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t trs rs rs t rt s rs rs s

  1. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s ❙③❛❜♦❧❝s ▼és③ár♦s ❈❡♥tr❛❧ ❊✉r♦♣❡❛♥ ❯♥✐✈❡rs✐t②✱ ❇✉❞❛♣❡st✱ ❍✉♥❣❛r② ✷✶t❤ ❏✉♥❡✱ ✷✵✶✼ ✶✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  2. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ▲❡t q ∈ k × ❛♥❞ O q ( M N ) := C � t ij | ✶ ≤ i , j ≤ N � / ( Rel ) ✇❤❡r❡ Rel ✿ ✳ ✳ t ik t il − qt il t ik   ✳ ✳ ✳ ✳ t ik t jk − qt jk t ik   ... t ik → t il ...   t jk t il − t il t jk  �  ↓ ↓ t ik t jl − t jl t ik − ( q − q − ✶ ) t il t jk     ... → ... t jk t jl   ✳ ✳   ✳ ✳ ✳ ✳ ( ∀ i < j , k < l ) R ∈ C N ✷ × N ✷ s✉❝❤ t❤❛t ˆ ∃ ˆ R s❛t✐s✜❡s t❤❡ ❨❛♥❣✲❇❛①t❡r ❊q✳ ❛♥❞ � � R ij ˆ kl t km t ln − t ik t jl ˆ R kl ∑ ( Rel ) = mn | i , j , m , n ≤ N k , l ✷✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  3. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ▲❡t q ∈ k × ❛♥❞ O q ( M N ) := C � t ij | ✶ ≤ i , j ≤ N � / ( Rel ) ✇❤❡r❡ Rel ✿ ✳ ✳ t ik t il − qt il t ik   ✳ ✳ ✳ ✳ t ik t jk − qt jk t ik   ... t ik → t il ...   t jk t il − t il t jk  �  ↓ ↓ t ik t jl − t jl t ik − ( q − q − ✶ ) t il t jk     ... → ... t jk t jl   ✳ ✳   ✳ ✳ ✳ ✳ ( ∀ i < j , k < l ) R ∈ C N ✷ × N ✷ s✉❝❤ t❤❛t ˆ ∃ ˆ R s❛t✐s✜❡s t❤❡ ❨❛♥❣✲❇❛①t❡r ❊q✳ ❛♥❞ � � R ij ˆ kl t km t ln − t ik t jl ˆ R kl ∑ ( Rel ) = mn | i , j , m , n ≤ N k , l ✷✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  4. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ▲❡t q ∈ k × ❛♥❞ O q ( M N ) := C � t ij | ✶ ≤ i , j ≤ N � / ( Rel ) ✇❤❡r❡ Rel ✿ ✳ ✳ t ik t il − qt il t ik   ✳ ✳ ✳ ✳ t ik t jk − qt jk t ik   ... t ik → t il ...   t jk t il − t il t jk  �  ↓ ↓ t ik t jl − t jl t ik − ( q − q − ✶ ) t il t jk     ... → ... t jk t jl   ✳ ✳   ✳ ✳ ✳ ✳ ( ∀ i < j , k < l ) R ∈ C N ✷ × N ✷ s✉❝❤ t❤❛t ˆ ∃ ˆ R s❛t✐s✜❡s t❤❡ ❨❛♥❣✲❇❛①t❡r ❊q✳ ❛♥❞ � � R ij ˆ kl t km t ln − t ik t jl ˆ R kl ∑ ( Rel ) = mn | i , j , m , n ≤ N k , l ✷✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  5. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s Pr♦♣❡rt✐❡s ♦❢ O q ( M N ) ❆ss✉♠♣t✐♦♥✿ q ∈ k × ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ O q ( M N ) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O ( M N ) ✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s O q ( M N ) coc ✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N ✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ O q ( M N ) coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳ ✸✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  6. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s Pr♦♣❡rt✐❡s ♦❢ O q ( M N ) ❆ss✉♠♣t✐♦♥✿ q ∈ k × ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ O q ( M N ) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O ( M N ) ✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s O q ( M N ) coc ✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N ✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ O q ( M N ) coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳ ✸✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  7. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s Pr♦♣❡rt✐❡s ♦❢ O q ( M N ) ❆ss✉♠♣t✐♦♥✿ q ∈ k × ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ O q ( M N ) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O ( M N ) ✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s O q ( M N ) coc ✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N ✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ O q ( M N ) coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳ ✸✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  8. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s Pr♦♣❡rt✐❡s ♦❢ O q ( M N ) ❆ss✉♠♣t✐♦♥✿ q ∈ k × ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ O q ( M N ) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O ( M N ) ✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s O q ( M N ) coc ✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N ✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ O q ( M N ) coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳ ✸✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

  9. ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s Pr♦♣❡rt✐❡s Pr♦♣❡rt✐❡s ♦❢ O q ( M N ) ❆ss✉♠♣t✐♦♥✿ q ∈ k × ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ O q ( M N ) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O ( M N ) ✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s O q ( M N ) coc ✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N ✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ O q ( M N ) coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳ ✸✴✶✸ ❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

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