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Journal articles on the topic 'Base du Gröbner'

Author: Grafiati

Published: 25 May 2024

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1

Bokut, L. A., Yuqun Chen, and Zerui Zhang. "Gröbner–Shirshov bases method for Gelfand–Dorfman–Novikov algebras." Journal of Algebra and Its Applications 16, no. 01 (2017): 1750001. http://dx.doi.org/10.1142/s0219498817500013.

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We establish Gröbner–Shirshov base theory for Gelfand–Dorfman–Novikov algebras over a field of characteristic [Formula: see text]. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand–Dorfman–Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand–Dorfman–Novikov algebra which is not free.

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2

Steiner, Matthias Johann. "Solving Degree Bounds for Iterated Polynomial Systems." IACR Transactions on Symmetric Cryptology 2024, no. 1 (2024): 357–411. http://dx.doi.org/10.46586/tosc.v2024.i1.357-411.

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For Arithmetization-Oriented ciphers and hash functions Gröbner basis attacks are generally considered as the most competitive attack vector. Unfortunately, the complexity of Gröbner basis algorithms is only understood for special cases, and it is needless to say that these cases do not apply to most cryptographic polynomial systems. Therefore, cryptographers have to resort to experiments, extrapolations and hypotheses to assess the security of their designs. One established measure to quantify the complexity of linear algebra-based Gröbner basis algorithms is the so-called solving degree. Cam

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3

Jha, Ranjan, Damien Chablat, and Luc Baron. "Influence of design parameters on the singularities and workspace of a 3-RPS parallel robot." Transactions of the Canadian Society for Mechanical Engineering 42, no. 1 (2018): 30–37. http://dx.doi.org/10.1139/tcsme-2017-0011.

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This paper presents variations in the workspace, singularities, and joint space with respect to design parameter k, which is the ratio of the dimensions of the mobile platform to the dimensions of the base of a 3-RPS parallel manipulator. The influence of the design parameters on parasitic motion, which is important when selecting a manipulator for a desired task, is also studied. The cylindrical algebraic decomposition method and Gröbner-based computations are used to model the workspace and joint space with parallel singularities in 2R1T (two rotational and one translational) and 3T (three t

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4

Yoshida, Hiroshi. "A model for analyzing phenomena in multicellular organisms with multivariable polynomials: Polynomial life." International Journal of Biomathematics 11, no. 01 (2018): 1850007. http://dx.doi.org/10.1142/s1793524518500079.

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Most of life maintains itself through turnover, namely cell proliferation, movement and elimination. Hydra’s cells, for example, disappear continuously from the ends of tentacles, but these cells are replenished by cell proliferation within the body. Inspired by such a biological fact, and together with various operations of polynomials, I here propose polynomial-life model toward analysis of some phenomena in multicellular organisms. Polynomial life consists of multicells that are expressed as multivariable polynomials. A cell is expressed as a term of polynomial, in which point [Formula: see

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5

Gräbe, Hans-Gert, and Franz Pauer. "A remark on Hodge algebras and Gröbner bases." Czechoslovak Mathematical Journal 42, no. 2 (1992): 331–38. http://dx.doi.org/10.21136/cmj.1992.128327.

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6

Bellini, Emanuele, Massimiliano Sala, and Ilaria Simonetti. "Nonlinearity of Boolean Functions: An Algorithmic Approach Based on Multivariate Polynomials." Symmetry 14, no. 2 (2022): 213. http://dx.doi.org/10.3390/sym14020213.

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We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves the second one by avoiding the Gröbner basis computation. We also estimate the complexity of the algorithms, and, in particular, we show that the third method reaches an asymptotic worst-case complexity of O(n2n) operations over the integers, that is, sums and doublings. This way, with a different approach, the same asymptotic complexit

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7

Eder, Christian. "Improving incremental signature-based Gröbner basis algorithms." ACM Communications in Computer Algebra 47, no. 1/2 (2013): 1–13. http://dx.doi.org/10.1145/2503697.2503699.

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8

Eder, Christian, and Jean-Charles Faugère. "A survey on signature-based Gröbner basis computations." ACM Communications in Computer Algebra 49, no. 2 (2015): 61. http://dx.doi.org/10.1145/2815111.2815156.

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9

Francis, Maria, and Thibaut Verron. "A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains." Mathematics in Computer Science 14, no. 2 (2019): 515–30. http://dx.doi.org/10.1007/s11786-019-00432-5.

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AbstractSignature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and

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10

Eder, Christian. "An analysis of inhomogeneous signature-based Gröbner basis computations." Journal of Symbolic Computation 59 (December 2013): 21–35. http://dx.doi.org/10.1016/j.jsc.2013.08.001.

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11

Zheng, Licui, Jinwang Liu, Weijun Liu, and Dongmei Li. "A new signature-based algorithms for computing Gröbner bases." Journal of Systems Science and Complexity 28, no. 1 (2015): 210–21. http://dx.doi.org/10.1007/s11424-015-2260-z.

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12

Roanes-Lozano, E., L. M. Laita, E. Roanes-Macı́as, et al. "A Gröbner bases-based shell for rule-based expert systems development." Expert Systems with Applications 18, no. 3 (2000): 221–30. http://dx.doi.org/10.1016/s0957-4174(99)00064-0.

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13

Piury, Josefina, Luis M. Laita, Eugenio Roanes-Lozano, et al. "A Gröbner bases-based rule based expert system for fibromyalgia diagnosis." Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 106, no. 2 (2012): 443–56. http://dx.doi.org/10.1007/s13398-012-0064-8.

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14

Galkin, V. V. "Simple signature based iterative algorithm for calculation of Gröbner bases." Moscow University Mathematics Bulletin 68, no. 5 (2013): 231–36. http://dx.doi.org/10.3103/s0027132213050033.

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15

Schauenburg, Peter. "A Gröbner-based treatment of elimination theory for affine varieties." Journal of Symbolic Computation 42, no. 9 (2007): 859–70. http://dx.doi.org/10.1016/j.jsc.2007.06.003.

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16

Eder, Christian, and Jean-Charles Faugère. "A survey on signature-based algorithms for computing Gröbner bases." Journal of Symbolic Computation 80 (May 2017): 719–84. http://dx.doi.org/10.1016/j.jsc.2016.07.031.

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17

Shirayanagi, Kiyoshi, and Hiroshi Sekigawa. "A new Gröbner basis conversion method based on stabilization techniques." Theoretical Computer Science 409, no. 2 (2008): 311–17. http://dx.doi.org/10.1016/j.tcs.2008.09.007.

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18

Maletzky, Alexander. "A generic and executable formalization of signature-based Gröbner basis algorithms." Journal of Symbolic Computation 106 (September 2021): 23–47. http://dx.doi.org/10.1016/j.jsc.2020.12.001.

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19

He, Honghui, and Jinzhao Wu. "A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances Method." Information 11, no. 5 (2020): 246. http://dx.doi.org/10.3390/info11050246.

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In generating invariants for hybrid systems, a main source of intractability is that transition relations are first-order assertions over current-state variables and next-state variables, which doubles the number of system variables and introduces many more free variables. The more variables, the less tractability and, hence, solving the algebraic constraints on complete inductive conditions by a comprehensive Gröbner basis is very expensive. To address this issue, this paper presents a new, complete method, called the Citing Instances Method (CIM), which can eliminate the free variables and d

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20

Sakata, Kosuke. "An efficient reduction strategy for signature-based algorithms to compute Gröbner basis." ACM Communications in Computer Algebra 53, no. 3 (2019): 81–92. http://dx.doi.org/10.1145/3377006.3377007.

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21

Sun, Yao, Dongdai Lin, and Dingkang Wang. "On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI." ACM Communications in Computer Algebra 49, no. 2 (2015): 63–64. http://dx.doi.org/10.1145/2815111.2815161.

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22

Ioakimidis, N. I., and E. G. Anastasselou. "Computer-based manipulation of systems of equations in elasticity problems with Gröbner bases." Computer Methods in Applied Mechanics and Engineering 110, no. 1-2 (1993): 103–11. http://dx.doi.org/10.1016/0045-7825(93)90022-p.

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23

Fukasaku, Ryoya, Shutaro Inoue, and Yosuke Sato. "On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems." Mathematics in Computer Science 9, no. 3 (2015): 267–81. http://dx.doi.org/10.1007/s11786-015-0237-x.

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24

Shany, Yaron, and Amit Berman. "A Gröbner-Bases Approach to Syndrome-Based Fast Chase Decoding of Reed–Solomon Codes." IEEE Transactions on Information Theory 68, no. 4 (2022): 2300–2318. http://dx.doi.org/10.1109/tit.2022.3140678.

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25

Matsumoto, Ryutaroh, Diego Ruano, and Olav Geil. "List decoding algorithm based on voting in Gröbner bases for general one-point AG codes." Journal of Symbolic Computation 79 (March 2017): 384–410. http://dx.doi.org/10.1016/j.jsc.2016.02.015.

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26

Eder, Christian, Pierre Lairez, Rafael Mohr, and Mohab Safey El Din. "Towards signature-based gröbner basis algorithms for computing the nondegenerate locus of a polynomial system." ACM Communications in Computer Algebra 56, no. 2 (2022): 41–45. http://dx.doi.org/10.1145/3572867.3572872.

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Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[ x 1 ,..., x n ] over K and a finite sequence of polynomials f 1 ,..., f c in R with c ≤ n. Let V ⊂ K n be the algebraic set defined by the simultaneous vanishing of the f i 's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set V c which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f 1 ,..., f c .

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27

Liu, Ruixian, Philippe Serré, Jean-François Rameau, and André Clément. "Generic Approach for the Generation of Symbolic Dimensional Variations Based on Gröbner Basis for Over-constrained Mechanical Assemblies." Procedia CIRP 27 (2015): 223–29. http://dx.doi.org/10.1016/j.procir.2015.04.070.

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28

Jangisarakul, P., and C. Charoenlarpnopparut. "Algebraic decoder of multidimensional convolutional code: constructive algorithms for determining syndrome decoder and decoder matrix based on Gröbner basis." Multidimensional Systems and Signal Processing 22, no. 1-3 (2010): 67–81. http://dx.doi.org/10.1007/s11045-010-0139-7.

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29

Khan, Muhammad Fahad, Khalid Saleem, Tariq Shah, Mohammad Mazyad Hazzazi, Ismail Bahkali, and Piyush Kumar Shukla. "Block Cipher’s Substitution Box Generation Based on Natural Randomness in Underwater Acoustics and Knight’s Tour Chain." Computational Intelligence and Neuroscience 2022 (May 20, 2022): 1–17. http://dx.doi.org/10.1155/2022/8338508.

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The protection of confidential information is a global issue, and block encryption algorithms are the most reliable option for securing data. The famous information theorist, Claude Shannon, has given two desirable characteristics that should exist in a strong cipher which are substitution and permutation in their fundamental research on “Communication Theory of Secrecy Systems.” block ciphers strictly follow the substitution and permutation principle in an iterative manner to generate a ciphertext. The actual strength of the block ciphers against several attacks is entirely based on its subst

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30

WALKLING, ADRIAN P., and G. RUSSELL COOPE. "Climatic reconstructions from the Eemian/Early Weichselian transition in Central Europe based on the coleopteran record from Gröbern, Germany." Boreas 25, no. 3 (2008): 145–59. http://dx.doi.org/10.1111/j.1502-3885.1996.tb00843.x.

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31

Mohammadi, Fatemeh, and Farbod Shokrieh. "Divisors on graphs, Connected flags, and Syzygies." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (2013). http://dx.doi.org/10.46298/dmtcs.2351.

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International audience We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.

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32

Zhao, Xiangui, and Yang Zhang. "A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings." Open Mathematics 13, no. 1 (2015). http://dx.doi.org/10.1515/math-2015-0028.

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AbstractSignature-based algorithms are efficient algorithms for computing Gröbner-Shirshov bases in commutative polynomial rings, and some noncommutative rings. In this paper, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings over fields.

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33

Arikawa, Keisuke. "Kinematic Analysis of Mechanisms Based on Parametric Polynomial System: Basic Concept of a Method Using Gröbner Cover and Its Application to Planar Mechanisms." Journal of Mechanisms and Robotics 11, no. 2 (2019). http://dx.doi.org/10.1115/1.4042475.

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Many kinematic problems in mechanisms can be represented by polynomial systems. By algebraically analyzing the polynomial systems, we can obtain the kinematic properties of the mechanisms. Among these algebraic methods, approaches based on Gröbner bases are effective. Usually, the analyses are performed for specific mechanisms; however, we often encounter phenomena for which, even within the same class of mechanisms, the kinematic properties differ significantly. In this research, we consider the cases where the parameters are included in the polynomial systems. The parameters are used to expr

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34

Roanes-Lozano, Eugenio, and Eugenio Roanes-Macias. "Maple-based introductory visual guide to Gröbner bases." Maple Transactions 2, no. 1 (2022). http://dx.doi.org/10.5206/mt.v2i1.14425.

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In 1975 the Consejo Superior de Investigaciones Científicas (the main Spanish institution for scientific research) published the monograph [14] by the second author (by the way, father and Ph.D. advisor of the first author). Its title could be translated as "Geometric Interpretation of Ideal Theory" (nowadays Ideal Theory is not normally used, in favour of Commutative Algebra). It somehow illustrated the geometric ideas underlying the basics of the classic books of the period (like [2, 11, 16]) and was a success: although written in Spanish, the edition was sold out.Of course there are much mo

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35

G Karpuz, Eylem, Firat Ates, A. Sinan Çevik, and I. Naci Cangul. "The graph based on Gröbner-Shirshov bases of groups." Fixed Point Theory and Applications 2013, no. 1 (2013). http://dx.doi.org/10.1186/1687-1812-2013-71.

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36

Lyskov, Denis. "A Generalization of Operads Based on Subgraph Contractions." International Mathematics Research Notices, May 20, 2024. http://dx.doi.org/10.1093/imrn/rnae096.

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Abstract We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\operatorname{\overline{{\mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröb

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37

Bhayani, Snehal, Janne Heikkilä, and Zuzana Kukelova. "Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix." Journal of Mathematical Imaging and Vision, March 23, 2024. http://dx.doi.org/10.1007/s10851-024-01182-1.

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AbstractMany computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants based on the Newton polytopes has

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38

Zhu, Ganmin, Shimin Wei, Duanling Li, Yingli Wang, and Qizheng Liao. "CGA-based geometric modeling method for forward displacement analysis of 6-4 Stewart platforms." Journal of Mechanisms and Robotics, September 27, 2023, 1–19. http://dx.doi.org/10.1115/1.4063501.

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Abstract This paper presents a novel geometric modeling method for direct displacement analysis of 6-4 Stewart platforms based on the conformal geometric algebra (CGA). Firstly, a geometric constraint relationship of four lines and a plane intersecting at a point is first published. Secondly, a new coordinate-invariant geometric constraint equation of 6-4 Stewart platforms is deduced by CGA operation. Thirdly, five polynomial equations are established by CGA theory. Fourthly, Based on the above six equations, a 5 × 5 Sylvester's matrix is formulated by using Sylvester's Dialytic elimination me

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39

Kong, Xianwen. "Classification of a 3-RER Parallel Manipulator Based on the Type and Number of Operation Modes." Journal of Mechanisms and Robotics, August 31, 2020, 1–17. http://dx.doi.org/10.1115/1.4048262.

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Abstract The type/number of operation modes of a parallel manipulator (PM) may vary with the link parameters of the PM. This paper presents a systematic classification of a 3-RER PM based on the type/number of operation modes. The 3-RER PM was proposed as a 4-DOF (degree-of-freedom) 3T1R PM in the literature. Using the proposed method, the classification of a PM based on the type/number of operation modes can be carried out in four steps, including formulation of constraint equations of the PM, preliminary classification of the PM using Gröbner Cover, operation mode analysis of all the types o

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