Alizadeh, B.M., Rahmany, S., Basiri, A. Interval gröbner system and its applications (2018) Reliable Computing, 26, pp. 67-96.
ABSTRACT In this paper, the concept of the interval Gröbner system together with an algorithm for its computation are proposed to analyze algebraic polynomial systems with interval coefficients (interval polynomial systems). These systems appear in many computational problems arising from both the engineering and mathematical sciences. As opposed to linear interval polynomial systems, there is no method to solve and/or analyze a nonlinear interval polynomial system. Interval Gröbner systems enable us to determine whether an interval polynomial system has any solutions or not. If so, a finite decomposition of the solution set will be constructed by the elements of the computed interval Gröbner system. Furthermore, this concept allows us to verify whether two interval polynomial systems share a common solution or not. The concept of the interval Gröbner system is based on elimination tools on the set of interval polynomials. It is worth noting that this is not a trivial extension of usual techniques, since the set of interval polynomials does not satisfy the distributivity and additive inverse axioms of a ring with usual interval arithmetic. In doing so, we introduce the concept of the ideal family associated to an interval polynomial system which contains an infinite number of (non-interval) polynomial ideals. Then we analyze all of these ideals using an equivalence relation with a finite number of equivalence classes. This method is based on a novel computational algebraic tool, the concept of comprehensive Gröbner system, equipped with an interval-based criterion to omit unnecessary computations. We also provide some applications of interval Grobner systems to analyze interval polynomial systems, nding multiple roots and solving the divisibility problem of interval polynomials. Our algorithm for the computation of interval Grobner systems has been implemented in both Maple and Magma software packages. © 2018 Springer Netherlands. All rights reserved.
INDEX KEYWORDS: Equivalence classes; Inverse problems, Algebraic polynomials; Computational problem; Divisibility problem; Equivalence relations; Interval arithmetic; Interval coefficients; Interval polynomials; Mathematical science, Polynomials PUBLISHER: Springer Netherlands
Boroujeni, M., Basiri, A., Rahmany, S., Valibouze, A. Finding solutions of fuzzy polynomial equations systems by an Algebraic method (2016) Journal of Intelligent and Fuzzy Systems, 30 (2), pp. 791-800.
DOI: 10.3233/IFS-151801
ABSTRACT Finding solutions for fuzzy polynomial systems has recently received much attention and many efforts have been made to make the available algorithms for solving such problems more and more efficient. In the present paper, Wu's algorithm is introduced as a solution procedure to obtain fuzzy polynomial systems solutions. In this approach, the parametric form of the problem is first obtained from the computation of r-cuts of fuzzy polynomials. Wu's algorithm is then applied in order to convert the parametric form of a fuzzy polynomial system into a finite number of characteristic sets. We then have the right relation between solutions of these sets and those of the polynomial system. The most outstanding advantage of the proposed method lies in the fact that it leads to solve triangular sets amenable to easy solution. © 2016 - IOS Press and the authors. All rights reserved.
AUTHOR KEYWORDS: characteristic sets; fuzzy polynomial systems; Polynomial systems; Wu's algorithm INDEX KEYWORDS: Algebra; Algorithms; Parameter estimation, Algebraic method; Characteristic set; Finding solutions; Parametric forms; Polynomial equation; Polynomial systems; Solution procedure; Triangular sets, Polynomials PUBLISHER: IOS Press
Boroujeni, M., Basiri, A., Rahmany, S., Valibouze, A. Solving Fuzzy Systems in Dual Form Using Wu's Method (2015) International Journal of Fuzzy Systems, 17 (2), pp. 170-180.
DOI: 10.1007/s40815-015-0033-4
ABSTRACT In this paper, fuzzy polynomial systems in dual form are considered and an algebraic approach for finding their solutions is presented. A dual fuzzy polynomial system in the form AX + B = CX + D, where A, B, C, and D are fuzzy matrices, is converted to a system with real coefficients and variables first. Then, Wu's algorithm is used as a solution procedure for solving this system. This algorithm leads to solving characteristic sets that are amenable to easy solution. Finally, the accuracy of the presented algorithm is shown via some examples. © 2015 Taiwan Fuzzy Systems Association and Springer-Verlag.
AUTHOR KEYWORDS: Characteristic sets; Fuzzy polynomial systems; Polynomial systems; Wu's algorithm INDEX KEYWORDS: Algorithms, Algebraic approaches; Characteristic set; Dual form; Fuzzy matrix; Polynomial systems; Real coefficients; Solution procedure; Wu's method, Polynomials PUBLISHER: Springer Berlin Heidelberg
Boroujeni, M., Basiri, A., Rahmany, S., Valibouze, A. F4-invariant algorithm for computing SAGBI-Gröbner bases (2015) Theoretical Computer Science, 573, pp. 54-62.
DOI: 10.1016/j.tcs.2015.01.045
ABSTRACT This paper introduces a new algorithm for computing SAGBI-Gröbner bases for ideals of invariant rings of permutation groups. This algorithm is based on F4 algorithm. A first implementation of this algorithm has been made in SAGE and MAPLE computer algebra systems and has been successfully tried on a number of examples. © 2015 Elsevier B.V.
AUTHOR KEYWORDS: F4 algorithm; F5 algorithm; Invariant ring; SAGBI-Gröbner basis INDEX KEYWORDS: Algebra, Computer algebra systems; Invariant algorithms; Invariant ring; Permutation group, Algorithms PUBLISHER: Elsevier
Farahani, H., Rahmany, S., Basiri, A. Determining of level sets for a fuzzy surface using Gröbner basis (2015) International Journal of Fuzzy System Applications, 4 (2), pp. 1-14.
DOI: 10.4018/IJFSA.2015040101
ABSTRACT In this paper, a manner to determine the level sets of a fuzzy surface using the benefits of Gröbner basis is presented. Fuzzy surfaces are constructed from incomplete datasets or from data that contain uncertainty which has not statistical nature. The authors firstly define the concept of level sets for the fuzzy surfaces. Then, employing Gröbner bases benefits a criterion is proposed for when the level sets of the fuzzy surface are nonempty sets. Moreover, a new algorithm is designed to determine the level sets. The big advantage of the proposed method lies in the fact that it attains all members of the level sets of the fuzzy surface at a time. Finally, some applied numerical examples are illustrated to demonstrate the proficiency of the given approach. Copyright © 2015, IGI Global.
AUTHOR KEYWORDS: Fuzzy numbers; Fuzzy surfaces; Gröbner bases; Level sets PUBLISHER: IGI Global
Farahani, H., Rahmany, S., Basiri, A., Abbasi Molai, A. Resolution of a system of fuzzy polynomial equations using eigenvalue method (2014) Soft Computing, 19 (2), pp. 283-291.
DOI: 10.1007/s00500-014-1249-1
ABSTRACT In this paper, a system of fuzzy polynomial equations is studied. Two solution types are defined for this system, called solution and (Formula presented.)-cut solution. Then sufficient and necessary conditions are proposed for existence of solution and (Formula presented.)-cut solution of the system, respectively. The solution set of the system is also determined. Moreover, a new algorithm is designed to find all the solutions and all the (Formula presented.)-cut solutions of the system based on the eigenvalue method. Finally, some examples are given to illustrate the concepts and the algorithm. © 2014, Springer-Verlag Berlin Heidelberg.
AUTHOR KEYWORDS: Eigenvalue method; Fuzzy numbers; Gröbner basis; Real solution; System of fuzzy polynomial equations INDEX KEYWORDS: Eigenvalues and eigenfunctions; Fuzzy sets, Eigenvalue methods; Existence of Solutions; Fuzzy numbers; Polynomial equation; Real solutions; Solution set; Solution types; Sufficient and necessary condition, Polynomials PUBLISHER: Springer Verlag
Rahmany, S., Basiri, A., Farahani, H., Borzabadi, A.H. A Gröbner basis approach for finding positive solution of fully fuzzy polynomial equations systems (2013) Journal of Intelligent and Fuzzy Systems, 25 (2), pp. 395-402.
DOI: 10.3233/IFS-2012-0646
ABSTRACT This paper presents a novel approach for finding positive solution of fully fuzzy polynomial equations systems employing Gröbner bases benefits. First, the system of fully fuzzy polynomial equations is converted to an equivalent crisp polynomial equations system. Then, using Gröbner basis properties a criteria for determining existence and finding positive fuzzy solutions of original system is introduced. The big advantage of the proposed method lies in the fact that it attains all positive fuzzy solutions of problem at a time. Finally, some applied numerical examples are illustrated to demonstrate the proficiency of the given approach. © 2013 - IOS Press and the authors. All rights reserved.
AUTHOR KEYWORDS: affine varieties; fully fuzzy polynomial equations systems; Fuzzy numbers; Gröbner bases INDEX KEYWORDS: affine varieties; Basis property; Fuzzy numbers; Numerical example; Original systems; Polynomial equation; Positive solution, Fuzzy sets, Polynomials
Abbasi Molai, A., Basiri, A., Rahmany, S. Resolution of a system of fuzzy polynomial equations using the Gröbner basis (2013) Information Sciences, 220, pp. 541-558.
DOI: 10.1016/j.ins.2012.07.029
ABSTRACT The occurrence of imprecision in the real world is inevitable due to some unexpected situations. The imprecision is often involved in any engineering design process. The imprecision and uncertainty are often interpreted as fuzziness. Fuzzy systems have an essential role in the uncertainty modelling, which can formulate the uncertainty in the actual environment. In this paper, a new approach is proposed to solve a system of fuzzy polynomial equations based on the Gröbner basis. In this approach, first, the h-cut of a system of fuzzy polynomial equations is computed, and a parametric form for the fuzzy system with respect to the parameter of h is obtained. Then, a Gröbner basis is computed for the ideal generated by the h-cuts of the system with respect to the lexicographical order using Faugère's algorithm, i.e., F 4 algorithm. The Gröbner basis of the system has an upper triangular structure. Therefore, the system can be solved using the forward substitution. Hence, all the solutions of the system of fuzzy polynomial equations can easily be obtained. Finally, the proposed approach is compared with the current numerical methods. Some theorems together with some numerical examples and applications are presented to show the efficiency of our method with respect to the other methods. © 2012 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: F 4 algorithm; Faugère's algorithm; Fuzzy number; Gröbner basis; Lexicographical order; System of fuzzy polynomial equation INDEX KEYWORDS: Actual environments; Engineering design process; Fuzzy numbers; Lexicographical order; Numerical example; Parametric forms; Polynomial equation; Triangular structures; Uncertainty modelling, Algorithms; Fuzzy sets; Fuzzy systems; Uncertainty analysis, Polynomials
Chaharbashloo, M.S., Basiri, A., Rahmany, S., Zarrinkamar, S. An Application of Grobner Basis in Differential Equations of Physics (2013) Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences, 68 (10-11), pp. 646-650.
DOI: 10.5560/zna.2013-0044
ABSTRACT We apply the Grobner basis to the ansatz method in quantum mechanics to obtain the energy eigenvalues and the wave functions in a very simple manner. There are important physical potentials such as the Cornell interaction which play significant roles in particle physics and can be treated via this technique. As a typical example, the algorithm is applied to the semi-relativistic spinless Salpeter equation under the Cornell interaction. Many other applications of the idea in a wide range of physical fields are listed as well. © 2013, Verlag der Zeitschrift für Naturforschung. All rights reserved.
AUTHOR KEYWORDS: Ansatz Method; Gröbner Basis; Quantum Mechanics
Noori, H., Basiri, A., Rahmany, S. Some applications of gr̈obner bases (2011) World Academy of Science, Engineering and Technology, 77, pp. 141-144.
ABSTRACT In this paper we will introduce a brief introduction to theory of Gr̈obner bases and some applications of Gr̈obner bases to graph coloring problem, automatic geometric theorem proving and cryptography.
AUTHOR KEYWORDS: Application of gr̈obner bases; Automatic geometric theorem proving; Cryptography; Gr̈obner bases; Graph coloring INDEX KEYWORDS: Automatic geometric theorem proving; Geometric theorem proving; Graph coloring problem; Graph colorings; Geometric theorem proving; Graph coloring problem; Graph colorings, Cryptography; Problem solving; Theorem proving; Cryptography; Theorem proving, Graph theory; Graph theory
Rahmany, S., Basiri, A. Computing SAGB-Gröbner basis of ideals of invariant rings by using Gaussian elimination (2010) World Academy of Science, Engineering and Technology, 62, pp. 55-58.
ABSTRACT Abstract-The link between Gröbner basis and linear algebra was described by Lazard [4,5] where he realized the Gröbner basis computation could be archived by applying Gaussian elimination over Macaulay's matrix. In this paper, we indicate how same technique may be used to SAGBI- Gröbner basis computations in invariant rings.
AUTHOR KEYWORDS: Gröbner basis; Invariant ring; Permutation groups; Reduction; SAGBI- Gröbner basis INDEX KEYWORDS: Gaussian elimination; Invariant ring; matrix; Permutation group, Engineering; Reduction; Technology, Linear algebra
Basiri, A., Rahmany, S., Khatibi, D. A new implementation of Miura-Arita algorithm for Miura curves (2010) World Academy of Science, Engineering and Technology, 62, pp. 51-52.
ABSTRACT The aim of this paper is to review some of standard fact on Miura curves. We give some easy theorem in number theory to define Miura curves, then we present a new implementation of Arita algorithm for Miura curves.
AUTHOR KEYWORDS: Algebraic curve cryptography; Discrete logarithm problem; Jacobian group; Miura curve INDEX KEYWORDS: Algebraic curve cryptography; Discrete logarithm problems; Jacobian group; Miura curve, Algorithms; Cryptography; Number theory, Algebra; Algorithms
Basiri, A., Rahmany, S., Khatibi, D. A new implementation of Miura-Arita algorithm for Miura curves (2010) World Academy of Science, Engineering and Technology, 62, pp. 137-138.
ABSTRACT The aim of this paper is to review some of standard fact on Miura curves. We give some easy theorem in number theory to define Miura curves, then we present a new implementation of Arita algorithm for Miura curves.
AUTHOR KEYWORDS: Algebraic curve cryptography; Discrete logarithm problem; Jacobian group; Miura curve INDEX KEYWORDS: Algebraic curve cryptography; Discrete logarithm problems; Jacobian group; Miura curve, Algorithms; Cryptography; Number theory, Algebra
Basiri, A., Enge, A., Faugère, J.-C., Gürel, N. The arithmetic of Jacobian groups of superelliptic cubics (2005) Mathematics of Computation, 74 (249), pp. 389-410.
DOI: 10.1090/S0025-5718-04-01699-0
ABSTRACT We present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor's reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data.
AUTHOR KEYWORDS: Arithmetic; Cab curve; Gröbner basis; Jacobian; Superelliptic curve
Basiri, A., Enge, A., Faugère, J.-C., Gürel, N. Implementing the arithmetic of C3,4 curves (2004) Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 3076, pp. 87-101.
ABSTRACT We provide explicit formulae for realising the group law in Jacobians of superelliptic curves of genus 3 and C3,4 curves. It is shown that two distinct elements in the Jacobian of a C3,4 curve can be added with 150 multiplications and 2 inversions in the field of definition of the curve, while an element can be doubled with 174 multiplications and 2 inversions. In superelliptic curves, 10 multiplications are saved. © Springer-Verlag Berlin Heidelberg 2004.
INDEX KEYWORDS: Artificial intelligence; Computer science; Computers, Distinct elements; Explicit formula; Jacobians; Superelliptic curves, Number theory