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◮ ❆❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣ ❂ ❛♣♣❧②✐♥❣ r❡✇r✐t✐♥❣ ♠❡t❤♦❞s t♦ st✉❞② ✐♥tr✐♥s❡q✉❡ ♣r♦♣❡rt✐❡s ♦❢
❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ s②③②❣✐❡s ✭r❡❧❛t✐♦♥s ❛♠♦♥❣ r❡❧❛t✐♦♥s✮ ❊①❡♠♣❧❡✳ ❋♦r t❤❡ ❣r♦✉♣
✸
✶ ✶ ✶ ✱ t❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t② ✶ ✐s s✉❝❤ ❛ s②③②❣②✱ ✇✐t❤ ❛♥❞ ✳ ❋♦r ♠♦♥♦✐❞s ♦r ❝❛t❡❣♦r✐❡s✱ ❙q✉✐❡r✬s t❤❡♦r❡♠ ❣✐✈❡s ❛ ❣❡♥❡r❛t✐♥❣ ❢❛♠✐❧② ❢♦r s②③②❣✐❡s ❢r♦♠ ❛ ✜♥✐t❡ ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥✱ ●✉✐r❛✉❞✲▼❛❧❜♦s ✬✵✾✱ ●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼❛❧❜♦s ✬✶✹✳ ■❢ ❛ ❣r♦✉♣ ✐s ♣r❡s❡♥t❡❞ ❛s ❛ ♠♦♥♦✐❞ ✶ ✶ ✱ t❤❡ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠
✶
✐s ❛♥ ❛rt❡❢❛❝t ✐♥❞✉❝❡❞ ❜② t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s②③②❣②✳
◮ ❆❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣ ❂ ❛♣♣❧②✐♥❣ r❡✇r✐t✐♥❣ ♠❡t❤♦❞s t♦ st✉❞② ✐♥tr✐♥s❡q✉❡ ♣r♦♣❡rt✐❡s ♦❢
❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s✳
◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ s②③②❣✐❡s ✭r❡❧❛t✐♦♥s ❛♠♦♥❣ r❡❧❛t✐♦♥s✮
❊①❡♠♣❧❡✳ ❋♦r t❤❡ ❣r♦✉♣ Z✸ = x, y, z | [x, y] = ✶, [y, z] = ✶, [z, x] = ✶✱ t❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t② [xy, [y, z]][yz, [z, x]][zx, [x, y]] = ✶ ✐s s✉❝❤ ❛ s②③②❣②✱ ✇✐t❤ [x, y] = xyx−y− ❛♥❞ xy = y−xy✳ ❋♦r ♠♦♥♦✐❞s ♦r ❝❛t❡❣♦r✐❡s✱ ❙q✉✐❡r✬s t❤❡♦r❡♠ ❣✐✈❡s ❛ ❣❡♥❡r❛t✐♥❣ ❢❛♠✐❧② ❢♦r s②③②❣✐❡s ❢r♦♠ ❛ ✜♥✐t❡ ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥✱ ●✉✐r❛✉❞✲▼❛❧❜♦s ✬✵✾✱ ●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼❛❧❜♦s ✬✶✹✳ ■❢ ❛ ❣r♦✉♣ ✐s ♣r❡s❡♥t❡❞ ❛s ❛ ♠♦♥♦✐❞ ✶ ✶ ✱ t❤❡ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠
✶
✐s ❛♥ ❛rt❡❢❛❝t ✐♥❞✉❝❡❞ ❜② t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s②③②❣②✳
◮ ❆❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣ ❂ ❛♣♣❧②✐♥❣ r❡✇r✐t✐♥❣ ♠❡t❤♦❞s t♦ st✉❞② ✐♥tr✐♥s❡q✉❡ ♣r♦♣❡rt✐❡s ♦❢
❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s✳
◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ s②③②❣✐❡s ✭r❡❧❛t✐♦♥s ❛♠♦♥❣ r❡❧❛t✐♦♥s✮
❊①❡♠♣❧❡✳ ❋♦r t❤❡ ❣r♦✉♣ Z✸ = x, y, z | [x, y] = ✶, [y, z] = ✶, [z, x] = ✶✱ t❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t② [xy, [y, z]][yz, [z, x]][zx, [x, y]] = ✶ ✐s s✉❝❤ ❛ s②③②❣②✱ ✇✐t❤ [x, y] = xyx−y− ❛♥❞ xy = y−xy✳
◮ ❋♦r ♠♦♥♦✐❞s ♦r ❝❛t❡❣♦r✐❡s✱ ❙q✉✐❡r✬s t❤❡♦r❡♠ ❣✐✈❡s ❛ ❣❡♥❡r❛t✐♥❣ ❢❛♠✐❧② ❢♦r s②③②❣✐❡s ❢r♦♠ ❛
✜♥✐t❡ ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥✱ ●✉✐r❛✉❞✲▼❛❧❜♦s ✬✵✾✱ ●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼❛❧❜♦s ✬✶✹✳ ■❢ ❛ ❣r♦✉♣ ✐s ♣r❡s❡♥t❡❞ ❛s ❛ ♠♦♥♦✐❞ ✶ ✶ ✱ t❤❡ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠
✶
✐s ❛♥ ❛rt❡❢❛❝t ✐♥❞✉❝❡❞ ❜② t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s②③②❣②✳
◮ ❆❧❣❡❜r❛✐❝ r❡✇r✐t✐♥❣ ❂ ❛♣♣❧②✐♥❣ r❡✇r✐t✐♥❣ ♠❡t❤♦❞s t♦ st✉❞② ✐♥tr✐♥s❡q✉❡ ♣r♦♣❡rt✐❡s ♦❢
❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s✳
◮ ❈♦♠♣✉t❛t✐♦♥ ♦❢ s②③②❣✐❡s ✭r❡❧❛t✐♦♥s ❛♠♦♥❣ r❡❧❛t✐♦♥s✮
❊①❡♠♣❧❡✳ ❋♦r t❤❡ ❣r♦✉♣ Z✸ = x, y, z | [x, y] = ✶, [y, z] = ✶, [z, x] = ✶✱ t❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t② [xy, [y, z]][yz, [z, x]][zx, [x, y]] = ✶ ✐s s✉❝❤ ❛ s②③②❣②✱ ✇✐t❤ [x, y] = xyx−y− ❛♥❞ xy = y−xy✳
◮ ❋♦r ♠♦♥♦✐❞s ♦r ❝❛t❡❣♦r✐❡s✱ ❙q✉✐❡r✬s t❤❡♦r❡♠ ❣✐✈❡s ❛ ❣❡♥❡r❛t✐♥❣ ❢❛♠✐❧② ❢♦r s②③②❣✐❡s ❢r♦♠ ❛
✜♥✐t❡ ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥✱ ●✉✐r❛✉❞✲▼❛❧❜♦s ✬✵✾✱ ●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼❛❧❜♦s ✬✶✹✳
◮ ■❢ ❛ ❣r♦✉♣ G = X | R ✐s ♣r❡s❡♥t❡❞ ❛s ❛ ♠♦♥♦✐❞ M = X X | R ∪ {xx− αx
⇒ ✶, x−x
αx
⇒ ✶}✱ t❤❡ ❝♦♥✢✉❡♥❝❡ ❞✐❛❣r❛♠ x xx−✶x
αx x
✐s ❛♥ ❛rt❡❢❛❝t ✐♥❞✉❝❡❞ ❜② t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s②③②❣②✳
◮ ❖❜❥❡❝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✉❞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✉❞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✉❞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✉❞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✉❞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 = =
❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳
◮ ❖❜❥❡❝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 = =
◮ ❆❧❣❡❜r❛✐❝ ❛①✐♦♠s ❛r❡ ♥♦t r❡✇r✐t✐♥❣ r✉❧❡s✱ ❜✉t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ r❡✇r✐t✐♥❣✳
◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s ✳ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
❘❡✇r✐t✐♥❣ ✇✐t❤ ♠♦❞✉❧♦ ✱ ❍✉❡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✐t✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E✳
✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
❘❡✇r✐t✐♥❣ ✇✐t❤ ♠♦❞✉❧♦ ✱ ❍✉❡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✐t✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡✇r✐t✐♥❣ s②st❡♠ R ❛♥❞ ❛ s❡t ♦❢ ❡q✉❛t✐♦♥s E✳ ◮ ✸ ♣❛r❛❞✐❣♠s ♦❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
❘❡✇r✐t✐♥❣ ✇✐t❤ ♠♦❞✉❧♦ ✱ ❍✉❡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✐t✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ 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 ✬✽✵
u
R E
u′
R w E
R v ′ R w ′
❈✉❜✐❝❛❧ ✿ ❘❡✇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✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ 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 ✬✽✵
u
R E
u′
R w E
R v ′ R w ′
❈✉❜✐❝❛❧ ✿ ❘❡✇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✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ 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 ✬✽✵
u
R E
u′
R w E
R v ′ R w ′
❈✉❜✐❝❛❧
◮ E RE ✿ ❘❡✇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✐t✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ 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 ✬✽✵
u
R E
u′
R w E
R v ′ R w ′
❈✉❜✐❝❛❧
◮ E RE ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♦♥ E✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s
u
E RE E
v
E
R v ′
❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ s✉❝❤ t❤❛t ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳
▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ ❂
◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ 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 ✬✽✵
u
R E
u′
R w E
R v ′ R w ′
❈✉❜✐❝❛❧
◮ E RE ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♦♥ E✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s
u
E RE E
v
E
R v ′ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ S s✉❝❤ t❤❛t R ⊆ S ⊆ E RE ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳
▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r ✳ ❂
◮ ❘❡✇r✐t✐♥❣ s②st❡♠ R✿ ◮ ❈♦❤❡r❡♥❝❡ r❡s✉❧ts ✐♥ n✲❝❛t❡❣♦r✐❡s✳
◮ ■♥ 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 ✬✽✵
u
R E
u′
R w E
R v ′ R w ′
❈✉❜✐❝❛❧
◮ E RE ✿ ❘❡✇r✐t✐♥❣ ✇✐t❤ R ♦♥ E✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s
u
E RE E
v
E
R v ′ ◮ ❘❡✇r✐t✐♥❣ ✇✐t❤ ❛♥② s②st❡♠ S s✉❝❤ t❤❛t R ⊆ S ⊆ E RE ✱ ❏♦✉❛♥♥❛✉❞ ✲ ❑✐r❝❤♥❡r ✬✽✹✳ ◮ ▼❛✐♥ ✐♥t❡r❡st ❛♥❞ r❡s✉❧ts ❢♦r E R✳
u
E R
E
v ❂
R v
◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡
❣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
✶ ✷ ✶ ✷
◮ ❲❡ ✐♥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
✶ ✷ ✶ ✷
◮ ❲❡ ✐♥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
✶ ✷ ✶ ✷
◮ ❲❡ ✐♥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
✶ ✷ ✶ ✷
◮ ❲❡ ✐♥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
✶ ✷ ✶ ✷
◮ ❲❡ ✐♥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
✶ ✷ ✶ ✷
◮ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❝✉❜✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❝♦❤❡r❡♥❝❡✱ r❡❧❛t❡❞ t♦ n✲❝❛t❡❣♦r✐❡s ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡
❣r♦✉♣♦✐❞s✳
◮ ❆ ❞♦✉❜❧❡ ❝❛t❡❣♦r② ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② (❈✶, ❈✵, ∂❈
−, ∂❈ +, ◦❈, i❈) ✐♥ ❈❛t✳
❊❤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♠❛♥♥ ✬✻✹ (❈✵)✵
(❈✶)✵ (❈✵)✶
(❈✵)✵
(❈✵)✶
(❈✶)✶
x✶
f
x✶
e
◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s
·
∂h
−,✶(A)
∂v
−,✶(A)
∂v
+,✶(A)
∂h
+,✶(A)
A
✶
✵ ✶
✷
✵ ✷
✶ ✷
✶ ✵ ✵ ✶
❈♦♠♣♦s✐t✐♦♥s
✶
✶ ✶
✷
✷ ✷
✸
✸
✶
✶
✷
✷
✸ ✶
✶ ✷ ✶
✸
✸
✶
✶ ✷
✸ ✶
✶ ✶
✷
✷
✶
✷ ✶
✷
✷
✶
✸
✷ ✶
✶ ✶ ✶
✷
✷ ✷
✶
✸
✷
❢♦r ❛❧❧ ✱ ✱ ♣♦✐♥t ❝❡❧❧s✱ ✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s✱ ✱ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ ✱ ✱ sq✉❛r❡ ❝❡❧❧s✳
◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s
·
∂h
−,✶(A)
∂v
−,✶(A)
∂v
+,✶(A)
∂h
+,✶(A)
A
x✶
f
✵(x✶)
iv
✵(x✷)
f
ih
✶(f )
ih
✵(x)
e
e
ih
✵(y)
iv
✶(e)
✶
✶ ✶
✷
✷ ✷
✸
✸
✶
✶
✷
✷
✸ ✶
✶ ✷ ✶
✸
✸
✶
✶ ✷
✸ ✶
✶ ✶
✷
✷
✶
✷ ✶
✷
✷
✶
✸
✷ ✶
✶ ✶ ✶
✷
✷ ✷
✶
✸
✷
❢♦r ❛❧❧ ✱ ✱ ♣♦✐♥t ❝❡❧❧s✱ ✱ ❤♦r✐③♦♥t❛❧ ❝❡❧❧s✱ ✱ ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ ✱ ✱ sq✉❛r❡ ❝❡❧❧s✳
◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s
·
∂h
−,✶(A)
∂v
−,✶(A)
∂v
+,✶(A)
∂h
+,✶(A)
A
x✶
f
✵(x✶)
iv
✵(x✷)
f
ih
✶(f )
ih
✵(x)
e
e
ih
✵(y)
iv
✶(e)
x✶
f✶
e✷
e✸
g✶
A
B
f✶f✷
e✸
g✶g✷
A⋄v B
✶ ✶
✷
✷
✶
✷ ✶
✷
✷
✶
✸
✷ ✶
✶ ✶ ✶
✷
✷ ✷
✶
✸
✷
❢♦r ❛❧❧ xi✱yi✱zi ♣♦✐♥t ❝❡❧❧s✱ fi✱ gi ❤♦r✐③♦♥t❛❧ ❝❡❧❧s✱ ei✱e′
i ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ A✱ A′✱B sq✉❛r❡ ❝❡❧❧s✳
◮ ❚❤❡r❡ ❛r❡ sq✉❛r❡ ❝❡❧❧s
·
∂h
−,✶(A)
∂v
−,✶(A)
∂v
+,✶(A)
∂h
+,✶(A)
A
x✶
f
✵(x✶)
iv
✵(x✷)
f
ih
✶(f )
ih
✵(x)
e
e
ih
✵(y)
iv
✶(e)
x✶
f✶
e✷
e✸
g✶
A
B
f✶f✷
e✸
g✶g✷
A⋄v B
f✶
e✷
f✷
✶
y✷
e′
✷
f✸
A′
f✶
✶
e✷e′
✷
f✸
A⋄hA′
i ✈❡rt✐❝❛❧ ❝❡❧❧s ❛♥❞ A✱ A′✱B sq✉❛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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
✷ ✷
✸
✸
✷
✷
✸ ✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸ ✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸ ✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸
❉♦✉❜❧❡ ❣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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
✷ ✷
✸
✸
✷
✷
✸
⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
✷ ✷
✸
✸
✷
✷
✸ ✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸ ✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸
❉♦✉❜❧❡ ❣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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
f✷
x✸
e✸
g✷
B
⋄v ⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
g✷
✷
y✸
e′
✸
h✷
B′
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸ ✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸
❉♦✉❜❧❡ ❣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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
f✷
x✸
e✸
g✷
B
⋄v ⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
g✷
✷
y✸
e′
✸
h✷
B′
✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸ ✶
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸
❉♦✉❜❧❡ ❣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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
f✷
x✸
e✸
g✷
B
⋄v ⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
g✷
✷
y✸
e′
✸
h✷
B′
x✶
f✶
x✷
e✷
g✶
A
x✷
f✷
x✸
e✸
g✷
B
✶ ✶
✷
✷
✶
✶
✷ ✷
✷ ✷
✸
✸
✷
✷
✸
❉♦✉❜❧❡ ❣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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
f✷
x✸
e✸
g✷
B
⋄v ⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
g✷
✷
y✸
e′
✸
h✷
B′
x✶
f✶
x✷
e✷
g✶
A
x✷
f✷
x✸
e✸
g✷
B
y✶
g✶
✶
y✷
e′
✷
h✶
A′
y✷
g✷
✷
y✸
e′
✸
h✷
B′
❈ ❈ ❈ ❈
✐♥ ✇❤✐❝❤ ❈✶ ❛♥❞ ❈✵ ❛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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
f✷
x✸
e✸
g✷
B
⋄v ⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
g✷
✷
y✸
e′
✸
h✷
B′
x✶
f✶
x✷
e✷
g✶
A
x✷
f✷
x✸
e✸
g✷
B
y✶
g✶
✶
y✷
e′
✷
h✶
A′
y✷
g✷
✷
y✸
e′
✸
h✷
B′
−, ∂❈ +, ◦❈, i❈) ✐♥ ✇❤✐❝❤ ❈✶ ❛♥❞ ❈✵ ❛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✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
f✷
x✸
e✸
g✷
B
⋄v ⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
g✷
✷
y✸
e′
✸
h✷
B′
x✶
f✶
x✷
e✷
g✶
A
x✷
f✷
x✸
e✸
g✷
B
y✶
g✶
✶
y✷
e′
✷
h✶
A′
y✷
g✷
✷
y✸
e′
✸
h✷
B′
−, ∂❈ +, ◦❈, i❈) ✐♥ ✇❤✐❝❤ ❈✶ ❛♥❞ ❈✵ ❛r❡
❣r♦✉♣♦✐❞s✳
◮ n✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ n✲❝❛t❡❣♦r② C s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t Cn(x, y) ✐s ❛
❞♦✉❜❧❡ ❣r♦✉♣♦✐❞✳ ❍♦r✐③♦♥t❛❧ ✶ ✲❝❛t❡❣♦r② ✇✐❧❧ ❜❡ t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ r❡✇r✐t✐♥❣s❀ ✈❡rt✐❝❛❧ ✶ ✲❝❛t❡❣♦r② ✐s t❤❡ ✶ ✲❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧♦ r✉❧❡s✳
◮ ❚❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s s❛t✐s❢② t❤❡ ♠✐❞❞❧❡ ❢♦✉r ✐♥t❡r❝❤❛♥❣❡ ❧❛✇✿
x✶
f✶
x✷
e✷
g✶
A
f✷
x✸
e✸
g✷
B
⋄v ⋄h y✶
g✶
✶
y✷
e′
✷
h✶
A′
g✷
✷
y✸
e′
✸
h✷
B′
x✶
f✶
x✷
e✷
g✶
A
x✷
f✷
x✸
e✸
g✷
B
y✶
g✶
✶
y✷
e′
✷
h✶
A′
y✷
g✷
✷
y✸
e′
✸
h✷
B′
−, ∂❈ +, ◦❈, i❈) ✐♥ ✇❤✐❝❤ ❈✶ ❛♥❞ ❈✵ ❛r❡
❣r♦✉♣♦✐❞s✳
◮ n✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s ❂ n✲❝❛t❡❣♦r② C s✉❝❤ t❤❛t ❛♥② ❤♦♠s❡t Cn(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✳
◮ ❆ ❞♦✉❜❧❡ n✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ (Pv, Ph, Ps) ♠❛❞❡ ♦❢✿
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✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ (Pv, Ph, Ps) ♠❛❞❡ ♦❢✿ ◮ t✇♦ (n + ✶)✲♣♦❧②❣r❛♣❤s Pv ❛♥❞ Ph s✉❝❤ t❤❛t Pv k = Ph k ❢♦r k ≤ n✱
❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣❛✐r ♦❢ ✶ ✲❝❛t❡❣♦r✐❡s ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ✱ ✇✐t❤ ✱ ♠❛❦✐♥❣ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ Pv
n+✶
n
Ph
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✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ (Pv, Ph, Ps) ♠❛❞❡ ♦❢✿ ◮ t✇♦ (n + ✶)✲♣♦❧②❣r❛♣❤s Pv ❛♥❞ Ph s✉❝❤ t❤❛t Pv k = Ph k ❢♦r k ≤ n✱
❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣❛✐r ♦❢ ✶ ✲❝❛t❡❣♦r✐❡s ✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ♠❛♣s ✱ ✇✐t❤ ✱ ♠❛❦✐♥❣ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳ (Pv
n+✶)∗ ∂v −,n
+,n
n+✶)∗ ∂h +,n
−,n
n+✶
n
Ph
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✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ (Pv, Ph, Ps) ♠❛❞❡ ♦❢✿ ◮ t✇♦ (n + ✶)✲♣♦❧②❣r❛♣❤s Pv ❛♥❞ Ph s✉❝❤ t❤❛t Pv k = Ph k ❢♦r k ≤ n✱ ◮ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ t❤❡ ♣❛✐r ♦❢ (n + ✶)✲❝❛t❡❣♦r✐❡s ((Pv)∗, (Ph)∗)✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞
✇✐t❤ ❢♦✉r ♠❛♣s ∂µ
±,n✱ ✇✐t❤ µ ∈ {v, h}✱ ♠❛❦✐♥❣ Γ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳
Ps
∂h +,n+✶
−,n+✶
−,n+✶
+,n+✶
n+✶)∗ ∂v −,n
+,n
n+✶)∗ ∂h +,n
−,n
n+✶
n
Ph
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✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ (Pv, Ph, Ps) ♠❛❞❡ ♦❢✿ ◮ t✇♦ (n + ✶)✲♣♦❧②❣r❛♣❤s Pv ❛♥❞ Ph s✉❝❤ t❤❛t Pv k = Ph k ❢♦r k ≤ n✱ ◮ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ t❤❡ ♣❛✐r ♦❢ (n + ✶)✲❝❛t❡❣♦r✐❡s ((Pv)∗, (Ph)∗)✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞
✇✐t❤ ❢♦✉r ♠❛♣s ∂µ
±,n✱ ✇✐t❤ µ ∈ {v, h}✱ ♠❛❦✐♥❣ Γ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳
Ps
∂h +,n+✶
−,n+✶
−,n+✶
+,n+✶
n+✶)∗ ∂v −,n
+,n
n+✶)∗ ∂h +,n
−,n
n+✶
n
Ph
n+✶
♦♥ ((Pv)⊤, (Ph)⊤)✳ ❆ ❞♦✉❜❧❡ ✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ ✷ ✲♣♦❧②❣r❛♣❤✮ ❣❡♥❡r❛t❡s ❛ ❢r❡❡ ✶ ✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② ✳
◮ ❆ ❞♦✉❜❧❡ n✲♣♦❧②❣r❛♣❤ ✐s ❛ ❞❛t❛ (Pv, Ph, Ps) ♠❛❞❡ ♦❢✿ ◮ t✇♦ (n + ✶)✲♣♦❧②❣r❛♣❤s Pv ❛♥❞ Ph s✉❝❤ t❤❛t Pv k = Ph k ❢♦r k ≤ n✱ ◮ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ t❤❡ ♣❛✐r ♦❢ (n + ✶)✲❝❛t❡❣♦r✐❡s ((Pv)∗, (Ph)∗)✱ t❤❛t ✐s ❛ s❡t ❡q✉✐♣♣❡❞
✇✐t❤ ❢♦✉r ♠❛♣s ∂µ
±,n✱ ✇✐t❤ µ ∈ {v, h}✱ ♠❛❦✐♥❣ Γ ❛ ✷✲❝✉❜✐❝❛❧ s❡t✳
Ps
∂h +,n+✶
−,n+✶
−,n+✶
+,n+✶
n+✶)∗ ∂v −,n
+,n
n+✶)∗ ∂h +,n
−,n
n+✶
n
Ph
n+✶
♦♥ ((Pv)⊤, (Ph)⊤)✳
◮ ❆ ❞♦✉❜❧❡ n✲♣♦❧②❣r❛♣❤ ✭r❡s♣✳ ❞♦✉❜❧❡ (n + ✷, n)✲♣♦❧②❣r❛♣❤✮ (Pv, Ph, Ps) ❣❡♥❡r❛t❡s ❛ ❢r❡❡
(n − ✶)✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ✭r❡s♣✳ ✐♥ ❞♦✉❜❧❡ ❣r♦✉♣♦✐❞s✮✱ ❞❡♥♦t❡❞ ❜② (Pv, Ph, Ps) | = ✳
◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ ((Pv)⊤, (Ph)⊤) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡
S = ·
(Ph)⊤ (Pv )⊤
(Pv )⊤
(Ph)⊤ ·
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✐♦♥ ♦❢ ❈✳
◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ ((Pv)⊤, (Ph)⊤) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡
S = ·
(Ph)⊤ (Pv )⊤
(Pv )⊤
(Ph)⊤ · A
| = s✉❝❤ t❤❛t ∂(A) = S✳ ❆ ✷✲❢♦❧❞ ❝♦❤❡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✐♦♥ ♦❢ ❈✳
◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ ((Pv)⊤, (Ph)⊤) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡
S = ·
(Ph)⊤ (Pv )⊤
(Pv )⊤
(Ph)⊤ · A
| = s✉❝❤ t❤❛t ∂(A) = S✳
◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ (n + ✷, n)✲♣♦❧②❣r❛♣❤
(Pv, Ph, Ps) s✉❝❤ t❤❛t✿
◮ t❤❡ (n + ✶)✲♣♦❧②❣r❛♣❤ Pv Ph ♣r❡s❡♥ts ❈❀ ◮ Ps ✐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✐♦♥ ♦❢ ❈✳
◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ ((Pv)⊤, (Ph)⊤) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡
S = ·
(Ph)⊤ (Pv )⊤
(Pv )⊤
(Ph)⊤ · A
| = s✉❝❤ t❤❛t ∂(A) = S✳
◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ (n + ✷, n)✲♣♦❧②❣r❛♣❤
(Pv, Ph, Ps) s✉❝❤ t❤❛t✿
◮ t❤❡ (n + ✶)✲♣♦❧②❣r❛♣❤ Pv Ph ♣r❡s❡♥ts ❈❀ ◮ Ps ✐s ❛❝②❝❧✐❝ ◮ ❊①❛♠♣❧❡✿ ▲❡t E ❜❡ ❛ ❝♦♥✈❡r❣❡♥t (n + ✶)✲♣♦❧②❣r❛♣❤ ❛♥❞ ❈ t❤❡ n✲❝❛t❡❣♦r② ♣r❡s❡♥t❡❞ ❜② E✳
❈❞ ✿❂ 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✐♦♥ ♦❢ ❈✳
◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ ((Pv)⊤, (Ph)⊤) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡
S = ·
(Ph)⊤ (Pv )⊤
(Pv )⊤
(Ph)⊤ · A
| = s✉❝❤ t❤❛t ∂(A) = S✳
◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ (n + ✷, n)✲♣♦❧②❣r❛♣❤
(Pv, Ph, Ps) s✉❝❤ t❤❛t✿
◮ t❤❡ (n + ✶)✲♣♦❧②❣r❛♣❤ Pv Ph ♣r❡s❡♥ts ❈❀ ◮ Ps ✐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❡♠✱ ❈❞ ✐s ❛ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈✳
◮ ❆ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ Ps ♦❢ ((Pv)⊤, (Ph)⊤) ✐s ❛❝②❝❧✐❝ ✐❢ ❢♦r ❛♥② sq✉❛r❡
S = ·
(Ph)⊤ (Pv )⊤
(Pv )⊤
(Ph)⊤ · A
| = s✉❝❤ t❤❛t ∂(A) = S✳
◮ ❆ ✷✲❢♦❧❞ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ n✲❝❛t❡❣♦r② ❈ ✐s ❛ ❞♦✉❜❧❡ (n + ✷, n)✲♣♦❧②❣r❛♣❤
(Pv, Ph, Ps) s✉❝❤ t❤❛t✿
◮ t❤❡ (n + ✶)✲♣♦❧②❣r❛♣❤ Pv Ph ♣r❡s❡♥ts ❈❀ ◮ Ps ✐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✐♦♥ ♦❢ ❈✳
❆ n✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ (R, E, 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
✶ ✶
✳
❆ n✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ (R, E, S) ♠❛❞❡ ♦❢
◮ ❛♥ n✲♣♦❧②❣r❛♣❤ 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
✶ ✶
✳
❆ n✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ (R, E, S) ♠❛❞❡ ♦❢
◮ ❛♥ n✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t Ek = Rk ❢♦r k ≤ n − ✷ ❛♥❞ En−✶ ⊆ Rn−✶✱ ♦❢ ♠♦❞✉❧♦ 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
✶ ✶
✳
❆ n✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ (R, E, S) ♠❛❞❡ ♦❢
◮ ❛♥ n✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t Ek = Rk ❢♦r k ≤ n − ✷ ❛♥❞ En−✶ ⊆ Rn−✶✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ◮ S ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ R∗
n−✶ s✉❝❤ t❤❛t R ⊆ S ⊆ E RE ✱
✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ✐s ❞❡✜♥❡❞ ❜② ❙♣❤
✶ ✶
✇❤❡r❡ ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s ✐♥
✶
s✉❝❤ t❤❛t ❛♥❞ t❤❡ ♠❛♣ ✐s ❞❡✜♥❡❞ ❜②
✶ ✶
✳
❆ n✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ (R, E, S) ♠❛❞❡ ♦❢
◮ ❛♥ n✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t Ek = Rk ❢♦r k ≤ n − ✷ ❛♥❞ En−✶ ⊆ Rn−✶✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ◮ S ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ R∗
n−✶ s✉❝❤ t❤❛t R ⊆ S ⊆ E RE ✱ ✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ E RE ✐s ❞❡✜♥❡❞ ❜②
γ E RE : E RE → ❙♣❤n−✶(R∗
n−✶)
✇❤❡r❡ E RE ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s (e, f , e′) ✐♥ E ⊤ × R∗(✶) × E ⊤ s✉❝❤ t❤❛t u
e
✐s ❞❡✜♥❡❞ ❜②
✶ ✶
✳
❆ n✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ✐s ❛ ❞❛t❛ (R, E, S) ♠❛❞❡ ♦❢
◮ ❛♥ n✲♣♦❧②❣r❛♣❤ R ♦❢ ♣r✐♠❛r② r✉❧❡s✱ ◮ ❛♥ n✲♣♦❧②❣r❛♣❤ E s✉❝❤ t❤❛t Ek = Rk ❢♦r k ≤ n − ✷ ❛♥❞ En−✶ ⊆ Rn−✶✱ ♦❢ ♠♦❞✉❧♦ r✉❧❡s✱ ◮ S ✐s ❛ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ R∗
n−✶ s✉❝❤ t❤❛t R ⊆ S ⊆ E RE ✱ ✇❤❡r❡ t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ E RE ✐s ❞❡✜♥❡❞ ❜②
γ E RE : E RE → ❙♣❤n−✶(R∗
n−✶)
✇❤❡r❡ E RE ✐s t❤❡ s❡t ♦❢ tr✐♣❧❡s (e, f , e′) ✐♥ E ⊤ × R∗(✶) × E ⊤ s✉❝❤ t❤❛t u
e
◮ ❆ ❜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✿ u
f
u′ v
g v′
■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❛♥❝❤✐♥❣ ♠♦❞✉❧♦ 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✿ u
f
u′ v
g v′
◮ ■t ✐s ❧♦❝❛❧ ✐❢ f ✐s ❛♥ n✲❝❡❧❧ ♦❢ S∗(✶)✱ g ✐s ❛♥ n✲❝❡❧❧ ♦❢ S∗ ❛♥❞ e ❛♥ n✲❝❡❧❧ ♦❢ E ⊤ s✉❝❤ t❤❛t
ℓ(g) + ℓ(e) = ✶✳ ■t ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✐❢ t❤❡r❡ ❡①✐sts ✲❝❡❧❧s ✐♥ ❛♥❞ ✐♥ ✿ ✐s s❛✐❞ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✭r❡s♣✳ ❧♦❝❛❧❧② ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✮ ✐❢ ❛♥② ❜r❛♥❝❤✐♥❣ ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣✮ ♦❢ ♠♦❞✉❧♦ ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✳
◮ ❆ ❜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✿ u
f
u′ v
g 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 ⊤✿
u
f
u′
f ′ w e′
g v′ g′ w′
✐s s❛✐❞ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✭r❡s♣✳ ❧♦❝❛❧❧② ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✮ ✐❢ ❛♥② ❜r❛♥❝❤✐♥❣ ✭r❡s♣✳ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣✮ ♦❢ ♠♦❞✉❧♦ ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✳
◮ ❆ ❜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✿ u
f
u′ v
g 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 ⊤✿
u
f
u′
f ′ w e′
g v′ g′ w′
◮ S ✐s s❛✐❞ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E ✭r❡s♣✳ ❧♦❝❛❧❧② ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E✮ ✐❢ ❛♥② ❜r❛♥❝❤✐♥❣ ✭r❡s♣✳
❧♦❝❛❧ ❜r❛♥❝❤✐♥❣✮ ♦❢ S ♠♦❞✉❧♦ E ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E✳
◮ ❲❡ ❝♦♥s✐❞❡r Γ ❛ ✷✲sq✉❛r❡ ❡①t❡♥s✐♦♥ ♦❢ (E ⊤, S∗)✳
❆ ❜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❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿
✶ ✶ ✷
❛♥❞
✶ ✶ ✶ ✶ ✷
◮ ❲❡ ❝♦♥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 ⊤
❛♥❞ ❛♥ ✶ ✲❝❡❧❧ ✐♥ P❡✐✛ ✿ u
f
u′
f ′ w e′
g v′ g′ w′
✐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❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿
✶ ✶ ✷
❛♥❞
✶ ✶ ✶ ✶ ✷
◮ ❲❡ ❝♦♥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 ⊤
❛♥❞ ❛♥ (n + ✶)✲❝❡❧❧ A ✐♥ (E, S, E ⋊ Γ ∪ P❡✐✛(E, S)) | =
,v✿
u
f
u′
f ′ A
e′
g v′ g′ w′
✐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❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿
✶ ✶ ✷
❛♥❞
✶ ✶ ✶ ✶ ✷
◮ ❲❡ ❝♦♥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 ⊤
❛♥❞ ❛♥ (n + ✶)✲❝❡❧❧ A ✐♥ (E, S, E ⋊ Γ ∪ P❡✐✛(E, S)) | =
,v✿
u
f
u′
f ′ A
e′
g v′ g′ w′
◮ (E, S, −)
| =
,v ✐s t❤❡ ❢r❡❡ n✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② (E, S, −)✱ ✐♥
✇❤✐❝❤ ❛❧❧ ✈❡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❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿
✶ ✶ ✷
❛♥❞
✶ ✶ ✶ ✶ ✷
◮ ❲❡ ❝♦♥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 ⊤
❛♥❞ ❛♥ (n + ✶)✲❝❡❧❧ A ✐♥ (E, S, E ⋊ Γ ∪ P❡✐✛(E, S)) | =
,v✿
u
f
u′
f ′ A
e′
g v′ g′ w′
◮ (E, S, −)
| =
,v ✐s t❤❡ ❢r❡❡ n✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② (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∗✳ u ⋆i v
f ⋆i v u⋆i e
u′ ⋆i v
u′⋆i e
f ⋆i v′ u′ ⋆i v ′
w ⋆i u
w⋆i f e′⋆i u
w ⋆i u′
e′⋆i u′
w′⋆i f w ′ ⋆i u′
❊ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✿
✶ ✶ ✷
❛♥❞
✶ ✶ ✶ ✶ ✷
◮ ❲❡ ❝♦♥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 ⊤
❛♥❞ ❛♥ (n + ✶)✲❝❡❧❧ A ✐♥ (E, S, E ⋊ Γ ∪ P❡✐✛(E, S)) | =
,v✿
u
f
u′
f ′ A
e′
g v′ g′ w′
◮ (E, S, −)
| =
,v ✐s t❤❡ ❢r❡❡ n✲❝❛t❡❣♦r② ❡♥r✐❝❤❡❞ ✐♥ ❞♦✉❜❧❡ ❝❛t❡❣♦r✐❡s ❣❡♥❡r❛t❡❞ ❜② (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∗✳ u ⋆i v
f ⋆i v u⋆i e
u′ ⋆i v
u′⋆i e
f ⋆i v′ u′ ⋆i v ′
w ⋆i u
w⋆i f e′⋆i u
w ⋆i u′
e′⋆i u′
w′⋆i f w ′ ⋆i u′ ◮ ❊ ⋊ Γ ✐s t♦ ❛✈♦✐❞ ✧r❡❞✉♥❞❛♥t✧ ❡❧❡♠❡♥ts ✐♥ Γ ❢♦r ❞✐✛❡r❡♥t sq✉❛r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s❛♠❡
❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E✿ u
f
v
f ′
e′
g=e✶g✶e✷ w g′
❛♥❞ u
f
v
f ′
e′
g✶e✷ w g′
◮ 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 ✬✶✽❪ ■❢ ✐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✜❡❞✳
◮ 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 RE ✐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✜❡❞✳
◮ 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 RE ✐s t❡r♠✐♥❛t✐♥❣✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ S ✐s Γ✲❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E❀
✐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✜❡❞✳
◮ 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 RE ✐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❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳ s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳
❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐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 RE ✐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 ❛) ❛♥❞ ❜)✿
❛✮✿ u
S∗(✶)
❂
S∗ v ′ E⊤
R∗(✶) w S∗ w ′ A
u
S∗(✶) E⊤(✶)
v
S∗ v ′ E⊤
S∗
B
s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛ ❛♥❞ ❜ ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ ♠♦❞✉❧♦ ✳
❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐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 RE ✐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 ❛) ❛♥❞ ❜)✿
❛✮✿ u
S∗(✶)
❂
S∗ v ′ E⊤
R∗(✶) w S∗ w ′ A
u
S∗(✶) E⊤(✶)
v
S∗ v ′ E⊤
S∗
B
◮ S s❛t✐s✜❡s ♣r♦♣❡rt✐❡s ❛) ❛♥❞ ❜) ❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ S ♠♦❞✉❧♦ E✳
❋♦r ✱ ♣r♦♣❡rt② ❜ ✐s tr✐✈✐❛❧❧② s❛t✐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 RE ✐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 ❛) ❛♥❞ ❜)✿
❛✮✿ u
S∗(✶)
❂
S∗ v ′ E⊤
R∗(✶) w S∗ w ′ A
u
S∗(✶) E⊤(✶)
v
S∗ v ′ E⊤
S∗
B
◮ 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✜❡❞✳
◮ 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 RE ✐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 ❛) ❛♥❞ ❜)✿
❛✮✿ u
S∗(✶)
❂
S∗ v ′ E⊤
R∗(✶) w S∗ w ′ A
u
S∗(✶) E⊤(✶)
v
S∗ v ′ E⊤
S∗
B
◮ 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✜❡❞✳
u
f e
v v′
◮ 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 RE ✐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 ❛) ❛♥❞ ❜)✿
❛✮✿ u
S∗(✶)
❂
S∗ v ′ E⊤
R∗(✶) w S∗ w ′ A
u
S∗(✶) E⊤(✶)
v
S∗ v ′ E⊤
S∗
B
◮ 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✜❡❞✳
u
f e
v v′
e−·f
◮ 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 RE ✐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 ❛) ❛♥❞ ❜)✿
❛✮✿ u
S∗(✶)
❂
S∗ v ′ E⊤
R∗(✶) w S∗ w ′ A
u
S∗(✶) E⊤(✶)
v
S∗ v ′ E⊤
S∗
B
◮ 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✜❡❞✳
u
f e
v ❂
e−·f
◮ ❆ 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 RE ✐s t❡r♠✐♥❛t✐♥❣✱
t❤❡♥ Γ ∪ ❈❞(E) ✐s ❛❝②❝❧✐❝✳
◮ ❆ ❝♦❤❡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 Af ,g ❛♥❞ Bf ,e✿ u
f
f ′ Af ,g
e′
g v g′ w′
u
f
u′
f ′ Bf ,e
e′
g′
❢♦r ❛♥② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s (f , g) ❛♥❞ (f , e) ♦❢ S ♠♦❞✉❧♦ E✳ ❈♦r♦❧❧❛r②✳ ❬❉✳✲▼❛❧❜♦s ✬✶✽❪ ▲❡t ❜❡ ❛♥ ✲♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ s✉❝❤ t❤❛t
✐s ❝♦♥✈❡r❣❡♥t✱ ✐s ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✱ ■rr ✐s ✲♥♦r♠❛❧✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ✱ ✐s t❡r♠✐♥❛t✐♥❣✱
❚❤❡♥ ❈❞ ✐s ❛❝②❝❧✐❝✳ ❈♦r♦❧❧❛r②✿ ❯s✉❛❧ ❙q✉✐❡r✬s t❤❡♦r❡♠✳ ✭ ✮
◮ ❆ ❝♦❤❡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 Af ,g ❛♥❞ Bf ,e✿ u
f
f ′ Af ,g
e′
g v g′ w′
u
f
u′
f ′ Bf ,e
e′
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 RE ✐s t❡r♠✐♥❛t✐♥❣✱
❚❤❡♥ C(S) ∪ ❈❞(E) ✐s ❛❝②❝❧✐❝✳ ❈♦r♦❧❧❛r②✿ ❯s✉❛❧ ❙q✉✐❡r✬s t❤❡♦r❡♠✳ ✭ ✮
◮ ❆ ❝♦❤❡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 Af ,g ❛♥❞ Bf ,e✿ u
f
f ′ Af ,g
e′
g v g′ w′
u
f
u′
f ′ Bf ,e
e′
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 RE ✐s t❡r♠✐♥❛t✐♥❣✱
❚❤❡♥ C(S) ∪ ❈❞(E) ✐s ❛❝②❝❧✐❝✳
◮ ❈♦r♦❧❧❛r②✿ ❯s✉❛❧ ❙q✉✐❡r✬s t❤❡♦r❡♠✳ ✭E = ∅✮
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ✵ ✵
✱
✶ ✶
✱
✷ ✷ ✷ ✸
❢♦r ✐♥ ✵ ✶
✸
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ✶ ✶
✱
✷ ✷ ✷ ✸
❢♦r ✐♥ ✵ ✶
✸
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ◮ E✶ = R✶ = {∧, ∨}✱ ✷ ✷ ✷ ✸
❢♦r ✐♥ ✵ ✶
✸
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ◮ E✶ = R✶ = {∧, ∨}✱ ◮ E✷ =
, , , ,
✷ ✷ ✸
❢♦r ✐♥ ✵ ✶
✸
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ◮ E✶ = R✶ = {∧, ∨}✱ ◮ E✷ =
, , , ,
✷ ✷ ◮ E✸ =
⇛ •µ ,
⇛ •µ ,
⇛ •µ ,
⇛ •µ ❢♦r µ ✐♥ {✵, ✶} ,
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ◮ E✶ = R✶ = {∧, ∨}✱ ◮ E✷ =
, , , ,
R✷ = E✷
,
◮ E✸ =
⇛ •µ ,
⇛ •µ ,
⇛ •µ ,
⇛ •µ ❢♦r µ ✐♥ {✵, ✶} ,
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ◮ E✶ = R✶ = {∧, ∨}✱ ◮ E✷ =
, , , ,
R✷ = E✷
,
◮ E✸ =
⇛ •µ ,
⇛ •µ ,
⇛ •µ ,
⇛ •µ ❢♦r µ ✐♥ {✵, ✶} ,
α+
⇛
α−
⇛
β+
⇛
β−
⇛
γ
⇛
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ◮ E✶ = R✶ = {∧, ∨}✱ ◮ E✷ =
, , , ,
R✷ = E✷
,
◮ E✸ =
⇛ •µ ,
⇛ •µ ,
⇛ •µ ,
⇛ •µ ❢♦r µ ✐♥ {✵, ✶} ,
α+
⇛
α−
⇛
β+
⇛
β−
⇛
γ
⇛
❋❛❝t✿ ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ▲❡t E ❛♥❞ R ❜❡ t✇♦ ✸✲♣♦❧②❣r❛♣❤s ❞❡✜♥❡❞ ❜②✿ ◮ E✵ = R✵ = {∗}✱ ◮ E✶ = R✶ = {∧, ∨}✱ ◮ E✷ =
, , , ,
R✷ = E✷
,
◮ E✸ =
⇛ •µ ,
⇛ •µ ,
⇛ •µ ,
⇛ •µ ❢♦r µ ✐♥ {✵, ✶} ,
α+
⇛
α−
⇛
β+
⇛
β−
⇛
γ
⇛
◮ ❋❛❝t✿ E ✐s ❝♦♥✈❡r❣❡♥t✳
◮ ■❢ ♥♦ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
■❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
❆❧❧ ❜r❛♥❝❤✐♥❣s ❛r❡ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ ✳ ❞♦❡s ♥♦t ♦✈❡r❧❛♣ ✇✐t❤ ❛♥❞ ✳ ❇r❛♥❝❤✐♥❣s ❜❡t✇❡❡♥ ❛♥❞ ❛r❡ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ ✳
✐s t❡r♠✐♥❛t✐♥❣✳
❚❡r♠✐♥❛t✐♦♥ ♦r❞❡r ❢♦r ❞❡✜♥❡❞ ❜② ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❞✐❛❣r❛♠s✱ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ✳
■rr ✐s ✲♥♦r♠❛❧✐③✐♥❣✳
❆♥② ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ ✐♥ ❛ s♦✉r❝❡✴t❛r❣❡t ♦❢ ❛♥ ✲r❡✇r✐t✐♥❣ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧s ♦❢ ✳
◮ ■❢ ♥♦ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
◮ ■❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❆❧❧ ❜r❛♥❝❤✐♥❣s ( E R, R) ❛r❡ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ (R, R)✳ ◮ γ ❞♦❡s ♥♦t ♦✈❡r❧❛♣ ✇✐t❤ α± ❛♥❞ β±✳ ◮ ❇r❛♥❝❤✐♥❣s ❜❡t✇❡❡♥ α± ❛♥❞ β± ❛r❡ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E✳
✐s t❡r♠✐♥❛t✐♥❣✳
❚❡r♠✐♥❛t✐♦♥ ♦r❞❡r ❢♦r ❞❡✜♥❡❞ ❜② ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❞✐❛❣r❛♠s✱ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ✳
■rr ✐s ✲♥♦r♠❛❧✐③✐♥❣✳
❆♥② ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ ✐♥ ❛ s♦✉r❝❡✴t❛r❣❡t ♦❢ ❛♥ ✲r❡✇r✐t✐♥❣ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧s ♦❢ ✳
◮ ■❢ ♥♦ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
◮ ■❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❆❧❧ ❜r❛♥❝❤✐♥❣s ( E R, R) ❛r❡ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ (R, R)✳ ◮ γ ❞♦❡s ♥♦t ♦✈❡r❧❛♣ ✇✐t❤ α± ❛♥❞ β±✳ ◮ ❇r❛♥❝❤✐♥❣s ❜❡t✇❡❡♥ α± ❛♥❞ β± ❛r❡ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E✳ ◮
E RE ✐s t❡r♠✐♥❛t✐♥❣✳
◮ ❚❡r♠✐♥❛t✐♦♥ ♦r❞❡r ❢♦r E ❞❡✜♥❡❞ ❜② ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❞✐❛❣r❛♠s✱ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ R✳
■rr ✐s ✲♥♦r♠❛❧✐③✐♥❣✳
❆♥② ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ ✐♥ ❛ s♦✉r❝❡✴t❛r❣❡t ♦❢ ❛♥ ✲r❡✇r✐t✐♥❣ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧s ♦❢ ✳
◮ ■❢ ♥♦ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿
◮ ■❢ r❡✇r✐t✐♥❣ ♠♦❞✉❧♦✿ ◮ ❆❧❧ ❜r❛♥❝❤✐♥❣s ( E R, R) ❛r❡ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ (R, R)✳ ◮ γ ❞♦❡s ♥♦t ♦✈❡r❧❛♣ ✇✐t❤ α± ❛♥❞ β±✳ ◮ ❇r❛♥❝❤✐♥❣s ❜❡t✇❡❡♥ α± ❛♥❞ β± ❛r❡ ❝♦♥✢✉❡♥t ♠♦❞✉❧♦ E✳ ◮
E RE ✐s t❡r♠✐♥❛t✐♥❣✳
◮ ❚❡r♠✐♥❛t✐♦♥ ♦r❞❡r ❢♦r E ❞❡✜♥❡❞ ❜② ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❞✐❛❣r❛♠s✱ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ R✳ ◮ ■rr(E) ✐s E✲♥♦r♠❛❧✐③✐♥❣✳ ◮ ❆♥② ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ ✐♥ ❛ s♦✉r❝❡✴t❛r❣❡t ♦❢ ❛♥ R✲r❡✇r✐t✐♥❣ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧s
♦❢ E✳
◮ ❲❡ ♣r♦✈❡❞ ❛ ❝♦❤❡r❡♥❝❡ r❡s✉❧t ❢♦r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳
❍♦✇ t♦ ✇❡❛❦❡♥ ✲♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❄ ■s ❛♥② ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t t♦ ❛♥ ✲♥♦r♠❛❧✐③✐♥❣ ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❄
❊①♣❧✐❝✐t ❛ q✉♦t✐❡♥t ♦❢ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❜② ❛❧❧ ♠♦❞✉❧♦ r✉❧❡s ❲♦r❦ ✐♥ ♣r♦❣r❡ss✿
❊①t❡♥❞ t❤❡s❡ r❡s✉❧ts t♦ ❧✐♥❡❛r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ❖❜t❛✐♥ ❛ ❜❛s✐s t❤❡♦r❡♠ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❤②♣♦t❤❡s✐s ♦❢ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦✳
◮ ❲❡ ♣r♦✈❡❞ ❛ ❝♦❤❡r❡♥❝❡ r❡s✉❧t ❢♦r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ◮ ❍♦✇ t♦ ✇❡❛❦❡♥ E✲♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❄
■s ❛♥② ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t t♦ ❛♥ ✲♥♦r♠❛❧✐③✐♥❣ ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❄
❊①♣❧✐❝✐t ❛ q✉♦t✐❡♥t ♦❢ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❜② ❛❧❧ ♠♦❞✉❧♦ r✉❧❡s ❲♦r❦ ✐♥ ♣r♦❣r❡ss✿
❊①t❡♥❞ t❤❡s❡ r❡s✉❧ts t♦ ❧✐♥❡❛r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ❖❜t❛✐♥ ❛ ❜❛s✐s t❤❡♦r❡♠ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❤②♣♦t❤❡s✐s ♦❢ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦✳
◮ ❲❡ ♣r♦✈❡❞ ❛ ❝♦❤❡r❡♥❝❡ r❡s✉❧t ❢♦r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ◮ ❍♦✇ t♦ ✇❡❛❦❡♥ E✲♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❄ ◮ ■s ❛♥② ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t t♦ ❛♥ E✲♥♦r♠❛❧✐③✐♥❣ ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❄
❊①♣❧✐❝✐t ❛ q✉♦t✐❡♥t ♦❢ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❜② ❛❧❧ ♠♦❞✉❧♦ r✉❧❡s ❲♦r❦ ✐♥ ♣r♦❣r❡ss✿
❊①t❡♥❞ t❤❡s❡ r❡s✉❧ts t♦ ❧✐♥❡❛r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ❖❜t❛✐♥ ❛ ❜❛s✐s t❤❡♦r❡♠ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❤②♣♦t❤❡s✐s ♦❢ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦✳
◮ ❲❡ ♣r♦✈❡❞ ❛ ❝♦❤❡r❡♥❝❡ r❡s✉❧t ❢♦r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ◮ ❍♦✇ t♦ ✇❡❛❦❡♥ E✲♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❄ ◮ ■s ❛♥② ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t t♦ ❛♥ E✲♥♦r♠❛❧✐③✐♥❣ ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❄ ◮ ❊①♣❧✐❝✐t ❛ q✉♦t✐❡♥t ♦❢ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❜② ❛❧❧ ♠♦❞✉❧♦ r✉❧❡s
❲♦r❦ ✐♥ ♣r♦❣r❡ss✿
❊①t❡♥❞ t❤❡s❡ r❡s✉❧ts t♦ ❧✐♥❡❛r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ❖❜t❛✐♥ ❛ ❜❛s✐s t❤❡♦r❡♠ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❤②♣♦t❤❡s✐s ♦❢ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦✳
◮ ❲❡ ♣r♦✈❡❞ ❛ ❝♦❤❡r❡♥❝❡ r❡s✉❧t ❢♦r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ◮ ❍♦✇ t♦ ✇❡❛❦❡♥ E✲♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❄ ◮ ■s ❛♥② ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t t♦ ❛♥ E✲♥♦r♠❛❧✐③✐♥❣ ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❄ ◮ ❊①♣❧✐❝✐t ❛ q✉♦t✐❡♥t ♦❢ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❜② ❛❧❧ ♠♦❞✉❧♦ r✉❧❡s ◮ ❲♦r❦ ✐♥ ♣r♦❣r❡ss✿
❊①t❡♥❞ t❤❡s❡ r❡s✉❧ts t♦ ❧✐♥❡❛r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ❖❜t❛✐♥ ❛ ❜❛s✐s t❤❡♦r❡♠ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❤②♣♦t❤❡s✐s ♦❢ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦✳
◮ ❲❡ ♣r♦✈❡❞ ❛ ❝♦❤❡r❡♥❝❡ r❡s✉❧t ❢♦r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ◮ ❍♦✇ t♦ ✇❡❛❦❡♥ E✲♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❄ ◮ ■s ❛♥② ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t t♦ ❛♥ E✲♥♦r♠❛❧✐③✐♥❣ ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❄ ◮ ❊①♣❧✐❝✐t ❛ q✉♦t✐❡♥t ♦❢ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❜② ❛❧❧ ♠♦❞✉❧♦ r✉❧❡s ◮ ❲♦r❦ ✐♥ ♣r♦❣r❡ss✿ ◮ ❊①t❡♥❞ t❤❡s❡ r❡s✉❧ts t♦ ❧✐♥❡❛r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳
❖❜t❛✐♥ ❛ ❜❛s✐s t❤❡♦r❡♠ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❤②♣♦t❤❡s✐s ♦❢ ❝♦♥✢✉❡♥❝❡ ♠♦❞✉❧♦✳
◮ ❲❡ ♣r♦✈❡❞ ❛ ❝♦❤❡r❡♥❝❡ r❡s✉❧t ❢♦r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ◮ ❍♦✇ t♦ ✇❡❛❦❡♥ E✲♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❄ ◮ ■s ❛♥② ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t t♦ ❛♥ E✲♥♦r♠❛❧✐③✐♥❣ ♣♦❧②❣r❛♣❤ ♠♦❞✉❧♦ ❄ ◮ ❊①♣❧✐❝✐t ❛ q✉♦t✐❡♥t ♦❢ ❛ sq✉❛r❡ ❡①t❡♥s✐♦♥ ❜② ❛❧❧ ♠♦❞✉❧♦ r✉❧❡s ◮ ❲♦r❦ ✐♥ ♣r♦❣r❡ss✿ ◮ ❊①t❡♥❞ t❤❡s❡ r❡s✉❧ts t♦ ❧✐♥❡❛r ♣♦❧②❣r❛♣❤s ♠♦❞✉❧♦✳ ◮ ❖❜t❛✐♥ ❛ ❜❛s✐s t❤❡♦r❡♠ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❤②♣♦t❤❡s✐s ♦❢ ❝♦♥✢✉❡♥❝❡
♠♦❞✉❧♦✳