:: Schur's Theorem on the Stability of Networks
:: by Agnieszka Rowi\'nska-Schwarzweller and Christoph Schwarzweller
::
:: Received October 19, 2006
:: Copyright (c) 2006 Association of Mizar Users
Lm1:
for L being non empty right_complementable add-associative right_zeroed addLoopStr
for F being FinSequence of L
for G being FinSequence
for k being Nat st G = F | (Seg k) & len F = k + 1 holds
( G is FinSequence of L & dom G c= dom F & len G = k & F = G ^ <*(F /. (k + 1))*> )
theorem Th1: :: HURWITZ:1
theorem Th2: :: HURWITZ:2
theorem Th3: :: HURWITZ:3
Lm2:
( Im (1_ F_Complex ) = 0 & Im (- (1_ F_Complex )) = 0 & Im (0. F_Complex ) = 0 )
Lm3:
for z being Element of F_Complex st Im z = 0 holds
z *' = z
theorem Th4: :: HURWITZ:4
theorem Th5: :: HURWITZ:5
theorem Th6: :: HURWITZ:6
theorem Th7: :: HURWITZ:7
theorem Th8: :: HURWITZ:8
Lm4:
for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of L holds - (- p) = p
theorem Th9: :: HURWITZ:9
Lm5:
for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of L
for f being Element of (Polynom-Ring L) st f = p holds
- f = - p
theorem :: HURWITZ:10
theorem Th11: :: HURWITZ:11
theorem Th12: :: HURWITZ:12
:: deftheorem Def1 defines Coeff HURWITZ:def 1 :
theorem Th13: :: HURWITZ:13
Lm6:
for L being non empty right_complementable distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of L
for p2' being Element of (Polynom-Ring L) st p2' = p2 holds
for F being FinSequence of (Polynom-Ring L) st p1 = Sum F holds
p2 *' p1 = Sum (p2' * F)
theorem Th14: :: HURWITZ:14
theorem Th15: :: HURWITZ:15
theorem Th16: :: HURWITZ:16
theorem Th17: :: HURWITZ:17
theorem Th18: :: HURWITZ:18
theorem Th19: :: HURWITZ:19
:: deftheorem defines degree HURWITZ:def 2 :
Lm7:
for L being non empty ZeroStr
for s being AlgSequence of L st len s > 0 holds
s . ((len s) - 1) <> 0. L
theorem Th20: :: HURWITZ:20
Lm8:
for L being non empty ZeroStr
for p being Polynomial of L st deg p <> - 1 holds
p . (deg p) <> 0. L
Lm9:
for L being non empty ZeroStr
for p being Polynomial of L holds
( deg p is Element of NAT iff p <> 0_. L )
theorem :: HURWITZ:21
Lm10:
for L being non empty ZeroStr
for p being Polynomial of L holds deg p >= - 1
theorem Th22: :: HURWITZ:22
theorem Th23: :: HURWITZ:23
theorem Th24: :: HURWITZ:24
Lm11:
for L being non empty unital doubleLoopStr
for z being Element of L
for k being Element of NAT st k <> 0 holds
( ((0_. L) +* (0 ,k --> (- ((power L) . z,k)),(1_ L))) . 0 = - ((power L) . z,k) & ((0_. L) +* (0 ,k --> (- ((power L) . z,k)),(1_ L))) . k = 1_ L )
Lm12:
for L being non empty unital doubleLoopStr
for z being Element of L
for k being Element of NAT
for i being Nat st i <> 0 & i <> k holds
((0_. L) +* (0 ,k --> (- ((power L) . z,k)),(1_ L))) . i = 0. L
:: deftheorem defines rpoly HURWITZ:def 3 :
theorem :: HURWITZ:25
theorem :: HURWITZ:26
theorem Th27: :: HURWITZ:27
theorem Th28: :: HURWITZ:28
theorem Th29: :: HURWITZ:29
theorem Th30: :: HURWITZ:30
:: deftheorem Def4 defines qpoly HURWITZ:def 4 :
theorem :: HURWITZ:31
theorem Th32: :: HURWITZ:32
theorem Th33: :: HURWITZ:33
:: deftheorem Def5 defines div HURWITZ:def 5 :
:: deftheorem defines mod HURWITZ:def 6 :
:: deftheorem Def7 defines divides HURWITZ:def 7 :
theorem Th34: :: HURWITZ:34
theorem :: HURWITZ:35
theorem :: HURWITZ:36
:: deftheorem Def8 defines Hurwitz HURWITZ:def 8 :
theorem :: HURWITZ:37
theorem :: HURWITZ:38
theorem Th39: :: HURWITZ:39
theorem Th40: :: HURWITZ:40
Lm13:
for f, g, h being Polynomial of F_Complex st f = g *' h holds
for x being Element of F_Complex st ( x is_a_root_of g or x is_a_root_of h ) holds
x is_a_root_of f
theorem Th41: :: HURWITZ:41
:: deftheorem Def9 defines *' HURWITZ:def 9 :
theorem Th42: :: HURWITZ:42
theorem Th43: :: HURWITZ:43
theorem Th44: :: HURWITZ:44
theorem Th45: :: HURWITZ:45
theorem Th46: :: HURWITZ:46
theorem Th47: :: HURWITZ:47
Lm14:
for x, xv being Element of COMPLEX
for u, v, uv, vv being Element of REAL st x = u + (v * <i> ) & xv = uv + (vv * <i> ) holds
(|.(x + (xv *' )).| ^2 ) - (|.(x - xv).| ^2 ) = (4 * u) * uv
Lm15:
for x, xv being Element of COMPLEX
for u, v, uv, vv being Element of REAL st x = u + (v * <i> ) & xv = uv + (vv * <i> ) & uv < 0 holds
( ( u < 0 implies |.(x - xv).| < |.(x + (xv *' )).| ) & ( u > 0 implies |.(x - xv).| > |.(x + (xv *' )).| ) & ( u = 0 implies |.(x - xv).| = |.(x + (xv *' )).| ) )
theorem Th48: :: HURWITZ:48
theorem Th49: :: HURWITZ:49
theorem Th50: :: HURWITZ:50
:: deftheorem defines F* HURWITZ:def 10 :
theorem Th51: :: HURWITZ:51
theorem Th52: :: HURWITZ:52
theorem :: HURWITZ:53
theorem :: HURWITZ:54