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Prrs rs r Pr rs s s tr

  1. Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ P♦❧②❤②♣❡r❜♦❧✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❚✇♦ ❡①❛♠♣❧❡s ♦❢ ❣❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r② ◆ór❛ ❙③❛❦á❝s ❯♥✐✈❡rs✐t② ♦❢ ❙③❡❣❡❞ ❲✐♥t❡r ❖♥❡✲r❡❧❛t♦r ❲♦r❦s❤♦♣✱ ◆♦r✇✐❝❤✱ ✷✵✶✽ ◆ór❛ ❙③❛❦á❝s ●❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r②

  2. ❡✈❡r② ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ✶ ❀ ✐❞❡♠♣♦t❡♥t✿ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t♦♥ ♦❢ ✐s t❤❡ ✐♥✈❡rs❡ ✶ ❛✉t♦♠❛t♦♥ ❀ t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ ✐s ❞❡✜♥❡❞ ❜② ❢♦r s♦♠❡ ✐❞❡♠♣♦t❡♥t ❀ ❛ ✇♦r❞ ✐s ❛❝❝❡♣t❡❞ ❜② ❀ ❛♥❞ ✳ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ P♦❧②❤②♣❡r❜♦❧✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s M = � X � ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ ❘❡❝❛❧❧✿ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ♦❢ m ∈ M ✿ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ m ❀ ◆ór❛ ❙③❛❦á❝s ●❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r②

  3. t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t♦♥ ♦❢ ✐s t❤❡ ✐♥✈❡rs❡ ✶ ❛✉t♦♠❛t♦♥ ❀ t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ ✐s ❞❡✜♥❡❞ ❜② ❢♦r s♦♠❡ ✐❞❡♠♣♦t❡♥t ❀ ❛ ✇♦r❞ ✐s ❛❝❝❡♣t❡❞ ❜② ❀ ❛♥❞ ✳ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ P♦❧②❤②♣❡r❜♦❧✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s M = � X � ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ ❘❡❝❛❧❧✿ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ♦❢ m ∈ M ✿ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ m ❀ ◮ ❡✈❡r② ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ✐❞❡♠♣♦t❡♥t✿ mm − ✶ ❀ ◆ór❛ ❙③❛❦á❝s ●❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r②

  4. t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ ✐s ❞❡✜♥❡❞ ❜② ❢♦r s♦♠❡ ✐❞❡♠♣♦t❡♥t ❀ ❛ ✇♦r❞ ✐s ❛❝❝❡♣t❡❞ ❜② ❀ ❛♥❞ ✳ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ P♦❧②❤②♣❡r❜♦❧✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s M = � X � ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ ❘❡❝❛❧❧✿ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ♦❢ m ∈ M ✿ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ m ❀ ◮ ❡✈❡r② ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ✐❞❡♠♣♦t❡♥t✿ mm − ✶ ❀ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t♦♥ SA ( m ) ♦❢ m ✐s t❤❡ ✐♥✈❡rs❡ ❛✉t♦♠❛t♦♥ ( mm − ✶ , S ( m ) , m ) ❀ ◆ór❛ ❙③❛❦á❝s ●❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r②

  5. ❛ ✇♦r❞ ✐s ❛❝❝❡♣t❡❞ ❜② ❀ ❛♥❞ ✳ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ P♦❧②❤②♣❡r❜♦❧✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s M = � X � ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ ❘❡❝❛❧❧✿ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ♦❢ m ∈ M ✿ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ m ❀ ◮ ❡✈❡r② ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ✐❞❡♠♣♦t❡♥t✿ mm − ✶ ❀ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t♦♥ SA ( m ) ♦❢ m ✐s t❤❡ ✐♥✈❡rs❡ ❛✉t♦♠❛t♦♥ ( mm − ✶ , S ( m ) , m ) ❀ ◮ t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ M ✐s ❞❡✜♥❡❞ ❜② s ≤ t ⇐ ⇒ s = te ❢♦r s♦♠❡ ✐❞❡♠♣♦t❡♥t e ❀ ◆ór❛ ❙③❛❦á❝s ●❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r②

  6. ❛♥❞ ✳ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ P♦❧②❤②♣❡r❜♦❧✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s M = � X � ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ ❘❡❝❛❧❧✿ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ♦❢ m ∈ M ✿ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ m ❀ ◮ ❡✈❡r② ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ✐❞❡♠♣♦t❡♥t✿ mm − ✶ ❀ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t♦♥ SA ( m ) ♦❢ m ✐s t❤❡ ✐♥✈❡rs❡ ❛✉t♦♠❛t♦♥ ( mm − ✶ , S ( m ) , m ) ❀ ◮ t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ M ✐s ❞❡✜♥❡❞ ❜② s ≤ t ⇐ ⇒ s = te ❢♦r s♦♠❡ ✐❞❡♠♣♦t❡♥t e ❀ ◮ ❛ ✇♦r❞ w ✐s ❛❝❝❡♣t❡❞ ❜② SA ( m ) ⇐ ⇒ m ≤ w ❀ ◆ór❛ ❙③❛❦á❝s ●❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r②

  7. Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ P♦❧②❤②♣❡r❜♦❧✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s M = � X � ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ ❘❡❝❛❧❧✿ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ♦❢ m ∈ M ✿ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ m ❀ ◮ ❡✈❡r② ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ S ( m ) ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ✐❞❡♠♣♦t❡♥t✿ mm − ✶ ❀ ◮ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t♦♥ SA ( m ) ♦❢ m ✐s t❤❡ ✐♥✈❡rs❡ ❛✉t♦♠❛t♦♥ ( mm − ✶ , S ( m ) , m ) ❀ ◮ t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ M ✐s ❞❡✜♥❡❞ ❜② s ≤ t ⇐ ⇒ s = te ❢♦r s♦♠❡ ✐❞❡♠♣♦t❡♥t e ❀ ◮ ❛ ✇♦r❞ w ✐s ❛❝❝❡♣t❡❞ ❜② SA ( m ) ⇐ ⇒ m ≤ w ❀ ◮ = ⇒ u = v ⇐ ⇒ u ∈ SA ( v ) ❛♥❞ v ∈ SA ( u ) ✳ ◆ór❛ ❙③❛❦á❝s ●❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ t❤❡♦r②

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