مقدمه‌ای بر جبر غیرخطی

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