ΜΕΤΑΠΤΥΧΙΑΚΗ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. Ελευθερίου Β. Χρυσούλα. Επιβλέπων: Νικόλαος Καραμπετάκης Καθηγητής Α.Π.Θ.

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ΜΕΤΑΠΤΥΧΙΑΚΟ ΠΡΟΓΡΑΜΜΑ ΣΠΟΥ ΩΝ ΘΕΩΡΗΤΙΚΗ ΠΛΗΡΟΦΟΡΙΚΗ ΚΑΙ ΘΕΩΡΙΑ ΣΥΣΤΗΜΑΤΩΝ ΚΑΙ ΕΛΕΓΧΟΥ Αναγνώριση συστημάτων με δεδομένη συνεχή και κρουστική συμπεριφορά ΜΕΤΑΠΤΥΧΙΑΚΗ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ Ελευθερίου Β. Χρυσούλα Επιβλέπων: Νικόλαος Καραμπετάκης Καθηγητής Α.Π.Θ. Θεσσαλονίκη, εκέμβριος 24

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ΜΕΤΑΠΤΥΧΙΑΚΟ ΠΡΟΓΡΑΜΜΑ ΣΠΟΥ ΩΝ ΘΕΩΡΗΤΙΚΗ ΠΛΗΡΟΦΟΡΙΚΗ ΚΑΙ ΘΕΩΡΙΑ ΣΥΣΤΗΜΑΤΩΝ ΚΑΙ ΕΛΕΓΧΟΥ Αναγνώριση συστημάτων με δεδομένη συνεχή και κρουστική συμπεριφορά ΜΕΤΑΠΤΥΧΙΑΚΗ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ Ελευθερίου Β. Χρυσούλα Επιβλέπων: Νικόλαος Καραμπετάκης Καθηγητής Α.Π.Θ. Εγκρίθηκε από την τριμελή εξεταστική επιτροπή την η εκεμβρίου 24. Ν. Καραμπετάκης E. Αντωνίου Ο. Κοσμίδου Καθηγητής Α.Π.Θ. Επικ. Καθηγητής Α.Τ.Ε.Ι.Θ. Αναπλ. Καθηγήτρια.Π.Θ.

.. Ελευθερίου Β. Χρυσούλα Πτυχιούχος Μαθηματικός Α.Π.Θ. Copyright Ελευθερίου Β. Χρυσούλα, 24. Με επιφύλαξη παντός δικαιώματος. All rights reserved. Απαγορεύεται η αντιγραφή, αποθήκευση και διανομή της παρούσας εργασίας, εξ ολοκλήρου ή τμήματος αυτής, για εμπορικό σκοπό. Επιτρέπεται η ανατύπωση, αποθήκευση και διανομή για σκοπό μη κερδοσκοπικό, εκπαιδευτικής ή ερευνητικής φύσης, υπό την προϋπόθεση να αναφέρεται η πηγή προέλευσης και να διατηρείται το παρόν μήνυμα. Ερωτήματα που αφορούν τη χρήση της εργασίας για κερδοσκοπικό σκοπό πρέπει να απευθύνονται προς τον συγγραφέα. Οι απόψεις και τα συμπεράσματα που περιέχονται σε αυτό το έγγραφο εκφράζουν τον συγγραφέα και δεν πρέπει να ερμηνευτεί ότι εκφράζουν τις επίσημες θέσεις του Α.Π.Θ.

A(ρ)β(t) = ρ := d dt C

m n A =[a ij ] [a ij ] A(s) =[a ij (s)] a ij (s) R A(s) R m,n [s] A ν = ( aij) ν i =, 2,..., m j =, 2,..., n ν =,,..., k A(s) =A + A s +... + A k s k + A k s k A k A(s) s A(s) R p m [s] R [s] R A i R p m i =,,..., k p, m k A(s)

A(s) A(s) (A(s)) A(s) R p p [s] rank R A k = p ± A(s) R p p [s] A(s) R p p [s] A(s) A(s) =I p A(s) c r (A) (c c (A)) A(s) R p m [s] ranka(s) =p ranka(s) =m c r (A) c c (A) (A(s)) A(s) R p m [s]

A(s) i A(s) R i A(s) t(s) R[s] j A(s) A(s) R p m [s] I p (I m ) i, j A(s) i j i a

i a i A(s) j t(s) i t(s) j j i

A(s),B(s) R p m [s] U L (s) R p p [s] U R (s) R m m A(s) =U L (s)b(s)u R (s) U L (s) R p p [s] A(s) =U L (s)b(s) U R (s) R m m A(s) =B(s)U R (s) p m A(s) R p m [s] I P A(s)I m = A(s) A(s),B(s) R p m [s] U L (s) R p p [s] U R (s) R m m [s] U L (s)a(s)u R (s) =B(s) U L A(s)U R (s) =U L A(s) =U (s)b(s)u (s) L (s)u L(s)A(s)U R (s) =U (s)b(s) (s)b(s) A(s)U R(s)U (s) =U (s)b(s)u (s) R R L L R

U L (s) Rp p [s],,u R (s) Rm m [s] A(s),B(s) R p m [s] U L (s) R p p [s] U R (s) R m m [s] B(s),C(s) R p m [s] U L(s) R p p [s] U R(s) R m m [s] U L (s)a(s)u R (s) =B(s) U L(s)B(s)U R(s) =C(s) U L(s)B(s)U R (s) =C(s) U L (s)u L (s)b(s)u R(s) =U L (s)c(s) B(s)U R(s)U R (s) =U L (s)c(s)u R (s) B(s) =U L (s)c(s)u R (s) U L (s)a(s)u R (s) =B(s) U L (s)a(s)u R (s) =U L (s)c(s)u R (s) A(s) =U L (s)u L (s)c(s)u R (s)u R (s) A(s) =[U L(s)U L (s)] C(s)[U R (s)u R(s)] [U L(s)U L (s)] R p p [s] [U R (s)u R(s)] R m m [s] U L (s),u L(s),U R (s),u R(s)

A(s) R p m [s] rank R(s) A(s) =r r {p, m} A(s) S C A(s) (s) Rp m [s] S C A(s)(s) =block[diag { ε (s),ε 2 (s),ε 3 (s),..., ε r (s), p r, m r } ] C A(s) ε i (s) R [s] ε i (s) ε i+ (s) i r a ij A(s) a (s) A(s) a i (s) a j (s) a (s) a i (s) =a (s)q i (s)+r i (s),α j (s) =a (s)q j (s)+r j (s) i =, 2, 3,..., p j =, 2, 3,..., m

r i (s),r j (s) r j (s) j q j (s) a j (s) r j (s) a (s) s r 2,..., r 2p,r 2,..., r m i q i (s) j q j (s) a (s) a 22 (s) a 2m (s) a p2 (s) a pm (s) a ij (s),i=2, 3,..., p j =2, 3,..., m a (s) a (s) a (s)

ε (s) b 22 (s) b 2m (s) b p2 (s) b pm (s) b ij (s) ε (s) b ij (s) i =2,..., p j =2,..., m ε (s) ε 2 (s) c 33 (s) c 3m (s) c p3 (s) c pm (s) ε 2 (s) ε (s) c ij (s),i=3,..., p, j = 3,..., m ε 2 (s) ε (s) ε 2 (s) ε r (s) r < {p, m}

r ε i (s) A(s) ε i (s) ε i+ (s) i r ε i (s) ε i (s) = Δ i(s) Δ i (s),i=,..., r Δ (s) =, Δ i (s) = i i A(s) Δ i (s) A(s) A(s) λ s C ε i (s) i r λ s,s =,..., ν A(s) ε i (s) ε i (s) = ν s= m is (s λ s ) (s λ s ) m is m s m 2s... m rs,s=,..., ν A(s) λ s s + A(s) = s + R 2 2 [s] (s +2) 2 A(s) a = s +

a 2 (s) a 2 (s) a (s) a 2 (s) =a (s)q 2 (s)+=(s +) + 2 q 2 (s) s + s + = s + (s +2) 2 (s +2) 2 } {{ } V a 22 (s) a (s) (s +2) 2 = s 2 +4s +4=(s +)(s +3)+ 2 s +3 s + = s + (s +3) (s +2) 2 (s +)(s +3) (s +2) 2 } {{ } U 2 s + = s + s + (s +)(s +3) (s +2) 2 (s +)(s +3) } {{ } V 2 s + s + s + = s + (s +)(s +3) (s +)(s +3) } {{ } V 3

s + s + = (s +)(s +3) (s +)(s +3) s + s + } {{ } U 2 2 s +2 (s +)(s +3) = (s +)(s +3) (s +) s + s + (s +)(s +2) 2 } {{ } U 3 2 (s +)(s +3) (s +)(s +3) (s +)(s +3) = (s +)(s +2) 2 (s +)(s +2) 2 } {{ } V 4 U L (s) =U 3 (s)u 2 (s)u (s) = = (s +3) (s +) } {{ }} {{ }} {{ } U 3 U 2 U (s +) = (s +2) 2 (s +3)

U R (s) =V (s)v 2 (s)v 3 (s)v 4 (s) = = (s +)(s +3) (s +)(s +3) = } {{ }} {{ }} {{ }} {{ } V V 2 V 3 V 4 SA(s)(s) C =U L (s)a(s)u R (s) = (s +3) = s + s + = (s +2) 2 (s +) (s +2) 2 (s +)(s +3) (s +)(s +2) 2 (C λ R r n,j λ R n n ) J λ λ λ λ J λ = λ λ λ (C λ,j λ ) A(s) λ A(s) λ n

C λ C λ J λ rank = n C λ J n λ A k C λ J k λ +... + A C λ J λ + A C λ = λ i =,..., j (C λi R r n i,j λi R n i n i ) A(s) C = ( ) C λ,c λ2,..., C λj J = blockdiag (J,J 2..., J j ) n = n + n 2 +... + n j (C λ,j λ ) A(s) s + A(s) = s + R 2 2 [s] (s +2) 2 A(s) = s 2 + s + 4 4 } {{ } } {{ } } {{ } A 2 A A C =,J = 2 2 A(s) =(s +)(s +2) 2 2 A(s)

rank C CJ 2 = =2 2 2 A 2 CJ 2 + A CJ + A C = 2,2 A(s) R r r [s] l C λ,λ 2..., λ l SA(s)(s) C =diag,,,...,,f }{{} k (s),f k+ (s),..., f r (s) k A(s) C k r f i (s) R [s] A(s) f j (s) f j+ (s) j = k, k +,..., r f k (s),..., f r (s) f k (s) =(s λ ) σ k (s λ 2 ) σ 2k...(s λ l ) σ lk f k+ (s) =(s λ ) σ k+ (s λ 2 ) σ 2k+...(s λ l ) σ lk+ f r (s) =(s λ ) σ r (s λ 2 ) σ 2r...(s λ l ) σ lr σ ik σ ik+... σ ir,i =, 2,..., l f j (s) = (s λ i ) σ ij f j (s),j = k, k +,..., r f j (s) (s λ i ) σ ij A(s) s = λ i [ r ] n := f j (s) = j=k l r i=j j=k σ ij

n U L (s),u R (s) U L (s) A (s) U R (s) =S C A(s) u j (s) R r [s] j = k, k +,..., r U R (s) u q j (s) :=( ) d q ds q uj (s) f j (s) j = k, k +,..., r β i jq := q! u(q) j (λ i ) j = k, k +,..., r q =,,..., σ ij λ i C, i l A(s) σ ik σ ik+... σ ir i l j = k, k +,..., r β i j,β i j,..., β i jσ ij λ i A(s) σ ij C ij := ( ) βj,β i j, i..., βjσ i ij 2,βjσ i ij R r σ ij λ i λ i J ij := λ i λ i

i l j = k, k +,..., r C ij J ij C i := (C ik,c ik+,..., C ir ) R r m i J i := (J ik,j ik+,..., J ir ) R m i m i m i := σ ik + σ ik+ +... + σ ir A(s) C := (C,C 2,..., C l ) R r n J := blockdiag (J,J 2,..., J l ) R n n [ r ] n := m + m 2 +... + m l = f j (s) = j=k l r i= j=k σ ij s + A(s) = s + (s +2) 2 A(s) A(s) S C A(s)(s) =U L (s) A (s) U R (s)

(s +3) s + s + = +(s +)(s +3) (s +) (s +2) 2 (s +)(s +3) = (s +)(s +2) 2 λ =,λ 2 = 2 l =2 SA(s) C (s) k =2 k =2=r j = k, k +,..., r j 2 σ ij i l =2 σ 2,σ 22 σ 2 =,σ 22 =2 n =2+=3 βjq i := q! u(q) j (λ i ) q =,..., σ ij U R (s) = = (s +)(s +3) ( u (s) ) u 2 (s) u (s) = u 2 (s) = (s +)(s +3) β2,β 2,β 2 2 2 β2 = u 2 ( ) =,β2 2 = u 2 ( 2) =,β2 2 = u 2 ( 2) = C 2 =(β 2),C 22 =(β 2 2,β 2 22)

σ 2 =,σ 22 =2 J 2 =( ),J 22 = 2 2 A(s) C =(C 2,C 22 )= J = blockdiag (J 2,J 22 )= 2 2

A (ρ) β (t) =,t A (ρ) ρ := d dt A (ρ) R r r [s] β (t) :(, ) R r β (t) β (q) ( ) =β (q) (+) = β (q) (),q =,, 2 β (q) (t) q β (t) t A (ρ) =A + A ρ +... + A k ρ k + A k ρ k A i R r r,i=,,..., k β ( ),β () ( ),..., β (k ) ( ) β (t), 2,..., k t = λ C

A (ρ) A (ρ) = [ t μ β (t) = μ! β + tμ (μ )! β +... + t ]! β μ + β μ e λ t β i C, i =,,..., μ, β β (t) A (λ ) β = A () (λ ) β + A (λ ) β = μ! A(μ) (λ ) β + (μ )! A(μ ) (λ ) β +... + A () (λ ) β μ + A (λ ) β μ = β,β,..., β μ A (ρ) λ C β R r,β λ C A (ρ) β,..., β μ λ C A (ρ) β (t) μ> β,β,..., β μ

β β,β β,β,..., β μ, 2,..., μ λ C A (ρ) β j (t),j =,,..., μ β j (t) =[ρi r λ I r ] j β (t) = [ t μ j (μ j)! β t μ j + (μ j )! β +... + t ]! β μ j + β μ j β j (t) {β μ (t),β μ (t),..., β (t),β (t)} e λ t [β μ (t),β μ (t),..., β (t) β (t)] = t t μ! (μ )! t μ 2 (μ 2)! [β,β,..., β μ,β μ ] t! t μ μ! t μ (μ )! e λ t

Ψ(t) :=[β μ (t),β μ (t),..., β (t) β (t)] C := [β,β,..., β μ,β μ ] R r (μ+) λ λ λ J := R (μ+) (μ+) λ λ Ψ=Ce Jt t t μ! (μ )! t μ 2 (μ 2)! e Jt = t! t μ μ! t μ (μ )! A(s) SA(s) C (s) A(s) A(s)

[ βjq i t σ ij q t σ ij 2 q := (σ ij q)! βi j + (σ ij 2 q)! βi j +... + t ]! βi j,σ ij 2 q + βj,σ i ij q e λ it i l, j = k, k +,..., r, q =,,..., σ ij [ ] Ψ ij (t) := βj,σ i ij (t),βj,σ i ij 2 (t),..., βj i (t),βj i (t) [ ] C ij := βj,β i j, i..., βj,σ i ij R r σ ij i l, j = k, k +,..., r λ i λ i λ i J ij := R σ ij σ ij λ i λ i λ i Ψ i (t) :=[Ψ ik (t), Ψ i,k+ (t),..., Ψ ir (t)] C i := [C ik,c i,k+,..., C ir ] R r m i J i := blockdiag [J ik,j i,k+,..., J ir ] R m i m i m i = σ ik + σ i,k+ +... + σ ir

Ψ:=[Ψ (t), Ψ 2 (t),..., Ψ l (t)] C := [C,C 2,..., C l ] R r n J := blockdiag [J,J 2,..., J l ] R n n [ r ] n := m + m 2 +... + m l = f j (s) = S C A(s) = A(s) j=k Ψ Ψ X A (ρ) β (t) =,t s + s + A(s) = R 2 2 [s] (s +2) 2, 2 Ψ=Ce Jt e t J = blockdiag (J 2,J 22 )= 2 ejt = e 2t te 2t 2 e 2t

C =(C 2,C 22 )= Ψ=Ce Jt = e t e 2t te 2t = e t e 2t te 2t e 2t te 2t e 2t A (ρ) β (t) =,t B C = e t, e 2t e 2t, te 2t te 2t β (t)+ β 2 (t) = β (t) β 2 (t) = β 2 (t) t β (t) =[β (t),β 2 (t)] ρ ρ + ρ3 β (t) = A (ρ) β (t) = ρ + β 2 (t) SA(s) C A(s) = s + s3 s +

s + s3 (s2 s +) = s + s + s + }{{} s + = s + s + (s +) }{{} V 2 s + = s + (s +) (s +) } {{ } V 3 V s + = s + s + (s +) (s +) 2 }{{} U s + (s +) = (s +) 2 (s +) 2 } {{ } V 4 U R (s) =V V 2 V 3 V 4 = s2 s + s 3 s + U L (s) =U = s +

SA(s)(s) C = (s +) 2 λ =,l =,j = k, k +,..., r =2=r σ 2 =2,n=2,q =, u 2 (s) = s3 s + βjq i = { } β2,β 2 2 βjq i = q! u(q) j (λ i ) β2 = u 2 ( ) =,β22 = u () 2 ( ) = 3 C 2 =(β2,β 22) =C = C J 2 = = J = J

Ψ=Ce Jt = 3 e t = 3 e t te t = e t e t (t 3) e t e t B C = e t (t 3) e t, = (t 3) e t, e t e t

A(s) =A + A s +... + A k s k + A k s k A i R r r,i=,,..., k,a k rank R(s) A(s) =r ( ) A (s) =s k A = A k + A k s +... + A s k + A s k R r r [s] s SA(s) (s) A(s) R r r [s] s = {}}{ U L (s) A (s) U R (s) =SA(s) = blockdiag s q,..., s q k,i }{{} ν κ,,..., sˆq ν+ sˆq r }{{} k r ν ν r, q q 2... q k, < ˆq ν+ <... < ˆq r U L (s),u R (s) R r r [s] q i, (ˆq i ) A(s) s = ˆq := r i=ν+ ˆq i A(s) A(s) ν

Ã (s) s = A (s) Ã (s) s = S C (s) Ã (s) Ã(s) S (s) s = Ã(s) SA(s) (s) s = S Ã(s) (s) =diag [sμ,s μ 2,..., s μ j ],μ j >,j =, 2,..., r s =S (s) Ã(s) S A(s) (s) SA(s) (s) =s q S Ã(s) ( ) ( S = Ã(s) s s ),q = k ( s ) q SA(s) (s) ( ) Ũ L (s) =U L s ( ) Ũ R (s) =U R s A (s) s μ j,j =2, 3,..., r μ j := q q j >,j =2, 3,..., k μ j := q +ˆq j >,j = ν +,ν+2,..., r

q i,i=, 2,..., k A(s) s = μ j,j =2, 3,..., k s = j = μ j = ˆq i,i = ν +,ν +2,..., r s = μ j,j = ν +,ν +2,...r s = A(s) A (s) = s s2 + s + R 2 2 [s] s SA(s) (s) A (s) + Ã (s) =s 2 + s s 2 s = s s2 + s + s 2 s s 2 s S C (s) Ã (s) Ã(s)

s s2 + s + = s s2 + s + s s 2 s s 2 s 3 }{{}}{{}}{{} U Ã(s) Ã (s) s s2 + s + (s +) = s s 3 s 3 }{{}}{{}}{{} Ã (s) V Ã 2 (s) s = s s 3 } {{ } Ã 2 (s) } {{ } V 2 s 3 } {{ } Ã 3 (s) s = s s 3 s 3 s 4 } {{ }} {{ } } {{ } U 2 Ã 3 (s) Ã 4 (s) s s = s 4 }{{} Ã 4 (s) } {{ } V 3 s 4 }{{} Ã 5 (s) = = S C s 4 s 4 Ã(s) } {{ }} {{ } Ã 5 (s) V 4 S C (s) = Ã(s) }{{} S (s) = = s2 2 s 4 Ã(s) s= s 4 s 2+2

A (s) μ 2 = q +ˆq 2 =2+2=4 s = μ 2 = q q =2 2= SA(s) = s 2 = s2 s 4 s 2 A(s) s = q =2 s = ˆq 2 =2 ( ) Ũ R (s) =U R s Ũ R (s) =V V 2 V 3 V 4 = (s +) s = = (s +) (s2 + s +) s U R (s) = ( +) ( + +) s s 2 s = = s+ s s +s+s2 s 2 s = s+ s +s+s2 s 2 s

( ) Ũ L (s) =U L s Ũ L (s) =U U 2 = = s 3 s s 3 + s U L (s) = = = + s 3 s s 2 s 3 s 2 s 3

(C R r μ,j R μ μ ) J λ = J = (C,J ) A(s) λ = Ã (s) λ = μã (s) Ã (s) =A k + A k s +... + A s k + A s k R r r [s]

C C J rank = μ C J μ A C J k +... + A k C J + A k C = r,μ λ i,i=, 2,..., r Ã (s) (C i R r μ i,j i R μ i μ i ) A(s) (C R r μ,j R μ μ ) C =(C,C 2,..., C r ) R r μ J = blockdiag (J,J 2,..., J r ) R μ μ μ = μ + μ 2 +... + μ r A (s) (C,J ) A (s) λ = Ã (s) A (s) = s s2 + s + R 2 2 [s] s Ã (s)

Ã (s) = s s2 + s + s 2 s s 2 A (s) C =,J = Ã (s) Ã (s) = s2 s 3 + s 4 + s 3 + s 2 = s 4 Ã (s) λ = 4 C C J rank = C J 2 =4 C J μ =3 A C J 2 + A C J + A 2 C = 2,4 Ã (s) s =

S C (s) Ã (s) s = Ã(s) S Ã(s) (s) ŨL (s), ŨR (s) ŨR (s) = [ũ (s), ũ 2 (s),..., ũ r (s)], ũ j (s) R r (s) Ũ L (s) Ã (s) ŨR (s) =S Ã(s) Ã (s)ũ Rj (s) =ũ Lj (s) s μ j,j = ν +,ν +2,..., r μ j = q +ˆq j,j = ν +,..., r ũ Lj (s) j ŨL(s) ũ (q) j (s), Ã(q) (s) q ũ j (s), Ã (s) s = q =,, 2,..., μ j,j = ν +,..., r x jq := q!ũ(q) j () x j,x j,..., x jr R r,j = ν +,..., r Ã (s) s = x j,x j,..., x jr R r,j = ν +,..., r Ã () x j = Ã () () x j + Ã () x j = )!Ã(μ j ) () x j + 2)!Ã(μ j 2) () x j +... + (μ j (μ j Ã () x j,μ j = (C,J ) A (s)

A(s) = s s2 + s + R 2 2 [s] s A (s) Ã (s) s = S C (s) = S = Ã(s) s 4 Ã(s) s 4 q SA(s)(s) =s 2 = s2 s 4 s 2 Ũ R (s) = (s +) (s2 + s +) s = k = 2 j = ν +,..., r = 2 A(s) q q =,,..., μ j μ j =2+2=4 x jq 4 s = Ã (s) x jq = {x 2,x 2,x 22,x 23 } x 2 = () = (2 ++) = ũ!ũ2 2 (s) ŨR (s) x 2 =!ũ 2 () =,x 22 =,x 23 =

(C,J ) A(s) C = C 2 =(x 2,x 2,x 22,x 23 )= R r μ i = R 2 4 J = R 4 4

C j =(x j,x j,..., x j,μ ) R r μ,j j = R μ μ β ( ),β () ( ),..., β (q ) ( ) β ( ) = x jq+,β () ( ) = x jq+2,β (q ) ( ) = x jq+q q =,,..., ˆq j,j = ν +,..., r A (ρ) β (t) =,t β jq (t) =x j δ (q) (t)+x j δ (q ) (t)+... + x jq δ () (t)+x jq δ (t)

x jq δ (t) ˆq r β (t) δ (t) ˆq r s = SA(s) (s) B A (ρ) β (t) =,t β i (t) B S C (s) Ã (s) s = Ã(s) S (s) Ã(s) S A(s) (s) A(s) ŨL (s), ŨR (s) Ũ R (s) =[ũ (s), ũ 2 (s),..., ũ r (s)],u j (s) R r (s) Ũ L (s) Ã (s) ŨR (s) =S Ã(s) (s) x jq := q!ũ(q) j () Ã (s) s = (C,J ) A(s) q =,,..., μ j,j = ν +,..., r

(C,J ) [ C j := C ν+, C ν+2,..., C ] r R r μ j [ J j := J ν+, J ν+2,..., J ] r R μ j μ j j = ν +,..., r ˆq j k ˆq j ˆq j ˆq j ( C j J j ) B Ψ = ˆq r i= C j J (i) j δ(i) (t) J μ j = q +ˆq j,j = ν +,..., r ˆq j B = μ j = r j=ν+ ˆq j j = ν +,..., r A(s) A(s) = s s2 + s + R 2 2 [s] s A (ρ) β (t) = A(s) s =

(C,J )=, ˆq j =2 C J 2 2 j =2 Ψ = ˆq r i= 2 = (i) C j J j δ(i) (t) = i= C 2 = J 2 = C 2 J (i) 2δ (i) (t) = C 2 J () 2δ () (t)+ C 2 J () 2δ () (t) = = δ (t)+ δ () (t) δ (t) δ (t) = + δ() (t) δ (t) = δ (t) δ() (t) δ (t) δ (t) ˆq r s = r 2 δ (t) B =, δ (t) δ() (t) δ (t)

A(s) = s + s2 s B k =2 s 2 +=s ( s)+,s= s Ã (s) =s 2 s s 2 = s + s2 s 2 s 2 s s s + s2 s 2 = s2 s + s 2 s 2 s s s 2 } {{ }} {{ } } {{ } Ã(s) V Ã (s) s2 s + s 2 = s s2 s s 2 s 2 s + s 2 } {{ }} {{ } } {{ } U Ã (s) Ã 2 (s) s s2 = s s2 s s 2 s + s 2 s + s 2 + s 3 } {{ }} {{ } } {{ } U 2 Ã 2 (s) Ã 3 (s)

s s2 s = s s + s 2 + s 3 s 3 + s 2 } {{ }} {{ } } {{ } Ã 3 (s) V 2 Ã 4 (s) s = s3 + s 2 s 3 + s 2 s } {{ }} {{ } } {{ } U 3 Ã 4 (s) Ã 5 (s) s3 + s 2 = s3 + s 2 s s s 4 s 3 } {{ }} {{ } } {{ } U 4 Ã 5 (s) Ã 6 (s) s3 + s 2 s3 s 2 = s 4 s 3 s 4 s 3 } {{ }} {{ } } {{ } Ã 6 (s) V 3 Ã 7 (s) = = = S C s 4 s 3 s 4 + s 3 s 3 Ã(s) ( + s) } {{ }} {{ } Ã 7 (s) V 4 S (s) = S Ã(s) A(s)(s) =s 2 = s2 s 3 s 3 s A(s) s = q =2 ˆq 2 = s = μ 2 = q +ˆq 2 =2+=3 μ 2 =3 s = s = Ã (s)

x jq = {x 2,x 2,x 22 },q =,,..., μ j =,, 2 j =2 ˆq 2 = Ũ R (s) =V V 2 V 3 V 4 = s 3 + s 2 + s x 2 = ũ 2 () =,x 2 =ũ () 2 () =,x 22 = 2!ũ(2) 2 () = C =,J = C 2 = R 2, J 2 =() R (C,J ) Ψ = ˆq r = i= C j J (i) j δ(i) (t) =C 2 J () 2δ (t) = δ (t) B = δ (t) δ (t) = t β (t)+ d3 β 2 (t) = dt 3 β 2 (t)+ dβ 3 (t) = dt β 3 (t) =

B ρ ρ 3 β (t) ρ β 2 (t) = β 3 (t) Ã (s) =s3 A ( ) s Ã (s) =s 3 s 3 s 3 s = s 3 s 2 s 3 s 3 s 3 s s 2 = s s 2 s 3 s 3 } {{ }} {{ } } {{ } Ã(s) V Ã (s) s 3 s 3 s s s 2 = s s 2 s 3 s 3 } {{ }} {{ } } {{ } U Ã (s) Ã 2 (s) s 3 s 3 s s 2 = s s 2 s 3 s 3 } {{ }} {{ } } {{ } Ã 2 (s) V 2 Ã 3 (s)

s s 2 = s s 6 s 3 s 3 } {{ }} {{ } } {{ } Ã 3 (s) V 3 Ã 4 (s) s s 6 = s s 6 s s 3 s 7 } {{ }} {{ } } {{ } U 2 Ã 4 (s) Ã 5 (s) s s 6 s = s = SC Ã(s) s 7 s 7 } {{ }} {{ } Ã 5 (s) V 4 k =3 S (s) = Ã(s) s 2 s 7 s 3 SA(s) (s) =s 3 s 2 = s s 7 s 4 A(s) s = q =3,q 2 = ˆq 3 =4 4 μ 3 = q +ˆq j = 3+4 = 7

s = Ã (s) μ =3 3= μ 2 = q q 2 =3 =2 s = s = μ 3 = q +ˆq 3 = 3+4=7 x jq = {x 3,x 3,x 32,x 33,x 34,x 35,x 36 },q =,,..., μ j =,,..., 7 =,,..., 6 j = 3 ˆq 3 =4 Ũ R (s) =V V 2 V 3 V 4 = s 3 s 4 x 3 = () =!ũ3,x 3 =!ũ() 3 () = 3s 2 = 4s 3 s= x 32 = 2!ũ(2) 3 () = 6s = 2,x 33 = 3!ũ(3) 3 () = 6 6 2s 2 24s s= x 34 = 4!ũ(4) 3 () = = 24,x 35 = 5!ũ(5) 3 () = 2 24 s= s= s= = = = x 36

C = R3 7,J = R 7 7 C 3 =, J 3 = 3 4 4 4

Ψ = ˆq r =4 i= C 3 J (i) 3δ (i) (t) = = C () 3 J 3δ (t)+ C () 3 J 3δ () (t)+ C (2) 3 J 3δ (2) (t)+ C (3) 3 J 3δ (3) (t) = = δ (t)+ δ () (t) + δ (2) (t)+ + δ (3) (t) = δ (t) + δ() (t)+ δ(2) (t)+ δ (t) δ () (t) δ (2) (t) δ (3) (t) + δ(3) (t) = δ (t)

B = δ (t), δ () (t), δ (2) (t), δ (3) (t) δ (t)

A(ρ)β (t) = B C = βjk i (t) = σ ij k= β jk t k e λ i(t) k =,,..., σ ij i =, 2,..., l i j = z,z +,..., r q N A (ρ) =A + A ρ +... + A q ρ q A i R r r,i =, 2,..., q β i jk (t) BC A(ρ)β (t) =

C jk = ( ) βj i βj i... βjσ i ij R r σ ij λ j λ i λ i J ij = R σ ij σ ij λ i λ i β i jk (t) BC β i jk = ( t σ ij σ ij βi j+ ) tσ ij 2 σ ij 2 βi t j+...! βi jσ ij 2 + βjσ i ij e λ it i l, j = z,z+,..., r, k =,,..., σ ij A (λ ) β i j = A () (λ ) β i j + A (λ ) β i j = A(σij ) (λ ) β i j + (σ ij )! (σ ij 2)! A(σ ij 2) (λ ) βj i +... + A () (λ ) β σij 2 + A (λ ) β σij =

q λ q (σ ij ) I q(q )λ q 2 I qλ q I λ q I σ ij (q )λ q 2 I λ q I ) (A ij λ q (σ ij ) I q A A σ ij I 2I 2λ I λ 2 I I λ I I } {{ } Q i βj i βj i βj i βj i = βjσ i ij 2 βj i βjσ i ij βjσ i ij 2 βj i βj i }{{} W i ( ) A (s) q =

Q i,w i Q i,w i Q, W ) ( ) (A q A A QW = A i R r r,i =,,..., q Q i,w i 2 r q = q + A(s) A (ρ) β (t) = β (t) = 3 3 e 2t }{{} β β 2 (t) = 3 e 2t + 3 }{{} β 3 3 }{{} β te 2t q = A (s) = A s + A λ =2,r =2 ) (A A I 2I β ( ) = I β β } {{ }} {{ } Q W

A i = a i a i3 a i2 a i4,i=, ) (A A β +2β 2β = β β ( ) Q, W A,A A (s) β = 3 3 ) (A A,β = 3 3 2 3 2 3 3 3 3 3 = 3 ( ) c {{ 8 }, 38,,, { 58, 8 }},, A = 8 3 8,A = 5 8 8 A (s) = s 8 3s + 8 5s + 8 8

A (s) = (s 4 2)2 q = q =2 A(s)

A (ρ) β (t) = A (ρ) R r r [ρ],ρ = d dt A (ρ) β (t) = ˆq j B = β j (t) = x jk δ (ˆq j k) (t) k= x jk C r, k ˆq j, j l q N A (ρ) =A + A ρ +... + A q ρ q A i R r r,i =, 2,q

β j (t) B A (ρ) β (t) = C j = ( x j,x j,..., x j,q+ˆqj ) R r q+ˆq j,j j = R q+ˆq j q+ˆq j A(s) =A q s q +... + A s + A A q A q A q A A 2 A 3 A q A A A 2 A q A q A A A q 2 A q A q A A A 2 A q A q }{{} rμ j rμ j x j x j x jq x jq x jq+ x jμj + }{{} rμ j = }{{} rμ j μ j = q +ˆq j n =,,..., q q

A(s) q = q Ã (n) () = n!a q q,n=,, 2,..., q Ã (n) () =,n= q +,q +2,..., q +ˆq =μ j Ã () x j = Ã () () x j + Ã () x j =!Ã(q) () x j + )!Ã(q ) () x j +... + q (q Ã () x jq =!Ã(q) () x j + )!Ã(q ) () x j2 +... + q (q Ã() () x jq + Ã () x jq + =!Ã(q) () x jμj (q q +) + )!Ã(q ) () x jμj q (q +... + Ã() () x jμj 2 + Ã () x jμ j =

j = ν +,..., r ν A q x j = A q x j + A q x j = A q 2x j + A q x j + A q x j2 = A x j + A x j +... + A q x jq = A x j + A x j2 +... + A q x jq + A q x jq + = A x j ˆqj + A x j ˆqj +... + A q x jμj 2 + A q x jμj = q = q,μ j = q +ˆq j A q A q A q A A 2 A 3 A q A A A 2 A q A q A A A q 2 A q A q A A A 2 A q A q }{{} rμ j rμ j x j x j x j ˆqj x j ˆqj x j ˆqj + x jq+ˆqj }{{} rμ j = }{{} rμ j β jl = x j δ (l) (t)+x j δ (l ) (t)+... + x jl δ () (t)+x jl δ (t) l =,,..., ˆq j

β ( ) = x jl+,β () ( ) = x jl+2,..., β (q ) ( ) = x jl+q μ j = q +ˆq j ˆq j ) (A q A q A x jl x j x j x jq x jq x jq+ x j,q+ˆqj x j x jq 2 x jq x jq x j,q+ˆqj 2 x jq 2 x jq ) x (A q A q A A jq 2 = }{{} r r(q+) x j x j ˆqj x j x j x j ˆqj }{{}}{{} r(q+) μ j rμ j

x j x j x j ˆqj x j ˆqj x j ˆqj + x j,q+ˆqj x j x j ˆqj 2 x j ˆqj x j ˆqj x j,q+ˆqj 2 x j ˆqj 2 x j ˆqj ) x (A q A q A A j ˆqj 2 = }{{} r r(q+) x j x j ˆqj x j x j x j ˆqj }{{}}{{} r(q+) μ j rμ j ˆq j μ j = q +ˆq j A (s) q = A i A i R r r,i =,,..., q a ij μ j = q +ˆq j,j =, 2,..., l x j ˆqj,x j ˆqj +,..., x j,q+ˆqj, lq 2 A(s) = q q = q + 2

A i A (s) A (s) q = x j x j x j ˆqj x j ˆqj x j ˆqj + x j,q+ˆqj x j x j ˆqj 2 x j ˆqj x j ˆqj x j,q+ˆqj 2 x j ˆqj 2 x j ˆqj ) x (A q A q A A j ˆqj 2 = }{{} r r(q+) x j x j ˆqj x j x j x j ˆqj }{{}}{{} rμ j Q R r(q+) μ j A i A i R r r,i=,,..., q A i A i R r r,i=,,..., q Q q = q + 2 A(s) A (ρ) β (t) =

A(s) A(s) A (ρ) β (t) = β (t) = δ (t) =x δ (t) β 2 (t) = δ (t)+ δ () (t) =x δ (t)+x δ () (t) q = A(s) = A + A s A A A A A x x x 2 = ˆq =2 μ = q +ˆq =+2=3 r =2 x ji A i = a i a i3 x =,x =,x 2 = x x 2 a i2 a i4,i=, rμ = r (q +ˆq )=2(+2)=6 (q +)r 2 + lqr =(+)2 2 +

2= A x = A x + A x = A x + A x 2 = a a 2 = a = a =,a 3 = a 3 a 4 a 3 a a 2 + a a 2 = a + a a 2 = a 3 a 4 a 3 a 4 a 3 a 3 a 4 a + a a 2 = }{{} a = a 2,a 3 = a 4 a 3 + a 3 a 4 a a 2 + a a 2 x = a a 2 + a 2 x 2 }{{} = }{{} a 3 a 4 a 3 a 4 x 2 a 3 a 4 + a 4 x 2 () a 2 a 2 + a 2 x 2 = a 4 a 4 + a 4 x 2 = a 2 = a 2 + a 2 x 2 x 2 = a 2 a 2 a 2 a 4 = a 4 + a 4 x 2 a 4 = a 4 + a 4 a 2 a 2 a 2 = a 4a 2 a 2

A (s) a 2 = a 4 = A (s) = a 2 =,a 4 A =,A = a 2 }{{} a 4 a 4 a 4 a 4 a 4 A (s) =A + A s = a 4 a 4 + a 4 s a 2,a 4 = A = a 2,A = a 2 a 2 a 2 a 2 }{{} a 4 A (s) =A + A s = a 2 a 2 + a 2 s a 2,a 4 A = a 2,A = a 2 a 2 a 2 a 2 }{{} a a 4 a 4 a 4 a 2 a 4 4 a 2 A (s) =A + A s = a 2 a 2 + a 2 s + a 4 s a 4 a 2 a 4 a 2

A(s) SA(s) = s ˆq = s 2 q = q +=2 A 2 x A A 2 x = A A A 2 x 2 A A A 2 x 3 ˆq =2 μ = q +ˆq =2+2=4 r =2 x ji A i = a i a i3 a i2 a i4,i=,, 2 x =,x =,x 2 = x,x 3 = x 3 x 2 x 4

(A + A s + A 2 s 2 ) A 2 x = A x + A 2 x = A x + A x + A 2 x 2 = A x + A x 2 + A 2 x 3 = a 2 a 22 = a 2 =,a 23 = a 23 a 24 a a 2 + a 2 a 22 = a + a 2 a 22 = }{{} a 3 a 4 a 23 a 24 a 3 + a 23 a 24 a = a 22 a 3 = a 24 a a 2 + a a 2 + a 2 a 22 x = a 3 a 4 a 3 a 4 a 23 a 24 x 2 a + a a 2 + a 2 x + a 22 x 2 = a 3 + a 3 a 4 + a 23 x + a 24 x 2 = }{{} a + a 22 a 2 + a 22 x 2 =,a 3 + a 24 a 4 + a 24 x 2 =, x 2 = a 2 a 22 a a 22 a 3 a 4 + a 24a 2 a 22 a 24a a 22 =

a a 2 + a a 2 x + a 2 a 22 x 3 = a 3 a 4 a 3 a 4 x 2 a 23 a 24 x 4 a a 2 + a x + a 2 x 2 + a 2 x 3 + a 22 x 4 = a 3 a 4 + a 3 x + a 4 x 2 + a 23 x 3 + a 24 x 4 = }{{} a a 2 + a 22 x + a 2 x 2 + a 22 x 4 =,a 3 a 4 + a 24 x + a 4 x 2 + a 24 x 4 =, x 4 = a + a 2 a 22 x a 2 x 2 a 22 a 3 a 4 + a 4 x 2 a 24a a 22 + a 24a 2 a 22 + a 24a 2 a 22 x 2 = x 4 x 4 = a 3 + a 4 a 4 x 2 a 24 x a 24 a a 2 + a 2 x 2 a 22a 3 a 24 + a 22a 4 a 24 a 22a 4 x 2 a 24 = a 22,a 24 a 2,a 2,a 4 a 2 = a a 3 = a 4 x 2 = a 4 = a 2a 24 a 22 x 4 = a 2 a 22 x a 22 x 4 = a 4 a 24 x a 24 A 2 = a 22,A = a 22 a 2,A = a 2 a 2 a 24 a 24 a 4 a 4 a 2 a 24 a 22

A(s) = a 2 + sa 22 a 2 + sa 2 + s 2 a 22 a 4 + sa 24 a 2 a 24 a 22 + sa 4 + s 2 a 24 A(s) = a 2 a 22 (a 2 a 24 a 22 a 4 ) SA(s) = s2 s 2 β (t) A (ρ) β (t) = A(ρ)β(t) = A (ρ) β (s) = a 2 + sa 22 s 2 a 22 + sa 2 + a 2 = a 2 sa 22 a 4 + sa 24 s 2 a 24 + sa 4 + a 2a 24 a 22 a 4 sa 24 a 22 = s a 24 ) = (si s a 22 a 2 a 22 x 2 I 2 A 2 x A A 2 x 2 a 24 a 4 a 24 ) = (si 2 I 2 A 2 β ( ) A A 2 β () ( )

A (ρ) β 2 (s) = a 2 + sa 22 s 2 a 22 + sa 2 + a 2 + s = a 4 + sa 24 s 2 a 24 + sa 4 + a 2a 24 a 22 a = 2 a 2 + sa 22 = a 22 (a 22 a 4 a 2 a 24 + sa 2 22a 24 ) a 22 x = s a 24 = s a 22 a 2 a 22 x 3 a a 24 a 4 a 4 a 24 x 24 a 24 ) = (si 2 I 2 A 2 x 2 = A A 2 x 3 ) = (si 2 I 2 A 2 β ( ) A A 2 β () ( ) a 22 =,a 24 a 2,a 4,a 4 x 2 = a = a 2 a 3 = a 4 a 2 = x 4 = a 4 a 24 x a 24 A 2 =,A = a 2,A = a 2 a 24 a 24 a 4 a 4 a 4 A (s) = a 2 sa 2 a 4 + sa 24 a 4 + sa 4 + s 2 a 24 A (s) =a 2 a 4

SA(s) = s2 s 2 a 22 =,a 24 a 2,a 4,a 4 x 2 a = a 2 x 2 = a 4 a 24 a 3 a 24 a 2 a a 2 + a 2a 4 a 2 a 2a 3 = a 24 a 24 a 2 a 3 = a 2 + a 2a 4 a 2 a 3 = a 2 a 24 + a 2 a 4 a 24 a 24 a 3 = a 2a 24 + a 2 a 4 a 2 x 4 = a 4 a 24 a2 4 + a a 2 2 24 a 2 + a 3a 4 a 2 24 A 2 =,A = a 2 a 24 a 24 a 4 x a,a = 2 a 2 a 2 a 24 +a 2 a 4 a 2 a 4 A (s) = a 2 a 2 + sa 2 a 2 a 24 +a 2 a 4 a 2 + sa 24 a 4 + sa 4 + s 2 a 24 A (s) =a 2 a 4 + a2 2a 24 a 2 a 2 a 4 SA(s) = s2 s 2 a 22,a 24 =, a 2,a 4,a 4 x 2 = a 3 = a 4 a = a 2 a 4 =x 4 = a 2 a 22 x a 22 A 2 = a 22,A = a 22 a 2,A = a 4 a 2 a 2 a 4

A (s) = a 2 + sa 22 a 2 + sa 2 + s 2 a 22 a 4 sa 4 A (s) = a 4 a 2 SA(s) = s2 s 2 a 22,a 24 = a 2,a 4,a 2 x 2 a 3 = a 4 a x 2 = a 2 a 22 a a 22 a 4 a 3 a 4 + a 4a 2 a 22 a 4 a 4a a 22 = a 4 = a 4a 2 a 4 a a 4 x 4 = a a 22 + a2 2 + a 2+a 2 a 2 22 a 22 + a 2a x a 2 22 A 2 = a 22,A = a 22 a 2,A = a a 2 a 4 a 4 a 4 a 2 a 4 a a 4 A (s) = a + sa 22 a 2 + sa 2 + s 2 a 22 a 4 a 4 a 2 a 4 a a 4 + sa 4 A (s) = a 4 a 2 a2 a 4 a 22 + a a 2 a 4 a 22 SA(s) = s2 s 2 a 22,a 24 a,a 2,a 2,a 4

x 2 = a 2 a 22 a a 22 a 3 = a 4 a 24 a 24a 2 a 22 + a 24 + a 24a a 22 = a 4 a 24a 2 a 22 + a 24a a 22 = a 4a 22 a 24 a 2 + a 24 a a 22 a 4 = a 4a 2 a 4a + a 2a 24 a2 2a 24 + a 2a 24 a a 22 a 22 a 22 a 2 22 a 2 22 = a 4a 2 a 22 a 4 a a 22 + a 2 a 24 a 22 a 2 2a 24 + a 2 a 24 a a 2 22 = a 2a 3 a 22 a 4 a a 22 + a 2 a 24 a 22 a 2 22 = a 2a 3 a 4 a + a 2 a 24 a 22 A 2 = a 22,A = a 22 a 2,A = a 24 a 24 a 4 a a 2 a 4 a 22 a 24 a 2 +a 24 a a 22 a 2 a 3 a 4 a +a 2 a 24 a 22 A(s) = a + sa 22 a 2 + sa 2 + s 2 a 22 a 4 a 22 a 24 a 2 +a 24 a a 22 + sa 24 a 2 a 3 a 4 a +a 2 a 24 a 22 + sa 4 + s 2 a 24 A(s) = a a 2 a 3 a 4 a 2 a 2 a 4 a 22 + a 4 a 2 a 2 SA(s) = s2 a 22 s 2

a 24,a 22 = a 2 = A(s) = a 4 + sa 24 a 4 + sa 4 + s 2 a 24 a 24 =,a 22, a 4 = A(s) = a 2 + sa 22 a 2 + sa 2 + s 2 a 22 q = μ = q +ˆq =+2=3 r =2 ) (A A x x x 2 ( ) = }{{} x x r r(q+) } {{ } r(q+) μ j }{{} r μ j A i = a i a i2,i =, x =,x =,x 2 = x a i3 a i4 x 2

x ji,i =,, 2 A i,i =, x 2 = {,,, } r = A = ( ),A =(, ) A(s) = ( ) s q =2 μ = q +ˆq =2+2=4,r =2 A (s) x ) x x 2 x 3 ( ) (A 2 A A x x x 2 = x x x ji,i=,, 2, 3 A i,i=,, 2 x 2 =,x 3 = {{,,,,, 2}, {,,,,, }} A 2 =,A =,A = 2

A (s) = s s2 +2 s A (s) = 2 A R p m A R m p AA A = A A AA = A ( AA ) T = AA ( A A ) T = A A A T A A A = A β j (t) = ˆq j k= x jk δ ˆq j +k (t) x jk C r k ˆq j, j l ˆq j C j = ( x j,x j,..., x j ˆqj,x j ˆqj,..., x j,q+ˆqj 2,x j,q+ˆqj ),j =, 2,..., l

J C =(C,C 2,..., C l ) R r μ J 2,J = R μ μ J l l μ = μ j,μ j = q +ˆq j j= a Ã(s) =I r C(J ai n ) { (s a)v +(s a) 2 V 2 +... +(s a) q V q } q = ind (C, J) C CJ (V,..., V q ) CJ q C C(J ai n ) S q = C(J ai n ) q q+ˆq j k t q+ˆq j k β j (t) = x jk (q +ˆq j k )! k= j =, 2,..., l Ã(ρ) β(t) ( = Ã(ρ) =ρq A ρ )

q = ind (C, J) C j = ( x j,x j,..., x j ˆqj,x j ˆqj,..., x j,q+ˆqj 2,x j,q+ˆqj ),j =, 2,..., l J C =(C,C 2,..., C l ) R r μ J 2,J = R μ μ J l l μ = μ j,μ j = q +ˆq j j= C CJ S q = rank(s q )=n CJ q V =(V,..., V q ) C C(J ai n ) S q = C(J ai n ) q

Ã(s) =I r C(J ai n ) { (s a)v +(s a) 2 V 2 +... +(s a) q V q } A(s) =s q Ã ( ) s A(s) A(ρ)β(t) = β (t) = δ (t) =x δ (t) β 2 (t) = δ (t)+ δ () (t) =x δ (t)+x δ () (t) q = ˆq =2 μ = q +ˆq = +2=3 x ji = (x x x 2 ),i =,,..., 3 r =2 q = C = x,j = x 2 S q = S =(C) = x x 2 q = q =2 ˆq =2 μ = q+ˆq =2+2=4x ji =(x x x 2 x 3 ),i=

,..., 4 q =2 C = x x 3,J = x 2 x 4 S q = S 2 = = C CJ 2 x x 3 = C x 2 x 4 = CJ x x 2 S q = x 2 x x 2 2 x 4 n =4 CJ = x x 3 = x x 2 x 4 x 2 q = ind (CJ)=2 a = Ã(s) =I 2 C(J ai 4 ) { (s a)v +(s a) 2 } V 2 Ã(s) =I 2 C(J I 4 ) { (s )V +(s ) 2 } V 2 V =(V,V 2 ) C C(J I 4 )

x x 3 C (V,V 2 )= x 2 x 4 = C(J I 4 ) 2 x 2 x x 3 2 x 2 x 4 x 2 C(J I 4 ) = C = C = x x 3 = x 2 x 4 = 2 x 2 x 3 x 2 x 2 x 4 x 2

Ã(s) = x x x 3 x 2 x 2 x 4 x x 3 (s ) +(s ) 2 x 2 x 4 2 x 2 x x 3 2 x 2 x 4 x 2 = s(sx 2 2 +x 2 s+sx +sx 4 +) x 2 2 +x 2+x +x 4 (s )(x 2 2s+sx 2sx 2 +sx 4 +) x 2 2 +x 2+x +x 4 (s )s (s 2 x 2 2 3s sx 2+s 2 x +2s 2 x 2 +s 2 x 4 +2s 2 +) x 2 2 +x 2+x +x 4 x 2 2 +x 2+x +x 4 A(s) =s 2 A ( s) x 2 2 +x 2+x +x 4 (x 2 (s ) 2 + sx 2 + s + x + x 4 ) x 2 2 +x 2+x +x 4 ( 2x 2 s + x + x 2 + x 4 2) s x 2 2 +x 2+x +x 4 x 2 2 +x 2+x +x 4 (s 2 sx 2 3s + x 2 2 +2x 2 + x + x 4 +2)