Introduction to Non-Linear Algebra

✍ Scribed by V. Dolotin, A Morozov

• Publisher: World Scientific Publishing Company
• Year: 2007
• Tongue: English
• Leaves: 127
• Category: Library

✦ Synopsis

Concise introduction to a relatively new subject of non-linear algebra: literal extension of text-book linearalgebra to the case of non-linear equations and maps. This powerful science is based on the notions of discriminant(hyperdeterminant) and resultant, which today can be effectively studied both analytically and by modern computerfacilities. The paper is mostly focused on resultants of non-linear maps. First steps are described in direction ofMandelbrot-set theory, which is direct extension of the eigenvalue problem from linear algebra, and is related byrenormalization group ideas to the theory of phase transitions and dualities

✦ Table of Contents

Formulation of the problem......Page 3
Comparison of linear and non-linear algebra......Page 5
Tensors......Page 9
Tensor algebra......Page 10
Solutions to poly-linear and non-linear equations......Page 13
Non-homogeneous equations......Page 16
Homogeneous non-linear equations......Page 17
Solution of systems of non-homogeneous equations: generalized Craemer rule......Page 19
Tensors, possessing a resultant: generalization of square matrices......Page 20
Multiplicativity w.r.t. composition: generalization of detAB = detA detB for determinants......Page 21
Resultant for matrix-like maps: a more interesting generalization of det(to1.5.diag ajj )to1.5. = j=1n ajj for matrices......Page 22
Additive decomposition: generalization of detA = (-)i Ai(i) for determinants......Page 23
Linear equations......Page 24
On the origin of extra factors in......Page 26
An example of cubic equation......Page 27
Koszul complex. I. Definitions......Page 28
Linear maps (the case of s1=…=sn=1)......Page 29
A triple of polynomials (the case of n=3)......Page 30
Koszul complex. II. Explicit expression for determinant of exact complex......Page 31
Koszul complex. IV. Formulation through -tensors......Page 34
Not only Koszul and not only complexes......Page 36
Operators......Page 38
Rectangular tensors and linear maps......Page 39
Generalized Vieta formula for solutions of non-homogeneous equations......Page 40
Discriminants of polylinear forms......Page 45
Degree of discriminant......Page 46
Discriminant as an k=1r SL(nk) invariant......Page 47
Symmetric, diagonal and other specific tensors......Page 48
Relation to resultants......Page 49
Degeneracy condition in terms of det......Page 50
Example......Page 51
Koshul complexes, associated with poly-linear and symmetric functions......Page 52
Reductions of Koshul complex for poly-linear tensor......Page 53
Reduced complex for generic bilinear nn tensor: discriminant is determinant of the square matrix......Page 55
Complex for generic symmetric discriminant......Page 56
Iterated discriminant......Page 57
Discriminants from diagrams......Page 58
The case of rank r=1 (vectors)......Page 59
The case of rank r=2 (matrices)......Page 60
The 222 case (Cayley hyperdeterminant Cay)......Page 63
Generalities......Page 66
The n|r = 2|2 case......Page 67
The n|r = 2|3 case......Page 70
The n|r = 2|4 case......Page 71
Direct evaluation of Z(T)......Page 74
Gaussian integrations: specifics of cases n=2 and r=2......Page 78
Alternative partition functions......Page 79
Pure tensor-algebra (combinatorial) partition functions......Page 82
Tensorial exponent......Page 86
From linear to non-linear case......Page 87
Generalities......Page 88
Number of eigenvectors cn|s as compared to the dimension Mn|s of the space of symmetric functions......Page 89
Decomposition (6.8) of characteristic equation: example of diagonal map......Page 90
Decomposition (6.8) of characteristic equation: non-diagonal example for n|s = 2|2......Page 93
Generalities......Page 94
Examples for diagonal maps......Page 95
The map f(x) = x2 + c:......Page 97
Map from its eigenvectors: the case of n|s = 2|2......Page 98
Appropriately normalized eigenvectors and elimination of -parameters......Page 99
Iterated maps......Page 101
Relation between Rn|s2(s+1|A2) and Rn|s(|A)......Page 102
Unit maps and exponential of maps: non-linear counterpart of algebra group relation......Page 104
Exponential maps for n|s=2|2......Page 105
Examples of exponential maps for 2|s......Page 106
Potential applications......Page 107
Number of solutions......Page 108
Index of projective map......Page 110
Bifurcations of maps, Julia and Mandelbrot sets......Page 111
Relation between discrete and continuous dynamics: iterated maps, RG-like equations and effective actions......Page 112
Taking integrals......Page 116
Integrals of polylinear forms......Page 117
Multiplicativity of integral discriminants......Page 118
Acknowledgements......Page 119

✦ Subjects

Математика;Общая алгебра;