▲✐♥❡❛r ❉✐s❝r✐♠✐♥❛♥t ❋✉♥❝t✐♦♥ ✿ t❤❡ ❧✐♥❡❛r ❞✐s❝r✐♠✐♥❛♥t ❢✉♥❝t✐♦♥✿ g ( x ) = w t x + ω 0 x ✐s t❤❡ ♣♦✐♥t✱ w ✐s t❤❡ ✇❡✐❣❤t ✈❡❝t♦r✱ ❛♥❞ ω 0 ✐s t❤❡ ❜✐❛s ✭ t ✐s t❤❡ tr❛♥s♣♦s❡✮✳ ❚✇♦ ❈❛t❡❣♦r② ❈❛s❡ ✿ ■♥ t❤❡ t✇♦ ❝❛t❡❣♦r② ❝❛s❡✱ ✇❡ ❤❛✈❡ t✇♦ ❝❧❛ss✐✜❡rs ✭s❛❧♠♦♥ ❛♥❞ s❡❛ ❜❛ss✮✳ ❲❡ ❞❡❝✐❞❡ ✇❤❡t❤❡r ✐t ❜❡❧♦♥❣s t♦ ❡❛❝❤ ❝❧❛ss✐✜❡r ❜② t❛❦✐♥❣ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ❢✉♥❝t✐♦♥✱ ❛♥❞ ❛ss✐❣♥✐♥❣ ♣♦✐♥ts t♦ ω 1 ♦r ω 2 ✱ ❜❛s❡❞ ✉♣♦♥ ✇❤❡t❤❡r g ( x ) > 0 ♦r g ( x ) < 0 ✳ ❲❤❛t t❤✐s ❛❝t✉❛❧❧② ❜♦✐❧s ❞♦✇♥ t♦ ✐s ✇❤❡t❤❡r w t x ✐s ❣r❡❛t❡r ♦r ❧❡ss t❤❛♥ ω 0 ✱ ♦r ✇❤❡t❤❡r t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣♦✐♥t ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ tr❛♥s♣♦s❡ ♦❢ t❤❡ ✇❡✐❣❤t ✈❡❝t❡r ✐s ❣r❡❛t❡r ♦r ❧❡ss t❤❛♥ t❤❡ ❜✐❛s✳ ❚❤❡ ❛r❡❛ ✇❤❡r❡ g ( x ) = 0 ✐s t❤❡ ❞❡❝✐s✐♦♥ s✉r❢❛❝❡✱ ❝r❡❛t✐♥❣ t✇♦ r❡❣✐♦♥s ✭ R 1 ❛♥❞ R 2 ✮✳ ■❢ ❛ ♣♦✐♥t t❤❛t ✐s ❧♦❝❛t❡❞ ♦♥ t❤❡ ❞❡❝✐s✐♦♥ s✉r❢❛❝❡ H ✭✇❤❡r❡ w t x = ω 0 ✮✱ t❤❡ ♣♦✐♥t ✐s tr❡❛t❡❞ ❛s ❛♠❜✐❣✉♦✉s✱ ❛♥❞ ♥♦t ♥❡❝❛ss❛r✐❧② ❜❡❧♦♥❣✐♥❣ t♦ ❡✐t❤❡r s❡t✳ ■❢ t✇♦ ♣♦✐♥ts ❛r❡ ♦♥ t❤❡ ❞❡❝✐s✐♦♥ s✉r❢❛❝❡ t❤❡♥ t❤❡② ❜♦t❤ ❤❛✈❡ ❛♥ ❡q✉❛❧ ♦✉t♣✉t ✭ w t x 1 + ω 0 = w t x 0 + ω 0 ♦r✱ w t ( x 1 − x 2 ) = 0 ✮✳ g ( x ) ❛❧s♦ ❛❝ts ❛s ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❞❡❝✐s✐♦♥ s✉r❢❛❝❡✳ ❳ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ x = x p + r ( w/ || w || ) ❲❤❡r❡ x p ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ① ♦♥t♦ t❤❡ ❤②♣❡r♣❧❛♥❡ H ✱ ❛♥❞ r ✐s t❤❡ ❞✐st❛♥❝❡ ❞❡s✐r❡❞ ✭♣♦s✐t✐✈❡ ❢♦r ♠❡♠❜❡rs❤✐♣ ✐♥ R 1 ✱ ♥❡❣❛t✐✈❡ ❢♦r ♠❡♠❜❡rs❤✐♣ ✐♥ R 2 ✮✳ ❇❡❝❛✉s❡ x p ✐s ♣r♦❥❡❝t❡❞ ♦♥ H ✱ g ( x p ) = 0 ✳ ❆s ❛ r❡s✉❧t✱ g ( x ) = r || w || ✱ ✐♥ ♦t❤❡r ✇♦r❞s✿ r = g ( x ) / || w || ❆ ✈❡❝t♦r ❢r♦♠ t❤❡ ♦r✐❣✐♥ t♦ H ✱ V ❝♦✉❧❞ ❜❡ ❡①♣r❡ss❡❞ ✉s✐♥❣✿ V = ω 0 / || w || ❆s ②♦✉ ❝❛♥ s❡❡✱ w s❡ts t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ H ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r✐❣✐♥✱ ❛♥❞ ω 0 s❡ts t❤❡ ❞✐st❛♥❝❡ ♦❢ H ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ▼✉❧t✐❝❛t❡❣♦r② ❈❛s❡ ✿ ♦♥❡ ✐❞❡❛ ✐s t♦ s♣❧✐t ❡❛❝❤ r❡❣✐♦♥ R i ✐♥t♦ ♠❡♠❜❡rs❤✐♣ ✐♥ ❛ s❡t✳ ❋♦r ❡①❛♠♣❧❡✱ R i ✐s s♣❧✐t ✐t ✐♥t♦ s❛❧♠♦♥✱ tr♦✉t✱ ❛♥❞ s❡❛ ❜❛ss✱ ✇✐t❤ r❡❣✐♦♥s ❢♦r ♥♦t tr♦✉t✱ tr♦✉t✱ s❛❧♠♦♥✱ ♥♦t s❛❧♠♦♥✱ s❡❛ ❜❛ss✱ ❛♥❞ ♥♦t s❡❛❜❛ss✱ ❛❧♦♥❣ ✇✐t❤ ❛♥ ❛♠❜✐❣♦✉s r❡❣✐♦♥✳ ❛ ❝❧❛ss✐✜❡r H i s❡♣❛r❛t✐♥❣ R i ✐♥t♦ ω i ❛♥❞ ! ω i ❆♥♦t❤❡r ❛♣♣r♦❛❝❤ ✐s t♦ ❤❛✈❡ s♣❧✐t t❤❡♠ ❜② ♣❛✐rs✱ s♦ t❤❛t ✇❤❡♥ tr♦✉t ❜♦r❞❡rs s❛❧♠♦♥✱ ✇❡ s♣❧✐t t❤❡ r❡❣✐♦♥ ✐♥t♦ tr♦✉t ❛♥❞ s❡❛ ❜❛ss✱ ❜✉t st✐❧❧ ❤❛✈❡ t❤❡ ❛♠❜✐❣✉♦✉s r❡❣✐♦♥ ✇❤❡r❡ ❛❧❧ ♦❢ t❤❡ ❜♦✉♥❞❛r✐❡s ♠❡❡t✳ ■♥ t❤✐s ❛♣♣r♦❛❝❤✱ ✇❡ ✇✐❧❧ ❤❛✈❡ c ( c − 1) / 2 ❝❧❛ss✐✜❡rs✱ ♦♥❡ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ s✉❜s♣❛❝❡s s♦ ✇❡ ❣❡t H ij = t❤❡ ❧✐♥❡❛r ❞✐s❝r✐♠✐♥❛♥t ❜❡t✇❡❡♥ ω i ❛♥❞ ω j ✶
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