rt rrt - PowerPoint PPT Presentation

▼❛✐♥ q✉❡st✐♦♥s ✿ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s❀ ❈❛t❡❣♦r✐✜❝❛t✐♦♥ ❝♦♥str✉❝t✐✈❡ r❡s✉❧ts❀ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❜❛s❡s ❢♦r t❤❡s❡ ❛❧❣❡❜r❛s ❜② r❡✇r✐t✐♥❣✳ ❙tr✉❝t✉r❛❧ r✉❧❡s ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s ♠❛❦❡ t❤❡ st✉❞② ♦❢ ❧♦❝❛❧ ❝♦♥✢✉❡♥❝❡ ❝♦♠♣❧✐❝❛t❡❞✳ ❊①❛♠♣❧❡✿ ■s♦t♦♣② r❡❧❛t✐♦♥s ❲❡ ✉s❡ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✳ ❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳ ▼♦t✐✈❛t✐♦♥✿ ♦❜❥❡❝t✐✈❡s ◮ ❖❜❥❡❝t✐✈❡✿ ❙t✉❞② ❞✐❛❣r❛♠♠❛t✐❝ ❛❧❣❡❜r❛s ❛r✐s✐♥❣ ✐♥ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣✳ ◮ ❑❤♦✈❛♥♦✈✲▲❛✉❞❛✲❘♦✉q✉✐❡r ✭❑▲❘✮ ❛❧❣❡❜r❛s ❢♦r ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❣r♦✉♣s❀ ◮ ❚❡♠♣❡r❧❡②✲▲✐❡❜ ❛❧❣❡❜r❛s ✐♥ st❛t✐st✐❝❤❛❧ ♠❡❝❤❛♥✐❝s❀ ◮ ❇r❛✉❡r ❛❧❣❡❜r❛s ❛♥❞ ❇✐r♠❛♥✲❲❡♥③❧ ❛❧❣❡❜r❛s ✐♥ ❦♥♦t t❤❡♦r②✳

❙tr✉❝t✉r❛❧ r✉❧❡s ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s ♠❛❦❡ t❤❡ st✉❞② ♦❢ ❧♦❝❛❧ ❝♦♥✢✉❡♥❝❡ ❝♦♠♣❧✐❝❛t❡❞✳ ❊①❛♠♣❧❡✿ ■s♦t♦♣② r❡❧❛t✐♦♥s ❲❡ ✉s❡ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✳ ❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳ ▼♦t✐✈❛t✐♦♥✿ ♦❜❥❡❝t✐✈❡s ◮ ❖❜❥❡❝t✐✈❡✿ ❙t✉❞② ❞✐❛❣r❛♠♠❛t✐❝ ❛❧❣❡❜r❛s ❛r✐s✐♥❣ ✐♥ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣✳ ◮ ❑❤♦✈❛♥♦✈✲▲❛✉❞❛✲❘♦✉q✉✐❡r ✭❑▲❘✮ ❛❧❣❡❜r❛s ❢♦r ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❣r♦✉♣s❀ ◮ ❚❡♠♣❡r❧❡②✲▲✐❡❜ ❛❧❣❡❜r❛s ✐♥ st❛t✐st✐❝❤❛❧ ♠❡❝❤❛♥✐❝s❀ ◮ ❇r❛✉❡r ❛❧❣❡❜r❛s ❛♥❞ ❇✐r♠❛♥✲❲❡♥③❧ ❛❧❣❡❜r❛s ✐♥ ❦♥♦t t❤❡♦r②✳ ◮ ▼❛✐♥ q✉❡st✐♦♥s ✿ ◮ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s❀ ◮ ❈❛t❡❣♦r✐✜❝❛t✐♦♥ ❝♦♥str✉❝t✐✈❡ r❡s✉❧ts❀ ◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❜❛s❡s ❢♦r t❤❡s❡ ❛❧❣❡❜r❛s ❜② r❡✇r✐t✐♥❣✳

❙tr✉❝t✉r❛❧ r✉❧❡s ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s ♠❛❦❡ t❤❡ st✉❞② ♦❢ ❧♦❝❛❧ ❝♦♥✢✉❡♥❝❡ ❝♦♠♣❧✐❝❛t❡❞✳ ❊①❛♠♣❧❡✿ ■s♦t♦♣② r❡❧❛t✐♦♥s ❲❡ ✉s❡ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✳ ❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳ ▼♦t✐✈❛t✐♦♥✿ ♦❜❥❡❝t✐✈❡s ◮ ❖❜❥❡❝t✐✈❡✿ ❙t✉❞② ❞✐❛❣r❛♠♠❛t✐❝ ❛❧❣❡❜r❛s ❛r✐s✐♥❣ ✐♥ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣✳ ◮ ❑❤♦✈❛♥♦✈✲▲❛✉❞❛✲❘♦✉q✉✐❡r ✭❑▲❘✮ ❛❧❣❡❜r❛s ❢♦r ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❣r♦✉♣s❀ ◮ ❚❡♠♣❡r❧❡②✲▲✐❡❜ ❛❧❣❡❜r❛s ✐♥ st❛t✐st✐❝❤❛❧ ♠❡❝❤❛♥✐❝s❀ ◮ ❇r❛✉❡r ❛❧❣❡❜r❛s ❛♥❞ ❇✐r♠❛♥✲❲❡♥③❧ ❛❧❣❡❜r❛s ✐♥ ❦♥♦t t❤❡♦r②✳ ◮ ▼❛✐♥ q✉❡st✐♦♥s ✿ ◮ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s❀ ◮ ❈❛t❡❣♦r✐✜❝❛t✐♦♥ ❝♦♥str✉❝t✐✈❡ r❡s✉❧ts❀ ◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❜❛s❡s ❢♦r t❤❡s❡ ❛❧❣❡❜r❛s ❜② r❡✇r✐t✐♥❣✳

❊①❛♠♣❧❡✿ ■s♦t♦♣② r❡❧❛t✐♦♥s ❲❡ ✉s❡ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✳ ❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳ ▼♦t✐✈❛t✐♦♥✿ ♦❜❥❡❝t✐✈❡s ◮ ❖❜❥❡❝t✐✈❡✿ ❙t✉❞② ❞✐❛❣r❛♠♠❛t✐❝ ❛❧❣❡❜r❛s ❛r✐s✐♥❣ ✐♥ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣✳ ◮ ❑❤♦✈❛♥♦✈✲▲❛✉❞❛✲❘♦✉q✉✐❡r ✭❑▲❘✮ ❛❧❣❡❜r❛s ❢♦r ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❣r♦✉♣s❀ ◮ ❚❡♠♣❡r❧❡②✲▲✐❡❜ ❛❧❣❡❜r❛s ✐♥ st❛t✐st✐❝❤❛❧ ♠❡❝❤❛♥✐❝s❀ ◮ ❇r❛✉❡r ❛❧❣❡❜r❛s ❛♥❞ ❇✐r♠❛♥✲❲❡♥③❧ ❛❧❣❡❜r❛s ✐♥ ❦♥♦t t❤❡♦r②✳ ◮ ▼❛✐♥ q✉❡st✐♦♥s ✿ ◮ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s❀ ◮ ❈❛t❡❣♦r✐✜❝❛t✐♦♥ ❝♦♥str✉❝t✐✈❡ r❡s✉❧ts❀ ◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❜❛s❡s ❢♦r t❤❡s❡ ❛❧❣❡❜r❛s ❜② r❡✇r✐t✐♥❣✳ ◮ ❙tr✉❝t✉r❛❧ r✉❧❡s ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s ♠❛❦❡ t❤❡ st✉❞② ♦❢ ❧♦❝❛❧ ❝♦♥✢✉❡♥❝❡ ❝♦♠♣❧✐❝❛t❡❞✳

❲❡ ✉s❡ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✳ ❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳ ▼♦t✐✈❛t✐♦♥✿ ♦❜❥❡❝t✐✈❡s ◮ ❖❜❥❡❝t✐✈❡✿ ❙t✉❞② ❞✐❛❣r❛♠♠❛t✐❝ ❛❧❣❡❜r❛s ❛r✐s✐♥❣ ✐♥ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣✳ ◮ ❑❤♦✈❛♥♦✈✲▲❛✉❞❛✲❘♦✉q✉✐❡r ✭❑▲❘✮ ❛❧❣❡❜r❛s ❢♦r ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❣r♦✉♣s❀ ◮ ❚❡♠♣❡r❧❡②✲▲✐❡❜ ❛❧❣❡❜r❛s ✐♥ st❛t✐st✐❝❤❛❧ ♠❡❝❤❛♥✐❝s❀ ◮ ❇r❛✉❡r ❛❧❣❡❜r❛s ❛♥❞ ❇✐r♠❛♥✲❲❡♥③❧ ❛❧❣❡❜r❛s ✐♥ ❦♥♦t t❤❡♦r②✳ ◮ ▼❛✐♥ q✉❡st✐♦♥s ✿ ◮ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s❀ ◮ ❈❛t❡❣♦r✐✜❝❛t✐♦♥ ❝♦♥str✉❝t✐✈❡ r❡s✉❧ts❀ ◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❜❛s❡s ❢♦r t❤❡s❡ ❛❧❣❡❜r❛s ❜② r❡✇r✐t✐♥❣✳ ◮ ❙tr✉❝t✉r❛❧ r✉❧❡s ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s ♠❛❦❡ t❤❡ st✉❞② ♦❢ ❧♦❝❛❧ ❝♦♥✢✉❡♥❝❡ ❝♦♠♣❧✐❝❛t❡❞✳ ❊①❛♠♣❧❡✿ ■s♦t♦♣② r❡❧❛t✐♦♥s = = = • = • •

▼♦t✐✈❛t✐♦♥✿ ♦❜❥❡❝t✐✈❡s ◮ ❖❜❥❡❝t✐✈❡✿ ❙t✉❞② ❞✐❛❣r❛♠♠❛t✐❝ ❛❧❣❡❜r❛s ❛r✐s✐♥❣ ✐♥ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣✳ ◮ ❑❤♦✈❛♥♦✈✲▲❛✉❞❛✲❘♦✉q✉✐❡r ✭❑▲❘✮ ❛❧❣❡❜r❛s ❢♦r ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❣r♦✉♣s❀ ◮ ❚❡♠♣❡r❧❡②✲▲✐❡❜ ❛❧❣❡❜r❛s ✐♥ st❛t✐st✐❝❤❛❧ ♠❡❝❤❛♥✐❝s❀ ◮ ❇r❛✉❡r ❛❧❣❡❜r❛s ❛♥❞ ❇✐r♠❛♥✲❲❡♥③❧ ❛❧❣❡❜r❛s ✐♥ ❦♥♦t t❤❡♦r②✳ ◮ ▼❛✐♥ q✉❡st✐♦♥s ✿ ◮ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s❀ ◮ ❈❛t❡❣♦r✐✜❝❛t✐♦♥ ❝♦♥str✉❝t✐✈❡ r❡s✉❧ts❀ ◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❜❛s❡s ❢♦r t❤❡s❡ ❛❧❣❡❜r❛s ❜② r❡✇r✐t✐♥❣✳ ◮ ❙tr✉❝t✉r❛❧ r✉❧❡s ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s ♠❛❦❡ t❤❡ st✉❞② ♦❢ ❧♦❝❛❧ ❝♦♥✢✉❡♥❝❡ ❝♦♠♣❧✐❝❛t❡❞✳ ❊①❛♠♣❧❡✿ ■s♦t♦♣② r❡❧❛t✐♦♥s = = = • = • • ◮ ❲❡ ✉s❡ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✳ ◮ ❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳

❈✉❜✐❝❛❧ ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♦♥ ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s ✳ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♠♦❞✉❧♦ ✱ ❍✉❡t ✬✽✵ ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r

❈✉❜✐❝❛❧ ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♦♥ ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♠♦❞✉❧♦ ✱ ❍✉❡t ✬✽✵ ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳

❈✉❜✐❝❛❧ ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♦♥ ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♠♦❞✉❧♦ ✱ ❍✉❡t ✬✽✵ ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿

❈✉❜✐❝❛❧ ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♦♥ ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ � ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♠♦❞✉❧♦ E ✱ ❍✉❡t ✬✽✵ R � R � w u ′ u E � E R � v ′ R � w ′ v

✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ ♦♥ ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ � ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♠♦❞✉❧♦ E ✱ ❍✉❡t ✬✽✵ R � R � w u ′ u E � E ❈✉❜✐❝❛❧ R � v ′ R � w ′ v

❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ � ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♠♦❞✉❧♦ E ✱ ❍✉❡t ✬✽✵ R � R � w u ′ u E � E ❈✉❜✐❝❛❧ R � v ′ R � w ′ v E R E ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♦♥ E ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ◮

❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ � � ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♠♦❞✉❧♦ E ✱ ❍✉❡t ✬✽✵ R � R � w u ′ u E � E ❈✉❜✐❝❛❧ R � v ′ R � w ′ v E RE � u v E � E E R E ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♦♥ E ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ◮ u ′ R � v ′

❂ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ � � ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♠♦❞✉❧♦ E ✱ ❍✉❡t ✬✽✵ R � R � w u ′ u E � E ❈✉❜✐❝❛❧ R � v ′ R � w ′ v E RE � u v E � E E R E ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♦♥ E ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ◮ u ′ R � v ′ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ S s✉❝❤ t❤❛t R ⊆ S ⊆ E R E ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳

� � � ❚❤r❡❡ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R ✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n ✲❝❛t❡❣♦r✐❡s✳ ●❧♦❜✉❧❛r ◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E ✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♠♦❞✉❧♦ E ✱ ❍✉❡t ✬✽✵ R � R � w u ′ u E � E ❈✉❜✐❝❛❧ R � v ′ R � w ′ v E RE � u v E � E E R E ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♦♥ E ✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ◮ u ′ R � v ′ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ S s✉❝❤ t❤❛t R ⊆ S ⊆ E R E ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ E R � u v E � ◮ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r E R ✳ ❂ R � v u ′

■■✳ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s

❈ ❈ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹ ❈ ❈ ❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② ✶ ✶ ✷ ✷ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳

❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② ✶ ✶ ✷ ✷ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳ ◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹

❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② ✶ ✶ ✷ ✷ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳ ◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹ ( ❈ ✵ ) ✵

❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② ✶ ✶ ✷ ✷ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳ ◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹ ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✶ � ( ❈ ✵ ) ✵

❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② ✶ ✶ ✷ ✷ � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳ ◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹ ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✶ � ( ❈ ✵ ) ✶ ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵

❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② ✶ ✶ ✷ ✷ � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳ ◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹ ( ❈ ✶ ) ✵ � ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✶ � ( ❈ ✵ ) ✶ � ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵ ( ❈ ✶ ) ✵

❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② ✶ ✶ ✷ ✷ � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳ ◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹ ( ❈ ✶ ) ✵ � ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✶ � ( ❈ ✶ ) ✶ ( ❈ ✵ ) ✶ � ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵ ( ❈ ✶ ) ✵

� � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n ✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✳ ◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ❈❛t✳ ❊❤r❡s♠❛♥♥ ✬✻✹ ( ❈ ✶ ) ✵ � ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✶ � ( ❈ ✶ ) ✶ ( ❈ ✵ ) ✶ � ( ❈ ✵ ) ✵ ( ❈ ✵ ) ✵ ( ❈ ✶ ) ✵ ◮ ❚❤❡r❡ ❛r❡ ♣♦✐♥t ❝❡❧❧s✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s ❛♥❞ ✈❡rt✐❝❛❧ ❝❡❧❧s r❡s♣❡❝t✐✈❡❧② ♣✐❝t✉r❡❞ ❜② x ✶ e f � x ✷ x ✶ x ✷

✵ ✶ ✷ ✱ ✇✐t❤ ✐❞❡♥t✐t✐❡s ✶ ✷ ✵ ✵ ✶ ✶ ✶ ✷ ✵ ❈♦♠♣♦s✐t✐♦♥s ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✶ ✸ ✶ ✷ ✸ ✶ ✸ ✶ ✷ ✸ ✶ ✸ ✶ ✷ ✶ ✷ ✶ ✶ ✶ ✷ ✶ ✷ ✶ ✷ ✶ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✸ ❢♦r ❛❧❧ ✱ ✱ ♣♦✐♥t ❝❡❧❧s✱ ✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s✱ ✱ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ ✱ ✱ sq✉❛r❡ ❝❡❧❧s✳ � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s ∂ h − , ✶ ( A ) � · · ∂ v ∂ v − , ✶ ( A ) + , ✶ ( A ) A � · · ∂ h + , ✶ ( A )

❈♦♠♣♦s✐t✐♦♥s ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✶ ✸ ✶ ✷ ✸ ✶ ✸ ✶ ✷ ✸ ✶ ✸ ✶ ✷ ✶ ✷ ✶ ✶ ✶ ✷ ✶ ✷ ✶ ✷ ✶ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✸ ❢♦r ❛❧❧ ✱ ✱ ♣♦✐♥t ❝❡❧❧s✱ ✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s✱ ✱ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ ✱ ✱ sq✉❛r❡ ❝❡❧❧s✳ � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s ∂ h − , ✶ ( A ) � i h ✵ ( x ) � f · · x ✶ x ✷ x x ∂ v ∂ v i v i v i v − , ✶ ( A ) + , ✶ ( A ) ✱ ✇✐t❤ ✐❞❡♥t✐t✐❡s i h A ✵ ( x ✶ ) ✶ ( f ) ✵ ( x ✷ ) e ✶ ( e ) e � · � y · � x ✷ y x ✶ ∂ h f i h + , ✶ ( A ) ✵ ( y )

✶ ✶ ✶ ✷ ✶ ✷ ✶ ✷ ✶ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✸ � � � � � � � � � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s ∂ h − , ✶ ( A ) � i h ✵ ( x ) � f · · x ✶ x ✷ x x ∂ v ∂ v i v i v i v − , ✶ ( A ) + , ✶ ( A ) ✱ ✇✐t❤ ✐❞❡♥t✐t✐❡s i h A ✵ ( x ✶ ) ✶ ( f ) ✵ ( x ✷ ) e ✶ ( e ) e � · � y · � x ✷ y x ✶ ∂ h f i h + , ✶ ( A ) ✵ ( y ) ◮ ❈♦♠♣♦s✐t✐♦♥s f ✶ f ✷ f ✶ f ✷ � x ✸ x ✶ x ✷ x ✶ x ✸ A ⋄ v B e ✶ e ✷ e ✸ e ✶ e ✸ A B � � y ✷ � y ✸ � y ✸ y ✶ y ✶ g ✶ g ✷ g ✶ g ✷ ❢♦r ❛❧❧ x i ✱ y i ✱ z i ♣♦✐♥t ❝❡❧❧s✱ f i ✱ g i ❤♦r✐③♦♥t❛❧ ❝❡❧❧s✱ e i ✱ e ′ i ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ A ✱ A ′ ✱ B sq✉❛r❡ ❝❡❧❧s✳

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s ∂ h − , ✶ ( A ) � i h ✵ ( x ) � f · · x ✶ x ✷ x x ∂ v ∂ v i v i v i v − , ✶ ( A ) + , ✶ ( A ) ✱ ✇✐t❤ ✐❞❡♥t✐t✐❡s i h A ✵ ( x ✶ ) ✶ ( f ) ✵ ( x ✷ ) e ✶ ( e ) e � · � y · � x ✷ y x ✶ ∂ h f i h + , ✶ ( A ) ✵ ( y ) ◮ ❈♦♠♣♦s✐t✐♦♥s f ✶ f ✷ f ✶ f ✷ � x ✸ x ✶ x ✷ x ✶ x ✸ A ⋄ v B e ✶ e ✷ e ✸ e ✶ e ✸ A B � � y ✷ � y ✸ � y ✸ y ✶ y ✶ g ✶ g ✷ g ✶ g ✷ f ✶ f ✶ x ✶ x ✷ x ✶ x ✷ e ✶ e ✷ A e ✶ e ′ e ✷ e ′ � y ✶ y ✷ A ⋄ h A ′ ✶ ✷ f ✷ e ′ e ′ A ′ ✶ � ✷ � z ✷ z ✶ � z ✷ z ✶ f ✸ f ✸ ❢♦r ❛❧❧ x i ✱ y i ✱ z i ♣♦✐♥t ❝❡❧❧s✱ f i ✱ g i ❤♦r✐③♦♥t❛❧ ❝❡❧❧s✱ e i ✱ e ′ i ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ A ✱ A ′ ✱ B sq✉❛r❡ ❝❡❧❧s✳

✶ ✷ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ❈ ❈ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❈ ❈ ❣r♦✉♣♦✐❞s✳ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿

✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✷ ✸ ✶ ✷ ✷ ✸ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ❈ ❈ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❈ ❈ ❣r♦✉♣♦✐❞s✳ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ x ✶ x ✷ e ✶ � e ✷ A � y ✷ y ✶ g ✶

✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✷ ✸ ✶ ✷ ✷ ✸ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✷ ✸ ✶ ✷ ✷ ✸ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ❈ ❈ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❈ ❈ ❣r♦✉♣♦✐❞s✳ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ x ✶ x ✷ e ✶ � e ✷ A � y ✷ y ✶ g ✶ ⋄ h g ✶ y ✶ y ✷ e ′ e ′ A ′ ✶ � ✷ � z ✷ z ✶ h ✶

✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ❈ ❈ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❈ ❈ ❣r♦✉♣♦✐❞s✳ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ f ✷ x ✶ x ✷ x ✷ x ✸ e ✶ � e ✷ e ✷ � e ✸ A B � y ✷ � y ✸ y ✶ g ✶ y ✷ g ✷ ⋄ h ⋄ v ⋄ h g ✶ g ✷ y ✶ y ✷ y ✷ y ✸ e ′ e ′ e ′ e ′ A ′ B ′ ✶ � ✷ � ✷ ✸ � z ✷ � z ✸ z ✶ z ✷ h ✶ h ✷

✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ❈ ❈ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❈ ❈ ❣r♦✉♣♦✐❞s✳ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ f ✷ x ✶ x ✷ x ✷ x ✸ e ✶ � e ✷ e ✷ � e ✸ A B � y ✷ � y ✸ y ✶ g ✶ y ✷ g ✷ ⋄ h ⋄ v ⋄ h = g ✶ g ✷ y ✶ y ✷ y ✷ y ✸ e ′ e ′ e ′ e ′ A ′ B ′ ✶ � ✷ � ✷ ✸ � z ✷ � z ✸ z ✶ z ✷ h ✶ h ✷

✶ ✷ ✶ ✷ ✷ ✸ ✶ ✷ ✷ ✸ ✶ ✶ ✷ ✷ ✷ ✸ ❈ ❈ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❈ ❈ ❣r♦✉♣♦✐❞s✳ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � � � � � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ f ✷ f ✶ f ✷ x ✶ x ✷ x ✷ x ✸ x ✶ x ✷ x ✷ x ✸ ⋄ v e ✶ � e ✷ e ✷ � e ✸ e ✶ � e ✷ e ✷ � e ✸ A B A B � y ✷ � y ✸ � y ✷ � y ✸ y ✶ g ✶ y ✷ g ✷ y ✶ g ✶ y ✷ g ✷ ⋄ h ⋄ v ⋄ h = g ✶ g ✷ y ✶ y ✷ y ✷ y ✸ e ′ e ′ e ′ e ′ A ′ B ′ ✶ � ✷ � ✷ ✸ � z ✷ � z ✸ z ✶ z ✷ h ✶ h ✷

❈ ❈ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ❈ ✶ ❈ ✵ ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❈ ❈ ❣r♦✉♣♦✐❞s✳ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � � � � � � � � � � � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ f ✷ f ✶ f ✷ x ✶ x ✷ x ✷ x ✸ x ✶ x ✷ x ✷ x ✸ ⋄ v e ✶ � e ✷ e ✷ � e ✸ e ✶ � e ✷ e ✷ � e ✸ A B A B � y ✷ � y ✸ � y ✷ � y ✸ y ✶ g ✶ y ✷ g ✷ y ✶ g ✶ y ✷ g ✷ ⋄ h ⋄ v ⋄ h = ⋄ h g ✶ g ✷ g ✶ g ✷ y ✶ y ✷ y ✷ y ✸ y ✶ y ✷ y ✷ y ✸ ⋄ v e ′ e ′ e ′ e ′ e ′ e ′ e ′ e ′ A ′ B ′ A ′ B ′ ✶ � ✷ � ✶ � ✷ � ✷ ✸ ✷ ✸ � z ✷ � z ✸ � z ✷ � z ✸ z ✶ z ✷ z ✶ z ✷ h ✶ h ✷ h ✶ h ✷

✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ ✲❝❛t❡❣♦r② s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � � � � � � � � � � � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ f ✷ f ✶ f ✷ x ✶ x ✷ x ✷ x ✸ x ✶ x ✷ x ✷ x ✸ ⋄ v e ✶ � e ✷ e ✷ � e ✸ e ✶ � e ✷ e ✷ � e ✸ A B A B � y ✷ � y ✸ � y ✷ � y ✸ y ✶ g ✶ y ✷ g ✷ y ✶ g ✶ y ✷ g ✷ ⋄ h ⋄ v ⋄ h = ⋄ h g ✶ g ✷ g ✶ g ✷ y ✶ y ✷ y ✷ y ✸ y ✶ y ✷ y ✷ y ✸ ⋄ v e ′ e ′ e ′ e ′ e ′ e ′ e ′ e ′ A ′ B ′ A ′ B ′ ✶ � ✷ � ✶ � ✷ � ✷ ✸ ✷ ✸ � z ✷ � z ✸ � z ✷ � z ✸ z ✶ z ✷ z ✶ z ✷ h ✶ h ✷ h ✶ h ✷ ◮ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❣r♦✉♣♦✐❞s✳

❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳ � � � � � � � � � � � � � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ f ✷ f ✶ f ✷ x ✶ x ✷ x ✷ x ✸ x ✶ x ✷ x ✷ x ✸ ⋄ v e ✶ � e ✷ e ✷ � e ✸ e ✶ � e ✷ e ✷ � e ✸ A B A B � y ✷ � y ✸ � y ✷ � y ✸ y ✶ g ✶ y ✷ g ✷ y ✶ g ✶ y ✷ g ✷ ⋄ h ⋄ v ⋄ h = ⋄ h g ✶ g ✷ g ✶ g ✷ y ✶ y ✷ y ✷ y ✸ y ✶ y ✷ y ✷ y ✸ ⋄ v e ′ e ′ e ′ e ′ e ′ e ′ e ′ e ′ A ′ B ′ A ′ B ′ ✶ � ✷ � ✶ � ✷ � ✷ ✸ ✷ ✸ � z ✷ � z ✸ � z ✷ � z ✸ z ✶ z ✷ z ✶ z ✷ h ✶ h ✷ h ✶ h ✷ ◮ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❣r♦✉♣♦✐❞s✳ ◮ n ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ n ✲❝❛t❡❣♦r② C s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t C n ( x , y ) ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳

� � � � � � � � � � � � � � � � � � � � � � � � ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞s ◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿ f ✶ f ✷ f ✶ f ✷ x ✶ x ✷ x ✷ x ✸ x ✶ x ✷ x ✷ x ✸ ⋄ v e ✶ � e ✷ e ✷ � e ✸ e ✶ � e ✷ e ✷ � e ✸ A B A B � y ✷ � y ✸ � y ✷ � y ✸ y ✶ g ✶ y ✷ g ✷ y ✶ g ✶ y ✷ g ✷ ⋄ h ⋄ v ⋄ h = ⋄ h g ✶ g ✷ g ✶ g ✷ y ✶ y ✷ y ✷ y ✸ y ✶ y ✷ y ✷ y ✸ ⋄ v e ′ e ′ e ′ e ′ e ′ e ′ e ′ e ′ A ′ B ′ A ′ B ′ ✶ � ✷ � ✶ � ✷ � ✷ ✸ ✷ ✸ � z ✷ � z ✸ � z ✷ � z ✸ z ✶ z ✷ z ✶ z ✷ h ✶ h ✷ h ✶ h ✷ ◮ ❉♦✉❜❧❡ ❣r♦✉♣♦✐❞ ❂ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ( ❈ ✶ , ❈ ✵ , ∂ ❈ − , ∂ ❈ + , ◦ ❈ , i ❈ ) ✐♥ ✇❤✐❝❤ ❈ ✶ ❛♥❞ ❈ ✵ ❛r❡ ❣r♦✉♣♦✐❞s✳ ◮ n ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ n ✲❝❛t❡❣♦r② C s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t C n ( x , y ) ✐s ❛ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ◮ ❍♦r✐③♦♥t❛❧ ( n + ✶ ) ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ( n + ✶ ) ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ( n + ✶ ) ✲❝❛t❡❣♦r② ✐s t❤❡ ( n + ✶ ) ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳

t✇♦ ✶ ✲♣♦❧②❣r❛♣❤s ❛♥❞ s✉❝❤ t❤❛t ❢♦r ✱ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣❛✐r ♦❢ ✶ ✲❝❛t❡❣♦r✐❡s ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ✱ ✇✐t❤ ✱ ♠❛❦✐♥❣ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ ❆ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✇❤♦s❡ sq✉❛r❡ ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ ✳ ❆ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤✮ ❣❡♥❡r❛t❡s ❛ ❢r❡❡ ✶ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② ✳ ❉♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ ( P v , P h , P s ) ♠❛❞❡ ♦❢✿

❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣❛✐r ♦❢ ✶ ✲❝❛t❡❣♦r✐❡s ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ✱ ✇✐t❤ ✱ ♠❛❦✐♥❣ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ ❆ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✇❤♦s❡ sq✉❛r❡ ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ ✳ ❆ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤✮ ❣❡♥❡r❛t❡s ❛ ❢r❡❡ ✶ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② ✳ �� ❉♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ ( P v , P h , P s ) ♠❛❞❡ ♦❢✿ ◮ t✇♦ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤s P v ❛♥❞ P h s✉❝❤ t❤❛t P v k = P h k ❢♦r k ≤ n ✱ P v �� P ∗ P h n + ✶ n n + ✶

❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣❛✐r ♦❢ ✶ ✲❝❛t❡❣♦r✐❡s ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ✱ ✇✐t❤ ✱ ♠❛❦✐♥❣ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ ❆ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✇❤♦s❡ sq✉❛r❡ ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ ✳ ❆ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤✮ ❣❡♥❡r❛t❡s ❛ ❢r❡❡ ✶ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② ✳ � � �� � � �� � � ❉♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ ( P v , P h , P s ) ♠❛❞❡ ♦❢✿ ◮ t✇♦ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤s P v ❛♥❞ P h s✉❝❤ t❤❛t P v k = P h k ❢♦r k ≤ n ✱ ( P v n + ✶ ) ∗ ( P h n + ✶ ) ∗ ∂ v ∂ h − , n − , n ∂ v ∂ h + , n + , n P ∗ P v P h n + ✶ n + ✶ n

❆ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✇❤♦s❡ sq✉❛r❡ ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ ✳ ❆ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤✮ ❣❡♥❡r❛t❡s ❛ ❢r❡❡ ✶ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② ✳ � � � � � �� � � �� � � � ❉♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ ( P v , P h , P s ) ♠❛❞❡ ♦❢✿ ◮ t✇♦ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤s P v ❛♥❞ P h s✉❝❤ t❤❛t P v k = P h k ❢♦r k ≤ n ✱ ◮ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ t❤❡ ♣❛✐r ♦❢ ( n + ✶ ) ✲❝❛t❡❣♦r✐❡s (( P v ) ∗ , ( P h ) ∗ ) ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ∂ µ ± , n ✱ ✇✐t❤ µ ∈ { v , h } ✱ ♠❛❦✐♥❣ Γ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ P s ∂ h ∂ v + , n + ✶ + , n + ✶ ∂ v ∂ h − , n + ✶ − , n + ✶ n + ✶ ) ∗ n + ✶ ) ∗ ( P v ( P h ∂ v ∂ h − , n − , n ∂ v ∂ h + , n + , n P v P ∗ P h n + ✶ n n + ✶

❆ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤✮ ❣❡♥❡r❛t❡s ❛ ❢r❡❡ ✶ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② ✳ � � �� � �� � � � � � � � ❉♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ ( P v , P h , P s ) ♠❛❞❡ ♦❢✿ ◮ t✇♦ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤s P v ❛♥❞ P h s✉❝❤ t❤❛t P v k = P h k ❢♦r k ≤ n ✱ ◮ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ t❤❡ ♣❛✐r ♦❢ ( n + ✶ ) ✲❝❛t❡❣♦r✐❡s (( P v ) ∗ , ( P h ) ∗ ) ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ∂ µ ± , n ✱ ✇✐t❤ µ ∈ { v , h } ✱ ♠❛❦✐♥❣ Γ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ P s ∂ h ∂ v + , n + ✶ + , n + ✶ ∂ v ∂ h − , n + ✶ − , n + ✶ n + ✶ ) ∗ n + ✶ ) ∗ ( P v ( P h ∂ v ∂ h − , n − , n ∂ v ∂ h + , n + , n P v P ∗ P h n + ✶ n n + ✶ ◮ ❆ ❞♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✇❤♦s❡ sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ✐s ❞❡✜♥❡❞ ♦♥ (( P v ) ⊤ , ( P h ) ⊤ ) ✳

� � �� � � � � � � �� � � ❉♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ ( P v , P h , P s ) ♠❛❞❡ ♦❢✿ ◮ t✇♦ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤s P v ❛♥❞ P h s✉❝❤ t❤❛t P v k = P h k ❢♦r k ≤ n ✱ ◮ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ t❤❡ ♣❛✐r ♦❢ ( n + ✶ ) ✲❝❛t❡❣♦r✐❡s (( P v ) ∗ , ( P h ) ∗ ) ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ∂ µ ± , n ✱ ✇✐t❤ µ ∈ { v , h } ✱ ♠❛❦✐♥❣ Γ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ P s ∂ h ∂ v + , n + ✶ + , n + ✶ ∂ v ∂ h − , n + ✶ − , n + ✶ n + ✶ ) ∗ n + ✶ ) ∗ ( P v ( P h ∂ v ∂ h − , n − , n ∂ v ∂ h + , n + , n P v P ∗ P h n + ✶ n n + ✶ ◮ ❆ ❞♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✇❤♦s❡ sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ✐s ❞❡✜♥❡❞ ♦♥ (( P v ) ⊤ , ( P h ) ⊤ ) ✳ ◮ ❆ ❞♦✉❜❧❡ n ✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤✮ ( P v , P h , P s ) ❣❡♥❡r❛t❡s ❛ ❢r❡❡ ( n − ✶ ) ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② | ( P v , P h , P s ) = ✳

t❤❡r❡ ❡①✐sts ❛ sq✉❛r❡ ✶ ✲❝❡❧❧ ✐♥ s✉❝❤ t❤❛t ✳ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ ✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤ s✉❝❤ t❤❛t✿ t❤❡ ✶ ✲♣♦❧②❣r❛♣❤ ♣r❡s❡♥ts ❈ ❀ ✐s ❛❝②❝❧✐❝ ❊①❛♠♣❧❡✿ ▲❡t ❜❡ ❛ ❝♦♥✈❡r❣❡♥t ✶ ✲♣♦❧②❣r❛♣❤ ❛♥❞ ❈ t❤❡ ✲❝❛t❡❣♦r② ♣r❡s❡♥t❡❞ ❜② ✳ ✿❂ sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ✶ ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡s ❈❞ ✶ ✶ ✶ ✷ ✶ ✷ ❢♦r ❛ ❝❤♦✐❝❡ ♦❢ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ✳ ✶ ✷ ❋r♦♠ ❙q✉✐❡r✬s t❤❡♦r❡♠✱ ❈❞ ✐s ❛ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ � � ❆❝②❝❧✐❝✐t② ◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ (( P v ) ⊤ , ( P h ) ⊤ ) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡ ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ S = ( P h ) ⊤ � · ·

❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ ✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤ s✉❝❤ t❤❛t✿ t❤❡ ✶ ✲♣♦❧②❣r❛♣❤ ♣r❡s❡♥ts ❈ ❀ ✐s ❛❝②❝❧✐❝ ❊①❛♠♣❧❡✿ ▲❡t ❜❡ ❛ ❝♦♥✈❡r❣❡♥t ✶ ✲♣♦❧②❣r❛♣❤ ❛♥❞ ❈ t❤❡ ✲❝❛t❡❣♦r② ♣r❡s❡♥t❡❞ ❜② ✳ ✿❂ sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ✶ ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡s ❈❞ ✶ ✶ ✶ ✷ ✶ ✷ ❢♦r ❛ ❝❤♦✐❝❡ ♦❢ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ✳ ✶ ✷ ❋r♦♠ ❙q✉✐❡r✬s t❤❡♦r❡♠✱ ❈❞ ✐s ❛ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ � � � ❆❝②❝❧✐❝✐t② ◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ (( P v ) ⊤ , ( P h ) ⊤ ) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡ ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = t❤❡r❡ ❡①✐sts ❛ sq✉❛r❡ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( P v , P h , P s ) s✉❝❤ t❤❛t ∂ ( A ) = S ✳

❊①❛♠♣❧❡✿ ▲❡t ❜❡ ❛ ❝♦♥✈❡r❣❡♥t ✶ ✲♣♦❧②❣r❛♣❤ ❛♥❞ ❈ t❤❡ ✲❝❛t❡❣♦r② ♣r❡s❡♥t❡❞ ❜② ✳ ✿❂ sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ✶ ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡s ❈❞ ✶ ✶ ✶ ✷ ✶ ✷ ❢♦r ❛ ❝❤♦✐❝❡ ♦❢ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ✳ ✶ ✷ ❋r♦♠ ❙q✉✐❡r✬s t❤❡♦r❡♠✱ ❈❞ ✐s ❛ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ � � � ❆❝②❝❧✐❝✐t② ◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ (( P v ) ⊤ , ( P h ) ⊤ ) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡ ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = t❤❡r❡ ❡①✐sts ❛ sq✉❛r❡ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( P v , P h , P s ) s✉❝❤ t❤❛t ∂ ( A ) = S ✳ ◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n ✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤ ( P v , P h , P s ) s✉❝❤ t❤❛t✿ ◮ t❤❡ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤ P v � P h ♣r❡s❡♥ts ❈ ❀ ◮ P s ✐s ❛❝②❝❧✐❝

✿❂ sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ✶ ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡s ❈❞ ✶ ✶ ✶ ✷ ✶ ✷ ❢♦r ❛ ❝❤♦✐❝❡ ♦❢ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ✳ ✶ ✷ ❋r♦♠ ❙q✉✐❡r✬s t❤❡♦r❡♠✱ ❈❞ ✐s ❛ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ � � � ❆❝②❝❧✐❝✐t② ◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ (( P v ) ⊤ , ( P h ) ⊤ ) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡ ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = t❤❡r❡ ❡①✐sts ❛ sq✉❛r❡ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( P v , P h , P s ) s✉❝❤ t❤❛t ∂ ( A ) = S ✳ ◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n ✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤ ( P v , P h , P s ) s✉❝❤ t❤❛t✿ ◮ t❤❡ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤ P v � P h ♣r❡s❡♥ts ❈ ❀ ◮ P s ✐s ❛❝②❝❧✐❝ ◮ ❊①❛♠♣❧❡✿ ▲❡t E ❜❡ ❛ ❝♦♥✈❡r❣❡♥t ( n + ✶ ) ✲♣♦❧②❣r❛♣❤ ❛♥❞ ❈ t❤❡ n ✲❝❛t❡❣♦r② ♣r❡s❡♥t❡❞ ❜② E ✳

❋r♦♠ ❙q✉✐❡r✬s t❤❡♦r❡♠✱ ❈❞ ✐s ❛ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ � � � � ❆❝②❝❧✐❝✐t② ◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ (( P v ) ⊤ , ( P h ) ⊤ ) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡ ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = t❤❡r❡ ❡①✐sts ❛ sq✉❛r❡ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( P v , P h , P s ) s✉❝❤ t❤❛t ∂ ( A ) = S ✳ ◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n ✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤ ( P v , P h , P s ) s✉❝❤ t❤❛t✿ ◮ t❤❡ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤ P v � P h ♣r❡s❡♥ts ❈ ❀ ◮ P s ✐s ❛❝②❝❧✐❝ ◮ ❊①❛♠♣❧❡✿ ▲❡t E ❜❡ ❛ ❝♦♥✈❡r❣❡♥t ( n + ✶ ) ✲♣♦❧②❣r❛♣❤ ❛♥❞ ❈ t❤❡ n ✲❝❛t❡❣♦r② ♣r❡s❡♥t❡❞ ❜② E ✳ ❈❞ ( E ) ✿❂ sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , ✶ ) ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡s = � · · e ✶ ⋆ n − ✶ e ′ e ✷ ⋆ n − ✶ e ′ ✶ � ✷ = � · · ❢♦r ❛ ❝❤♦✐❝❡ ♦❢ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ( e ✶ , e ✷ ) ♦❢ E ✳

� � � � ❆❝②❝❧✐❝✐t② ◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ P s ♦❢ (( P v ) ⊤ , ( P h ) ⊤ ) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡ ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = t❤❡r❡ ❡①✐sts ❛ sq✉❛r❡ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( P v , P h , P s ) s✉❝❤ t❤❛t ∂ ( A ) = S ✳ ◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n ✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ ( n + ✷ , n ) ✲♣♦❧②❣r❛♣❤ ( P v , P h , P s ) s✉❝❤ t❤❛t✿ ◮ t❤❡ ( n + ✶ ) ✲♣♦❧②❣r❛♣❤ P v � P h ♣r❡s❡♥ts ❈ ❀ ◮ P s ✐s ❛❝②❝❧✐❝ ◮ ❊①❛♠♣❧❡✿ ▲❡t E ❜❡ ❛ ❝♦♥✈❡r❣❡♥t ( n + ✶ ) ✲♣♦❧②❣r❛♣❤ ❛♥❞ ❈ t❤❡ n ✲❝❛t❡❣♦r② ♣r❡s❡♥t❡❞ ❜② E ✳ ❈❞ ( E ) ✿❂ sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , ✶ ) ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡s = � · · e ✶ ⋆ n − ✶ e ′ e ✷ ⋆ n − ✶ e ′ ✶ � ✷ = � · · ❢♦r ❛ ❝❤♦✐❝❡ ♦❢ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ( e ✶ , e ✷ ) ♦❢ E ✳ ◮ ❋r♦♠ ❙q✉✐❡r✬s t❤❡♦r❡♠✱ ( E , ∅ , ❈❞ ( E )) ✐s ❛ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳

■■■✳ P♦❧②❣r❛♣❤s ♠♦❞✉❧♦

❛♥ ✲♣♦❧②❣r❛♣❤ ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ❛♥ ✲♣♦❧②❣r❛♣❤ s✉❝❤ t❤❛t ❢♦r ✷ ❛♥❞ ✶ ✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ✶ ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ ✶ s✉❝❤ t❤❛t ✱ ✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ❜② ❙♣❤ ✶ ✶ ✶ ✇❤❡r❡ ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s ✐♥ s✉❝❤ t❤❛t ❛♥❞ t❤❡ ♠❛♣ ✐s ❞❡✜♥❡❞ ❜② ✳ ✶ ✶ P♦❧②❣r❛♣❤s ♠♦❞✉❧♦ ❆ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ ( R , E , S ) ♠❛❞❡ ♦❢

❛♥ ✲♣♦❧②❣r❛♣❤ s✉❝❤ t❤❛t ❢♦r ✷ ❛♥❞ ✶ ✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ✶ ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ ✶ s✉❝❤ t❤❛t ✱ ✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ❜② ❙♣❤ ✶ ✶ ✶ ✇❤❡r❡ ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s ✐♥ s✉❝❤ t❤❛t ❛♥❞ t❤❡ ♠❛♣ ✐s ❞❡✜♥❡❞ ❜② ✳ ✶ ✶ P♦❧②❣r❛♣❤s ♠♦❞✉❧♦ ❆ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ ( R , E , S ) ♠❛❞❡ ♦❢ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱

✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ ✶ s✉❝❤ t❤❛t ✱ ✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ❜② ❙♣❤ ✶ ✶ ✶ ✇❤❡r❡ ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s ✐♥ s✉❝❤ t❤❛t ❛♥❞ t❤❡ ♠❛♣ ✐s ❞❡✜♥❡❞ ❜② ✳ ✶ ✶ P♦❧②❣r❛♣❤s ♠♦❞✉❧♦ ❆ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ ( R , E , S ) ♠❛❞❡ ♦❢ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t E k = R k ❢♦r k ≤ n − ✷ ❛♥❞ E n − ✶ ⊆ R n − ✶ ✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱

✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ❜② ❙♣❤ ✶ ✶ ✶ ✇❤❡r❡ ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s ✐♥ s✉❝❤ t❤❛t ❛♥❞ t❤❡ ♠❛♣ ✐s ❞❡✜♥❡❞ ❜② ✳ ✶ ✶ P♦❧②❣r❛♣❤s ♠♦❞✉❧♦ ❆ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ ( R , E , S ) ♠❛❞❡ ♦❢ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t E k = R k ❢♦r k ≤ n − ✷ ❛♥❞ E n − ✶ ⊆ R n − ✶ ✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ◮ S ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ R ∗ n − ✶ s✉❝❤ t❤❛t R ⊆ S ⊆ E R E ✱

❛♥❞ t❤❡ ♠❛♣ ✐s ❞❡✜♥❡❞ ❜② ✳ ✶ ✶ � � � � � P♦❧②❣r❛♣❤s ♠♦❞✉❧♦ ❆ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ ( R , E , S ) ♠❛❞❡ ♦❢ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t E k = R k ❢♦r k ≤ n − ✷ ❛♥❞ E n − ✶ ⊆ R n − ✶ ✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ◮ S ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ R ∗ n − ✶ s✉❝❤ t❤❛t R ⊆ S ⊆ E R E ✱ ✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ E R E ✐s ❞❡✜♥❡❞ ❜② γ E R E : E R E → ❙♣❤ n − ✶ ( R ∗ n − ✶ ) ✇❤❡r❡ E R E ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s ( e , f , e ′ ) ✐♥ E ⊤ × R ∗ ( ✶ ) × E ⊤ s✉❝❤ t❤❛t e u � � v f e ′

� �� � � � P♦❧②❣r❛♣❤s ♠♦❞✉❧♦ ❆ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ ( R , E , S ) ♠❛❞❡ ♦❢ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n ✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t E k = R k ❢♦r k ≤ n − ✷ ❛♥❞ E n − ✶ ⊆ R n − ✶ ✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ◮ S ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ R ∗ n − ✶ s✉❝❤ t❤❛t R ⊆ S ⊆ E R E ✱ ✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ E R E ✐s ❞❡✜♥❡❞ ❜② γ E R E : E R E → ❙♣❤ n − ✶ ( R ∗ n − ✶ ) ✇❤❡r❡ E R E ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s ( e , f , e ′ ) ✐♥ E ⊤ × R ∗ ( ✶ ) × E ⊤ s✉❝❤ t❤❛t e u � � v f e ′ ❛♥❞ t❤❡ ♠❛♣ γ E R E ✐s ❞❡✜♥❡❞ ❜② γ E R E ( e , f , e ′ ) = ( ∂ − , n − ✶ ( e ) , ∂ + , n − ✶ ( e ′ )) ✳

✶ ✱ ■t ✐s ❧♦❝❛❧ ✐❢ ✐s ❛♥ ✲❝❡❧❧ ♦❢ ✐s ❛♥ ✲❝❡❧❧ ♦❢ ❛♥❞ ❛♥ ✲❝❡❧❧ ♦❢ s✉❝❤ t❤❛t ✶✳ ■t ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✐❢ t❤❡r❡ ❡①✐sts ✲❝❡❧❧s ✐♥ ❛♥❞ ✐♥ ✿ ✐s s❛✐❞ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✭r❡s♣✳ ❧♦❝❛❧❧② ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✮ ✐❢ ❛♥② ❜r❛♥❝❤✐♥❣ ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣✮ ♦❢ ♠♦❞✉❧♦ ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✳ � ❇r❛♥❝❤✐♥❣s ❛♥❞ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ♦❢ t❤❡ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ S ✐s ❛ tr✐♣❧❡ ( f , e , g ) ✇❤❡r❡ f ❛♥❞ g ❛r❡ n ✲❝❡❧❧s ♦❢ S ∗ ✇✐t❤ f ♥♦♥ tr✐✈✐❛❧ ❛♥❞ e ✐s ❛♥ n ✲❝❡❧❧ ♦❢ E ⊤ ✱ s✉❝❤ t❤❛t✿ f u ′ u e � g � v ′ v

■t ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✐❢ t❤❡r❡ ❡①✐sts ✲❝❡❧❧s ✐♥ ❛♥❞ ✐♥ ✿ ✐s s❛✐❞ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✭r❡s♣✳ ❧♦❝❛❧❧② ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✮ ✐❢ ❛♥② ❜r❛♥❝❤✐♥❣ ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣✮ ♦❢ ♠♦❞✉❧♦ ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✳ � ❇r❛♥❝❤✐♥❣s ❛♥❞ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ♦❢ t❤❡ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ S ✐s ❛ tr✐♣❧❡ ( f , e , g ) ✇❤❡r❡ f ❛♥❞ g ❛r❡ n ✲❝❡❧❧s ♦❢ S ∗ ✇✐t❤ f ♥♦♥ tr✐✈✐❛❧ ❛♥❞ e ✐s ❛♥ n ✲❝❡❧❧ ♦❢ E ⊤ ✱ s✉❝❤ t❤❛t✿ f u ′ u e � g � v ′ v ◮ ■t ✐s ❧♦❝❛❧ ✐❢ f ✐s ❛♥ n ✲❝❡❧❧ ♦❢ S ∗ ( ✶ ) ✱ g ✐s ❛♥ n ✲❝❡❧❧ ♦❢ S ∗ ❛♥❞ e ❛♥ n ✲❝❡❧❧ ♦❢ E ⊤ s✉❝❤ t❤❛t ℓ ( g ) + ℓ ( e ) = ✶✳

✐s s❛✐❞ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✭r❡s♣✳ ❧♦❝❛❧❧② ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✮ ✐❢ ❛♥② ❜r❛♥❝❤✐♥❣ ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣✮ ♦❢ ♠♦❞✉❧♦ ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✳ � � � ❇r❛♥❝❤✐♥❣s ❛♥❞ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ♦❢ t❤❡ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ S ✐s ❛ tr✐♣❧❡ ( f , e , g ) ✇❤❡r❡ f ❛♥❞ g ❛r❡ n ✲❝❡❧❧s ♦❢ S ∗ ✇✐t❤ f ♥♦♥ tr✐✈✐❛❧ ❛♥❞ e ✐s ❛♥ n ✲❝❡❧❧ ♦❢ E ⊤ ✱ s✉❝❤ t❤❛t✿ f u ′ u e � g � v ′ v ◮ ■t ✐s ❧♦❝❛❧ ✐❢ f ✐s ❛♥ n ✲❝❡❧❧ ♦❢ S ∗ ( ✶ ) ✱ g ✐s ❛♥ n ✲❝❡❧❧ ♦❢ S ∗ ❛♥❞ e ❛♥ n ✲❝❡❧❧ ♦❢ E ⊤ s✉❝❤ t❤❛t ℓ ( g ) + ℓ ( e ) = ✶✳ ◮ ■t ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✐❢ t❤❡r❡ ❡①✐sts n ✲❝❡❧❧s f ′ , g ′ ✐♥ S ∗ ❛♥❞ e ′ ✐♥ E ⊤ ✿ f ′ � w f u ′ u e ′ e � g � v ′ g ′ � w ′ v

� � � ❇r❛♥❝❤✐♥❣s ❛♥❞ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ♦❢ t❤❡ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ S ✐s ❛ tr✐♣❧❡ ( f , e , g ) ✇❤❡r❡ f ❛♥❞ g ❛r❡ n ✲❝❡❧❧s ♦❢ S ∗ ✇✐t❤ f ♥♦♥ tr✐✈✐❛❧ ❛♥❞ e ✐s ❛♥ n ✲❝❡❧❧ ♦❢ E ⊤ ✱ s✉❝❤ t❤❛t✿ f u ′ u e � g � v ′ v ◮ ■t ✐s ❧♦❝❛❧ ✐❢ f ✐s ❛♥ n ✲❝❡❧❧ ♦❢ S ∗ ( ✶ ) ✱ g ✐s ❛♥ n ✲❝❡❧❧ ♦❢ S ∗ ❛♥❞ e ❛♥ n ✲❝❡❧❧ ♦❢ E ⊤ s✉❝❤ t❤❛t ℓ ( g ) + ℓ ( e ) = ✶✳ ◮ ■t ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✐❢ t❤❡r❡ ❡①✐sts n ✲❝❡❧❧s f ′ , g ′ ✐♥ S ∗ ❛♥❞ e ′ ✐♥ E ⊤ ✿ f ′ � w f u ′ u e ′ e � g � v ′ g ′ � w ′ v ◮ S ✐s s❛✐❞ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ❜r❛♥❝❤✐♥❣ ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣✮ ♦❢ S ♠♦❞✉❧♦ E ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳

■❱✳ ❈♦❤❡r❡♥❝❡ ♠♦❞✉❧♦

❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ ✐s ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✐❢ t❤❡r❡ ❡①✐st ✲❝❡❧❧s ✐♥ ✱ ✐♥ ❛♥❞ ❛♥ ✶ ✲❝❡❧❧ ✐♥ P❡✐✛ ✿ ✐s t❤❡ ❢r❡❡ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② ✱ ✐♥ ✇❤✐❝❤ ❛❧❧ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛r❡ ✐♥✈❡rt✐❜❧❡✳ P❡✐✛ ✐s t❤❡ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ sq✉❛r❡s ❢♦r ❛❧❧ ✱ ❛♥❞ ✳ ❊ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿ ❛♥❞ ✶ ✶ ✶ ✶ ✶ ✷ ✶ ✷ ❈♦❤❡r❡♥t ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❲❡ ❝♦♥s✐❞❡r Γ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , S ∗ ) ✳

❛♥❞ ❛♥ ✶ ✲❝❡❧❧ ✐♥ P❡✐✛ ✿ ✐s t❤❡ ❢r❡❡ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② ✱ ✐♥ ✇❤✐❝❤ ❛❧❧ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛r❡ ✐♥✈❡rt✐❜❧❡✳ P❡✐✛ ✐s t❤❡ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ sq✉❛r❡s ❢♦r ❛❧❧ ✱ ❛♥❞ ✳ ❊ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿ ❛♥❞ ✶ ✶ ✶ ✶ ✶ ✷ ✶ ✷ � � ❈♦❤❡r❡♥t ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❲❡ ❝♦♥s✐❞❡r Γ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , S ∗ ) ✳ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✐❢ t❤❡r❡ ❡①✐st n ✲❝❡❧❧s f ′ , g ′ ✐♥ S ∗ ✱ e ′ ✐♥ E ⊤ f ′ � w f u ′ u e ′ e � g � v ′ g ′ � w ′ v

✐s t❤❡ ❢r❡❡ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② ✱ ✐♥ ✇❤✐❝❤ ❛❧❧ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛r❡ ✐♥✈❡rt✐❜❧❡✳ P❡✐✛ ✐s t❤❡ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ sq✉❛r❡s ❢♦r ❛❧❧ ✱ ❛♥❞ ✳ ❊ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿ ❛♥❞ ✶ ✶ ✶ ✶ ✶ ✷ ✶ ✷ � � � ❈♦❤❡r❡♥t ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❲❡ ❝♦♥s✐❞❡r Γ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , S ∗ ) ✳ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✐❢ t❤❡r❡ ❡①✐st n ✲❝❡❧❧s f ′ , g ′ ✐♥ S ∗ ✱ e ′ ✐♥ E ⊤ | = , v ✿ ❛♥❞ ❛♥ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( E , S , E ⋊ Γ ∪ P❡✐✛ ( E , S )) f ′ � f u ′ u w e ′ e � A g � v ′ g ′ � w ′ v

P❡✐✛ ✐s t❤❡ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ sq✉❛r❡s ❢♦r ❛❧❧ ✱ ❛♥❞ ✳ ❊ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿ ❛♥❞ ✶ ✶ ✶ ✶ ✶ ✷ ✶ ✷ � � � ❈♦❤❡r❡♥t ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❲❡ ❝♦♥s✐❞❡r Γ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , S ∗ ) ✳ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✐❢ t❤❡r❡ ❡①✐st n ✲❝❡❧❧s f ′ , g ′ ✐♥ S ∗ ✱ e ′ ✐♥ E ⊤ | = , v ✿ ❛♥❞ ❛♥ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( E , S , E ⋊ Γ ∪ P❡✐✛ ( E , S )) f ′ � f u ′ u w e ′ e � A g � v ′ g ′ � w ′ v | , v ✐s t❤❡ ❢r❡❡ n ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② ( E , S , − ) ✱ ✐♥ = ◮ ( E , S , − ) ✇❤✐❝❤ ❛❧❧ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛r❡ ✐♥✈❡rt✐❜❧❡✳

❊ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿ ❛♥❞ ✶ ✶ ✶ ✶ ✶ ✷ ✶ ✷ � � � � � ❈♦❤❡r❡♥t ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❲❡ ❝♦♥s✐❞❡r Γ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , S ∗ ) ✳ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✐❢ t❤❡r❡ ❡①✐st n ✲❝❡❧❧s f ′ , g ′ ✐♥ S ∗ ✱ e ′ ✐♥ E ⊤ | = , v ✿ ❛♥❞ ❛♥ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( E , S , E ⋊ Γ ∪ P❡✐✛ ( E , S )) f ′ � f u ′ u w e ′ e � A g � v ′ g ′ � w ′ v | , v ✐s t❤❡ ❢r❡❡ n ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② ( E , S , − ) ✱ ✐♥ = ◮ ( E , S , − ) ✇❤✐❝❤ ❛❧❧ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛r❡ ✐♥✈❡rt✐❜❧❡✳ ◮ P❡✐✛ ( E , S ) ✐s t❤❡ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ sq✉❛r❡s ❢♦r ❛❧❧ e ✱ e ′ ∈ E ⊤ ❛♥❞ f ∈ S ∗ ✳ f ⋆ i v � w ⋆ i f � u ′ ⋆ i v w ⋆ i u ′ u ⋆ i v w ⋆ i u u ⋆ i e � u ′ ⋆ i e e ′ ⋆ i u � e ′ ⋆ i u ′ f ⋆ i v ′ � u ′ ⋆ i v ′ w ′ ⋆ i u w ′ ⋆ i f � w ′ ⋆ i u ′ u ⋆ i v ′

� � � � � � � � � ❈♦❤❡r❡♥t ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦ ◮ ❲❡ ❝♦♥s✐❞❡r Γ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ ( E ⊤ , S ∗ ) ✳ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✐❢ t❤❡r❡ ❡①✐st n ✲❝❡❧❧s f ′ , g ′ ✐♥ S ∗ ✱ e ′ ✐♥ E ⊤ | = , v ✿ ❛♥❞ ❛♥ ( n + ✶ ) ✲❝❡❧❧ A ✐♥ ( E , S , E ⋊ Γ ∪ P❡✐✛ ( E , S )) f ′ � f u ′ u w e ′ e � A g � v ′ g ′ � w ′ v | , v ✐s t❤❡ ❢r❡❡ n ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② ( E , S , − ) ✱ ✐♥ = ◮ ( E , S , − ) ✇❤✐❝❤ ❛❧❧ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛r❡ ✐♥✈❡rt✐❜❧❡✳ ◮ P❡✐✛ ( E , S ) ✐s t❤❡ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ sq✉❛r❡s ❢♦r ❛❧❧ e ✱ e ′ ∈ E ⊤ ❛♥❞ f ∈ S ∗ ✳ f ⋆ i v � w ⋆ i f � u ′ ⋆ i v w ⋆ i u ′ u ⋆ i v w ⋆ i u u ⋆ i e � u ′ ⋆ i e e ′ ⋆ i u � e ′ ⋆ i u ′ f ⋆ i v ′ � u ′ ⋆ i v ′ w ′ ⋆ i u w ′ ⋆ i f � w ′ ⋆ i u ′ u ⋆ i v ′ ◮ ❊ ⋊ Γ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ Γ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✿ f ′ f ′ f f � v ′ � v ′ u v u v e � e ′ ❛♥❞ e ⋆ n − ✶ e ✶ � e ′ u ′ g = e ✶ g ✶ e ✷ � w � w ′ g ✶ e ✷ � w � w ′ u ✶ g ′ g ′

❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✐s ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ❀ ✐s ❧♦❝❛❧❧② ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ❀ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ✿ ✶ ✶ ❂ ✶ ❛✮ ✿ ❜✮ ✿ ✶ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ ❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳

✐s ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ❀ ✐s ❧♦❝❛❧❧② ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ❀ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ✿ ✶ ✶ ❂ ✶ ❛✮ ✿ ❜✮ ✿ ✶ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ ❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿

✐s ❧♦❝❛❧❧② ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ❀ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ✿ ✶ ✶ ❂ ✶ ❛✮ ✿ ❜✮ ✿ ✶ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ ❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀

s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ✿ ✶ ✶ ❂ ✶ ❛✮ ✿ ❜✮ ✿ ✶ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ ❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S ✐s ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀

s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ ❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ � � � � � ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S ✐s ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ✿ S ∗ � v ′ S ∗ � v ′ S ∗ ( ✶ ) � S ∗ ( ✶ ) � u v u v E ⊤ ( ✶ ) � E ⊤ ❂ E ⊤ A B ❛✮ ✿ ❜✮ ✿ S ∗ � w ′ R ∗ ( ✶ ) � w u ′ � w u S ∗ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳

❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ � � � � � ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S ✐s ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ✿ S ∗ � v ′ S ∗ � v ′ S ∗ ( ✶ ) � S ∗ ( ✶ ) � u v u v E ⊤ ( ✶ ) � E ⊤ ❂ E ⊤ A B ❛✮ ✿ ❜✮ ✿ S ∗ � w ′ R ∗ ( ✶ ) � w u ′ � w u S ∗ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳

� � � � � ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S ✐s ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ✿ S ∗ � v ′ S ∗ � v ′ S ∗ ( ✶ ) � S ∗ ( ✶ ) � u v u v E ⊤ ( ✶ ) � E ⊤ ❂ E ⊤ A B ❛✮ ✿ ❜✮ ✿ S ∗ � w ′ R ∗ ( ✶ ) � w u ′ � w u S ∗ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ ❋♦r S = E R ✱ ♣r♦♣❡rt② ❜ ) ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳

� � � � � ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S ✐s ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ✿ S ∗ � v ′ S ∗ � v ′ S ∗ ( ✶ ) � S ∗ ( ✶ ) � u v u v E ⊤ ( ✶ ) � E ⊤ E ⊤ ❂ A B ❛✮ ✿ ❜✮ ✿ R ∗ ( ✶ ) � w S ∗ � w ′ u ′ � w u S ∗ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ f � ◮ ❋♦r S = E R ✱ ♣r♦♣❡rt② ❜ ) ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ u v e � v ′

� � � � � ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S ✐s ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ✿ S ∗ � v ′ S ∗ � v ′ S ∗ ( ✶ ) � S ∗ ( ✶ ) � u v u v E ⊤ ( ✶ ) � E ⊤ ❂ E ⊤ A B ❛✮ ✿ ❜✮ ✿ S ∗ � w ′ u ′ R ∗ ( ✶ ) � w � w u S ∗ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ f � ◮ ❋♦r S = E R ✱ ♣r♦♣❡rt② ❜ ) ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ u v e � v ′ � v e − · f

� � � � � � ❈♦❤❡r❡♥t ◆❡✇♠❛♥ ❛♥❞ ❝r✐t✐❝❛❧ ♣❛✐r ❧❡♠♠❛s ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✮ ✐❢ ❛♥② ♦❢ ✐ts ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♠♦❞✉❧♦ E ✮ ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✳ ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ■❢ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S ✐s ❧♦❝❛❧❧② Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ❀ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ✿ S ∗ � v ′ S ∗ � v ′ S ∗ ( ✶ ) � S ∗ ( ✶ ) � u v u v E ⊤ ( ✶ ) � E ⊤ ❂ E ⊤ A B ❛✮ ✿ ❜✮ ✿ S ∗ � w ′ u ′ R ∗ ( ✶ ) � w � w u S ∗ ❢♦r ❛♥② ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ) ❛♥❞ ❜ ) ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E ✳ f � ◮ ❋♦r S = E R ✱ ♣r♦♣❡rt② ❜ ) ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ u v e � ❂ v ′ � v e − · f

❈♦❤❡r❡♥❝❡ ♠♦❞✉❧♦ ◮ ❆ s❡t X ♦❢ ( n − ✶ ) ✲❝❡❧❧s ✐♥ R ∗ n − ✶ ✐s E ✲♥♦r♠❛❧✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ S ✐❢ ❢♦r ❛♥② u ✐♥ X ✱✯ ◆❋ ( S , u ) ∩ ■rr ( E ) � = ∅ . ◮ ❚❤❡♦r❡♠✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ▲❡t ( R , E , S ) ❜❡ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦✱ ❛♥❞ Γ ❜❡ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣❛✐r ♦❢ ( n + ✶ , n ) ✲❝❛t❡❣♦r✐❡s ( E ⊤ , S ⊤ ) s✉❝❤ t❤❛t ◮ E ✐s ❝♦♥✈❡r❣❡♥t✱ ◮ S ✐s Γ ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✱ ◮ ■rr ( E ) ✐s E ✲♥♦r♠❛❧✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ S ✱ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ ◮ t❤❡♥ Γ ∪ ❈❞ ( E ) ✐s ❛❝②❝❧✐❝✳

❈♦r♦❧❧❛r②✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ▲❡t ❜❡ ❛♥ ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ s✉❝❤ t❤❛t ✐s ❝♦♥✈❡r❣❡♥t✱ ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✱ ✐s ✲♥♦r♠❛❧✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ✱ ■rr ✐s t❡r♠✐♥❛t✐♥❣✱ ❚❤❡♥ ✐s ❛❝②❝❧✐❝✳ ❈❞ ❈♦r♦❧❧❛r②✿ ❯s✉❛❧ ❙q✉✐❡r✬s t❤❡♦r❡♠✳ ✭ ✮ � � � � � � � ❈♦❤❡r❡♥t ❡①t❡♥s✐♦♥s ◮ ❆ ❝♦❤❡r❡♥t ❝♦♠♣❧❡t✐♦♥ ♠♦❞✉❧♦ E ♦❢ S ✐s ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❞❡♥♦t❡❞ ❜② C ( S ) ♦❢ t❤❡ ♣❛✐r ♦❢ ( n + ✶ , n ) ✲❝❛t❡❣♦r✐❡s ( E ⊤ , S ⊤ ) ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡ ❝❡❧❧s A f , g ❛♥❞ B f , e ✿ f ′ � f ′ � f f u ′ u ′ u w u w e ′ e ′ ❂ A f , g e � B f , e g � v g ′ � w ′ � w ′ u v g ′ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s ( f , g ) ❛♥❞ ( f , e ) ♦❢ S ♠♦❞✉❧♦ E ✳

❈♦r♦❧❧❛r②✿ ❯s✉❛❧ ❙q✉✐❡r✬s t❤❡♦r❡♠✳ ✭ ✮ � � � � � � � ❈♦❤❡r❡♥t ❡①t❡♥s✐♦♥s ◮ ❆ ❝♦❤❡r❡♥t ❝♦♠♣❧❡t✐♦♥ ♠♦❞✉❧♦ E ♦❢ S ✐s ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❞❡♥♦t❡❞ ❜② C ( S ) ♦❢ t❤❡ ♣❛✐r ♦❢ ( n + ✶ , n ) ✲❝❛t❡❣♦r✐❡s ( E ⊤ , S ⊤ ) ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡ ❝❡❧❧s A f , g ❛♥❞ B f , e ✿ f ′ � f ′ � f f u ′ u ′ u w u w e ′ e ′ ❂ A f , g e � B f , e g � v g ′ � w ′ � w ′ u v g ′ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s ( f , g ) ❛♥❞ ( f , e ) ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ ❈♦r♦❧❧❛r②✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ▲❡t ( R , E , S ) ❜❡ ❛♥ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ s✉❝❤ t❤❛t ◮ E ✐s ❝♦♥✈❡r❣❡♥t✱ ◮ S ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✱ ◮ ■rr ( E ) ✐s E ✲♥♦r♠❛❧✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ S ✱ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ ◮ ❚❤❡♥ C ( S ) ∪ ❈❞ ( E ) ✐s ❛❝②❝❧✐❝✳

� � � � � � � ❈♦❤❡r❡♥t ❡①t❡♥s✐♦♥s ◮ ❆ ❝♦❤❡r❡♥t ❝♦♠♣❧❡t✐♦♥ ♠♦❞✉❧♦ E ♦❢ S ✐s ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❞❡♥♦t❡❞ ❜② C ( S ) ♦❢ t❤❡ ♣❛✐r ♦❢ ( n + ✶ , n ) ✲❝❛t❡❣♦r✐❡s ( E ⊤ , S ⊤ ) ❝♦♥t❛✐♥✐♥❣ sq✉❛r❡ ❝❡❧❧s A f , g ❛♥❞ B f , e ✿ f ′ � f ′ � f f u ′ u ′ u w u w e ′ e ′ ❂ A f , g e � B f , e g � v g ′ � w ′ � w ′ u v g ′ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s ( f , g ) ❛♥❞ ( f , e ) ♦❢ S ♠♦❞✉❧♦ E ✳ ◮ ❈♦r♦❧❧❛r②✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ▲❡t ( R , E , S ) ❜❡ ❛♥ n ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ s✉❝❤ t❤❛t ◮ E ✐s ❝♦♥✈❡r❣❡♥t✱ ◮ S ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✱ ◮ ■rr ( E ) ✐s E ✲♥♦r♠❛❧✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ S ✱ E R E ✐s t❡r♠✐♥❛t✐♥❣✱ ◮ ❚❤❡♥ C ( S ) ∪ ❈❞ ( E ) ✐s ❛❝②❝❧✐❝✳ ◮ ❈♦r♦❧❧❛r②✿ ❯s✉❛❧ ❙q✉✐❡r✬s t❤❡♦r❡♠✳ ✭ E = ∅ ✮

✱ ✵ ✵ ✱ ✶ ✶ ✷ ✷ ✷ ❢♦r ✐♥ ✵ ✶ ✸ ✸ ❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿

✱ ✶ ✶ ✷ ✷ ✷ ❢♦r ✐♥ ✵ ✶ ✸ ✸ ❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱

✷ ✷ ✷ ❢♦r ✐♥ ✵ ✶ ✸ ✸ ❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱ ◮ E ✶ = R ✶ = {∧ , ∨} ✱

✷ ✷ ❢♦r ✐♥ ✵ ✶ ✸ ✸ ❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱ ◮ E ✶ = R ✶ = {∧ , ∨} ✱       ◮ E ✷ = • , , , , • ,    

✷ ✷ ✸ ❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱ ◮ E ✶ = R ✶ = {∧ , ∨} ✱       ◮ E ✷ = • , , , , • ,        ⇛ • µ , ⇛ • µ , ⇛ • µ ❢♦r µ ✐♥ { ✵ , ✶ } , ◮ E ✸ = • • ⇛ • µ , • µ µ µ µ •   � • • , • • • • , • • , ⇛ ⇛ ⇛ ⇛

✸ ❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱ ◮ E ✶ = R ✶ = {∧ , ∨} ✱             ◮ E ✷ = • , , , , • , R ✷ = E ✷ � ,            ⇛ • µ , ⇛ • µ , ⇛ • µ ❢♦r µ ✐♥ { ✵ , ✶ } , ◮ E ✸ = • • ⇛ • µ , • µ µ µ µ •   � • • , • • • • , • • , ⇛ ⇛ ⇛ ⇛

❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱ ◮ E ✶ = R ✶ = {∧ , ∨} ✱             ◮ E ✷ = • , , , , • , R ✷ = E ✷ � ,            ⇛ • µ , ⇛ • µ , ⇛ • µ ❢♦r µ ✐♥ { ✵ , ✶ } , ◮ E ✸ = • • ⇛ • µ , • µ µ µ µ •   � • • , • • • • , • • , ⇛ ⇛ ⇛ ⇛   α − β −  α + β +  γ   ◮ R ✸ = ⇛ ⇛ ⇛ ⇛ ⇛    

❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳ ❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱ ◮ E ✶ = R ✶ = {∧ , ∨} ✱             ◮ E ✷ = • , , , , • , R ✷ = E ✷ � ,            ⇛ • µ , ⇛ • µ , ⇛ • µ ❢♦r µ ✐♥ { ✵ , ✶ } , ◮ E ✸ = • • ⇛ • µ , • µ µ µ µ •   � • • , • • • • , • • , ⇛ ⇛ ⇛ ⇛   α − β −  α + β +  γ   ◮ R ✸ = ⇛ ⇛ ⇛ ⇛ ⇛    

❚♦② ❡①❛♠♣❧❡✿ ❉✐❛❣r❛♠♠❛t✐❝ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦ ✐s♦t♦♣② ◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E ✵ = R ✵ = {∗} ✱ ◮ E ✶ = R ✶ = {∧ , ∨} ✱             ◮ E ✷ = • , , , , • , R ✷ = E ✷ � ,            ⇛ • µ , ⇛ • µ , ⇛ • µ ❢♦r µ ✐♥ { ✵ , ✶ } , ◮ E ✸ = • • ⇛ • µ , • µ µ µ µ •   � • • , • • • • , • • , ⇛ ⇛ ⇛ ⇛   α − β −  α + β +  γ   ◮ R ✸ = ⇛ ⇛ ⇛ ⇛ ⇛     ◮ ❋❛❝t✿ E ✐s ❝♦♥✈❡r❣❡♥t✳