BASES OF RECURRENT SEQUENCES
https://doi.org/10.22405/2226-8383-2015-16-2-155-185
Abstract
This paper provides an overview of the results (with varying degrees of detail) in three different directions. The main Central direction refers to recurrent sequences, primarily to their base (in a different sense) sets. Another direction is related to new combinatorial objects (v, k1, k2)-confi gurations encountered on the way of weakening the determinants of well-known combinatorial objects (v, k, λ)-configuration. The third direction deals with invariant differentials of higher orders from several smooth functions of one real variable. In each of these themes the issues associated with combinatorial configurations in the form of finite planes, and the results obtained through the same type of views, points of the corresponding configurations of points in multidimensional locally Euclidean spaces. In the case of invariant differentials of these representations arise naturally, and in the case of recurrent sequences and (v, k1, k2)-configurations are introduced by analogy, but in an artificial way.
References
1. Kostrikin, A. I. & Manin, Yu. I. 1986, "Lineinaya algebra i geometriya" (Russian) [Linear algebra and geometry] Second edition. “Nauka”, Moscow, 304 pp.
2. Kon, P. 1968, "Universal’naya algebra" (Russian) [Universal algebra] Translated from the English by T. M. Baranovic. Edited by A. G. Kurosh “MIR”, Moscow 351 pp.
3. Veblen, O. 1928, "Differential invariants and geometry" , Atti del Congr., Int. Mat., Bologna.
4. Kirillov, A. A. 1980, "Invariant operators over geometric quantities" (Russian) Current problems in mathematics, Vol. 16 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, pp. 3–29, 228.
5. Malyshev, F. M. 1980, "Simpletsialnye system of linear equations" , Algebra, Moscow, Moscow University Press, pp. 53–56.
6. Kartesi, F. 1980, "Vvedenie v konechnye geometrii" (Russian) [Introduction to finite geometries] Translated from the English by F. L. Varpahovskii and A. S. Solodovnikov., “Nauka”, Moscow, 320 pp.
7. Malyshev, F. M. 2008, "Generating sets of elements of recurrent sequences" , Tr. discr. Mat., Fizmatlit, Moscow, vol. 11, № 2, pp. 86 – 111.
8. Malyshev, F. M. 2011, "Bases of the set of integers with respect to multi-shift operations" , Mat. Issues. kriptogr., vol. 2, no. 1, pp. 29–73.
9. Malyshev, F. M. 2012, "Metric properties of the nested set of integers into a cylinder" , Mat. Issues. kriptogr., vol. 3, no. 3, pp. 57–79.
10. Delone, B. N. & Sandakova, N. N. 1961, "Theory of stereohedra" (Russian) Trudy Mat. Inst. Steklov., vol. 64, pp. 28—51.
11. Wolfram, S. 1988, "Cellular Automaton Supercomputing" , In High-Speed Computing. University of Illinois Press., pp. 40–48.
12. Malyshev, F. M. & Kutyreva, E. V. 2006, "On the distribution of the number of ones in a Boolean Pascal’s triangle" (Russian) Diskret. Mat., vol. 18, no. 2, pp. 123–131; translation in Discrete Math. Appl., vol. 16 (2006), no. 3, pp. 271—279.
13. Malyshev, F. M. & Tarakanov, V. E. 2001, "On (v,k)-configurations" (Russian) Mat. Sb., vol. 192, no. 9, pp. 85–108; translation in Sb. Math., vol. 192 (2001), no. 9–10, pp. 1341—1364.
14. Holl, M. 1970, "Kombinatorika" (Russian) [Combinatorial theory] Translated from the English by S. A. Shirokova. Edited by A. O. Gel’fond and V. E. Tarakanov, “MIR”, Moscow, 424 pp.
15. Harary, F. 1973, "Teoriya grafov" (Russian) [Graph theory] Translated from the English by V. P. Kozyrev. Edited by G. P. Gavrilov., “MIR”, Moscow, 300 pp.
16. Cameron, Peter J. & van Lint, Jacobus Hendricus 1980, "Teoriya grafov, teoriya kodirovaniya i blok-skhemy" (Russian) [Graph theory, coding theory and block designs] Translated from the English by B. S. Steckin. “Nauka”, Moscow, 140 pp.
17. Sachkov, V. N. 1977, "Kombinatornye metody diskretnoi matematiki" (Russian) [Combinatorial methods of discrete mathematics], “Nauka”, Moscow, 320 pp.
18. Tarakanov, V. E. 1985, "Kombinatornye zadachi i (0,1)-matritsy" (Russian) [Combinatorial problems and (0,1)-matrices], Problemy Nauki i Tekhnicheskogo Progressa. [Problems of Science and Technological Progress], “Nauka”, Moscow, 192 pp.
19. Malyshev, F. M. & Frolov, A. A. 2012, "Classification of (v,3)-configurations" , Translation of Mat. Zametki, vol. 91, no. 5, pp. 741—749. Math. Notes, vol. 91 (2012), no. 5-6, pp. 689—696.
20. Trishin, A. E. 2004, "Classification circulant (v, 5)-matrices" , Review of Applied and Industrial Mathematics, vol. 11, № 2. pp. 258 – 259.
21. Frolov, A. A. 2008, "Classification of indecomposable abelian (v,5)-groups" (Russian) Diskret. Mat., vol. 20, no. 1, pp. 94–108; translation in Discrete Math. Appl., vol. 18, no. 1, pp. 99—114.
22. Broslavsky, M. V. 2010, "Examples of (v, k1, k2)-configurations" Thesis. Moscow, v/ch 33965 (Russian).
23. Nikulin, V. V. & Shafarevich, I. R. 1983, "Geometrii i gruppy" (Russian) [Geometries and groups], “Nauka”, Moscow, 240 pp.
24. Ryshkov, S. S. & Baranovskii, E. P. 1979, "Classical methods of the theory of lattice packings" (Russian) Uspekhi Mat. Nauk, vol. 34, no. 4(208), pp. 3-–63, 256.
25. Kurakin, V. L., Kuzmin, A. S., Mikhalev, A. V. & Nechaev, A. A. 1995, "Linear recurring sequences over rings and modules" , Algebra, 2. J. Math. Sci., vol. 76, №. 6, pp. 2793 – 2915.
26. Kuzmin, A. S.; Kurakin, V. L.; Nechaev, A. A. 1997, "Pseudorandom and polylinear sequences" (Russian) Proceedings in discrete mathematics, Tr. Diskretn. Mat., Nauchn. Izd. TVP, Moscow, vol. 1, pp. 139—202.
27. National Institute of Standards and Technology, U.S.A., Advanced Encryption Standard (AES) FIPS – 197, 2001.
28. Specification of ARIA. National Security Research Institute (NSRI). January, 2005. http://www.nsri.re.kr/ARIA/index-e.html.
29. SEED Algorithm Specification. Korea Information Security Agency. 2005. https:// tools.ietf.org/draft-park-seed-01.
30. Lu J., Ji W., Hu L., Ding J., Pyshkin A., Weinmann R. Analysis of the SMS4 Block Cipher. Procedings of ACISP’07. LNCS 4586, 2007. P. 306 – 318.
31. Vinberg, E. B. & Shvartsman, O. V. 1988, "Discrete groups of motions of spaces of constant curvature" (Russian), Current problems in mathematics. Fundamental directions, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, vol. 29, pp. 147 – 259.
32. Malyshev, F. M. 1977, "Closed subsets of roots and the cohomology of regular subalgebras" (Russian), Mat. Sb. (N.S.), vol. 104(146), no. 1, pp. 140—150, 176.
33. Burbaki, N. 1972, "Gruppy i algebry Li. Gruppy Kokstera i sistemy Titsa. Gruppy, porozhdennye otrazheniyami. Sistemy kornei." (Russian) [Lie groups and algebras. Coxeter groups and Tits systems. Groups generated by reflections. Root systems] Translated from the French by A. I. Kostrikin and A. N. Tjurin. Edited by A. I. Kostrikin. “MIR”, Moscow, 334 pp.
34. Morgado, I. 1972, "Note on quasi-orders, partial orders and orders" , Notes comuus mat., № 43, pp. 31 – 40.
35. Krishnamurthy, V. 1966, "On the number of topologies on a finite set" , Amer. Math. Monthly., vol. 73. pp. 154 – 157.
36. Tolpygo, A. K. 1972, "The cohomology of parabolic Lie algebras" (Russian) Mat. Zametki, vol. 12, № 3, pp. 251–255.
37. Lidl, R. & Niderraiter, G. 1988, "Konechnye polya. Tom 1" (Russian) [Finite fields. Vol. 1] Translated from the English by A. E. Zhukov and V. I. Petrov. Translation edited and with a preface by V. I. Nechaev. “MIR”, Moscow, 430 pp.; Lidl, R. & Niderraiter, G. 1988, "Konechnye polya. Tom 2" (Russian) [Finite fields. Vol. 2] Translated from the English by A. E. Zhukov and V. I. Petrov. Translation edited by V. I. Nechaev. “MIR”, Moscow, pp. 433—822.
38. Lupanov, O. B. 2007, "Introduction to mathematical logic. Lecture notes." , Moscow, Mekh – mat. MGU M. V. Lomonosov, 199 p.
39. Vorob’ev, N. N. 1978, "Chisla Fibonachchi" (Russian) [Fibonacci numbers] Fourth edition. Populyarnye Lektsii po Matematike [Popular Lectures on Mathematics], 6. “Nauka”, Moscow, 142 pp.
Review
For citations:
Malyshev F.M. BASES OF RECURRENT SEQUENCES. Chebyshevskii Sbornik. 2015;16(2):155-185. (In Russ.) https://doi.org/10.22405/2226-8383-2015-16-2-155-185