛❞✐✉s ♦❢ t✐♠❡✲✈❛r②✐♥❣ s②s✲ t❡♠s✧✱ ❏✳ ❉✐✛❡r✳ ❊q✉✳✱ ✶✹✷✱ ♣♣✳ ✶✻✼✕✶✽✼✳ ✶✷✷ ❬✹✹❪ ❏▼✳ ❏✐❛♦ ✭✷✵✶✷✮✱ ✧❘♦❜✉st ❙t❛❜✐❧✐t② ❛♥❞ ❙t❛❜✐❧✐③❛t✐♦♥ ♦❢ ❉✐s❝r❡t❡ ❙✐♥❣✉❧❛r ❙②st❡♠s ✇✐t❤ ■♥t❡r✈❛❧ ❚✐♠❡✲✈❛r②✐♥❣ ❉❡❧❛② ❛♥❞ ▲✐♥❡❛r ❋r❛❝t✐♦♥❛❧ ❯♥❝❡r✲ t❛✐♥t②✧✱ ■♥t✳ ❏✳ ❆✉t♦♠✳ ❈♦♠♣✉t✳✱ ❱♦❧✉♠❡ ✾✱ ■ss✉❡ ✶✱ ♣♣ ✽✕✶✺✳ ❬✹✺❪ ▲✳ ❏♦❞❛r✱ P✳ ▼❡r❡❧❧♦ ✭✷✵✶✵✮✱ ✧❙♦❧✈✐♥❣ ❛♥ ❛♥❛❧②t✐❝ ❞②♥❛♠✐❝ ▲❡♦♥t✐❡❢ ♠♦❞❡❧ ✇✐t❤ t✐♠❡ ❞❡♣❡♥❞❡♥t ❝❛♣✐t❛❧ ♠❛tr✐①✧✱ ❡❧❧✐♥❣✱ ✺✶✱ ♣♣✳ ✹✵✵✕✹✵✹✳ ▼❛t❤❡♠❛t✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ▼♦❞✲ ❬✹✻❪ ▲✳ ❏♦❞❛r✱ P✳ ▼❡r❡❧❧♦ ✭✷✵✶✵✮✱ ✧P♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ ❞✐s❝r❡t❡ ❞②♥❛♠✐❝ ▲❡♦♥✲ t✐❡❢ ✐♥♣✉t✕♦✉t♣✉t ♠♦❞❡❧ ✇✐t❤ ♣♦ss✐❜❧② s✐♥❣✉❧❛r ❝❛♣✐t❛❧ ♠❛tr✐①✧✱ ♠❛t✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ▼♦❞❡❧❧✐♥❣✱ ✺✷✱ ♣♣✳ ✶✵✽✶✕✶✵✽✼✳ ❬✹✼❪ P✳ ❑✉♥❦❡❧ ❛♥❞ ❱✳ ▼❡❤r♠❛♥♥ ✭✷✵✵✻✮✱ ▼❛t❤❡✲ ❉✐✛❡r❡♥t✐❛❧✲ ❆❧❣❡❜r❛✐❝ ❊q✉❛t✐♦♥s✳ ❆♥❛❧②s✐s ❛♥❞ ◆✉♠❡r✐❝❛❧ ❙♦❧✉t✐♦♥✱ ❊▼❙ P✉❜❧✐s❤✐♥❣ ❍♦✉s❡✱ ❩ür✐❝❤✱ ❙✇✐t③❡r✲ ❧❛♥❞✳ ❬✹✽❪ ❘✳ ▲❛♠♦✉r✱ ❘✳ ▼är③✱ ❛♥❞ ❈✳❚✐s❝❤❡♥❞♦r❢ ✭✷✵✶✸✮✱ ❊q✉❛t✐♦♥s✿ ❆ Pr♦❥❡❝t♦r ❇❛s❡❞ ❆♥❛❧②s✐s✱ ❙♣r✐♥❣❡r✳ ❉✐✛❡r❡♥t✐❛❧✲❆❧❣❡❜r❛✐❝ ❬✹✾❪ ❱✳❍✳ ▲✐♥❤ ❛♥❞ ❱✳ ▼❡❤r♠❛♥♥ ✭✷✵✵✾✮✱ ✧▲②❛♣✉♥♦✈✱ ❇♦❤❧ ❛♥❞ ❙❛❝❦❡r✲❙❡❧❧ s♣❡❝tr❛❧ ✐♥t❡r✈❛❧s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✧✱ t✐♦♥s✱ ✷✶✱ ♣♣✳ ✶✺✸✕✶✾✹✳ ❬✺✵❪ ❱✳❍✳ ▲✐♥❤✱ ◆✳◆✳ ❚✉❛♥ ✭✷✵✶✹✮✱ ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✧✱ ✧❆s②♠♣t♦t✐❝ ❏✳ ❉②♥✳ ❉✐✛✳ ❊q✉❛✲ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❊❧❡❝tr♦♥✳ ❏✳ ◗✉❛❧✳ ❚❤❡♦r② ♦❢ ❉✐✛❡r✳ ❊q✉✳✱ ◆♦✳ ✶✷✱ ✶✼ ♣♣✳ ❬✺✶❪ ▲✳❈✳ ▲♦✐✱ ◆✳❍✳ ❉✉✱ ❛♥❞ P✳❑✳ ❆♥❤ ✭✷✵✵✷✮✱ ❛✉t♦♥♦♠♦✉s s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ✧❖♥ ❧✐♥❡❛r ✐♠♣❧✐❝✐t ♥♦♥✲ ❏✳ ❉✐✛❡r❡♥❝❡ ❊q✳ ❆♣♣❧✳✱ ✽✱ ♣♣✳ ✶✵✽✺✕✶✶✵✺✳ ❬✺✷❪ ❉✳●✳ ▲✉❡♥❜❡r❣❡r ✭✶✾✼✼✮✱ ✧❉②♥❛♠✐❝ ❡q✉❛t✐♦♥s ✐♥ ❞❡s❝r✐♣t♦r ❢♦r♠✧✱ ❚r❛♥s✳ ❆✉t♦♠❛t✳ ❈♦♥tr♦❧✱ ✷✷✱ ♣♣✳ ✸✶✷✕✸✷✷✳ ■❊❊❊ ❬✺✸❪ ❉✳●✳ ▲✉❡♥❜❡r❣❡r ✭✶✾✽✻✮✱ ✧❈♦♥tr♦❧ ♦❢ ❧✐♥❡❛r ❞②♥❛♠✐❝ ♠❛r❦❡t s②st❡♠s✧✱ ❊❝♦✳ ❉②♥✳ ❈♦♥tr♦❧✱ ✶✵✱ ♣♣✳ ✸✸✾✕✸✺✶✳ ❏✳ ❬✺✹❪ ❉✳●✳ ▲✉❡♥❜❡r❣❡r✱ ❆✳ ❆r❜❡❧ ✭✶✾✼✼✮✱ ✧❙✐♥❣✉❧❛r ❞②♥❛♠✐❝ ▲❡♦♥t✐❡❢ s②st❡♠s✧✱ ❊❝♦♥♦♠❡tr✐❝❛✱ ✹✺✱ ♣♣✳ ✾✾✶✕✾✾✺✳ ❬✺✺❪ ❱✳ ▼❡❤r♠❛♥♥ ❛♥❞ ❈✳ ❙❤✐ ✭✷✵✵✻✮✱ ✧❚r❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❤✐❣❤ ♦r❞❡r ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝ s②st❡♠s t♦ ✜rst ♦r❞❡r✧✱ ✷✽✶✕✸✵✼✳ ✶✷✸ ◆✉♠❡r✳ ❆❧❣♦r✐t❤♠s✱ ✹✷✱ ♣♣✳ ❬✺✻❪ ❱✳ ▼❡❤r♠❛♥♥✱ ❉✳❉✳ ❚❤✉❛♥ ✭✷✵✶✺✮✱ ✧❙t❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✉♥❞❡r r❡str✐❝t❡❞ ♣❡rt✉r❜❛t✐♦♥s✧✱ ❙■❆▼ ❏✳ ▼❛tr✐① ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✻✱ ♥♦✳✶✱ ♣♣✳ ✶✼✽✕✷✵✷✳ ❬✺✼❪ ▲✳ ▼♦②s✐s✱ ◆✳ ❑❛r❛♠♣❡t❛❦✐s ❛♥❞ ❊✳ ❆♥t♦♥✐♦✉ ✭✷✵✶✼✮✱ ✧❖❜s❡r✈❛❜✐❧✐t② ♦❢ ❧✐♥❡❛r ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s ♦❢ ❛❧❣❡❜r❛✐❝ ❛♥❞ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❈♦♥tr♦❧✱ ❉❖■✿ ✶✵✳✶✵✽✵✴✵✵✷✵✼✶✼✾✳✷✵✶✼✳✶✸✺✹✸✾✾ ■♥t❡r✲ ❬✺✽❪ ▼✉♦✐✱ ◆✳❍✳✱ ❘❛❥❝❤❛❦✐t✱ ●✳ ❛♥❞ P❤❛t✱ ❱✳◆✳ ✭✷✵✶✻✮✱✧▲▼■ ❆♣♣r♦❛❝❤ t♦ ❋✐♥✐t❡✲❚✐♠❡ ❙t❛❜✐❧✐t② ❛♥❞ ❙t❛❜✐❧✐③❛t✐♦♥ ♦❢ ❙✐♥❣✉❧❛r ▲✐♥❡❛r ❉✐s❝r❡t❡ ❉❡❧❛② ❙②st❡♠s✧✱ ❆❝t❛✳ ❆♣♣❧✳ ▼❛t❤✳✱ ✶✹✻✿ ✽✶✳ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴s✶✵✹✹✵✲ ✵✶✻✲✵✵✺✾✲✵ ❬✺✾❪ ◆✳ ❚✳ ❚✳ ◆❣❛ ✭✷✵✶✺✮✱ ✧❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ❢♦r ❧✐♥❡❛r ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥❞❡① ✶✧✱ P❤②s✐❝s✱ ❱♦❧✳✸✶✱ ◆♦✳✸✱ ♣♣✳ ✸✾✕✹✽✳ ❱◆❯ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡✿ ▼❛t❤❡♠❛t✐❝s✲ ❬✻✵❪ P✳ ❍✳ ❆✳ ◆❣♦❝ ❛♥❞ ▲✳ ❚✳ ❍✐❡✉✱ ✧◆❡✇ ❝r✐t❡r✐❛ ❢♦r ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ ♥♦♥❧✐♥❡❛r ❞✐✛❡r❡♥❝❡ s②st❡♠s ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ❞❡❧❛②✧✱ ■♥t✳ ❏✳ ❈♦♥tr♦❧✱ ✽✻✭✷✵✶✸✮✱ ♣♣✳ ✶✻✹✻✕✶✻✺✶✳ ❬✻✶❪ ❇✳ ❘♦❞❥❛♥❛❞✐❞✱ ◆✳ ❱✳ ❙❛♥❤✱ ◆✳ ❚✳ ❍❛ ❛♥❞ ◆✳ ❍✳ ❉✉ ✭✷✵✵✾✮✱ ✧❙t❛❜✐❧✐t② r❛❞✐✐ ❢♦r ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ❆s✐❛♥✲❊✉r♦♣❡❛♥ ❏✳ ▼❛t❤✳✱ ✷✱ ♣♣✳ ✾✺✕✶✶✺✳ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✱ ✷♥❞ ❊❞✐t✐♦♥✱ ▼❝●r❛✇ ❍✐❧❧✳ ❬✻✷❪ ❲❛❧t❡r ❘✉❞✐♥ ✭✶✾✾✶✮✱ ❬✻✸❪ ◆✳❍✳ ❙❛✉✱ P✳ ◆✐❛♠s✉♣✱ ❛♥❞ ❱✳◆✳ P❤❛t ✭✷✵✶✻✮✱ ✧P♦s✐t✐✈✐t② ❛♥❞ st❛❜✐❧✲ ✐t② ❛♥❛❧②s✐s ❢♦r ❧✐♥❡❛r ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❞❡❧❛② ❡q✉❛t✐♦♥s✧✱ ❆♣♣❧✳✱ ✺✶✵✱ ♣♣✳ ✷✺✕✹✶✳ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❬✻✹❪ ◆✳❑✳ ❙♦♥ ❛♥❞ ❉✳ ❍✐♥r✐❝❤s❡♥ ✭✶✾✾✻✮✱ ✧❘♦❜✉st st❛❜✐❧✐t② ♦❢ ♣♦s✐t✐✈❡ ❝♦♥t✐♥✉✲ ♦✉s t✐♠❡ s②st❡♠s✧✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✶✼✱ ♣♣✳ ✻✹✾✕✻✺✾✳ ❬✻✺❪ ◆✳❑✳ ❙♦♥ ❛♥❞ P✳❍✳❆✳ ◆❣♦❝ ✭✶✾✾✽✮✱ ✧❚❤❡ ❝♦♠♣❧❡① st❛❜✐❧✐t② r❛❞✐✉s ♦❢ ❧✐♥❡❛r t✐♠❡✲❞❡❧❛② s②st❡♠s✧✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳✱ ✷✻✱ ♣♣✳ ✸✼✾✕✸✽✹✳ ❬✻✻❪ ◆✳❑✳ ❙♦♥ ❛♥❞ P✳❍✳❆✳ ◆❣♦❝ ✭✶✾✾✾✮✱ ✧❘♦❜✉st st❛❜✐❧✐t② ♦❢ ♣♦s✐t✐✈❡ ❧✐♥❡❛r t✐♠❡ ❞❡❧❛② s②st❡♠s ✉♥❞❡r ❛✣♥❡ ♣❛r❛♠❡t❡r ♣❡rt✉r❜❛t✐♦♥s✧✱ ❱✐❡t♥❛♠✳✱ ✷✹✱ ♣♣✳ ✸✺✸✕✸✼✷✳ ✶✷✹ ❆❝t❛ ▼❛t❤✳ ❬✻✼❪ ◆✳❑✳ ❙♦♥ ❛♥❞ ❉✳❉✳ ❚❤✉❛♥ ✭✷✵✶✶✮✱ ✧❖♥ t❤❡ r❛❞✐✉s ♦❢ s✉r❥❡❝t✐✈✐t② ❢♦r r❡❝t✲ ❛♥❣✉❧❛r ♠❛tr✐❝❡s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ t♦ ♠❡❛s✉r✐♥❣ st❛❜✐❧✐③❛❜✐❧✐t② ♦❢ ❧✐♥❡❛r s②st❡♠s ✉♥❞❡r str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s✧✱ ❏✳ ◆♦♥❧✐♥✳ ❈♦♥✈❡① ❆♥❛❧✳✱ ✶✷✱ ♣♣✳ ✹✹✶✕✹✺✸✳ ❬✻✽❪ ◆✳❑✳ ❙♦♥ ❛♥❞ ❉✳❉✳ ❚❤✉❛♥ ✭✷✵✶✷✮✱ ✧❚❤❡ str✉❝t✉r❡❞ ❞✐st❛♥❝❡ t♦ ♥♦♥✲ s✉r❥❡❝t✐✈✐t② ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s t♦ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② r❛❞✐✉s ♦❢ ❞❡s❝r✐♣t♦r s②st❡♠s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✽✽✱ ♣♣✳ ✷✼✷✕✷✽✶✳ ❬✻✾❪ ▲❧♦②❞ ◆✳ ❚r❡❢❡t❤❡♥ ❛♥❞ ❉❛✈✐❞ ❇❛✉ ■■■ ✭✶✾✾✼✮✱ ◆✉♠❡r✐❝❛❧ ❧✐♥❡❛r ❛❧❣❡❜r❛✱ P❤✐❧❛❞❡❧♣❤✐❛✿ ❙♦❝✐❡t② ❢♦r ■♥❞✉str✐❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✳ ■❙❇◆ ✾✼✽✲ ✵✲✽✾✽✼✶✲✸✻✶✲✾✳ ❬✼✵❪ ▼✳ P✐t✉❦ ✭✷✵✵✹✮✱ ✧❆ ❝r✐t❡r✐♦♥ ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ ❧✐♥❡❛r ❞✐✛❡r✲ ❡♥❝❡ ❡q✉❛t✐♦♥s✧✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt✳✱ ✶✼✱ ♣♣✳ ✼✼✾✕✼✽✸✳ ❬✼✶❪ ▲✳ ◗✐✉ ❛♥❞ ❊✳❏✳ ❉❛✈✐s♦♥ ✭✶✾✾✷✮✿ ✧❚❤❡ st❛❜✐❧✐t② r♦❜✉st♥❡ss ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❡✐❣❡♥✈❛❧✉❡s✧✱ ■❊❊❊ ❚r❛♥s✳ ❆✉t♦♠✳ ❈♦♥tr♦❧✱ ✸✼✱ ♣♣✳ ✽✽✻✕✽✾✶✳ ❬✼✷❪ ▲✳ ◗✐✉✱ ❇✳ ❇❡r♥❤❛r❞ss♦♥✱ ❆✳ ❘❛♥t③❡r✱ ❊✳❏✳ ❉❛✈✐s♦♥✱ P✳▼✳ ❨♦✉♥❣✱ ❏✳❈✳ ❉♦②❧❡ ✭✶✾✾✺✮✱ ✧❆ ❢♦r♠✉❧❛ ❢♦r ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ r❡❛❧ st❛❜✐❧✐t② r❛❞✐✉s✧✱ ❆✉t♦♠❛t✐❝❛ ✸✶✱ ♣♣✳ ✽✼✾✕✽✾✵✳ ❬✼✸❪ ❋✳ ❚✐ss❡✉r ❛♥❞ ◆✳ ❍✐❣❤❛♠ ✭✷✵✵✶✮✱ ✧❙tr✉❝t✉r❡❞ ♣s❡✉❞♦s♣❡❝tr❛ ❢♦r ♣♦❧②♥♦✲ ♠✐❛❧ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s✱ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❙■❆▼ ❏✳ ▼❛tr✐① ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✸✱ ♣♣✳ ✶✽✼✕✷✵✽✳ ❬✼✹❪ ❙✳ ❳✉ ❛♥❞ ❏✳ ▲❛♠ ✭✷✵✵✹✮✱ ✧❘♦❜✉st st❛❜✐❧✐t② ❛♥❞ st❛❜✐❧✐③❛t✐♦♥ ♦❢ ❞✐s❝r❡t❡ s✐♥❣✉❧❛r s②st❡♠s✿ ❛♥ ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥✧✱ ❆✉t♦♠❛t✐❝ ❈♦♥tr♦❧✱ ❱♦❧✉♠❡✿ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ✹✾✱ ■ss✉❡✿ ✹✳ ❬✼✺❪ ❲✳ ❳✉❡ ❛♥❞ ❲✳ ▼❛♦ ✭✷✵✶✸✮✱ ✧❆❞♠✐ss✐❜❧❡ ❋✐♥✐t❡✲❚✐♠❡ ❙t❛❜✐❧✐t② ❛♥❞ ❙t❛✲ ❜✐❧✐③❛t✐♦♥ ♦❢ ❯♥❝❡rt❛✐♥ ❉✐s❝r❡t❡ ❙✐♥❣✉❧❛r ❙②st❡♠s✧✱ ❏✳ ❉②♥✳ ❙②s✳✱ ▼❡❛s✳✱ ❈♦♥tr♦❧✱ ❀✶✸✺✭✸✮✿✵✸✶✵✶✽✲✵✸✶✵✶✽✲✻✳ ❞♦✐✿✶✵✳✶✶✶✺✴✶✳✹✵✷✸✷✶✸✳ ✶✷✺ ... ♦❜t❛✐♥❡❞✳ ✐ Tóm tắt Trong cơng trình chúng tơi nghiên cứu phương trình sai phân suy biến tuyến tính cấp cấp hai Đối với phương trình sai phân suy biến tuyến tính cấp 1, sử dụng cách tiếp cận phép... {{(Ek,i , Ak,i )}k n0 }i∈N si := rf,i − rf,i+1 , hf,−1 := 0, vi := m − rf,i − hf,i ✭✶✳✷✶✮ ❲❡ ❤❛✈❡ t❤❛t rf,i rf,i+1 , hf,i hf,i+1 , vi+1 vi , si si+1 , si , vi ❢♦r ❛❧❧ i ∈ N ❋✉rt❤❡r♠♦r❡✱ t❤❡r❡... hóa tính ổn định hệ tác động nhiễu; thiết lập mối quan hệ tính ổn định mũ/ ổn định hệ tính chất nghiệm hệ không nhất; mở rộng khái niệm số mũ Bohl cho hệ sai phân suy biến số tính chất Đối với phương