Journal articles: 'System of Diophantine equations and calculus'

1

Cartiere, Carmelo R. "An Analytical Study of Diophantine Equations of Pythagorean Form: Causal Inferences on Hypothesized Relations between Quadratic and Non-quadratic Triples." Athens Journal of Education 12, no. 3 (2025): 527–46. https://doi.org/10.30958/aje.12-3-10.

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In XVII century, presumably between 1637 and 1638, with a note in the margin of Diophantus’ “Arithmetica”, Pierre de Fermat stated that Diophantine equations of the Pythagorean form, , have no integer solutions for , and . Of this statement, however, Fermat never provided a proof. Only after more than 350 years, in 1994, Prof. Andrew J. Wiles was finally successful in demonstrating it (Wiles, 1995; Taylor & Wiles, 1995; Boston, 2008). However, Wiles’ proof adopts calculus techniques far beyond Fermat’s knowledge. Our aim is to show an analytical method to attempt a proof to Fermat’s last theorem with the only use of elementary calculus techniques. Keywords: number theory, Diophantine equations, Pythagorean Theorem, Fermat’s last theorem, numerical analysis

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Pocherevin, R. V. "Multidimensional system of Diophantine equations." Moscow University Mathematics Bulletin 72, no. 1 (2017): 41–43. http://dx.doi.org/10.3103/s0027132217010089.

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3

V, Pandichelvi, and Vanaja R. "A Paradigm for Two Classes of Simultaneous Exponential Diophantine Equations." Indian Journal of Science and Technology 16, no. 40 (2023): 3514–21. https://doi.org/10.17485/IJST/v16i40.1643.

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Abstract Objectives: The goal of this article is to find integer solutions to two distinct kinds of simultaneous exponential Diophantine equations in three variables. Methods: The system of exponential Diophantine equations is translated into the eminent form of Thue equations, and then their generalised solutions satisfying certain conditions are applied. Findings: The finite set of integer solutions for two disparate categories of simultaneous exponential Diophantine equations consisting of three unknowns is scrutinized. In some circumstances, there is no solution in this analysis for both the dissimilar simultaneous Diophantine equations. Novelty: The motivation is considered to be two types of simultaneous exponential Diophantine equations are first converted into specific system of Pell equations, then into Thue equations for the possibilities of the sum of the exponents, such as or . If then, the equations are transformed into a cubic equation, which is not in the form of Pell equations. So, such cases are discarded for exploring solutions to the necessary equations. Keywords: Simultaneous Exponential Equations, Simultaneous Pell Equations, Thue Equations, Integer Solutions, Divisibility

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4

Acewicz, Marcin, and Karol Pąk. "Basic Diophantine Relations." Formalized Mathematics 26, no. 2 (2018): 175–81. http://dx.doi.org/10.2478/forma-2018-0015.

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Summary The main purpose of formalization is to prove that two equations ya(z)= y, y = xz are Diophantine. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem. In our previous work [6], we showed that from the diophantine standpoint these equations can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities. In this formalization, we express these relations in terms of Diophantine set introduced in [7]. We prove that these relations are Diophantine and then we prove several second-order theorems that provide the ability to combine Diophantine relation using conjunctions and alternatives as well as to substitute the right-hand side of a given Diophantine equality as an argument in a given Diophantine relation. Finally, we investigate the possibilities of our approach to prove that the two equations, being the main purpose of this formalization, are Diophantine. The formalization by means of Mizar system [3], [2] follows Z. Adamowicz, P. Zbierski [1] as well as M. Davis [4].

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Pandichelvi, V., and S. Saranya. "Application of System Linear Diophantine Equations in Balancing Chemical Equations." International Journal for Research in Applied Science and Engineering Technology 10, no. 10 (2022): 917–20. http://dx.doi.org/10.22214/ijraset.2022.47111.

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Abstract: In this manuscript, the step by step procedure for how the system of linear Diophantine equations are applied to balance chemical equations acquired by the reactions of various chemical compounds and their products is scrutinized.

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Osipyan, V. O., K. I. Litvinov, and A. S. Zhuck. "Research and development of the mathematic models of cryptosystems based on the universal Diophantine language." SHS Web of Conferences 141 (2022): 01020. http://dx.doi.org/10.1051/shsconf/202214101020.

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This paper shows the objective necessity of improving the information security systems under the development of information and telecommunication technologies. The paper for the first time involves a new area of NP-complete problems from Diophantine analysis, namely, multi-degree systems of Diophantine equations of a given dimension and degree of Tarry-Escott type. Based on a fundamentally new number-theoretic method, a mathematical model of an alphabetic information security system (ISS) has been developed that generalizes the principle of building cryptosystems with a public key – the so called dissymmetric bigram cryptosystem. This implies to implement direct and inverse transformations according to a given algorithm based on a two-parameter solution of a multi-degree system of Diophantine equations. A formalized algorithm has been developed for the specified model of a dissymmetric bigram cryptosystem and a training example based on a normal multi-degree system of Diophantine equations of the fifth degree is presented.

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Ye, Yangbo, Ge Wang, and Jiehua Zhu. "Linear diophantine equations for discrete tomography." Journal of X-Ray Science and Technology: Clinical Applications of Diagnosis and Therapeutics 10, no. 1-2 (2001): 59–66. http://dx.doi.org/10.3233/xst-2001-00057.

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In this report, we present a number-theory-based approach for discrete tomography (DT),which is based on parallel projections of rational slopes. Using a well-controlled geometry of X-ray beams, we obtain a system of linear equations with integer coefficients. Assuming that the range of pixel values is $a(i,j)=0,1<FORMULA>, &mldr;, <FORMULA>M-1<FORMULA>, with <FORMULA>M$ being a prime number, we reduce the equations modulo $M$ . To invert the linear system, each algorithmic step only needs $log^2_2 M$ bit operations. In the case of a small $M$ , we have a greatly reduced computational complexity, relative to the conventional DT algorithms, which require $log^2_2 N$ bit operations for a real number solution with a precision of $1/N$ . We also report computer simulation results to support our analytic conclusions.

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IBRAHIM, M., F. ABD RABOU, and H. ZORKTA. "APPLICATION OF DIOPHANTINE EQUATIONS IN A CIPHER SYSTEM." International Conference on Electrical Engineering 1, no. 1 (1998): 313–24. http://dx.doi.org/10.21608/iceeng.1998.61085.

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9

Garaev, M. Z., and V. N. Chubarikov. "Concerning the Sierpinski-Schinzel system of Diophantine equations." Mathematical Notes 66, no. 2 (1999): 142–47. http://dx.doi.org/10.1007/bf02674869.

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10

Chilikov, A. A., and Alexey Belov-Kanel. "Exponential diophantine equations in rings of positive characteristic." Journal of Knot Theory and Its Ramifications 29, no. 02 (2020): 2040002. http://dx.doi.org/10.1142/s0218216520400027.

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In this paper, we prove an algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations [Formula: see text] where [Formula: see text] are constants from matrix ring of characteristic [Formula: see text], [Formula: see text] are indeterminates. For any solution [Formula: see text] of the system we construct a word (over an alphabet containing [Formula: see text] symbols) [Formula: see text] where [Formula: see text] is a [Formula: see text]-tuple [Formula: see text], [Formula: see text] is the [Formula: see text]th digit in the [Formula: see text]-adic representation of [Formula: see text]. The main result of this paper is following: the set of words corresponding in this sense to solutions of a system of exponential-Diophantine equations is a regular language (i.e., recognizable by a finite automaton). There exists an algorithm which calculates this language. This algorithm is constructed in the paper.

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11

Ladzoryshyn, N. B., V. M. Petrychkovych, and H. V. Zelisko. "Matrix Diophantine equations over quadratic rings and their solutions." Carpathian Mathematical Publications 12, no. 2 (2020): 368–75. http://dx.doi.org/10.15330/cmp.12.2.368-375.

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The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

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Shams, Mudassir, Nasreen Kausar, Naveed Khan, and Mohd Asif Shah. "Modified Block Homotopy Perturbation Method for solving triangular linear Diophantine fuzzy system of equations." Advances in Mechanical Engineering 15, no. 3 (2023): 168781322311595. http://dx.doi.org/10.1177/16878132231159519.

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Numerous real-world applications can be solved using the broadly adopted notions of intuitionistic fuzzy sets, Pythagorean fuzzy sets, and q-rung orthopair fuzzy sets. These theories, however, have their own restrictions in terms of membership and non-membership levels. Because it utilizes benchmark or control parameters relating to membership and non-membership levels, this theory is particularly valuable for modeling uncertainty in real-world problems. We propose the unique concept of linear Diophantine fuzzy set with benchmark parameters to overcome these restrictions. Different numerical, analytical, and semi-analytical techniques are used to solve linear systems of equations with several fuzzy numbers, such as intuitionistic fuzzy number, triangular fuzzy number, bipolar fuzzy number, trapezoidal fuzzy number, and hexagon fuzzy number. The purpose of this research is to solve a fuzzy linear system of equations with the most generalized fuzzy number, such as Triangular linear Diophantine fuzzy number, using an analytical technique called Homotopy Perturbation Method. The linear systems co-efficient are crisp when the right hand side vector is a triangular linear Diophantine fuzzy number. A numerical test examples demonstrates how our newly improved analytical technique surpasses other existing methods in terms of accuracy and CPU time. The triangular linear Diophantine fuzzy systems of equations’ strong and weak visual representations are explored.

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13

Moir, T. J. "Polynomial Wiener LQG Controllers based on Toeplitz Matrices." Journal of Physics: Conference Series 2224, no. 1 (2022): 012114. http://dx.doi.org/10.1088/1742-6596/2224/1/012114.

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Abstract This paper re-examines the discrete-time Linear Quadratic Gaussian (LQG) regulator problem. The normal approach to this problem is to use a Kalman filter state estimator and Kalman control state feedback. Though quite successful, an alternative approach in the frequency domain was employed later. That method used z-transfer functions or polynomials in the z-domain. The transfer function approach is similar to the method used in Wiener filtering and requires the use of Diophantine equations (sometimes bilateral) to find the optimal controller. The contribution here uses a similar approach but uses lower triangular Toeplitz matrices instead of polynomials to gain advantage of eliminating the use of Diophantine equations. This is because the single Diophantine equation approach fails when the system has non-relative prime polynomials and the need for bilateral Diophantine equations is computationally far more complex.

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14

Yeung, K. S., and T. M. Chen. "Solving Matrix Diophantine Equations by Inverting a Square Nonsingular System of Equations." IEEE Transactions on Circuits and Systems II: Express Briefs 51, no. 9 (2004): 488–95. http://dx.doi.org/10.1109/tcsii.2004.832775.

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15

Abobala, Mohammad. "A Short Contribution to Split-Complex Linear Diophantine Equations in Two Variables." Galoitica: Journal of Mathematical Structures and Applications 6, no. 2 (2023): 32–35. http://dx.doi.org/10.54216/gjmsa.060204.

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In this work, we study the split-complex integer solutions for the split-complex linear Diophantine equation in two variables where are split-complex integers. An algorithm for generating all solutions will be obtained by transforming the split-complex equation to a classical equivalent system of linear Diophantine equations in four variables.

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16

Cvetkovic, Dragos, and Vesna Todorcevic. "Cospectrality graphs of Smith graphs." Filomat 33, no. 11 (2019): 3269–76. http://dx.doi.org/10.2298/fil1911269c.

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Graphs whose spectrum belongs to the interval [-2,2] are called Smith graphs. The structure of a Smith graph with a given spectrum depends on a system of Diophantine linear algebraic equations. We have established in [1] several properties of this system and showed how it can be simplified and effectively applied. In this way a spectral theory of Smith graphs has been outlined. In the present paper we introduce cospectrality graphs for Smith graphs and study their properties through examples and theoretical consideration. The new notion is used in proving theorems on cospectrality of Smith graphs. In this way one can avoid the use of the mentioned system of Diophantine linear algebraic equations.

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17

Liu, F. B., and C. F. Hu. "On the resolution of the system of fuzzy Diophantine equations." Journal of Intelligent & Fuzzy Systems 26, no. 2 (2014): 751–58. http://dx.doi.org/10.3233/ifs-130765.

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18

Luo, JiaGui, and PingZhi Yuan. "On the solutions of a system of two Diophantine equations." Science China Mathematics 57, no. 7 (2014): 1401–18. http://dx.doi.org/10.1007/s11425-014-4800-8.

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19

Cusick, Thomas W. "Cryptanalysis of a public key system based on Diophantine equations." Information Processing Letters 56, no. 2 (1995): 73–75. http://dx.doi.org/10.1016/0020-0190(95)00124-u.

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20

Devi, P. Jamuna, та K. S. Araththi. "On the Ternary Cubic Diophantine Equation 𝑥3 + 𝑦3 = 2(𝑧 + 𝑤)2(𝑧 − 𝑤)". Indian Journal Of Science And Technology 17, № 33 (2024): 3473–80. http://dx.doi.org/10.17485/ijst/v17i33.2186.

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The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. Objectives: The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. Method: Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition: like , where and Z. By the decomposing method in primary terms of a, we achieve a countable number of decompositions in k full factors . Each decomposition of this kind leads to a system of equations similar to: , . We get multitude of solutions for a given equation, by determining the system of equations. Findings: By the method of linear transformations, the ternary cubic equation with four unknowns is solved for its integral solutions. The equation is researched for its attributes and correlation among the solutions for its non – zero unique integer points. In each of the transformations taken, the cubic equation yields different solutions. The properties of the solutions and their relationship with the special numbers are also exhibited. Novelty: Mathematician’s interest towards solving Pell’s equation has been so much not because they approximate with a value for . The main importance of the Pell’s equation is due to that most of the common questions have answers in this equation which can be sorted by 2 variables in the Quadratic equations. This document is about the research on higher degree Cubic Diophantine equation which gives the integral solutions of this equation, taken into consideration. Keywords: Integral solutions, Ternary Cubic, Oblong number, Polygonal number

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Jain, Pankaj, Chandrani Basu, and Vivek Panwar. "Reduced $pq$-Differential Transform Method and Applications." Journal of Inequalities and Special Functions 13, no. 1 (2022): 24–40. http://dx.doi.org/10.54379/jiasf-2022-1-3.

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In this paper, Reduced Differential Transform method in the framework of (p, q)-calculus, denoted by Rp,qDT , has been introduced and applied in solving a variety of differential equations such as diffusion equation, 2Dwave equation, K-dV equation, Burgers equations and Ito system. While the diffusion equation has been studied for the special case p = 1, i.e., in the framework of q-calculus, the other equations have not been studied even in q-calculus.

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Gopalan, M., S. Vidhyalakshmi, and N. Thiruniraiselvi. "CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES." International Journal of Engineering Technologies and Management Research 1, no. 1 (2020): 1–7. http://dx.doi.org/10.29121/ijetmr.v1.i1.2015.20.

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Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.

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Khaldi, Ahmad, Khadija Ben Othman, Shtawzen Oliver von, and Sarah Jalal Mosa. "On Some Algorithms for Solving Different Types of Symbolic 2-Plithogenic Algebraic Equations." Neutrosophic Sets and Systems 54 (April 11, 2023): 101–12. https://doi.org/10.5281/zenodo.7817673.

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The main goal of this paper is to study three different types of algebraic symbolic 2-plithogenic equations. The symbolic 2-plithogenic linear Diophantine equations, symbolic 2-plithogenic quadratic equations, and linear system of symbolic 2-plithgenic equations will be discussed and handled, where algorithms to solve the previous types will be presented and proved by transforming them to classical algebraic systems of equations.

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24

Irmak, Nurettin, and Murat Alp. "Pellans sequence and its diophantine triples." Publications de l'Institut Math?matique (Belgrade) 100, no. 114 (2016): 259–69. http://dx.doi.org/10.2298/pim1614259i.

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We introduce a novel fourth order linear recurrence sequence {Sn} using the two periodic binary recurrence. We call it ?pellans sequence? and then we solve the system ab+1=Sx, ac+1=Sy bc+1=Sz where a < b < c are positive integers. Therefore, we extend the order of recurrence sequence for this variant diophantine equations by means of pellans sequence.

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25

Pąk, Karol. "Formalization of the MRDP Theorem in the Mizar System." Formalized Mathematics 27, no. 2 (2019): 209–21. http://dx.doi.org/10.2478/forma-2019-0020.

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Summary This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem. In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2 − (a2 − 1)y2 = 1 [8] and its solutions considered as two sequences $\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$ . We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9]. In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form. The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].

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26

Razzaghi, Mohsen. "Hybrid approximations for fractional calculus." ITM Web of Conferences 29 (2019): 01001. http://dx.doi.org/10.1051/itmconf/20192901001.

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In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting ofblock-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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27

P, Jamuna Devi, та S. Araththi K. "On the Ternary Cubic Diophantine Equation 𝑥3 + 𝑦3 = 2(𝑧 + 𝑤)2(𝑧 − 𝑤)". Indian Journal of Science and Technology 17, № 33 (2024): 3473–80. https://doi.org/10.17485/IJST/v17i33.2186.

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Abstract The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. Objectives: The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. Method: Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition: like , where and Z. By the decomposing method in primary terms of a, we achieve a countable number of decompositions in k full factors . Each decomposition of this kind leads to a system of equations similar to: , . We get multitude of solutions for a given equation, by determining the system of equations. Findings: By the method of linear transformations, the ternary cubic equation with four unknowns is solved for its integral solutions. The equation is researched for its attributes and correlation among the solutions for its non – zero unique integer points. In each of the transformations taken, the cubic equation yields different solutions. The properties of the solutions and their relationship with the special numbers are also exhibited. Novelty: Mathematician’s interest towards solving Pell’s equation has been so much not because they approximate with a value for . The main importance of the Pell’s equation is due to that most of the common questions have answers in this equation which can be sorted by 2 variables in the Quadratic equations. This document is about the research on higher degree Cubic Diophantine equation which gives the integral solutions of this equation, taken into consideration. Keywords: Integral solutions, Ternary Cubic, Oblong number, Polygonal number

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Ward, A. J. B. "84.07 A Matrix Method for a System of Linear Diophantine Equations." Mathematical Gazette 84, no. 499 (2000): 81. http://dx.doi.org/10.2307/3621482.

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Lin, C. H., C. C. Chang, and R. C. T. Lee. "A new public-key cipher system based upon the diophantine equations." IEEE Transactions on Computers 44, no. 1 (1995): 13–19. http://dx.doi.org/10.1109/12.368013.

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30

Allakov, I., and B. Kh Erdonov. "On the solution of a system of linear Diophantine equations in prime numbers." UZBEK MATHEMATICAL JOURNAL 69, no. 1 (2025): 14–26. https://doi.org/10.29229/uzmj.2025-1-2.

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In the paper, it is proved that the system of linear Diophantine equations consisting of s equations with m unknowns (where s < m ≤ 2s) is solvable in prime numbers with some exceptions. A lower bound for the number of solutions of the system under consideration is also obtained for the exceptional set. The obtained results complement the corresponding results of Wu Fang, M. C. Liu, K. M. Tsang and I. Allakov

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Story, Troy L. "Exterior Calculus: Thermodynamics and Navier-Stokes Dynamics." Advanced Materials Research 747 (August 2013): 761–64. http://dx.doi.org/10.4028/www.scientific.net/amr.747.761.

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Mathematical models of dynamics employing exterior calculus are shown to be mathematical representations of the same unifying principle; namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus to a set of differential equations and a characteristic tangent vector which define transformations of the system [1-4]. This principle, whose origin is V. I. Arnolds use of exterior calculus to describe Hamiltonian mechanics and geometric optics, is applied to irreversible thermodynamics and Navier-Stokes dynamics. Results include (a) a set of equations for irreversible thermodynamics equivalent to Maxwells equations for reversible thermodynamics, (b) transformation of the incompressible Navier-Stokes equation into a pair of simpler equivalent equations, which are solved and (c) a characteristic tangent vector for each area of dynamics, which indicates the direction of phase flow.

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Haug, E. J., and M. K. McCullough. "A Variational-Vector Calculus Approach to Machine Dynamics." Journal of Mechanisms, Transmissions, and Automation in Design 108, no. 1 (1986): 25–30. http://dx.doi.org/10.1115/1.3260779.

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A variational-vector calculus approach is presented to define virtual displacements and rotations and position, velocity, and acceleration of individual components of a multibody mechanical system. A two-body subsystem with both Cartesian and relative coordinates is used to illustrate a systematic method of exploiting the linear structure of both vector and differential calculus, in conjunction with a variational formulation of the equations of motion of rigid bodies, to derive the matrix structure of governing multibody system equations of motion. A pattern for construction of the system mass matrix and generalized force terms is developed and applied to derivation of the equations of motion of a vehicle system. The development demonstrates an approach to multibody machine dynamics that closely parallels methods used in finite-element structural analysis.

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Voevoda, Aleksander, Vladislav Filiushov, and Viktor Shipagin. "Polynomial method for the synthesis of regulators for the special case of multichannel objects with one input variable and several output values." Digital technology security, no. 3 (September 30, 2021): 21–42. http://dx.doi.org/10.17212/2782-2230-2021-3-21-42.

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Currently, an urgent task in control theory is the synthesis of regulators for objects with a smaller number of input values compared to output ones, such objects are described by matrix transfer functions of a non-square shape. A particular case of a multichannel object with one input variable and two / three / four output variables is considered; the matrix transfer function of such an object has not a square shape, but one column and two / three / four rows. To calculate the controllers, a polynomial synthesis technique is used, which consists in using a polynomial matrix description of a closed-loop control system. A feature of this approach is the ability to write the characteristic matrix of a closed multichannel system through the polynomial matrices of the object and the controller in the form of a matrix Diophantine equation. By solving the Diophantine equation, the desired poles of the matrix characteristic polynomial of the closed system are set. There are many options for solving the Diophantine equation and one of them is to represent the polynomial matrix Diophantine equation as a system of linear algebraic equations in matrix form, where the matrix of the system is the Sylvester matrix. The choice of the order of the polynomial matrix controller and the order of the characteristic matrix is carried out on the basis of the theorem given in the works of Chi-Tsong Chen, which always holds for controlled objects. If the minimum order of the controller is chosen in accordance with this theorem, and the Sylvester matrix has not full rank, then this means that there are more unknown elements in the system of linear algebraic equations than there are equations. In this case, the solution corresponding to the selected basic minor has free parameters, which are the parameters of the regulators. Free parameters of regulators can be set arbitrarily, which is used to set or exclude some zeros in a closed system. Thus, using various examples of objects with a non-square matrix transfer function, a polynomial synthesis technique is illustrated, which allows not only specifying the poles of a closed system, but also some zeros, which is a significant advantage, especially when synthesizing controllers for multichannel objects.

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Sana, Gul, Pshtiwan Othman Mohammed, Dong Yun Shin, Muhmmad Aslam Noor, and Mohammad Salem Oudat. "On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus." Fractal and Fractional 5, no. 3 (2021): 60. http://dx.doi.org/10.3390/fractalfract5030060.

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Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods.

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Banchuin, Rawid. "Nonlocal fractal calculus based analyses of electrical circuits on fractal set." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 41, no. 1 (2021): 528–49. http://dx.doi.org/10.1108/compel-06-2021-0210.

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Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

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Isha Das and Ishan Das. "Beyond the Equations: A Visual Journey into Multivariable Calculus." World Journal of Advanced Research and Reviews 21, no. 2 (2024): 2083–87. https://doi.org/10.30574/wjarr.2024.21.2.0408.

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Multivariable calculus, a cornerstone of advanced mathematics, often confounds learners with its abstract, multidimensional concepts—surfaces, gradients, and vector fields that defy simple intuition. Traditional tools like static graphs and equations fall short in bridging this gap. This paper introduces a novel visualization system designed to transform these complex ideas into an accessible, interactive experience. Leveraging real-time 3D rendering and user-adjustable parameters, the system enables students to explore functions like z = f(x, y) dynamically, revealing the hidden geometry of calculus. Tested with a cohort of learners, it demonstrated significant improvements—participants reported a 40% increase in comprehension of key concepts, such as partial derivatives, and educators noted enhanced engagement. While dependent on computational resources, this tool marks a leap forward in mathematical education, offering a visual journey that transcends the limitations of equations alone. This study outlines the system’s design, its impact, and its potential to redefine how multivariable calculus is taught and understood.

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Tyszka, Apoloniusz. "A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions." Open Computer Science 8, no. 1 (2018): 109–14. http://dx.doi.org/10.1515/comp-2018-0012.

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Abstract We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

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Dods, Victor. "Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus." Mathematics 10, no. 18 (2022): 3231. http://dx.doi.org/10.3390/math10183231.

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In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.

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Razzaghi, M. "A numerical scheme for problems in fractional calculus." ITM Web of Conferences 20 (2018): 02001. http://dx.doi.org/10.1051/itmconf/20182002001.

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In this paper, a new numerical method for solving the fractional differential equations with boundary value problems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the boundary value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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Ossicin, Andrea. "SOME DIOPHANTUS-FERMAT DOUBLE EQUATIONS EQUIVALENT TO FREY’S ELLIPTIC CURVE." JOURNAL OF RAMANUJAN SOCIETY OF MATHEMATICS AND MATHEMATICAL SCIENCES 11, no. 01 (2023): 43–54. http://dx.doi.org/10.56827/jrsmms.2023.1101.3.

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In this work I demonstrate that a possible origin of the Frey elliptic curve derives from an appropriate use of the double equations of Diophantus-Fermat and from an isomorphism: a birational application between the double equations and an elliptic curve. From this origin I deduce a Fundamental Theorem which allows an exact reformulation of Fermat’s Last Theorem. A complete proof of this Theorem, consisting of a system of homogeneous ternary quadratic Diophantine equations, is certainly possible also through methods known and discovered by Fermat,in order to solve his extraordinary equation.

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Le, Maohua. "On the Diophantine System $x^2 - Dy^2 = 1-D$ and $x=2z^2-1$." MATHEMATICA SCANDINAVICA 95, no. 2 (2004): 171. http://dx.doi.org/10.7146/math.scand.a-14455.

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Let $D$ be a positive integer such that $D-1$ is an odd prime power. In this paper we give an elementary method to find all positive integer solutions $(x, y, z)$ of the system of equations $x^2-Dy^2=1-D$ and $x=2z^2-1$. As a consequence, we determine all solutions of the equations for $D=6$ and $8$.

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Tai, Yongpeng, Ning Chen, Lijin Wang, Zaiyong Feng, and Jun Xu. "A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control." Mathematics 8, no. 7 (2020): 1134. http://dx.doi.org/10.3390/math8071134.

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Fractional calculus is widely used in engineering fields. In complex mechanical systems, multi-body dynamics can be modelled by fractional differential-algebraic equations when considering the fractional constitutive relations of some materials. In recent years, there have been a few works about the numerical method of the fractional differential-algebraic equations. However, most of the methods cannot be directly applied in the equations of dynamic systems. This paper presents a numerical algorithm of fractional differential-algebraic equations based on the theory of sliding mode control and the fractional calculus definition of Grünwald–Letnikov. The algebraic equation is considered as the sliding mode surface. The validity of the present method is verified by comparing with an example with exact solutions. The accuracy and efficiency of the present method are studied. It is found that the present method has very high accuracy and low time consumption. The effect of violation corrections on the accuracy is investigated for different time steps.

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Das, Radhika, Manju Somanath, and V. A. Bindu. "ON THE SYSTEM OF DIOPHANTINE EQUATIONS x² − 7y² = 1 AND x = az² − b." jnanabha 54, no. 01 (2024): 270–73. http://dx.doi.org/10.58250/jnanabha.2024.54132.

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Trigeassou, Jean-Claude, and Nezha Maamri. "Optimal State Control of Fractional Order Differential Systems: The Infinite State Approach." Fractal and Fractional 5, no. 2 (2021): 29. http://dx.doi.org/10.3390/fractalfract5020029.

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Optimal control of fractional order systems is a long established domain of fractional calculus. Nevertheless, it relies on equations expressed in terms of pseudo-state variables which raise fundamental questions. So in order remedy these problems, the authors propose in this paper a new and original approach to fractional optimal control based on a frequency distributed representation of fractional differential equations called the infinite state approach, associated with an original formulation of fractional energy, which is intended to really control the internal system state. In the first step, the fractional calculus of variations is revisited to express appropriate Euler Lagrange equations. Then, the quadratic optimal control of fractional linear systems is formulated. Thanks to a frequency discretization technique, the previous theoretical equations are converted into an equivalent large dimension integer order system which permits the implementation of a feasible optimal solution. A numerical example illustrates the validity of this new approach.

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Tyszka, Apoloniusz. "Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?" Open Computer Science 7, no. 1 (2017): 17–23. http://dx.doi.org/10.1515/comp-2017-0003.

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Abstract Let Bn = {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈ {1, . . . , n}} denote the system of equations in the variables x1, . . . , xn. For a positive integer n, let _(n) denote the smallest positive integer b such that for each system of equations S ⊆ Bn with a unique solution in positive integers x1, . . . , xn, this solution belongs to [1, b]n. Let g(1) = 1, and let g(n + 1) = 22g(n) for every positive integer n. We conjecture that ξ (n) 6 g(2n) for every positive integer n. We prove: (1) the function ξ : N \ {0} → N \ {0} is computable in the limit; (2) if a function f : N \ {0} → N \ {0} has a single-fold Diophantine representation, then there exists a positive integer m such that f (n) < ξ (n) for every integer n > m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x1, . . . , xp) = 0 and returns a positive integer d with the following property: for every positive integers a1, . . . , ap, if the tuple (a1, . . . , ap) solely solves the equation D(x1, . . . , xp) = 0 in positive integers, then a1, . . . , ap 6 d; (4) the conjecture implies that if a set M ⊆ N has a single-fold Diophantine representation, then M is computable; (5) for every integer n > 9, the inequality ξ (n) < (22n−5 − 1)2n−5 + 1 implies that 22n−5 + 1 is composite.

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Saburov, Mansoor, Mohd Ahmad, and Murat Alp. "The study on general cubic equations over p-adic fields." Filomat 35, no. 4 (2021): 1115–31. http://dx.doi.org/10.2298/fil2104115s.

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A Diophantine problem means to find all solutions of an equation or system of equations in integers, rational numbers, or sometimes more general number rings. The most frequently asked question is whether a root of a polynomial equation with coefficients in a p-adic field Qp belongs to domains Z*p, Zp \ Z*p, Qp \ Zp, Qp or not. This question is open even for lower degree polynomial equations. In this paper, this problem is studied for cubic equations in a general form. The solvability criteria and the number of roots of the general cubic equation over the mentioned domains are provided.

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Spiridonova, Margarita. "Operational Methods in the Environment of a Computer Algebra System." Serdica Journal of Computing 3, no. 4 (2010): 381–424. http://dx.doi.org/10.55630/sjc.2009.3.381-424.

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The presented research is related to the operational calculus approach and its representative applications. Operational methods are considered, as well as their program implementation using the computer algebra system Mathematica. The Heaviside algorithm for solving Cauchy’s problems for linear ordinary differential equations with constant coefficients is considered in the context of the Heaviside-Mikusinski operational calculus. The program implementation of the algorithm is described and illustrative examples are given. An extension of the Heaviside algorithm, developed by I. Dimovski and S. Grozdev, is used for finding periodic solutions of linear ordinary differential equations with constant coefficients both in the non-resonance and in the resonance cases. The features of its program implementation are described and examples are given. An operational method for solving local and nonlocal boundary value problems for some equations of the mathematical physics (the heat equation, the wave equation and the equation of a free supported beam) is developed and the capabilities of the corresponding program packages for solving those problems are described. A comparison with other methods for solving the same types of problems is included and the advantages of the operational methods are marked.

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Tian, Zilu, Dan Olteanu, and Christoph Koch. "Using Process Calculus for Optimizing Data and Computation Sharing in Complex Stateful Parallel Computations." Proceedings of the ACM on Management of Data 3, no. 3 (2025): 1–28. https://doi.org/10.1145/3725421.

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We propose novel techniques that exploit data and computation sharing to improve the performance of complex stateful parallel computations, like agent-based simulations. Parallel computations are translated into behavioral equations, a novel formalism layered on top of the foundational process calculus π-calculus. Behavioral equations blend code and data, allowing a system to easily compose and transform parallel programs into specialized programs. We show how optimizations like merging programs, synthesizing efficient message data structures, eliminating local messaging, rewriting communication instructions into local computations, and aggregation pushdown can be expressed as transformations of behavioral equations. We have also built a system called OptiFusion that implements behavioral equations and the aforementioned optimizations. Our experiments showed that OptiFusion is over 10× faster than state-of-the-art stateful systems benchmarked via complex stateful workloads. Generating specialized instructions that are impractical to write by hand allows OptiFusion to outperform even the hand-optimized implementations by up to 2×.

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Ibănescu, M., and R. Ibănescu. "New approach in solving derivative causality problems in the bond-graph method." IOP Conference Series: Materials Science and Engineering 1235, no. 1 (2022): 012053. http://dx.doi.org/10.1088/1757-899x/1235/1/012053.

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Abstract From the bond-graph model of a dynamic system, the state equations can be obtained. When the system contains only energy storing elements in integral causality, the system of state equations is immediately and directly determined. When the model contains at least one energy storing element in derivative causality, the resulted system of differential-algebraic equations arises some difficulties in finding the final form of the system of state equations. The work presents a new method of deducing the state equations in case of bond-graph models with one, or several energy storing elements in derivative causality. This method is based on the kinetic energy of the system and offers the possibility to avoid a difficult mathematical calculus for the transition from a system of differential algebraic equations to the system of state equations.

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Long, Huan, and Suhui Ye. "Global well-posedness for the 2D MHD equations with only vertical velocity damping term." AIMS Mathematics 9, no. 12 (2024): 36371–84. https://doi.org/10.3934/math.20241725.

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This paper concerns two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations without magnetic diffusion with only vertical velocity damping term in the periodic domain. We prove the stability and decay rate for smooth solutions on perturbations near a background magnetic field of the system under the assumptions that the initial magnetic field satisfies the Diophantine condition.

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Journal articles on the topic 'System of Diophantine equations and calculus'

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