Geodesic
Bases of recurrent sequences
Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 155-185.

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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 $(v,k_1,k_2)$-configurations encountered on the way of weakening the determinants of well-known combinatorial objects $(v,k,\lambda)$-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 $(v,k_1,k_2)$-configurations are introduced by analogy, but in an artificial way. Bibliography: 39 titles.
Keywords: recurrent sequences, lattices, Torah, combinatorial configuration, invariant differential operators. 
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F. M. Malyshev. Bases of recurrent sequences. Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 155-185. http://geodesic.mathdoc.fr/item/CHEB_2015_16_2_a10/

[1] Kostrikin A. I., Manin Yu. I., Linear algebra and geometry, Second edition, Nauka, M., 1986, 304 pp. (Russian) | MR

[2] Kon P., Universal algebra, Translated from the English by T. M. Baranovic, ed. A. G. Kurosh, Mir, M., 1968, 351 pp. (Russian) | MR

[3] Veblen O., “Differential invariants and geometry”, Atti del Congr., Int. Mat. (Bologna, 1928)

[4] Kirillov A. A., “Invariant operators over geometric quantities”, Current problems in mathematics, 16, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, M., 1980, 3–29 (Russian) ; 228

[5] Malyshev F. M., “Simpletsialnye system of linear equations”, Algebra, Moscow University Press, M., 1980, 53–56

[6] Kartesi F., Introduction to finite geometries, Translated from the English by F. L. Varpahovskii, A. S. Solodovnikov, Nauka, M., 1980, 320 pp. (Russian) | MR

[7] Malyshev F. M., “Generating sets of elements of recurrent sequences”, Tr. discr. Mat., 11:2 (2008), 86–111

[8] Malyshev F. M., “Bases of the set of integers with respect to multi-shift operations”, Mat. Issues. kriptogr., 2:1 (2011), 29–73 | MR

[9] Malyshev F. M., “Metric properties of the nested set of integers into a cylinder”, Mat. Issues. kriptogr., 3:3 (2012), 57–79

[10] Delone B. N., Sandakova N. N., “Theory of stereohedra”, Trudy Mat. Inst. Steklov, 64, 1961, 28–51 (Russian) | MR

[11] Wolfram S., “Cellular Automaton Supercomputing”, High-Speed Computing, University of Illinois Press, 1988, 40–48

[12] Malyshev F. M., Kutyreva E. V., “On the distribution of the number of ones in a Boolean Pascal's triangle"”, Discrete Math. Appl., 16:3 (2006), 271–279 | DOI | DOI | MR | Zbl

[13] Malyshev F. M., Tarakanov V. E., “On $(v,k)$-configurations”, Sb. Math., 192:9–10 (2001), 1341–1364 | DOI | DOI | MR | Zbl

[14] Holl M., Combinatorial theory, Translated from the English by S. A. Shirokova, ed. A. O. Gel'fond, V. E. Tarakanov, Mir, M., 1970, 424 pp. (Russian) | MR

[15] Harary F., Graph theory, Translated from the English by V. P. Kozyrev, ed. G. P. Gavrilov, Mir, M., 1973, 300 pp. (Russian) | MR

[16] Cameron P. J., van Lint J. H., Graph theory, coding theory and block designs, Translated from the English by B. S. Steckin, Nauka, M., 1980, 140 pp. (Russian)

[17] Sachkov V. N., Combinatorial methods of discrete mathematics, Nauka, M., 1977, 320 pp. (Russian)

[18] Tarakanov V. E., Combinatorial problems and $(0,1)$-matrices, Problems of Science and Technological Progress, Nauka, M., 1985, 192 pp. (Russian)

[19] Malyshev F. M., Frolov A. A., “Classification of $(v,3)$-configurations”, Math. Notes, 91:5–6 (2012), 689–696 | DOI | DOI | MR | Zbl

[20] Trishin A. E., “Classification circulant $ (v, 5) $-matrices”, Review of Applied and Industrial Mathematics, 11:2 (2004), 258–259

[21] Frolov A. A., “Classification of indecomposable abelian $(v,5)$-groups”, Discrete Math. Appl., 18:1 (2008), 99–114 | DOI | DOI | MR | Zbl

[22] Broslavsky M. V., Examples of $ (v, k_1, k_2) $-configurations, Thesis, v/ch 33965, M., 2010 (Russian)

[23] Nikulin V. V., Shafarevich I. R., Geometries and groups, Nauka, M., 1983, 240 pp. (Russian) | MR

[24] Ryshkov S. S., Baranovskii E. P., “Classical methods of the theory of lattice packings”, Uspekhi Mat. Nauk, 34:4(208) (1979), 3–63 (Russia) ; 256 | MR | Zbl

[25] Kuzmin A. S., Kurakin V. L., Mikhalev A. V., Nechaev A. A., “Linear recurreces over rings and modules”, J. Math. Science (Contemporary Math. and Its Appl. Thematic surveys), 76:6, 2793–2915 | MR | Zbl

[26] Kuzmin A. S.; Kurakin V. L., Nechaev A. A., “Pseudorandom and polylinear sequences”, Proceedings in discrete mathematics, 1 (1997), 139–202 (Russian) | Zbl

[27] Advanced Encryption Standard (AES) FIPS-197, National Institute of Standards and Technology, U.S.A., 2001

[28] Specification of ARIA, , National Security Research Institute, 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, 306–318

[31] Vinberg E. B., Shvartsman O. V., “Discrete groups of motions of spaces of constant curvature”, Current problems in mathematics. Fundamental directions, Itogi Nauki i Tekhniki, 29, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., M., 1988, 147–259 (Russian)

[32] Malyshev F. M., “Closed subsets of roots and the cohomology of regular subalgebras”, Mat. Sb. (N.S.), 104(146):1 (1977), 140–150 (Russian) ; 176 | MR | Zbl | Zbl

[33] Burbaki N., Lie groups and algebras. Coxeter groups and Tits systems. Groups generated by reflections. Root systems, Translated from the French by A. I. Kostrikin, A. N. Tjurin, ed. A. I. Kostrikin, Mir, M., 1972, 334 pp. (Russian)

[34] Morgado I., “Note on quasi-orders, partial orders and orders”, Notes comuus mat., 1972, no. 43, 31–40

[35] Krishnamurthy, “On the number of topologies on a finite set”, Amer. Math. Monthly, 73 (1966), 154–157 | DOI | MR | Zbl

[36] Tolpygo A. K., “The cohomology of parabolic Lie algebras”, Mat. Zametki, 12:3 (1972), 251–255 (Russian) | MR | Zbl

[37] Lidl R., Niderraiter G., Finite fields, Translated from the English by A. E. Zhukov, V. I. Petrov, with a preface by V. I. Nechaev, v. 1, ed. V. I. Nechaev, Mir, M., 1988, 430 pp. (Russian) | MR | Zbl

[38] Lupanov O. B., Introduction to mathematical logic. Lecture notes, Mekh.–mat. MGU M. V. Lomonosov, M., 2007, 199 pp.

[39] Vorob'ev N. N., Fibonacci numbers, Popular Lectures on Mathematics, 6, Fourth edition, Nauka, M., 1978, 142 pp. (Russian) | MR