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Download Algebraic geometry IV (Enc.Math.55, Springer 1994) by A.N. Parshin, I.R. Shafarevich, V.L. Popov, T.A. Springer, PDF

By A.N. Parshin, I.R. Shafarevich, V.L. Popov, T.A. Springer, E.B. Vinberg

This quantity of the Encyclopaedia includes contributions on heavily similar topics: the idea of linear algebraic teams and invariant conception. the 1st half is written through T.A. Springer, a widely known professional within the first pointed out box. He offers a finished survey, which includes a number of sketched proofs and he discusses the actual positive factors of algebraic teams over targeted fields (finite, neighborhood, and global). The authors of half , E.B. Vinberg and V.L. Popov, are one of the so much energetic researchers in invariant concept. The final twenty years were a interval of full of life improvement during this box because of the impact of contemporary tools from algebraic geometry. The booklet may be very valuable as a reference and study consultant to graduate scholars and researchers in arithmetic and theoretical physics.

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Example text

R ♠❛② ♥♦t✮ ✜♥❞ ✐t ❤❡❧♣❢✉❧ t♦ t❤✐♥❦ ♦❢ t❤❡s❡ ♣❛r❛♠❡tr✐❝ ❡q✉❛t✐♦♥s ❛s ♠❛♣♣✐♥❣s ❢r♦♠ ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✇✐t❤ t❤❡ s✐♥❣❧❡ ❝♦♦r❞✐♥❛t❡ s♣❛❝❡✳ t ✐♥t♦ ♦✉r ✉s✉❛❧ t✇♦✲ ♦r t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❚❤✉s✱ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ t t♦ ❛ x, y, z t♦ ❛ ❣❡t ❢r♦♠ ❛♥② ✈❛❧✉❡ ♦❢ ✈❛❧✉❡ ♦❢ ❣❡t ❢r♦♠ ✈❛❧✉❡s ♦❢ ✈❛❧✉❡ ♦❢ ❤❛✈❡ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ ✶ ♣r♦✈✐❞❡s ❛ ✇❛② t♦ x✱ y ♦r z ✭❜✉t ♥♦t ❛ ✇❛② t♦ t✱ ❛s ♦♥❧② ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ t✮✳ ■❢ ✇❡ ✐♥tr♦❞✉❝❡ ❛ s❡❝♦♥❞ ♣❛r❛♠❡t❡r ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥s ❛❜♦✈❡✱ t❤❡♥ ✇❡ ❛r❡ ♠❛♣♣✐♥❣ ❢r♦♠ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ✭✇❡✬❧❧ ✉s❡ t ❛♥❞ u ❢♦r ✐ts t✇♦ ❞✐♠❡♥s✐♦♥s✮ ✐♥t♦ r❡❛❧ s♣❛❝❡✳ ❇✉t ✐❢ ✇❡ ❞r♦♣ ❛♥ ❡q✉❛t✐♦♥ ❛♥❞ ♠❛♣ ❢r♦♠ t❤❡ t, u s♣❛❝❡ ✐♥t♦ ❛♥♦t❤❡r t✇♦✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✱ t❤❡♥ ✇❤❛t ✇❡ ❤❛✈❡ ✐s ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ tr❛♥s❢♦r♠✿ ♦r✱ ✇✐t❤ t❤r❡❡ ❡q✉❛✲ t✐♦♥s ❛♥❞ t❤r❡❡ ♣❛r❛♠❡t❡rs✱ ❛ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ tr❛♥s❢♦r♠✳ ❨♦✉ ❝❛♥ t❤✐♥❦ ♦❢ ❈❤❛♣t❡r ✾ ✇❤❡♥ ②♦✉ ❣❡t t❤❡r❡ ✐♥ t❡r♠s ♦❢ ♣❛r❛♠❡tr✐❝ ❡q✉❛✲ t✐♦♥s ✐❢ ✐t ♠❛❦❡s ②♦✉ ❤❛♣♣✐❡r✳ ■t ♠❛❦❡s ♠♦st ♣❡♦♣❧❡ q✉✐t❡ ❛ ❧♦t ❧❡ss ❤❛♣♣②✱ s♦ ✇❡ ✇♦♥✬t ♣✉rs✉❡ t❤❛t ❛✈❡♥✉❡✳ ✶ ❖r ❝✉r✈❡✱ ✐❢ ✇❡ ❤❛✈❡ t2 , t 3 ❛♥❞ s♦ ♦♥ ✐♥ t❤❡ ❡q✉❛t✐♦♥✳ €❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✸✷ t, u ▲❡t✬s ❜❛❝❦ ✉♣ ❛ ❜✐t ❤❡r❡❀ ✐❢ ✇❡ ♠❛♣ ❢r♦♠ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✭✐✳❡✳ x, y, z s♣❛❝❡ ✐♥t♦ ❛ t❤r❡❡✲ ❛s ✉s✉❛❧✮✱ ✇❡ ❝r❡❛t❡ ❛ s✉r❢❛❝❡✳ ❚❤❡ ♣❛r❛♠❡tr✐❝ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ x = x0 + f1 t + f2 u y = y0 + g1 t + g2 u z = z0 + h1 t + h2 u ❤❛s ❛❧r❡❛❞② ❛♣♣❡❛r❡❞✳ ■t✬s ♣r❡tt② t❡♠♣t✐♥❣ t♦ tr② ♠♦r❡ ❛❞✈❡♥t✉r♦✉s ❡①♣r❡ss✐♦♥s ✐♥ t❡r♠s ♦❢ t ♦r t ❛♥❞ u ✐♥ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥s✳ ■t ✐s ❡s♣❡❝✐❛❧❧② ❛ttr❛❝t✐✈❡ ❜❡❝❛✉s❡ t❤❡ ✇❤♦❧❡ ❡ss❡♥❝❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❢♦r♠ ✐s t❤❛t ✐t ♠❛❦❡s ✐t ❣❡♥❡r❛t❡ ♣♦✐♥ts ❡❛s② t♦ ♦♥ t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡✱ ❛♥❞ t♦ ❜♦✉♥❞ t❤❡♠ t♦ ❛ ♣❛rt✐❝✉❧❛r r❛♥❣❡ ♦❢ ♣❛r❛♠❡t❡r ✈❛❧✉❡s✳ ■❢ ✇❡ ❝❛♥ ❡✈❛❧✉❛t❡ t❤❡ ❡①♣r❡ss✐♦♥s ✇❡ ❤❛✈❡ ❝r❡❛t❡❞✱ t❤❡♥ ✇❡ ❝❛♥ ❣❡t ♣♦✐♥ts❀ ❛♥❞ ✇❡ ❝❛♥ ❣❡t t❤❡♠ ✇✐t❤✐♥ ❛ ♣❛rt✐❝✉❧❛r r❛♥❣❡ ✭✐♥t❡r✈❛❧✮ ♦❢ ✈❛❧✉❡s ♦❢ r❡❝t❛♥❣❧❡ ✐♥ t❤❡ t, u t✱ ♦r ✇✐t❤✐♥ ❛ ♣❧❛♥❡✳ ❖❢ ❝♦✉rs❡ t❤❛t ❞♦❡s♥✬t ♠❡❛♥ t❤❛t ♦t❤❡r ♦♣❡r❛t✐♦♥s ❛r❡ s♦ ❡❛s② ✭♠♦r❡ ♦❢ t❤❛t ❧❛t❡r✮✳ ❆r❣✉❛❜❧② t❤❡ s✐♠♣❧❡st ✇❛② t♦ ❡①t❡♥❞ ♣❛r❛♠❡tr✐❝ ❡q✉❛t✐♦♥s ✐s t♦ ♠❛❦❡ t❤❡ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♣❛r❛♠❡t❡r✭s✮ ❛♥ ❛r❜✐tr❛r② ♣♦❧②♥♦♠✐❛❧✿ x = a0 + a1 t + a2 t2 + · · · y = ··· ✳ ✳ ✳ = ✳ ✳ ✳ ♦r✱ ✇✐t❤ t✇♦ ♣❛r❛♠❡t❡rs✱ x = a0 + a1 t + a2 u + a3 tu + a4 t2 + a5 u2 + a6 t2 u + a7 tu2 + a8 t3 + · · · y = ··· ✳ ✳ ✳ = ✳ ✳ ✳ ❆❧t❡r♥❛t✐✈❡❧② ✇❡ ♠✐❣❤t ❝♦♥s✐❞❡r r❛t✐♦♥❛❧ ♣♦❧②♥♦♠✐❛❧s❀ ❣♦✐♥❣ ❜❛❝❦ t♦ ♦♥❡ ♣❛r❛♠❡t❡r✖t♦ ❦❡❡♣ t❤✐♥❣s ❛ ❧✐tt❧❡ ❜✐t s✐♠♣❧❡r✖✇❡ ❤❛✈❡✿ a0 + a1 t + a2 t2 + · · · b0 + b1 t + b2 t2 + · · · ··· y = ··· x = ✳ ✳ ✳ = ✳ ✳ ✳ ■♥t❡r♣♦❧❛t✐♦♥ ✸✸ ✭❲♦♥❞❡r❢✉❧ t❤✐♥❣✱ t❤❡ ❡❧❧✐♣s✐s✳✳✳✮ ❘❛t✐♦♥❛❧s ❣✐✈❡ ✉s t❤❡ ♦♣♣♦rt✉♥✐t② θ t♦ r❡♣r❡s❡♥t ❝♦♥✐❝ s❡❝t✐♦♥s ❡①❛❝t❧②✱ ✉s✐♥❣ t❤❡ t = tan ♣❛r❛♠❡t❡r✐✲ 2 ③❛t✐♦♥ ✇❡ s❛✇ ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✳ ❚❤❡ ❛❧t❡r♥❛t✐✈❡ ✐s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡♠ ✇✐t❤ ❛ s✐♥❣❧❡ ❤✐❣❤✲❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ ✭t❤❡ ❲❡✐❡rstr❛ss t❤❡♦r❡♠✮ t❤❛t ✇❡ ❝❛♥ ❞♦ t❤✐s t♦ ❛r❜✐tr❛r② ♣r❡❝✐s✐♦♥✱ ❜✉t ✐♥ ❣❡♥❡r❛❧ ❤✐❣❤✲❞❡❣r❡❡ ❡q✉❛t✐♦♥s ❛r❡ ♥♦t ❛ttr❛❝t✐✈❡✳ ❍♦✇❡✈❡r✱ ✇❤✐❧❡ r❛t✐♦♥❛❧s ❛❧❧♦✇ ✉s t♦ ❞♦ ♠♦r❡ ✇✐t❤ ❧♦✇❡r✲❞❡❣r❡❡ ❡q✉❛t✐♦♥s✱ t❤❡ ❡①✐s✲ t❡♥❝❡ ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ❝❛✉s❡s ❛♥ ♦❜✈✐♦✉s ♣r♦❜❧❡♠✿ ✇❤❛t ❤❛♣♣❡♥s ✐❢ ✐t ✐s ❛❧❧♦✇❡❞ t♦ ❝♦♠❡ ❝❧♦s❡ t♦✱ ♦r t♦ ❝r♦ss✱ ③❡r♦❄ ❲❡ ❝❛♥ ❛❞♦♣t ❛ 2 ♣♦❧②♥♦♠✐❛❧ t❤❛t ✐s ❣✉❛r❛♥t❡❡❞ t♦ r❡♠❛✐♥ ♣♦s✐t✐✈❡✱ s✉❝❤ ❛s t❤❡ 1 + t t❡r♠ ✐♥ t❤❛t ❝✐r❝❧❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥✱ ♦r s✐♠♣❧② ❡♥s✉r❡ t❤❛t t❤❡ r❛♥❣❡ ♦❢ ♣❛r❛♠❡t❡rs ✇✐t❤✐♥ ✇❤✐❝❤ ✇❡ ❛r❡ ✇♦r❦✐♥❣ ❞♦❡s ♥♦t ❝❛✉s❡ tr♦✉❜❧❡✳ ■t ♠❛❦❡s t❤✐♥❣s ❧❡ss str❛✐❣❤t❢♦r✇❛r❞✱ t❤♦✉❣❤✱ ❛♥❞ ❢♦r t❤❡ r❡st ♦❢ t❤✐s ❝❤❛♣t❡r ✇❡ s❤❛❧❧ ✐❣♥♦r❡ r❛t✐♦♥❛❧ ❡q✉❛t✐♦♥s✳ ❆ ♠♦r❡ ✐♠♠❡❞✐❛t❡ ♣r♦❜❧❡♠ ✐s✱ ✇❤❛t ✈❛❧✉❡s ❛r❡ ✇❡ ❣♦✐♥❣ t♦ ✉s❡ ❢♦r ❛❧❧ t❤❡s❡ ❝♦♥st❛♥ts a✱ b ❡t❝✳❄ ❲❡ ✇✐❧❧ st❛rt t♦ ❛♥s✇❡r t❤❛t q✉❡st✐♦♥ ❜❡❧♦✇✱ ❝♦♠♠❡♥❝✐♥❣ ✇✐t❤ ❝✉r✈❡s✱ ❛♥❞ ♠♦✈✐♥❣ ♦♥ t♦ ❞✐s❝✉ss s✉r❢❛❝❡s✳ Interpolation ❲❤❡♥ ✇❡ ❧♦♦❦❡❞ ❛t t❤❡ str❛✐❣❤t ❧✐♥❡ ❛♥❞ t❤❡ ♣❧❛♥❡✱ ✇❡ ✉s❡❞ ♥♦t❛t✐♦♥ 2 ♦❢ t❤✐s s♦rt✿ x = x0 +f t❀ ♥♦✇ ✇❡ ❤❛✈❡ ❝❤❛♥❣❡❞ t♦ x = a1 +a2 t+a3 t + · · ·✳ ❨♦✉ ♠❛② t❤✐♥❦ t❤❛t ✇❡✬✈❡ ❥✉st r✉♥ ♦✉t ♦❢ ❧❡tt❡rs✱ ❛♥❞ t❤❛t✬s tr✉❡✱ ❜✉t t❤❡ ♠❛✐♥ ♣♦✐♥t ✐s t❤❛t✱ ✐♥ t❤❡ ❡q✉❛t✐♦♥s ❢♦r t❤❡ str❛✐❣❤t ❧✐♥❡ ❛♥❞ ♣❧❛♥❡✱ ❡❛❝❤ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ❤❛❞ ❛ ♠❡❛♥✐♥❣✳ ❚❛❦❡ t❤❡ str❛✐❣❤t ❧✐♥❡ ✐♥ s♣❛❝❡✱ ❢♦r ✐♥st❛♥❝❡❀ (f, g, h) (x0 , y0 , z0 ) ✐s t❤❡ ♣♦✐♥t ✇❤❡r❡ t=0 ❛♥❞ ✐s ❛ ✈❡❝t♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❧✐♥❡✳ ❲❤❡♥ ✇❡ st❛rt ❛❞❞✐♥❣ t❡r♠s ✐♥ ❤✐❣❤❡r ♣♦✇❡rs ♦❢ ❤❛✈❡ ♥♦ ✐♥t✉✐t✐✈❡ ♠❡❛♥✐♥❣ ✳ t✱ t❤❡ ❝♦❡✣❝✐❡♥ts ❲❡ ♠✉st t❤❡r❡❢♦r❡ ❝♦♥tr♦❧ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✐♥ ❛ ❧❡ss ❞✐r❡❝t ✇❛②✳ ❚❤❡ ♠♦st ❝♦♠♠♦♥ ♠❡t❤♦❞ ✐s t♦ ♠❛❦❡ t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡ ♦❜❡② ❝❡rt❛✐♥ ❝♦♥str❛✐♥ts✱ ❛♥❞ t❤❡ ♠♦st ❝♦♠✲ ♠♦♥ ❝♦♥str❛✐♥ts ❛r❡ ♣♦s✐t✐♦♥ ❛♥❞ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥✱ ❛❧t❤♦✉❣❤ ♦t❤❡r ❝♦♥str❛✐♥ts✱ s✉❝❤ ❛s ❤✐❣❤❡r ❞❡r✐✈❛t✐✈❡s ❛♥❞ ❝✉r✈❛t✉r❡✱ ❛r❡ ❛❧s♦ ✉s❡❞✳ ❈♦♥str✉❝t✐♥❣ ❛ ❣❡♦♠❡tr✐❝ ❡❧❡♠❡♥t t♦ ♦❜❡② ❝♦♥str❛✐♥ts ♦❢ t❤✐s s♦rt ✐s ❝❛❧❧❡❞ ✐♥t❡r♣♦❧❛t✐♦♥ ✳ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ♠♦r❡ ❝♦❡✣❝✐❡♥ts t❤❡r❡ ❛r❡ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ♦r s✉r❢❛❝❡✱ t❤❡ ♠♦r❡ ❝♦♥str❛✐♥ts ✐t ✐s ❛❜❧❡ t♦ ♠❡❡t✳ ❚❤❛t €❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✸✹ ♠❡❛♥s t✇♦ t❤✐♥❣s✿ ❲❡ ♥❡❡❞ ❡♥♦✉❣❤ ❝♦❡✣❝✐❡♥ts ❢♦r t❤❡ ❥♦❜ ✐♥ ❤❛♥❞ ✭✉♥❧❡ss ✇❡✬r❡ ❣♦✐♥❣ t♦ ✉s❡ ♠♦r❡ t❤❛♥ ♦♥❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡✖s❡❡ ❧❛t❡r✮✳ ❲❡ ❞♦♥✬t ✇❛♥t t♦♦ ♠❛♥② ❝♦❡✣❝✐❡♥ts✱ ❛s t❤❡ ❡①tr❛ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ t❤❛t t❤❡s❡ ♣r♦✈✐❞❡ ❢♦r ✉s ✭♦r ❡♥❝✉♠❜❡r ✉s ✇✐t❤✮ ❤❛✈❡ t♦ ❜❡ ♠♦♣♣❡❞ ✉♣ s♦♠❡ ♦t❤❡r ✇❛②✳ ❙♦✱ ❧❡t✬s ❧♦♦❦ ❛t t❤❡ s♦rt ♦❢ ✐♥t❡r♣♦❧❛t✐♦♥s ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ♣♦✐♥ts✱ str❛✐❣❤t ❧✐♥❡s ❛♥❞ ❝✐r❝❧❡s✳ ❆ str❛✐❣❤t ❧✐♥❡ ✐s ❛❜❧❡ t♦ ❢✉❧✜❧ t✇♦ ❝♦♥✲ str❛✐♥ts✿ ✐t ❝❛♥ ❣♦ t❤r♦✉❣❤ t✇♦ ♣♦✐♥ts✱ ♦r ❣♦ t❤r♦✉❣❤ ❛ ♣♦✐♥t ❛♥❞ ❜❡ t❛♥❣❡♥t t♦ ❛ ❝✐r❝❧❡✳ ❆ ❝✐r❝❧❡ ❤❛s t❤r❡❡ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ s♦ ✐t ❝❛♥ ❣♦ t❤r♦✉❣❤ t❤r❡❡ ♣♦✐♥ts✱ ❣♦ t❤r♦✉❣❤ t✇♦ ❛♥❞ ❜❡ t❛♥❣❡♥t t♦ ❛ str❛✐❣❤t ❧✐♥❡✱ ❜❡ t❛♥❣❡♥t t♦ t✇♦ ❝✐r❝❧❡s ❛♥❞ ❛ ❧✐♥❡✱ ❛♥❞ ❧♦❛❞s ♠♦r❡❀ ♦r ✇❡ ❝❛♥ ✜① ✐ts r❛❞✐✉s✱ ❛♥❞ ✐t ❤❛s t✇♦ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❧✐❦❡ ❛ ❧✐♥❡✳ ❆❧t❤♦✉❣❤ t❤❡ ❣❡♦♠❡tr② ✐♥ ♣♦✐♥t✱ str❛✐❣❤t✲❧✐♥❡ ❛♥❞ ❝✐r❝❧❡ ❝♦♥✲ str✉❝t✐♦♥s ✐s s✐♠♣❧❡✱ t❤❡ ❝♦♥str✉❝t✐♦♥s ❛r❡ ✐♥t❡r❡st✐♥❣ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♣♦s✐t✐♦♥ ❛♥❞ t❛♥❣❡♥t ❝♦♥str❛✐♥ts ✐♥ ♠❛♥② ❞✐✛❡r❡♥t ❝♦♠❜✐♥❛t✐♦♥s✳ ❲✐t❤ ♠♦r❡ ❣❡♥❡r❛❧ ♣❛r❛♠❡tr✐❝ ♣♦❧②♥♦♠✐❛❧s✱ ✇❡ ❛r❡ ♥♦r♠❛❧❧② r❡✲ str✐❝t❡❞ t♦ s♣❡❝✐❢②✐♥❣ ♣♦s✐t✐♦♥ ❛♥❞ t❛♥❣❡♥t ✈❛❧✉❡s ❛t ♣❛rt✐❝✉❧❛r ✈❛❧✲ ✉❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡r✳ Lagrange interpolation ▲❛❣r❛♥❣✐❛♥ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❛❦❡s t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡ ♣❛ss t❤r♦✉❣❤ ❛ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts✳ ■t ❝❛♥ ♣❛ss t❤r♦✉❣❤ ❛ ♣♦✐♥t ❢♦r ❡✈❡r② ❝♦❡✣❝✐❡♥t ✐♥ ❡❛❝❤ ❡q✉❛t✐♦♥✳ ❋♦r ✐♥st❛♥❝❡✱ ❛ q✉❛❞r❛t✐❝ ✇✐❧❧ ❣♦ t❤r♦✉❣❤ t❤r❡❡✳ ❲❡ ❞❡❝✐❞❡ ✇❤❛t t❤❡ ❛❝t✉❛❧ ✈❛❧✉❡s ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ✇✐❧❧ ❜❡ ❜② s♦❧✈✐♥❣ ❛ s❡t ♦❢ s✐♠✉❧t❛♥❡♦✉s ❡q✉❛t✐♦♥s ✐♥ t❤❡ ❝♦❡✣❝✐❡♥ts✳ ❚❤❡s❡ ❛r❡ ♦❜t❛✐♥❡❞ ❜② s✉❜st✐t✉t✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❡❛❝❤ ♣♦✐♥t✖ (x, y, .

T∂u ❙❡❡ ■❧❧✉str❛t✐♦♥ ✸✭✐✐✐✮ ❢♦r ❛ s❦❡t❝❤ ♦❢ ❜♦t❤ ♦❢ t❤❡s❡ ❝❛s❡s✳ ❙✉❣❣❡st❡❞ s♦❧✉t✐♦♥s t♦ t❤✐s ♣r♦❜❧❡♠ ❤❛✈❡ ♣❛❞❞❡❞ ♦✉t ♠❛♥② ❛ t❤❡s✐s✳ ✭❚❤❛t✬s ✇❤② t❤❡②✬r❡ ❝❛❧❧❡❞ ❤✐❣❤❡r✲❞❡❣r❡❡ ♣❛t❝❤❡s✳✳✳✳✮ ❋♦r ♥♦✇✱ ❧❡t✬s ❧♦♦❦ ❛t s♦♠❡t❤✐♥❣ s✐♠♣❧❡✳ ❍♦✇ ❞♦ ✇❡ ❞r❛✇ ❛ ♣❛t❝❤❄ ❚❤❡ s✐♠♣❧❡st ✇❛② ✐s t♦ ♠❛❦❡ ❛ ❧✐♥❡ ❞r❛✇✐♥❣ ♦❢ ❛♥ ✐s♦✲♣❛r❛♠❡tr✐❝ ❣r✐❞✳ ▲❡t✬s ❧♦♦❦ ❛t s♦♠❡ ❝♦❞❡ t♦ ❞♦ t❤❛t✳ ❲❡ ❝♦✉❧❞ s✐♠♣❧② ✇r✐t❡ ❛ r♦✉t✐♥❡ t♦ ❡✈❛❧✉❛t❡ x✱ y ❛♥❞ z ✐♥ t❤❡ ♦❜✈✐♦✉s ✇❛②✱ ❛♥❞ ❦❡❡♣ ❝❛❧❧✐♥❣ t❤❛t❀ ❜✉t ✐t✬s ♥♦t ✈❡r② ❡✣❝✐❡♥t ❜❡❝❛✉s❡✱ ❛❧♦♥❣ ✐s♦✲♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ❡✐t❤❡r ♦r u t ✐s ✜①❡❞✱ ❛♥❞ s♦ ✇❡ ✇♦✉❧❞ ❜❡ ❞♦✐♥❣ ❛ ❧♦t ♦❢ r❡❝❛❧❝✉❧❛t✐♦♥✳ ❚❤❡ ❝♦❞❡ t❤❛t ❢♦❧❧♦✇s ❞r❛✇s ❛♥ ✐s♦✲♣❛r❛♠❡tr✐❝ str❛✐❣❤t ❧✐♥❡ ❛t ❛ s♣❡❝✐✜❡❞ ✈❛❧✉❡ ♦❢ u✱ ✈❛r②✐♥❣ t ✐♥ n st❡♣s✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ❛s t❤r❡❡ s❡ts ♦❢ ✈❛r✐❛❜❧❡s ❛①❬✶✻❪✱ ❛②❬✶✻❪✱ ❛③❬✶✻❪ ✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ s✉❜s❝r✐♣ts ✐♥ t❤❡ ❡q✉❛t✐♦♥s ❛❜♦✈❡ ❛♥❞ t❤❡ t❤r❡❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳ ❍❡r❡✬s t❤❡ r❡s✉❧t❀ ♥♦t❡ t❤❡ ♥❡st❡❞ ♦r ❍♦r♥❡r ❢♦r♠s ✹ ♦❢ t❤❡ ❡q✉❛✲ ✷ ❋♦r s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝♦♥t✐♥✉✐t② ♦♥ ❤✐❣❤❡r ❞❡r✐✈❛t✐✈❡s ✐s r❡q✉✐r❡❞✳ ✸ ❈♦♥t✐♥✉✐t② ♦❢ ♣❛r❛♠❡tr✐❝ ❞❡r✐✈❛t✐✈❡ ✐s ♥♦t ❛♥ ❡ss❡♥t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r s♠♦♦t❤✲ ♥❡ss❀ ♣❛t❝❤❡s ♠✐❣❤t ❛❧s♦ ❥♦✐♥ s♠♦♦t❤❧② ✐❢ t❤❡ t❛♥❣❡♥t ❞❡r✐✈❛t✐✈❡s ❛t t❤❡ ♠✉t✉❛❧ ❡❞❣❡ ✇❡r❡ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥s ❜✉t ❤❛❞ ❞✐✛❡r❡♥t ♠❛❣♥✐t✉❞❡s❀ ♦r t❤❡ ❞❡r✐✈❛✲ t✐✈❡s ♠✐❣❤t ♥♦t ♠❛t❝❤ ❛t ❛❧❧✱ ❜✉t t❤❡r❡ ❝♦✉❧❞ st✐❧❧ ❜❡ ❛ ❝♦♠♠♦♥ ♥♦r♠❛❧ ✭✐✳❡✳ t❤❡ s✉r❢❛❝❡s ❝♦✉❧❞ ❜❡ ❧♦❝❛❧❧② ❝♦♣❧❛♥❛r✮ ❛t t❤❡ ❥♦✐♥t✳ ✹ ❚❤❡ ❍♦r♥❡r ❢♦r♠ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ a0 + a1 t + a2 t2 + a3 t3 + · · · ✐s a0 + t(a1 + ✹✹ €❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✸✭✐✐✐✮✖❚❤❡ ❡①tr❛ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ♦❢ ❛ ❝✉❜✐❝ ♣❛t❝❤ ✐♥ ✐♥t❡r♣♦❧❛t✐♥❣ ♦✈❡r ❛ ♥❡t✇♦r❦ ♦❢ ❝✉r✈❡s✿ t✇✐st ✈❡❝t♦rs ✐♥ t❤❡ ❍❡r♠✐t❡ ❝❛s❡✱ ❛♥❞ ❝♦♥str❛✐♥ts ♦♥ ✐♥♥❡r ♣♦✐♥t ♣♦s✐t✐♦♥s ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ✐♥t❡r♣♦❧❛t✐♦♥✳ ❊❛❝❤ ❝♦♥str❛✐♥t ✐s s❤❛r❡❞ ✇✐t❤ ❢♦✉r ❛❞❥❛❝❡♥t ♣❛t❝❤❡s✱ t❤✉s ②✐❡❧❞✐♥❣ ❛♥ ❛✈❡r❛❣❡ ♦❢ ♦♥❡ ♣❡r ♣❛t❝❤✱ ❡①❝❡♣t ❛t t❤❡ ❡❞❣❡ ♦❢ ❛ ♣❛t❝❤❡❞ s✉r❢❛❝❡✳ ❙✉r❢❛❝❡ ♣❛t❝❤❡s ✹✺ t✐♦♥s ❛♣♣❡❛r ❛❣❛✐♥✳ ■♥ t❤✐s ❝❛s❡ t❤❡② s❛✈❡ ♣r❡✲❝❛❧❝✉❧❛t✐♦♥ ♦❢ u3 ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❧♦♦♣ ❛♥❞ ♦❢ t2 ❛♥❞ t3 ✇✐t❤✐♥ ✐t✳ u2 ❛♥❞ €❧♦tt✐♥❣ ✐s ♣❡r❢♦r♠❡❞ ❜② t✇♦ s♦♠❡✇❤❛t ♥♦t✐♦♥❛❧ r♦✉t✐♥❡s ❝❛❧❧❡❞ ♠♦✈❡✭①✱②✱③✮ ❛♥❞ ♣❧♦t✭①✱②✱③✮✱ ✇❤✐❝❤ ♠♦✈❡ t❤❡ ❵♣❡♥✬ ✐♥ ❵♣❡♥ ✉♣✬ ❛♥❞ ❵♣❡♥ ❞♦✇♥✬ ♠♦❞❡ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡② ❛r❡ ❦✐♥❞❧② ❣♦✐♥❣ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❥❡❝t✐♦♥✱ ❝❧✐♣♣✐♥❣ ❛♥❞ s♦ ♦♥ ❢♦r ✉s✳ ❢❧♦❛t ❛①❬✶✻❪✱ ❛②❬✶✻❪✱ ❛③❬✶✻❪❀ ✴✯ ✯ ❚❤❡ ♥❡①t ❝♦❡❢❢✐❝✐❡♥ts ❛r❡ t♦ ❜❡ ✉s❡❞ ✐♥ t❤❡ ✐♥♥❡r ✯ ❧♦♦♣✱ s♦ ✇❡ ❞♦♥✬t ✇❛♥t ❧♦ts ♦❢ ❛rr❛②✲s✉❜s❝r✐♣t ✯ ❛r✐t❤♠❡t✐❝❀ ❤❡♥❝❡ ♥♦ ❝①❬✹❪ ❡t❝✳ ✯✴ ❢❧♦❛t ❝①❴✵✱❝①❴✶✱❝①❴✷✱❝①❴✸✱ ❝②❴✵✱❝②❴✶✱❝②❴✷✱❝②❴✸✱ ❝③❴✵✱❝③❴✶✱❝③❴✷✱❝③❴✸❀ ❢❧♦❛t t✱ ✉✱ ❞t✱ ①✱ ②✱ ③❀ ✐♥t ✐✱ ♥❀ ❝①❴✵ ❝①❴✶ ❝①❴✷ ❝①❴✸ ❂ ❂ ❂ ❂ ❛①❬✸❪ ❛①❬✼❪ ❛①❬✶✶❪ ❛①❬✶✺❪ ✰ ✰ ✰ ✰ ✉✯✭❛①❬✷❪ ✉✯✭❛①❬✻❪ ✉✯✭❛①❬✶✵❪ ✉✯✭❛①❬✶✹❪ ✰ ✰ ✰ ✰ ✉✯✭❛①❬✶❪ ✉✯✭❛①❬✺❪ ✉✯✭❛①❬✾❪ ✉✯✭❛①❬✶✸❪ ✰ ✰ ✰ ✰ ✉✯❛①❬✵❪ ✮✮❀ ✉✯❛①❬✹❪ ✮✮❀ ✉✯❛①❬✽❪ ✮✮❀ ✉✯❛①❬✶✷❪✮✮❀ ❝②❴✵ ❝②❴✶ ❝②❴✷ ❝②❴✸ ❂ ❂ ❂ ❂ ❛②❬✸❪ ❛②❬✼❪ ❛②❬✶✶❪ ❛②❬✶✺❪ ✰ ✰ ✰ ✰ ✉✯✭❛②❬✷❪ ✉✯✭❛②❬✻❪ ✉✯✭❛②❬✶✵❪ ✉✯✭❛②❬✶✹❪ ✰ ✰ ✰ ✰ ✉✯✭❛②❬✶❪ ✉✯✭❛②❬✺❪ ✉✯✭❛②❬✾❪ ✉✯✭❛②❬✶✸❪ ✰ ✰ ✰ ✰ ✉✯❛②❬✵❪ ✮✮❀ ✉✯❛②❬✹❪ ✮✮❀ ✉✯❛②❬✽❪ ✮✮❀ ✉✯❛②❬✶✷❪✮✮❀ ❝③❴✵ ❝③❴✶ ❝③❴✷ ❝③❴✸ ❂ ❂ ❂ ❂ ❛③❬✸❪ ❛③❬✼❪ ❛③❬✶✶❪ ❛③❬✶✺❪ ✰ ✰ ✰ ✰ ✉✯✭❛③❬✷❪ ✉✯✭❛③❬✻❪ ✉✯✭❛③❬✶✵❪ ✉✯✭❛③❬✶✹❪ ✰ ✰ ✰ ✰ ✉✯✭❛③❬✶❪ ✉✯✭❛③❬✺❪ ✉✯✭❛③❬✾❪ ✉✯✭❛③❬✶✸❪ ✰ ✰ ✰ ✰ ✉✯❛③❬✵❪ ✮✮❀ ✉✯❛③❬✹❪ ✮✮❀ ✉✯❛③❬✽❪ ✮✮❀ ✉✯❛③❬✶✷❪✮✮❀ t(a2 + t(a3 + · · ·)))✳ ■t s❛✈❡s ❛r✐t❤♠❡t✐❝ ✇❤❡♥ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✐s ❜❡✐♥❣ ❡✈❛❧✉❛t❡❞✳ €❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✹✻ t ❂ ✵✳✵❀ ❞t ❂ ✶✳✵✴✭❢❧♦❛t✮✭♥✲✶✮❀ ♠♦✈❡✭❝①❴✸✱❝②❴✸✱❝③❴✸✮❀ ❢♦r ✭✐❂✶❀ ✐❁♥❀ ✐✰✰✮ ✴✯ ❲❡ ✇❛♥t t♦ ❧♦♦♣ ♥✲✶ t✐♠❡s ✯✴ ④ t ❂ t ✰ ❞t❀ ① ❂ ❝①❴✸ ✰ t✯✭❝①❴✷ ✰ t✯✭❝①❴✶ ✰ t✯❝①❴✵✮✮❀ ② ❂ ❝②❴✸ ✰ t✯✭❝②❴✷ ✰ t✯✭❝②❴✶ ✰ t✯❝②❴✵✮✮❀ ③ ❂ ❝③❴✸ ✰ t✯✭❝③❴✷ ✰ t✯✭❝③❴✶ ✰ t✯❝③❴✵✮✮❀ ⑥ ♣❧♦t✭①✱②✱③✮❀ Splines ■t ✐s ♥♦t ❢❡❛s✐❜❧❡ t♦ ♠❛❦❡ ❛ ❧♦♥❣ ❝✉r✈❡ ♦r ❝♦♠♣❧✐❝❛t❡❞ s✉r❢❛❝❡ ✇✐t❤ ❛ s✐♥❣❧❡ ❤✐❣❤✲❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧✳ ❖♥❡ ♣r♦❜❧❡♠ ✐s t❤❛t ♦❢ ♣❛r❛♠❡t❡r✐✲ ③❛t✐♦♥❀ ♣♦♦r❧② ❝❤♦s❡♥ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t❤❡ ❞❛t❛ ♣♦✐♥ts ♠❛❦❡ t❤❡ ❝✉r✈❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ✇✐❣❣❧② ❛s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❝✉r✈❡ ✐♥❝r❡❛s❡s✳ ❆❧s♦✱ ❤✐❣❤✲❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧s ✭t❤✐s ❝❛t❡❣♦r② ✉s✉❛❧❧② st❛rts s♦♠❡✲ ✇❤❡r❡ ❜❡t✇❡❡♥ ❞❡❣r❡❡ ✻ ❛♥❞ ✶✵✮ ❛r❡ ❡①♣❡♥s✐✈❡ t♦ ❝♦♠♣✉t❡✖❜❡❝❛✉s❡ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ t❡r♠s✖❛♥❞ ❡①tr❡♠❡❧② s❡♥s✐t✐✈❡ t♦ ✐♥❛❝❝✉r❛❝✐❡s ✐♥ t❤❡✐r ❝♦❡✣❝✐❡♥ts✳ ✭❚❤❡r❡ ✐s ♠♦r❡ ❛❜♦✉t t❤✐s ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳✮ ❚❤❡ ♦❜✈✐♦✉s s♦❧✉t✐♦♥ ✭✇❡❧❧✱ ❢❛✐r❧② ♦❜✈✐♦✉s s♦❧✉t✐♦♥✮ ✐s t♦ ❦♥♦t ❛ ❧♦t ♦❢ s✐♠♣❧❡ ♣✐❡❝❡s ♦❢ ❝✉r✈❡ ♦r s✉r❢❛❝❡ t♦❣❡t❤❡r✱ ❛♥❞ ❛ ❧♦t ♦❢ ❡♥❡r❣② ❤❛s ❜❡❡♥ ❢r✐tt❡r❡❞ ❛✇❛② ♦✈❡r t❤✐s ✈❡r② ♠❛tt❡r✳ ❲❤②❄ ■t ✐s✱ ②♦✉ ♠✐❣❤t t❤✐♥❦✱ ❛s ❡❛s② t♦ ♠❛❦❡ t❡♥ ❝✉r✈❡s ❛s ♦♥❡✱ ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥s t❤❛t ❡❛❝❤ ♦♥❡ ✐s t♦ ♠❡❡t ❛r❡ ♣r❡❝✐s❡❧② ❛♥❞ ✐♥❞✐✈✐❞✉❛❧❧② ❞❡✜♥❡❞❀ ❜✉t ✉s✉✲ ❛❧❧② t❤❡② ❛r❡ ♥♦t✳ ▼♦r❡ ❢r❡q✉❡♥t❧② ✇❡ ❤❛✈❡ s♦♠❡ ❞❛t❛ s♣❡❝✐✜❡❞ ❢♦r ❡❛❝❤ s♣❛♥✱ t②♣✐❝❛❧❧② ✐ts ♣♦s✐t✐♦♥❀ ❛♥❞ ♦t❤❡r r❡q✉✐r❡♠❡♥ts✱ t②♣✐❝❛❧❧② t❛♥❣❡♥t ♦r ❝✉r✈❛t✉r❡ ❝♦♥t✐♥✉✐t②✱ ❛r❡ s♣❡❝✐✜❡❞ ♦✈❡r t❤❡ ✇❤♦❧❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡✳ ❙♦ ✇❡ ♥❡❡❞ t♦ ✐♥✈❡♥t ✈❛❧✉❡s ❢♦r ❧♦❝❛❧ ❞❛t❛ t❤❛t s❛t✐s❢② t❤❡ ❣❧♦❜❛❧ ❝♦♥str❛✐♥t✱ ❛♥❞ ♣♦ss✐❜❧② ❛r❡ ❛❧s♦ ♦♣t✐♠❛❧ ✐♥ s♦♠❡ ❞❡✜♥❡❞ ✇❛②✳ €❡r❤❛♣s ✇❡ ✇❛♥t t♦ ♠✐♥✐♠✐③❡ ♦r ♠❛①✐♠✐③❡ t❤❡ ✐♥t❡❣r❛❧ ♦❢ s♦♠❡ ❙♣❧✐♥❡s ✹✼ q✉❛♥t✐t② ♦✈❡r t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡✱ ♦r ♣❡r❤❛♣s ✇❡✬❧❧ ❜❡ s❛t✐s✜❡❞ ✇✐t❤ ❛ ❝✉r✈❡ t❤❛t ✐s ♦♣t✐♠❛❧ ✐♥ t❤❡ ❞❡s✐❣♥❡r✬s ♦♣✐♥✐♦♥✳ ❚❤❡ ✇♦r❞ s♣❧✐♥❡ ❝♦✈❡rs ❛❧❧ s✉❝❤ ♣✐❡❝❡✇✐s❡ ❝✉r✈❡ ❛♥❞ s✉r❢❛❝❡ t❡❝❤✲ ♥✐q✉❡s✱ ❛♥❞ ✇❡✬❧❧ ❝❧❛ss✐❢② s♣❧✐♥❡s ✐♥t♦ t❤r❡❡ t②♣❡s✿ Sequential splining ❚❤✐s ✐s ❛ t❡r♠ t❤❛t ✇❡✬✈❡ ❥✉st ♠❛❞❡ ✉♣✿ ❛ ♣❤r❛s❡ t♦ ❞❡s❝r✐❜❡ t❤❡ s✐♠♣❧❡st ❛♣♣r♦❛❝❤ t♦ t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ ♣✐❡❝❡✇✐s❡ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✳ ❲❡ st❛rt ✇✐t❤ ♦♥❡ ♣✐❡❝❡ ♦❢ t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡✖✉s✉❛❧❧② t❤❡ ❧❛r❣❡st ❛♥❞ ♠♦st ✐♠♣♦rt❛♥t✖❛♥❞ t❤❡♥ ❥♦✐♥ ❢✉rt❤❡r ♣✐❡❝❡s t♦ ✐t✳ ❚❤❡s❡ ❡①tr❛ ♣✐❡❝❡s ✇✐❧❧ ♥❡❡❞ ❛t ❧❡❛st ❡♥♦✉❣❤ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ t♦ ♠❡❡t t❤❡ ❝♦♥str❛✐♥ts ✐♠♣♦s❡❞ ❜② ✇❤❛t✬s ❛❧r❡❛❞② ✐♥ ♣❧❛❝❡✱ ❛♥❞ ②♦✉✬❞ ❜❡tt❡r ❤❛✈❡ s♦♠❡ ♠♦r❡ ✐♥ ❤❛♥❞✱ ♦t❤❡r✇✐s❡ t❤❡ ♥❡✇ ❣❡♦♠❡tr② ✐s t♦t❛❧❧② ❞❡t❡r♠✐♥❡❞✱ ❛♥❞ ✇✐❧❧ ♣r♦❜❛❜❧② st❛rt t♦ ♦s❝✐❧❧❛t❡ ✇✐❧❞❧② ❛ ❢❡✇ ❝✉r✈❡ ♦r s✉r❢❛❝❡ ❡❧❡♠❡♥ts ♦✉t ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ ♣✐❡❝❡ ♦❢ ❣❡♦♠❡tr②✳ ❚❤✐s ✐s r❛t❤❡r ❛ ❢❛✐♥t✲❤❡❛rt❡❞ ✇❛② t♦ ❣♦ ❛❜♦✉t ❞❡s✐❣♥✐♥❣ ❧♦♥❣ ❝✉r✈❡s✱ ❜✉t ✐t✬s ✈❡r② s✐♠♣❧❡❀ ❡❛❝❤ ♥❡✇ ❝✉r✈❡ s❡❣♠❡♥t ♠❡❡ts ♥♦✱ ♦♥❡✱ ♦r t✇♦ ❞❡✜♥❡❞ ❡♥❞✲❝♦♥❞✐t✐♦♥s✱ ❛♥❞ ❝❛♥ ❜❡ ❝r❡❛t❡❞ ❜② ❍❡r♠✐t❡ ✐♥t❡r✲ ♣♦❧❛t✐♦♥ ♦r s✐♠✐❧❛r ♣r♦❝❡ss❡s✳ ❋♦r s✉r❢❛❝❡ ♣❛t❝❤❡s✱ t❤✐s ♠❡t❤♦❞ ♦❢ ❝♦♥str✉❝t✐♦♥ ✐s ❛ ❝♦♠♠♦♥ ♦♥❡❀ ❛❞❞✐♥❣ ♥❡✇ ♣❛t❝❤❡s ✐s ♠♦r❡ ❝♦♠♣❧✐✲ ❝❛t❡❞✱ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥❞✐t✐♦♥s t♦ ❜❡ ♠❡t ❣r♦✇s ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s t❤❛t ❛r❡ s❤❛r❡❞ ✇✐t❤ ♣❛t❝❤❡s t❤❛t ❤❛✈❡ ❛❧r❡❛❞② ❜❡❡♥ ❝r❡❛t❡❞ ✭s❡❡ ■❧❧✉str❛t✐♦♥ ✸✭✐✈✮✮✳ ❚❤✉s✱ ❦♥♦✇✐♥❣ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t♦ ❝r❡❛t❡ ❛ ♣❛t❝❤❡❞ s✉r❢❛❝❡ ✐s ❛ s✐❣♥✐✜❝❛♥t ♣✐❡❝❡ ♦❢ ❞❡s✐❣♥ ❡①♣❡rt✐s❡✱ ❛♥❞ s♦ t❤❡ ♠❡t❤♦❞ ✐s ❞✐✣❝✉❧t t♦ ❛✉t♦♠❛t❡✳ Local splining ❚♦ ❛✈♦✐❞ ♦r❞❡r✲❞❡♣❡♥❞❡♥❝❡✱ ✇❡ s❤♦✉❧❞ ❧✐❦❡ t♦ ❞❡t❡r♠✐♥❡ ❛❧❧ t❤❡ s♣❛♥s ♦r ♣✐❡❝❡s ♦❢ s✉r❢❛❝❡ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❖♥❡ ✇❛② ♦❢ ❞♦✐♥❣ t❤❛t ✐s t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❛ ♥✉♠❜❡r ♦❢ ❞❛t❛ ♣♦✐♥ts t❤❛t ❛r❡ ♥❡❛r✖❜✉t ♥♦t ♦♥ ♦r ❛❞❥♦✐♥✐♥❣✖❛ ♣❛rt✐❝✉❧❛r s♣❛♥ ♦r ♣✐❡❝❡ ♦❢ s✉r❢❛❝❡✳ ❚❤❡ ❇✲s♣❧✐♥❡ ✇❡ s❤❛❧❧ ♠❡❡t ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r ❞♦❡s ❡①❛❝t❧② t❤✐s✱ ❛❧t❤♦✉❣❤ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ❇✲s♣❧✐♥❡ s✉r❢❛❝❡✱ ❥✉st ✇✐t❤✐♥ ♣❛t❝❤❡s✱ s♦ t❤❡ ♦✈❡r❛❧❧ s❝❤❡♠❡ ✐s ❛ ❤②❜r✐❞ ❜❡t✇❡❡♥ ❧♦❝❛❧ ❛♥❞ s❡q✉❡♥t✐❛❧ s♣❧✐♥❡ ✐♥t❡r♣♦❧❛t✐♦♥✳ ❇✉t t❤❡ s❡q✉❡♥t✐❛❧ st❛❣❡ ✐s ❡❛s✐❡r ❜❡❝❛✉s❡ t❤❡ ❇✲s♣❧✐♥❡ ♣❛t❝❤❡s✱ ❜❡✐♥❣ ❛ss❡♠❜❧❡❞ ❢r♦♠ ♣✐❡❝❡s ♦❢ s✐♠♣❧❡r s✉r❢❛❝❡✱ ❝❛♥ ❜❡ ❧❛r❣❡r✿ ✇❡❧❧✱ t❤❛t✬s t❤❡ s❛❧❡s t❛❧❦✳ €❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✹✽ ✸✭✐✈✮✖❲❤❡♥ ❝♦♥str✉❝t✐♥❣ ❛ ❧❛r❣❡ ♣❛t❝❤❡❞ s✉r❢❛❝❡✱ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❛✈♦✐❞ ♣❛t❝❤❡s t❤❛t ❥♦✐♥ ♦t❤❡rs ❛t ♦♥❡ ✭❜✮ ❡❞❣❡s✳ €❛t❝❤❡s t❤❛t ♠❡❡t ♦t❤❡rs ❛t ❢♦✉r ✭❞✮ ❡❞❣❡s ❛r❡ ❡♥❝♦✉♥t❡r❡❞ ✐♥ ❡❞✐t✐♥❣ ❛ t✇♦ t❤r❡❡ ✭❛✮ ✭❝✮ s✉r❢❛❝❡✳ ❛♥❞ ❛♥❞ ❚②♣❡s ♦❢ ❝♦♥t✐♥✉✐t② ✹✾ ■t ✐s ❡❛s② t♦ ✐♥✈❡♥t ✭❝♦♥❝❡♣t✉❛❧❧②✮ s✐♠♣❧❡r ❧♦❝❛❧ s♣❧✐♥❡s✿ ❛ ❢❛✈♦✉r✐t❡ ✐s t❤❡ ❖✈❡r❤❛✉s❡r ❝✉r✈❡ ✭s❡❡ ♦✉r ♦t❤❡r ❡✛♦rt ❡tr②✮✳ ❆ €r♦❣r❛♠♠❡r✬s ●❡♦♠✲ ❚❤❡ ✐❞❡❛ ✐s t♦ ✜t s♦♠❡ s✉❜s❡t ♦❢ t❤❡ ❞❛t❛ ❛✈❛✐❧❛❜❧❡ ✇✐t❤ ❛ s✐♠♣❧❡ ❝✉r✈❡ s❡❣♠❡♥t✳ ■♥ ❡✛❡❝t ✇❡ ❛r❡ ❣❡♥❡r❛t✐♥❣ t❛♥❣❡♥t ♦r ❤✐❣❤❡r✲ ❞❡r✐✈❛t✐✈❡ ❞❛t❛ ❢r♦♠ t❤❡ ♣♦s✐t✐♦♥ ❞❛t❛ s✉♣♣❧✐❡❞✳ ❚❤♦✉❣❤ ❞❡♣❡♥❞❡♥t ✉♣♦♥ t❤❡s❡ ❞❛t❛✱ t❤❡s❡ t❛♥❣❡♥ts ❛♥❞ s♦ ♦♥ ❛r❡ ♥♦♥✲✉♥✐q✉❡✖✇❡ ❝♦✉❧❞ ❥✉st ❛s ❡❛s✐❧② ✉s❡ ♦t❤❡rs ♦❢ ❞✐✛❡r✐♥❣ ♠❛❣♥✐t✉❞❡✱ ❢♦r ❡①❛♠♣❧❡✳ ❖♥❝❡ t❤❡ t❛♥❣❡♥ts ❛♥❞ ❤✐❣❤❡r ❞❡r✐✈❛t✐✈❡s ❛r❡ ❞❡✜♥❡❞✱ ❡❛❝❤ s♣❛♥ ❝❛♥ ❜❡ ✐♥t❡r♣♦❧❛t❡❞ s❡♣❛r❛t❡❧②✳ Global splining ❚❤✐s ✐s t❤❡ r❡❛❧ ▼❝❈♦②✳ ◆♦✇ ✇❡ tr② t♦ ❞❡t❡r♠✐♥❡ ❛ ✇❤♦❧❡ ❝✉r✈❡ ✺ ✐♥ ❛ s✐♥❣❧❡ ♣r♦❝❡ss✳ ❚❤❡ t❛♥❣❡♥ts✱ ♦r ♦t❤❡r ❝r✐t❡r✐❛ t❤❛t ♠✉st ❜❡ ♠❛t❝❤❡❞ ❛t t❤❡ ❦♥♦ts ❜❡t✇❡❡♥ t❤❡ s♣❛♥s✱ ❛r❡ ❡q✉❛t❡❞ t♦❣❡t❤❡r ✐♥ ❛♣♣r♦♣r✐❛t❡ ♣❛✐rs✱ ❛♥❞ t❤❡ ❧❛r❣❡ s❡t ♦❢ ❡q✉❛t✐♦♥s t❤❛t r❡s✉❧ts ✐s s♦❧✈❡❞✱ ②✐❡❧❞✐♥❣ t❤❡ ❡♥❞ ❝♦♥❞✐t✐♦♥s ❢r♦♠ ✇❤✐❝❤ t❤❡ s♣❛♥s ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞✳ ❚❤❡ ❜❡♥❡✜t ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s ❛t ❧❡❛st t❤❡ ♦♣♣♦rt✉♥✐t② t♦ ❣❡t ❛ ❵❜❡tt❡r✬ ❝✉r✈❡❀ ✇❤❛t❡✈❡r ✇❡ ❛r❡ ♦♣t✐♠✐③✐♥❣ ✇✐❧❧ ❜❡ ♦♣t✐♠✐③❡❞ ♦✈❡r t❤❡ ✇❤♦❧❡ ❝✉r✈❡✱ ♥♦t ❥✉st s♦♠❡ s❡❣♠❡♥ts✳ ❚❤❡ ❞✐s❛❞✈❛♥t❛❣❡ ✐s t❤❛t ❝❤❛♥❣❡s t♦ t❤❡ ❞❛t❛ ❞❡✜♥✐♥❣ ❛ ❣❧♦❜❛❧ s♣❧✐♥❡ ♣r♦❧✐❢❡r❛t❡✖❛t ❧❡❛st t♦ s♦♠❡ ❡①t❡♥t✖❛❧♦♥❣ t❤❡ ✇❤♦❧❡ ❝✉r✈❡✱ s♦ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t ❛♥ ✐♠♣r♦✈❡✲ ♠❡♥t t♦ ❛ ❝✉r✈❡ ✐♥ ♦♥❡ ♣❧❛❝❡ ✇✐❧❧ ✇r❡❝❦ ✐t ❡❧s❡✇❤❡r❡✳ ❚❤❡r❡ ✐s ❛❧s♦ t❤❛t ❧❛r❣❡ s❡t ♦❢ s✐♠✉❧t❛♥❡♦✉s ❡q✉❛t✐♦♥s t♦ s♦❧✈❡✱ ❢♦r ✇❤✐❝❤ ♠❛tr✐① ♠❡t❤♦❞s ❛r❡ ♣r❡❢❡rr❡❞❀ t❤❡ ♠❛tr✐❝❡s ❛r❡ str♦♥❣❧② ❞✐❛❣♦♥❛❧✐③❡❞ ❛♥❞ r❡❧❛t✐✈❡❧② ❡❛s② t♦ ❞❡❛❧ ✇✐t❤ ✭s❡❡ t❤❡ ❜♦♦❦ ◆✉♠❡r✐❝❛❧ ❘❡❝✐♣❡s ✮✳ Types of continuity ❆t t❤❡ ❦♥♦ts ♦❢ s♣❧✐♥❡ ❝✉r✈❡s ✭✇❤❡r❡ t✇♦ ♣✐❡❝❡s✖♦r s♣❛♥s✖❛r❡ ❥♦✐♥❡❞✮ ✇❡ ❞♦ ♥♦t ✉s✉❛❧❧② ❛tt❡♠♣t t♦ ♠❛t❝❤ ❛❧❧ t❤❡ ♥♦♥✲③❡r♦ ❞❡r✐✈❛✲ t✐✈❡s t❤❛t t❤❡ s♣❛♥s ♣♦ss❡ss✳ ❈✉❜✐❝ s♣❛♥s✱ ❢♦r ✐♥st❛♥❝❡✱ ❤❛✈❡ ❛ ♥♦♥✲③❡r♦ t❤✐r❞ ❞❡r✐✈❛t✐✈❡❀ ❜✉t ❝♦♠♠♦♥❧② ♦♥❧② ♣♦s✐t✐♦♥ ❛♥❞ ✜rst ❞❡r✐✈❛t✐✈❡✖❛♥❞ ♣♦ss✐❜❧② s❡❝♦♥❞ t♦♦✖❛r❡ ❡♥❢♦r❝❡❞ ❛❝r♦ss t❤❡ ❦♥♦ts✳ 0 ❚❤❡s❡ ❞❡❣r❡❡s ♦❢ ❝♦♥t✐♥✉✐t② ❛r❡ ❝♦♠♠♦♥❧② ✇r✐tt❡♥ C ✭♣♦s✐t✐♦♥✮✱ ✺ ❙✉r❢❛❝❡s ❛r❡ ♥♦t ❣❧♦❜❛❧❧② s♣❧✐♥❡❞✱ ❜✉t s♦♠❡ t❡❝❤♥✐q✉❡s ❞♦ ❡①✐st ❢♦r s♣❧✐♥✐♥❣ ❛ ♠❡s❤ ♦❢ ❝✉r✈❡s✱ ✇❤❡r❡ t❤❡ ❦♥♦ts ✐♥ t❤❡ t✇♦ s❡ts ❛r❡ ❝♦✐♥❝✐❞❡♥t ❛♥❞ s❤❛r❡ t❛♥❣❡♥t ♣❧❛♥❡s✳ €❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✺✵ C1 ✱ C2 ❡t❝✳ ❲❡ s❤♦✉❧❞ ❜❡❛r ✐♥ ♠✐♥❞ t❤❛t s♣❧✐♥❡s ✇❡r❡ ♦r✐❣✐♥❛❧❧② ❝♦♥❝❡✐✈❡❞ ❛s ❡①♣❧✐❝✐t ❝✉r✈❡s ✭r❡♠✐♥❞❡r✿ y = f (x)✮ ❛♥❞ t❤❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥✖ ✇❤✐❝❤ ✇❡ ♥❡❡❞ t♦ ❣❡t ❣❡♦♠❡tr✐❝ ✢❡①✐❜✐❧✐t②✖✐s ❛ ❧✐❛❜✐❧✐t②✳ ❚❤✉s✱ ❝♦♥t✐♥✉✐t② ✐♥ ♣❛r❛♠❡t❡r ❞❡r✐✈❛t✐✈❡ ✐s ♥♦t t❤❡ s❛♠❡ ❛s ❝♦♥t✐♥✉✐t② ✐♥ ❛ ❣❡♦♠❡tr✐❝ s❡♥s❡❀ t❤✐s ✐s ❛ ✈❛r✐❛♥t ♦❢ t❤❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ♣r♦❜✲ ❧❡♠s ✇❡ s❛✇ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❋✐rst✲ ❛♥❞ s❡❝♦♥❞✲❞❡❣r❡❡ ❣❡♦♠❡tr✐❝ ❝♦♥t✐♥✉✐t② ❛r❡ ❝♦♥t✐♥✉✐t② ♦❢ ✉♥✐t t❛♥❣❡♥t ✈❡❝t♦r✱ ❛♥❞ ❝♦♥t✐♥✉✐t② ♦❢ ❝✉r✈❛t✉r❡✳ ❚❤❡② ✇♦✉❧❞ ❜❡ t❤❡ s❛♠❡ ❛s ♣❛r❛♠❡tr✐❝ ❝♦♥t✐♥✉✐t② ✐❢ ✇❡ ❤❛❞ ❛ tr✉❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✐③❛t✐♦♥❀ s✐♥❝❡ ✇❡ ♥❡✈❡r ❞♦✱ t❤❛t✬s ♥♦t s✉❝❤ ❛♥ ✐♥s✐❣❤t ✭s❡❡ ❋❛r♦✉❦✐ ❛♥❞ ❙❛❦❦❛❧✐s✬ ♣❛♣❡r ♦♥ t❤❡ ✐♠♣r❡s✲ s✐✈❡❧② ♥❛♠❡❞ €②t❤❛❣♦r❡❛♥ ❤♦❞♦❣r❛♣❤s ✱ ✇❤✐❝❤ ❤❛✈❡ ❛t ❧❡❛st ❛ r❛t✐♦♥❛❧ ♣♦❧②♥♦♠✐❛❧ ❡①♣r❡ss✐♦♥ ❢♦r ❛r❝ ❧❡♥❣t❤✮✳ ❚❤❡ ✉♥✐t t❛♥❣❡♥t ✐s ❡❛s✐❧② ✇r✐tt❡♥ ❞♦✇♥✿ dQ dt dQ .

X = a2 t3 + a6 t2 + a10 t + a14 . ∂u u=0 ❙✉r❢❛❝❡ ♣❛t❝❤❡s ✹✸ ❆♥② ♦t❤❡r ♣❛t❝❤ ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❜♦✉♥❞❛r② ❝✉r✈❡ ❛♥❞ ❞❡r✐✈❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ ✷ ❛t ✐ts ❡❞❣❡ ✇✐❧❧ ♠❛t❝❤ t❤✐s ♣❛t❝❤ ❛t ✐ts ✸ u = 0 ❡❞❣❡❀ s✐♠✐❧❛r ❝♦♥str❛✐♥ts ❛♣♣❧② ❛t t❤❡ ♦t❤❡r ❡❞❣❡s ✳ ■♥ ❛ ❝♦♠♠♦♥ ❝❛s❡✱ ✇❡ ❤❛✈❡ ❛ ♥❡t✇♦r❦ ♦❢ s♣❛❝❡ ❝✉r✈❡s r❡❛❞②✲ ❞❡s✐❣♥❡❞✳ ❆♥♥♦②✐♥❣❧②✱ ✐t ✇♦r❦s ♦✉t t❤❛t ❜✐❝✉❜✐❝ ♣❛t❝❤❡s ❤❛✈❡ ❥✉st ♦♥❡ t♦♦ ♠❛♥② ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ✭✐♥ ❡❛❝❤ ❞✐♠❡♥s✐♦♥✮ t♦ s✉r❢❛❝❡ s✉❝❤ ❛ ♥❡t✇♦r❦ ✇✐t❤♦✉t t❤❡ s✉♣♣❧② ♦❢ ❛❞❞✐t✐♦♥❛❧ ❞❛t❛✳ ✭❍✐❣❤❡r✲❞❡❣r❡❡ ♣❛t❝❤❡s ❤❛✈❡ ❧♦ts ♦❢ ❡①tr❛ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ q✉❛❞r❛t✐❝s ❞♦♥✬t ❤❛✈❡ ❡♥♦✉❣❤✳✮ ■❢ t❤❡ ♣❛t❝❤❡s ❛r❡ ❜❡✐♥❣ ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❍❡r♠✐t❡ t❡❝❤✲ ♥✐q✉❡✱ ♦r ❛s ❛ ❣❡♦♠❡tr✐❝ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❛❧❧♦✇❛❜❧❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r♥❛❧ ♣♦✐♥ts ✐♥ ❛❞❥❛❝❡♥t ♣❛t❝❤❡s ✭♦r✖❧♦♦❦✐♥❣ ❛❤❡❛❞✖t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡rt✐❝❡s ♦❢ ❛ ❇é③✐❡r ❝♦♥tr♦❧ ♠❡s❤✮✱ t❤❡♥ t❤❡ ❡①tr❛ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❡♠❡r❣❡ ❛s s♦✲❝❛❧❧❡❞ t✇✐st ✈❡❝t♦rs ❛t t❤❡ ♣❛t❝❤ ❝♦r♥❡rs✿ ∂ 2 Q(t, u) .

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