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■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❈❛♥♦♥✐❝❛❧ ♣♦❧②❛❞✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤✐r❞✲♦r❞❡r t❡♥s♦rs✿ s♦♠❡ r❡s✉❧ts ♦♥ ✉♥✐q✉❡♥❡ss ❛♥❞ ❛❧❣❡❜r❛✐❝ ❛❧❣♦r✐t❤♠ ■❣♥❛t ❉♦♠❛♥♦✈✱ ▲✐❡✈❡♥ ❉❡ ▲❛t❤❛✉✇❡r ●r♦✉♣ ❙❝✐❡♥❝❡✱ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ❑❯ ▲❡✉✈❡♥✕❑✉❧❛❦ ❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✭❊❙❆❚✮✱ ❙❈❉✕❙❚❆❉■❯❙✱ ❑❯ ▲❡✉✈❡♥ ❋❡❜r✉❛r② ✹✱ ✷✵✶✺ ✱ ✱ ✶✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❱✐③✉❛❧✐③❛t✐♦♥ ♦❢ T ❛♥❞ r❛♥❦✲✶ t❡♥s♦r T ≡ ❛♥ I × J × K ❛rr❛② ♦❢ ♥✉♠❜❡rs r❛♥❦✲✶ t❡♥s♦rs ✫ ✫ ❛❧❧ ✜❜❡rs ❛❧❧ r♦✇s ❛❧❧ ❝♦❧✉♠♥s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ r❛♥❦✲✶ t❡♥s♦r ✿ ❛ ⊗ ❜ ⊗ ❝ ∈ C I × J × K ✱ ❛ , ❜ , ❝ ✲ ♥♦♥③❡r♦ ✈❡❝t♦rs ✱ ✱ ✷✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❉❡❝♦♠♣♦s✐t✐♦♥s ✐♥t♦ r❛♥❦✲✶ t❡r♠s✿ I × J × K t❡♥s♦rs r❛♥❦✲✶ t❡♥s♦r ❛❧✇❛②s ❡q✉❛❧s t❤❡ ♦✉t❡r ♣r♦❞✉❝t ♦❢ t❤r❡❡ ✈❡❝t♦rs c K a ✶ b ✶ c K a ✶ b ✶ c K a ✶ b J c K c ✶ a ✶ b J c ✶ a ✶ b ✶ c ✶ a ✶ b ✶ b J = a I b ✶ c K a I b J c K a I b ✶ c ✶ a I b J c ✶ a I ✱ ✱ ✸✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts P❛r❛❢❛❝✱ ❈P ❞❡❝♦♠♣♦s✐t✐♦♥✱✳✳✳ R � P♦❧②❛❞✐❝ ❉❡❝♦♠♣♦s✐t✐♦♥ ✭ P❉ ✮✿ T = ❛ r ⊗ ❜ r ⊗ ❝ r r = ✶  P❛r❛❢❛❝ ♦r   ❈❛♥❞❡❝♦♠♣ ♦r    ❈❛♥❞❡❝♦♠♣✴P❛r❛❢❛❝ ♦r ❛ P❉ ✇✐t❤ ♠✐♥✐♠❛❧ R � �� �  ❈P ❞❡❝♦♠♣♦s✐t✐♦♥ ♦r  r❛♥❦ ♦❢ T    ❈❛♥♦♥✐❝❛❧ P♦❧②❛❞✐❝ ❉❡❝♦♠♣♦s✐t✐♦♥ ✭ ❈P❉ ✮ ✱ ✱ ✹✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❱✐③✉❛❧✐③❛t✐♦♥ ♦❢ P❉ ♦❢ T ✸ ✶ ✻ ✵ ✵ ✶✷ ✶ ✸ ✸ ✶✷ ✵ = ✲✸ ✸ ✻ + ✶ ✸ ✲✻ + ✹ ✶ ✸ ✲✶✷ ✵ ✵ ✶✷ ✲✹ ✵ ✹ ◆❖❚❆❚■❖◆✿   � − ✸ � � ✸ � � ✶ �   ✶ ✹ ✸ ✶ ✸ ✸   T = = [ ❆ , ❇ , ❈ ] ✸ , ,   − ✹ ✵ ✹ ✻ − ✻ ✸ ✸ ✶ ✻ � �� �   � �� � � �� � � �� � ❢❛❝t♦r ♠❛tr✐❝❡s ❆ ❇ ❈ ✸ ✱ ✱ ✺✴✷✼

❥ ✲❥ ✵ ✵ ✶✷ ✶ ✶ ✶✷ ✵ ✲✻❥ ✻❥ ✶ ✻ ✶ ✻ ✲✶✷ ✵ ✵ ✶✷ ❥ ✲❥ ✶ ✶ ✻ ✻ ✶ ✶ ✻ ✻ ✷ ■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❈❛♥ ✇❡ ✉s❡ ❧❡ss t❤❛♥ ✸ t❡r♠s❄ ❲❤❛t ✐s t❤❡ r❛♥❦ ♦❢ T ❄ ✱ ✱ ✻✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❈❛♥ ✇❡ ✉s❡ ❧❡ss t❤❛♥ ✸ t❡r♠s❄ ❲❤❛t ✐s t❤❡ r❛♥❦ ♦❢ T ❄ ❥ ✲❥ ✵ ✵ ✶✷ ✶ ✶ ✶✷ ✵ ✲✻❥ ✻❥ = ✶ ✻ + ✶ ✻ ✲✶✷ ✵ ✵ ✶✷ ❥ ✲❥ � ✻ �� ✶ � � � ✶ �� ✶ ✻ ✶ = , , − j − ✻ j − j j ✻ j j ✷ ✱ ✱ ✻✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❈P❉ ❈P❉ ♦❢ T ✐s ✉♥✐q✉❡ � ❛❧❧ ❈P❉s ♦❢ T ❤❛✈❡ t❤❡ s❛♠❡ r❛♥❦ ♦♥❡ t❡r♠s ✸ ✶ ✻ ✵ ✵ ✶✷ ✶ ✸ ✸ ✶✷ ✵ = ✲✸ ✸ ✻ + ✶ ✸ ✲✻ + ✹ ✶ ✸ ✲✶✷ ✵ ✵ ✶✷ ✲✹ ✵ ✹ ❥ ✲❥ ✶ ✶ ✲✻❥ ✻❥ = ✶ ✻ + ✶ ✻ ✱ ✱ ❥ ✲❥ ✼✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❑r✉s❦❛❧✬s r❡s✉❧ts ♦♥ ✉♥✐q✉❡♥❡ss ♦❢ ❈P❉ ❉❡✜♥✐t✐♦♥✿ ❚❤❡ k ✲r❛♥❦ ♦❢ ❛ ♠❛tr✐① ❆ k ❆ = ♠❛① { k : ❛♥② k ❝♦❧✉♠♥s ♦❢ ❆ ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t } . ❊①❛♠♣❧❡✿ � ✶ � ✵ ✶ ❆ = r ❆ = ✷ , k ❆ = ✷ , ✵ ✶ ✶ � ✶ � ✵ ✵ ❇ = r ❇ = ✷ , k ❇ = ✶ , ✵ ✶ ✶ ❊①❛♠♣❧❡✿ ❆ ✐s ❣❡♥❡r✐❝ I × R ♠❛tr✐① ⇒ k ❆ = ♠✐♥ ( I , R ) ✱ ✱ ✽✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❑r✉s❦❛❧✬s r❡s✉❧ts ♦♥ ✉♥✐q✉❡♥❡ss ♦❢ ❈P❉ ❚❤❡♦r❡♠ ▲❡t T = [ ❆ , ❇ , ❈ ] R ✳ ■❢ � k ❆ + r ❇ + r ❈ ≥ ✷ R + ✷ , ✭✶✮ ≥ R + ✷ , ♠✐♥ ( r ❈ + k ❇ , k ❈ + r ❇ ) ♦r ✏♣❡r♠✉t❡ ❆ ✱ ❇ ✱ ❈ ✐♥ ✭✶✮ ✑ ♦r k ❆ + k ❇ + k ❈ ≥ ✷ R + ✷ , ✭✷✮ t❤❡♥ r T = R ❛♥❞ t❤❡ ❈P❉ ♦❢ T ✐s ✉♥✐q✉❡✳ ❖❜✈✐♦✉s❧②✿ k ❆ ≤ r ❆ ✱ k ❇ ≤ ... ⇒ ✭✷✮ ✐s ♠♦r❡ r❡str✐❝t✐✈❡ t❤❛♥ ✭✶✮✳ ❏✳❇✳ ❑r✉s❦❛❧✳ ❚❤r❡❡✲✇❛② ❛rr❛②s✿ ❘❛♥❦ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ tr✐❧✐♥❡❛r ❞❡❝♦♠♣♦s✐t✐♦♥s✳ ▲✐♥✳ ❆❧❣✳ ❆♣♣❧✳ ✱ ✶✽✭✷✮✿✾✺✖✶✸✽✱ ✶✾✼✼✳ ✱ ✱ ✾✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❈♦♥❞✐t✐♦♥s ❢♦r ✉♥✐q✉❡♥❡ss✿ ▲✐♠✕❈♦♠♦♥ ❈♦♠♣✉t❛t✐♦♥ ♦❢ k ✲r❛♥❦ ✐s ◆P ❤❛r❞✳ ✶ ✐t ✐s ❦♥♦✇♥ t❤❛t µ ❆ ≥ k ❆ ✱ ✇❤❡r❡ | ❛ T i ❛ j | ❝♦❤❡r❡♥❝❡ ♦❢ ❆ ✿ µ ❆ := ♠❛① � ❛ i �� ❛ j � ✶ ≤ i < j ≤ R ■❞❡❛✿ r❡♣❧❛❝❡ ✐♥ ✭✷✮ t❤❡ ❦✲r❛♥❦s ❜② ❝♦❤❡r❡♥❝❡s − ✶ ❚❤❡♦r❡♠ ✶ ✶ ✶ ▲❡t T = [ ❆ , ❇ , ❈ ] R ✳ ■❢ µ ❆ + µ ❇ + µ ❈ ≥ ✷ R + ✷ , t❤❡♥ r T = R ❛♥❞ t❤❡ ❈P❉ ♦❢ t❡♥s♦r T ✐s ✉♥✐q✉❡✳ ▲✳✲❍✳ ▲✐♠ ❛♥❞ P✳ ❈♦♠♦♥✳ ❇❧✐♥❞ ♠✉❧t✐❧✐♥❡❛r ✐❞❡♥t✐✜❝❛t✐♦♥✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r② ✻✵✭✷✮✿✶✷✻✵✕✶✷✽✵✱ ✷✵✶✹ ✱ ✱ ✶✵✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts ❲❡ ❝❛♥ ❛❧s♦ r❡♣❧❛❝❡ t❤❡ ❦✲r❛♥❦s ❜② ❝♦❤❡r❡♥❝❡s − ✶ ✐♥ ✭✶✮ ❚❤❡♦r❡♠ ▲❡t T = [ ❆ , ❇ , ❈ ] R ✳ ■❢ � ✶ ≥ ✷ R + ✷ , µ ❆ + r ❇ + r ❈ ✭✸✮ µ ❇ , ✶ ✶ ♠✐♥ ( r ❈ + µ ❈ + r ❇ ) ≥ R + ✷ , t❤❡♥ r T = R ❛♥❞ t❤❡ ❈P❉ ♦❢ T ✐s ✉♥✐q✉❡✳ ❘❊▼❆❘❑✿ ✏♣❡r♠✉t❡ ❆ ✱ ❇ ✱ ❈ ✐♥ ✭✸✮✑ ✕ t✇♦ ♦t❤❡r ❝♦♥❞✐t✐♦♥s ✱ ✱ ✶✶✴✷✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❆ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡ ❘❡s✉❧ts Pr♦❜❧❡♠ ✶✿ ✭ ❯♥✐q✉❡♥❡ss ✮ R � ●✐✈❡♥ ❈P❉ T = ❛ r ⊗ ❜ r ⊗ ❝ r ✳ ■s t❤❡ ❈P❉ ✉♥✐q✉❡❄ r = ✶ Pr♦❜❧❡♠ ✷✿ ✭ ✏❆❧❣❡❜r❛✐❝✑ ❛❧❣♦r✐t❤♠ ✮ R � ❈♦♠♣✉t❡ t❤❡ ❈P❉ T = ❛ r ⊗ ❜ r ⊗ ❝ r ❜② ♠❡❛♥s ♦❢ r = ✶ ▲✐♥❡❛r ❆❧❣❡❜r❛✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ r❛♥❦s ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♥✉❧❧ s♣❛❝❡ ❡✐❣❡♥✈❛❧✉❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ✱ ✱ ✶✷✴✷✼