Website Search
Find information on spaces, staff, and services.
Catalog Search
Books
Introduction to non-linear algebra
- Author / Creator
- Available as
- Physical
Details
Checking for availability...
- Creator
- V. Dolotin, A. Morozov
- Format
- Books
- Language
- English
- Contributors
- Publication
-
- Hackensack, NJ : World Scientific, [2007]
- ©2007
- Physical Details
-
- xvi, 269 pages : illustrations, portraits ; 24 cm
- ISBNs
- 9789812708007, 9812708006
- OCLC
- ocn213766058
- Includes bibliographical references (pages 267-269).
- 1.1 Comparison of linear and non-linear algebra 5 -- 1.2 Quantities, associated with tensors of different types 12 -- 1.2.1 A word of caution 1.2.2 Tensors 14 -- 1.2.3 Tensor algebra 16 -- 1.2.4 Solutions to poly-linear and non-linear equations 21 -- 2 Solving Equations. Resultants 29 -- 2.1 Linear algebra (particular case of s = 1) 2.1.1 Homogeneous equations 2.1.2 Non-homogeneous equations 30 -- 2.2 Non-linear equations 31 -- 2.2.1 Homogeneous non-linear equations 2.2.2 Solution of systems of iron-homogeneous equations: Generalized Cramer rule 34 -- 3 Evaluation of Resultants and Their Properties 39 -- 3.1 Summary of resultant theory 3.1.1 Tensors, possessing a resultant: Generalization of square matrices 3.1.2 Definition of the resultant: Generalization of condition det A = 0 for solvability of system of homogeneous linear equations 40 -- 3.1.3 Degree of the resultant: Generalization of d[subscript n/2] = deg[subscript A] (det A) = n for matrices 3.1.4 Multiplicativity w.r.t. composition: Generalization of det AB = det A det B for determinants 41 -- 3.1.5 Resultant for diagonal maps: Generalization of det [Characters not reproducible] for matrices 42 -- 3.1.6 Resultant for matrix-like maps: A more interesting generalization of det [Characters not reproducible] for matrices 3.1.7 Additive decomposition: Generalization of det A = [Characters not reproducible] for determinants 44 -- 3.1.8 Evaluation of resultants 45 -- 3.2 Iterated resultants and solvability of systems of non-linear equations 46 -- 3.2.1 Definition of iterated resultant R[subscript n/s] {A} 3.2.2 Linear equations 47 -- 3.2.3 On the origin of extra factors in R 49 -- 3.2.4 Quadratic equations 50 -- 3.2.5 An example of cubic equation 51 -- 3.2.6 More examples of 1-parametric deformations 52 -- 3.2.7 Iterated resultant depends on simplicial structure 3.3 Resultants and Koszul complexes 3.3.1 Koszul complex. I. Definitions 53 -- 3.3.2 Linear maps (the case of s[subscript 1] = ... = s[subscript n] = 1) 55 -- 3.3.3 A pair of polynomials (the case of n = 2) 56 -- 3.3.4 A triple of polynomials (the case of n = 3) 57 -- 3.3.5 Koszul complex. II. Explicit expression for determinant of exact complex 59 -- 3.3.6 Koszul complex. III. Bicomplex structure 62 -- 3.3.7 Koszul complex. IV. Formulation through [epsilon]-tensors 64 -- 3.3.8 Not only Koszul and not only complexes 66 -- 3.4 Resultants and diagram representation of tensor algebra 69 -- 3.4.1 Tensor algebras T(A) and T(T), generated by [Characters not reproducible] and T 70 -- 3.4.2 Operators 71 -- 3.4.3 Rectangular tensors and linear maps 73 -- 3.4.4 Generalized Vieta formula for solutions of non-homogeneous equations 74 -- 3.4.5 Coinciding solutions of non-homogeneous equations: Generalized discriminantal varieties 80 -- 4 Discriminants of Polylinear Forms 85 -- 4.1 Definitions 4.1.1 Tensors and polylinear forms 4.1.2 Discriminantal tensors 86 -- 4.1.3 Degree of discriminant 4.1.4 Discriminant as an [Characters not reproducible] invariant 88 -- 4.1.5 Diagram technique for the [Characters not reproducible] invariants 89 -- 4.1.6 Symmetric, diagonal and other specific tensors 90 -- 4.1.7 Invariants from group averages 92 -- 4.1.8 Relation to resultants 4.2 Discriminants and resultants: Degeneracy condition 94 -- 4.2.1 Direct solution to discriminantal constraints 4.2.2 Degeneracy condition in terms of det T 95 -- 4.2.3 Constraint on P[z] 96 -- 4.2.4 Example 4.2.5 Degeneracy of the product 97 -- 4.2.6 An example of consistency between (4.18) and (4.22) 98 -- 4.3 Discriminants and complexes 99 -- 4.3.1 Koszul complexes, associated with poly-linear and symmetric functions 4.3.2 Reductions of Koszul complex for poly-linear tensor 101 -- 4.3.3 Reduced complex for generic bilinear n x n tensor: Discriminant is determinant of the square matrix 105 -- 4.3.4 Complex for generic symmetric discriminant 107 -- 4.4 Other representations 4.4.1 Iterated discriminant 4.4.2 Discriminant through paths 109 -- 4.4.3 Discriminants from diagrams 110 -- 5 Examples of Resultants and Discriminants 111 -- 5.1 The case of rank r = 1 (vectors) 5.2 The case of rank r = 2 (matrices) 113 -- 5.3 The 2 x 2 x 2 case (Cayley hyperdeterminant) 119 -- 5.4 Symmetric hypercubic tensors 2[superscript xr] and polynomials of a single variable 125 -- 5.4.1 Generalities 5.4.2 The n/r = 2/2 case 132 -- 5.4.3 The n/r = 2/3 case 137 -- 5.4.4 The n/r = 2/4 case 138 -- 5.5 Functional integral (1.7) and its analogues in the n = 2 case 143 -- 5.5.1 Direct evaluation of Z(T) 5.5.2 Gaussian integrations: Specifics of cases n = 2 and r = 2 150 -- 5.5.3 Alternative partition functions 151 -- 5.5.4 Pure tensor-algebra (combinatorial) partition functions 154 -- 5.6 Tensorial exponent 160 -- 5.6.1 Oriented contraction 5.6.2 Generating operation ("exponent") 5.7 Beyond n = 2 161 -- 5.7.1 D[subscript 3/3], D[subscript 3/4] and D[subscript 4/3] through determinants 5.7.2 Generalization: Example of non-Koszul description of generic symmetric discriminants 164 -- 6 Eigenspaces, Eigenvalues and Resultants 173 -- 6.1 From linear to non-linear case 6.2 Eigenstate (fixed point) problem and characteristic equation 174 -- 6.2.1 6.2.2 Number of eigenvectors c[subscript n/s] as compared to the dimension M[subscript n/s] of the space of symmetric functions 176 -- 6.2.3 Decomposition (6.8) of characteristic equation: Example of diagonal map 178 -- 6.2.4 Decomposition (6.8) of characteristic equation: Non-diagonal example for n/s = 2/2 181 -- 6.2.5 Numerical examples of decomposition (6.8) for n > 2 183 -- 6.3 Eigenvalue representation of non-linear map 6.3.1 6.3.2 Eigenvalue representation of Plucker coordinates 185 -- 6.3.3 Examples for diagonal maps 6.3.4 The map f(x) = x[superscript 2] + c 188 -- 6.3.5 Map from its eigenvectors: The case of n/s = 2/2 189 -- 6.3.6 Appropriately normalized eigenvectors and elimination of A-parameters 192 -- 6.4 Eigenvector problem and unit operators 194 -- 7 Iterated Maps 197 -- 7.1 Relation between R[subscript n/s[superscript 2]] ([lambda][superscript s+1]/A[superscript o2]) and R[subscript n/s] ([lambda]/A) 198 -- 7.2 Unit maps and exponential of maps: Non-linear counterpart of algebra [leftrightarrow] group relation 201 -- 7.3 Examples of exponential maps 203 -- 7.3.1 Exponential maps for n/s = 2/2 7.3.2 Examples of exponential maps for 2/s 205 -- 7.3.3 Examples of exponential maps for n/s = 3/2 8 Potential Applications 209 -- 8.1 Solving equations 8.1.1 Cramer rule 8.1.2 Number of solutions 211 -- 8.1.3 Index of projective map 212 -- 8.1.4 Perturbative (iterative) solutions 215 -- 8.2 Dynamical systems theory 221 -- 8.2.1 Bifurcations of maps, Julia and Mandelbrot sets 8.2.2 The universal Mandelbrot set 223 -- 8.2.3 Relation between discrete and continuous dynamics: Iterated maps, RG-like equations and effective actions 225 -- 8.3 Jacobian problem 231 -- 8.4 Taking integrals 8.4.1 Basic example: Matrix case, n/r = n/2 232 -- 8.4.2 Basic example: Polynomial case, n/r = 2/r 8.4.3 Integrals of polylinear forms 233 -- 8.4.4 Multiplicativity of integral discriminants 234 -- 8.4.5 Cayley 2 x 2 x 2 hyperdeterminant as an example of coincidence between integral and algebraic discriminants 236 -- 8.5 Differential equations and functional integrals 8.6 Renormalization and Bogolubov's recursion formula 237 -- 9 Appendix: Discriminant D[subscript 3/3](S) 241