Introduction to non-linear algebra

Introduction to Non-Linear Algebra

Volumn , Issue , 2007, Pages 1-269

  Dolotin, V ^a   Morozov, A ^a  

^a INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84967351042     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/6508     Document Type: Book

References (57)

1. I.Gelfand, Lectures on Linear Algebra (1948) Moscow
2. A.Dolotin and A.Morozov, The Universal Mandelbrot Set. Beginning of the Story, World Scientific, 2006; Algebraic Geometry of Discrete Dynamics. The case of one variable, hep-th/0501235; On the Shape of Elemantary Domains or why Mandelbroit Set is made from Almost Ideal Circles?, hep-th/0701234
3. R. Thom, The bifurcation subspace of a set of maps, Ed. N.H. Kuiper, Manifolds (Amsterdam, 1970), Lect. notes in math., Vol. 197, Springer, 1971, pp. 202-208;
4. V.I. Arnol’d, S.M. Gusein-Zade, A.N. Varchenko, Singularities of differentiable maps, Vol. 1-2, Birkhauser, 1985-1988;
5. V.I.Arnold, Mathematical Methods of Classical Mechanics (1989) Springer;
6. H.Haken, Synergetics, an Introduction (1977) Springer;
7. R.L.Devaney, An Introduction to Chaotic Dynamical Systems, 2-nd ed. (1989) Addison-Wesley Publ.;
8. S.H.Strogatz, Non-Linear Dynamics and Chaos (1994) Addison-Wesley Publ.
9. A.Cayley, On the Theory of Linear Transformations, Camb.Math.J. 4 (1845) 193-209; see [14] for recent show-up of Cayley’s 2×2×2 hyperdeterminant in stringtheory literature, where it appears in the role of the SL(2)3 invariant
10. B.L. van der Waerden, Algebra, Vol. 1-2, Springer, 1967-1971
11. L. Schläfli, Über die Resultante eines Systems mehrerer algebraischen Gleichungen, Denk.der Keiser. Akad. der Wiss., math-naturwiss. Klasse, 4 Band, 1852; Gesammelte Abhandlungen, Band 2, s. 9-112, Birkhäuser Verlag, Basel, 1953
12. I.Gelfand, M.Kapranov and A.Zelevinsky, Discriminants, Resultants and Multidimensional Determinants (1994) Birkhauser
13. A. Gorodentsev and B. Shapiro, On associated discriminants for polynomials of one variable, Beitrage Algebra Geom. 39 (1998) 53-74
14. V.Dolotin, On Discriminants of Polylinear Forms, alg-geom/9511010
15. V.Dolotin, On Invariant Theory, alg-geom/9512011
16. V.Dolotin, QFT’s with Action of Degree 3 and Higher and Degeneracy of Tensors, hep-th/9706001
17. Yu.Chernyakov and V.Dolotin, Operations and Identities in Tensor Algebra, math.RA/0501206
18. A.Morozov, String Theory, What is It?, Sov.Phys.Uspekhi, 35 (1992) 671-714 (Usp.Fiz.Nauk 162 83-176)
19. A.Miyake and M.Wadati, Multiparticle Entaglement and Hyperdeterminants, quant-ph/02121146;
20. V.Coffman, J.Kundu and W.Wooters, Distributed Entaglement, Phys.Rev. A61 (2000) 52306, quant-ph/9907047;
21. M.J.Duff, String Triality, Black Hole Entropy and Cayley’s Hyperdeterminant, hep-th/0601134; Hidden Symmetries of the Nambu-Goto Action, hepth/0602160;
22. R.Kallosh and A.Linde, Strings, Black Holes and Quantum Information, hep-th/0602061
23. S.I. Gelfand, Yu.I. Manin, Methods of Homological Algebra, Springer, 1999
24. A.Connes and D.Kreimer, Comm.Math.Phys. 199 (1998) 203-242
25. A.Gerasimov, A.Morozov and K.Selivanov, Bogolubov’s Recursion and Integrability of Effective Actions, Int.J. Mod.Phys. A16 (2001) 1531-1558, hepth/0005053
26. A.Gerasimov, S.Khoroshkin, D.Lebedev, A.Mironov and A.Morozov, Generalized Hirota equations and representation theory. The case of SL(2) and SLq(2), Int.J.Mod.Phys.A 10 (1995) 2589-2614, hep-th/9405011
27. A.Morozov and A.Niemi, Can RG Flow End in a Big Mess?, Nucl.Phys. B666 (2003) 311-336, hep-th/0304178
28. A.Morozov, Integrability and Matrix Models, Sov.Phys.Usp. 37 (1994) 1 (Usp.Fiz.Nauk 164 3-62) hep-th/9303139; hep-th/9502091;
29. A.Mironov, Int.J.Mod.Phys. A9 (1994) 4335, hep-th/9312212; Phys. of Particles and Nuclei 33 (2002) 537
30. S.Lang, Algebra (1965) Addison-Wesley Publ.;
31. A.Kurosh, Course of High Algebra (1971) Moscow; MAPLE and Mathematica contain programs for evaluating resultants and discriminants of ordinary polynomials. Hopefully soon they will be extended to evaluation of resultants and discriminants for arbitrary systems of nonlinear and poly-linear equations.
32. N.Berkovits, JHEP 0004 (2000) 018, hep-th/0001035; JHEP 0409 (2004) 047, hep-th/0406055;
33. A.Gorodentsev and A.Losev, Lectures at ITEP-DIAS School, July 2004
34. A.Gorodentsev and A.Rudakov, On syzigies of Highest Weight Orbits, math.AG/062316;
35. D.Krotov and A.Losev, Quantum Field Theory as Effective BV Theory from Chern-Simons, hep-th/0603201
36. Sh.Shakirov, The coincident root loci and higher discriminants of polynomials, Theor.Math.Phys. (2007), math/0609524
37. V.Dolotin, A.Morozov, Sh.Shakirov, A-infinity Structure on Simplicial Complexes, Theor.Math.Phys.(2007), arxiv: 0704.2609;
38. Higher Nilpotent Analogues of A-infinity Structure, Phys.Lett.B (2007), arxiv: 0704.2884
39. Sh.Shakirov et al., Games with Discriminants, to be published
40. J.-P. Serre, A course in arithmetic, Springer, 1973;
41. S.Lang, Elliptic Functions (1973) Addison-Wesley Publ.;
42. A.Weil, Elliptic Functions according to Eisentstein and Kronecker (1976) Springer-Verlag (Mir, Moscow, 1978);
43. N.Koblitz, Introduction to Elliptic Curves and Modular Forms (1984) Springer-Verlag
44. A.Alexandrov, A.Mironov and A.Morozov, Int.J.Mod.Phys. A19 (2004) 4127-4163, Theor.Math.Phys. 142 (2005) 419-488, hep-th/0412099; hepth/0412205; M-Theory of Matrix Models, hep-th/0605171;
45. A.Morozov, Challenges of Matrix Models, Lectures at Cargese School 2004, hep-th/0502010
46. A.Grothendieck, Esquisse d’un Programme (Sketch of a Program), 1984;
47. G.Shabat and V.Voevodsky, Drawing Curves over Number Fields, The Grothendieck Festschrift, Birkhauser, III (1990) 199-227;
48. A.Levin and A.Morozov, On the Foundations of Random Lattice Approach to Quantum Gravity, Phys.Lett. B243 (1990) 207-214;
49. G.Shabat, Complex Analysis and Dessins d’Enfants, Complex Analysis in Modern Math., Phasis (1998) 257-268
50. E.Akhmedov, Towards the Theory of Non-Abelian Tensor Fields, Theor.Math.Phys. 145(2005), 1646-1665, hep-th/0503234
51. E.Akhmedov, V.Dolotin and A.Morozov, Comment on the Surface Exponential for Tensor Fields, JETP Letters 81 (2005) 639-643 (Pisma v Zh.Eksp.Teor.Fiz. 81 (2005) 776-779) hep-th/0504160
52. N.Bogolubov and O.Parasyuk, Acta Math. 97 (1957) 227;
53. N.Bogolubov and D.Shirkov, Introduction to the Theory of Quantum Fields, Moscow, 1957;
54. B.Stepanov and O.Zavyalov, Yad.Phys. 1 (1965) 922;
55. K.Hepp, Comm.Math.Phys. 2 (1966) 301; M.Zimmerman, Comm.Math.Phys. 15 (1969) 208
56. A.Connes and D.Kreimer, Comm.Math.Phys. 199 (1998) 203-242; 210 (2000) 249-273, hep-th/9912092
57. A.Morozov and M.Serbin, Non-Linear Algebra and Bogolubov’s Recursion, Theor.Math.Phys.(2007), hep-th/0703258