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Journal articles on the topic 'Algebraic system'

Author: Grafiati

Published: 6 July 2023

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1

Kravtsov, H. O., S. M. Hrechko, V. V. Nikitchenko, and A. M. Prymushko. "Cognitive Algebraic System." Èlektronnoe modelirovanie 44, no. 3 (June 10, 2022): 14–30. http://dx.doi.org/10.15407/emodel.44.03.014.

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2

Omarov, A. I. "Orthogonally complete algebraic system." Algebra and Logic 30, no. 2 (March 1991): 134–39. http://dx.doi.org/10.1007/bf01978833.

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3

Fu, Jun, Jinzhao Wu, and Hongyan Tan. "A Deductive Approach towards Reasoning about Algebraic Transition Systems." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/607013.

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Algebraic transition systems are extended from labeled transition systems by allowing transitions labeled by algebraic equations for modeling more complex systems in detail. We present a deductive approach for specifying and verifying algebraic transition systems. We modify the standard dynamic logic by introducing algebraic equations into modalities. Algebraic transition systems are embedded in modalities of logic formulas which specify properties of algebraic transition systems. The semantics of modalities and formulas is defined with solutions of algebraic equations. A proof system for this logic is constructed to verify properties of algebraic transition systems. The proof system combines with inference rules decision procedures on the theory of polynomial ideals to reduce a proof-search problem to an algebraic computation problem. The proof system proves to be sound but inherently incomplete. Finally, a typical example illustrates that reasoning about algebraic transition systems with our approach is feasible.

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4

Holcombe, M. "Algebraic Techniques of System Specification." Irish Mathematical Society Bulletin 0021 (1988): 13–28. http://dx.doi.org/10.33232/bims.0021.13.28.

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5

Cadzow, J., and O. Solomon. "Algebraic approach to system identification." IEEE Transactions on Acoustics, Speech, and Signal Processing 34, no. 3 (June 1986): 462–69. http://dx.doi.org/10.1109/tassp.1986.1164849.

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6

Jang, Youngho. "Algebraic Weyl system and application." Annales mathématiques Blaise Pascal 4, no. 2 (1997): 27–40. http://dx.doi.org/10.5802/ambp.95.

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7

HINDMAN, NEIL. "The Topological-Algebraic System(?N, +, ?)." Annals of the New York Academy of Sciences 704, no. 1 Papers on Gen (December 1993): 155–63. http://dx.doi.org/10.1111/j.1749-6632.1993.tb52519.x.

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8

Letichevskii, A. A., and V. G. Marinchenko. "Objects in algebraic programming system." Cybernetics and Systems Analysis 33, no. 2 (March 1997): 283–99. http://dx.doi.org/10.1007/bf02665902.

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9

Phusanga, D., and J. Koppitz. "Some varieties of algebraic systems of type ((n),(m))." Asian-European Journal of Mathematics 12, no. 01 (February 2019): 1950005. http://dx.doi.org/10.1142/s1793557119500050.

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In the present paper, we classify varieties of algebraic systems of the type [Formula: see text], for natural numbers [Formula: see text] and [Formula: see text], which are closed under particular derived algebraic systems. If we replace in an algebraic system the [Formula: see text]-ary operation by an [Formula: see text]-ary term operation and the [Formula: see text]-ary relation by the [Formula: see text]-ary relation generated by an [Formula: see text]-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

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10

ŚWIRSZCZ, GRZEGORZ. "AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS." International Journal of Bifurcation and Chaos 15, no. 03 (March 2005): 1033–44. http://dx.doi.org/10.1142/s0218127405012442.

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Given a system of two autonomous ordinary differential equations whose right-hand sides are polynomials, it is very hard to tell if any nonsingular trajectories of the system are contained in algebraic curves. We present an effective method of deciding whether a given system has an invariant algebraic curve of a given degree. The method also allows the construction of examples of polynomial systems with invariant algebraic curves of a given degree. We present the first known example of a degree 6 algebraic saddle-loop for polynomial system of degree 2, which has been found using the described method. We also present some new examples of invariant algebraic curves of degrees 4 and 5 with an interesting geometry.

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11

Lesfari, A. "Systèmes dynamiques algébriquement complètement intégrables et géométrie." Annals of West University of Timisoara - Mathematics and Computer Science 53, no. 1 (July 1, 2015): 109–36. http://dx.doi.org/10.1515/awutm-2015-0006.

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Résumé In this paper I present the basic ideas and properties of the complex algebraic completely integrable dynamical systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several examples of algebraic completely integrable systems : Kowalewski’s top, geodesic flow on SO(4), Hénon-Heiles system, Garnier potential, two coupled nonlinear Schrödinger equations and Yang-Mills system.

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12

Zeghib, A. "Sur une notion d'autonomie de systèmes dynamiques, appliquée aux ensembles invariants des flots d' Anosov algébriques." Ergodic Theory and Dynamical Systems 15, no. 1 (February 1995): 175–207. http://dx.doi.org/10.1017/s0143385700008300.

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AbstractWe introduce a notion of autonomous dynamical systems which generalizes algebraic dynamical systems. We show by giving examples and by describing some properties that this generalization is not a trivial one. We apply the methods then developed to algebraic Anosov systems. We prove that a C1-submanifold of finite volume, which is invariant by an algebraic Anosov system is ‘essentially’ algebraic.

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13

Essert, Mario, and Darko Žubrinić. "Šare’s algebraic systems." Acta mathematica Spalatensia 2 (December 1, 2022): 1–28. http://dx.doi.org/10.32817/ams.2.1.

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We study algebraic systems MΓ of free semigroup structure, where Γ is a well ordered finite alphabet, discovered in 1970s within the Theory of Electric Circuits by Miro Šare, and and finding recent recent applications in Multivalued Logic, as well as in Computational Linguistics. We provide three simple axioms (reversion axiom (5) and two compression axioms (6) and (7)), which generate the corresponding equivalence relation between words. We also introduce a class of incompressible words, as well as the quotient Šare system MΓ~. The main result is contained in Theorem 16, announced by Šare without proof, which characterizes the equivalence of two words by means of Šare sums. The proof is constructive. We describe an algorithm for compression of words, study homomorphisms between quotient Šare systems for various alphabets Γ (Theorem 38), and introduce two natural Šare categories ŠŠa(M) and ŠŠa(M~). Šare systems are not inverse semigroups.

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14

Wampler, Charles W., and Andrew J. Sommese. "Numerical algebraic geometry and algebraic kinematics." Acta Numerica 20 (April 28, 2011): 469–567. http://dx.doi.org/10.1017/s0962492911000067.

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In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

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15

Junttila, Tommi A. "Finding Symmetries of Algebraic System Nets." Fundamenta Informaticae 37, no. 3 (1999): 269–89. http://dx.doi.org/10.3233/fi-1999-37305.

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16

Führer, Claus. "ALGEBRAIC METHODS IN VEHICLE SYSTEM ANALYSIS." Vehicle System Dynamics 16, sup1 (January 1987): 315–28. http://dx.doi.org/10.1080/00423118708969180.

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17

Amirat, Youcef Aït, and Sette Diop. "Towards an Algebraic System Theory Toolbox." IFAC Proceedings Volumes 29, no. 1 (June 1996): 2697–702. http://dx.doi.org/10.1016/s1474-6670(17)58083-0.

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18

Abdali, S. Kamal, Guy W. Cherry, and Neil Soiffer. "A Smalltalk system for algebraic manipulation." ACM SIGPLAN Notices 21, no. 11 (November 1986): 277–83. http://dx.doi.org/10.1145/960112.28724.

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19

Chandrashekara A C. "System Hypothesis Implications of Algebraic Geometry." international journal of engineering technology and management sciences 7, no. 1 (February 28, 2023): 105–8. http://dx.doi.org/10.46647/ijetms.2023.v07i01.018.

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Certain pole-placement concepts, such as an enhanced form of pole location with output response, are proven using fundamental algebraic geometry equations. Illustrations that highlight the algebra-geometric equations drawbacks and its possible application to systems analysis are shown. This study and ones that may come after it may help to make the potent theorems of current algebraic geometry comprehensible and useful for solving technical hurdles.

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20

Valls, Claudia. "Invariant algebraic surfaces and algebraic first integrals of the Maxwell–Bloch system." Journal of Geometry and Physics 146 (December 2019): 103516. http://dx.doi.org/10.1016/j.geomphys.2019.103516.

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21

Beran, Zdeněk, and Sergej Čelikovský. "Analytical-Algebraic Approach to Solving Chaotic System." International Journal of Bifurcation and Chaos 26, no. 03 (March 2016): 1650051. http://dx.doi.org/10.1142/s0218127416500516.

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The aim of this paper is to present the application of the analytical series technique to study properties of the nonlinear chaotic dynamical systems. More specifically, Laplace–Adomian decomposition method is applied to Rössler system and the so-called generalized Lorenz system. Some advantages and possible applications of this approach are discussed. Results are illustrated by numerical computations.

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22

Muhafzan, Zulakmal, Y. M. Putri, and L. W. June. "Minimum energy control of fractional-order differential-algebraic system." Cybernetics and Physics, Volume 11, 2022, Number 3 (November 17, 2022): 151–56. http://dx.doi.org/10.35470/2226-4116-2022-11-3-151-156.

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This paper discusses the minimum energy control problem of fractional-order differential-algebraic system. The main aim of this paper is to find the minimum energy that drives an initial state of the fractional order differential-algebraic system to the zero state such that an index performance is minimized. The method of solving is to convert the minimum energy control problem of fractional-order differential-algebraic system into the standard fractional-order linear quadratic optimization problem by using a transformation and further solve the standard fractional-order linear quadratic optimization using the available theory in the literature. Under some particular conditions, we find the explicit formulas of the minimum energy control of fractional-order differential-algebraic system in Mittag-Leffler terms.

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23

Exman, Iaakov, and Harel Wallach. "Linear Software Models: An Occam’s Razor Set of Algebraic Connectors Integrates Modules into a Whole Software System." International Journal of Software Engineering and Knowledge Engineering 30, no. 10 (October 2020): 1375–413. http://dx.doi.org/10.1142/s0218194020400185.

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Well-designed software systems, with providers only modules, have been rigorously obtained by algebraic procedures from the software Laplacian Matrices or their respective Modularity Matrices. However, a complete view of the whole software system should display, besides provider relationships, also consumer relationships. Consumers may have two different roles in a system: either internal or external to modules. Composite modules, including both providers and internal consumers, are obtained from the joint providers and consumers Laplacian matrix, by the same spectral method which obtained providers only modules. The composite modules are integrated into a whole Software System by algebraic connectors. These algebraic connectors are a minimal Occam’s razor set of consumers external to composite modules, revealed through iterative splitting of the Laplacian matrix by Fiedler eigenvectors. The composite modules, of the respective standard Modularity Matrix for the whole software system, also obey linear independence of their constituent vectors, and display block-diagonality. The spectral method leading to composite modules and their algebraic connectors is illustrated by case studies. The essential novelty of this work resides in the minimal Occam’s razor set of algebraic connectors — another facet of Brooks’ Propriety principle leading to Conceptual Integrity of the whole Software System — within Linear Software Models, the unified algebraic theory of software modularity.

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24

LIU, WEI, YUXIAN CHEN, and CHAOJIN FU. "DYNAMIC BEHAVIOR ANALYSIS OF A DIFFERENTIAL-ALGEBRAIC PREDATOR–PREY SYSTEM WITH PREY HARVESTING." Journal of Biological Systems 21, no. 03 (September 2013): 1350022. http://dx.doi.org/10.1142/s0218339013500228.

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This paper studies a differential-algebraic predator–prey system with prey harvesting, which consists of two differential equations and an algebraic equation. By using the differential-algebraic system theory, bifurcation theory and formal series expansions, we investigate the Hopf bifurcation and center stability of the differential-algebraic predator–prey system. Some sufficient conditions on these issues are obtained. In addition, numerical simulations illustrate the effectiveness of our results and their biological implications are discussed.

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25

Shashkin, Sergej Yurevich. "The inclusion of additional topics to the program of linear algebra course for economists and managers." Development of education, no. 1 (3) (March 18, 2019): 21–25. http://dx.doi.org/10.31483/r-22217.

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The paper generalizes the concept of “solving a system of linear algebraic equations in order to formulate a unified approach to the analysis of incompatible, indefinite and unstable systems”. Examples of unstable systems of linear algebraic equations are considered, which solutions depend on small changes in the numerical coefficients in the equations. The reasons for the instability of linear systems and the regularization algorithm for finding the solution of any system of linear algebraic equations are discussed. As the author notes, the Tikhonov regulatory algorithm is the most popular and practically convenient for solving unstable SLAES.

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26

Jibetean, Dorina, and Jan H. van Schuppen. "An Algebraic Method for System Reduction of Stationary Gaussian Systems." IFAC Proceedings Volumes 36, no. 16 (September 2003): 1879–84. http://dx.doi.org/10.1016/s1474-6670(17)35034-6.

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27

Moysis, Lazaros, and Nicholas Karampetakis. "Algebraic Methods for the Construction of Algebraic-Difference Equations With Desired Behavior." Electronic Journal of Linear Algebra 34 (February 21, 2018): 1–17. http://dx.doi.org/10.13001/1081-3810.3741.

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For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the prescribed behavior. It is proved that this problem can be reduced either to a linear system of equations problem or to an interpolation problem and an algorithm is proposed for constructing a system satisfying a given forward and/or backward behavior.

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28

Zhang, Guang, and Liang Bai. "Existence of Solutions for a Nonlinear Algebraic System." Discrete Dynamics in Nature and Society 2009 (2009): 1–28. http://dx.doi.org/10.1155/2009/785068.

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As well known, the existence and nonexistence of solutions for nonlinear algebraic systems are very important since they can provide the necessary information on limiting behaviors of many dynamic systems, such as the discrete reaction-diffusion equations, the coupled map lattices, the compartmental systems, the strongly damped lattice systems, the complex dynamical networks, the discrete-time recurrent neural networks, and the discrete Turing models. In this paper, both the existence of nonzero solution pairs and the nonexistence of nontrivial or nonzero solutions for a nonlinear algebraic system will be considered by using the critical point theory and Lusternik-Schnirelmann category theory. The process of proofs on the obtained results is simple, the conditions of theorems are also easy to be verified, however, some of them improve the known ones even if the system is reduced to the precial cases, in particular, others of them are still new.

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29

Barotov, Dostonjon, Aleksey Osipov, Sergey Korchagin, Ekaterina Pleshakova, Dilshod Muzafarov, Ruziboy Barotov, and Denis Serdechnyy. "Transformation Method for Solving System of Boolean Algebraic Equations." Mathematics 9, no. 24 (December 18, 2021): 3299. http://dx.doi.org/10.3390/math9243299.

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In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in Kn is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined.

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30

Zyuzin, Nikita, and Aleksandar Nanevski. "Contextual modal types for algebraic effects and handlers." Proceedings of the ACM on Programming Languages 5, ICFP (August 22, 2021): 1–29. http://dx.doi.org/10.1145/3473580.

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Programming languages with algebraic effects often track the computations’ effects using type-and-effect systems. In this paper, we propose to view an algebraic effect theory of a computation as a variable context; consequently, we propose to track algebraic effects of a computation with contextual modal types . We develop ECMTT, a novel calculus which tracks algebraic effects by a contextualized variant of the modal □ (necessity) operator, that it inherits from Contextual Modal Type Theory (CMTT). Whereas type-and-effect systems add effect annotations on top of a prior programming language, the effect annotations in ECMTT are inherent to the language, as they are managed by programming constructs corresponding to the logical introduction and elimination forms for the □ modality. Thus, the type-and-effect system of ECMTT is actually just a type system. Our design obtains the properties of local soundness and completeness, and determines the operational semantics solely by β-reduction, as customary in other logic-based calculi. In this view, effect handlers arise naturally as a witness that one context (i.e., algebraic theory) can be reached from another, generalizing explicit substitutions from CMTT. To the best of our knowledge, ECMTT is the first system to relate algebraic effects to modal types. We also see it as a step towards providing a correspondence in the style of Curry and Howard that may transfer a number of results from the fields of modal logic and modal type theory to that of algebraic effects.

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31

Xeyrəbadi, Qəzalə, and smət Ağalarova İ. "DETERMINATION OF THE BOUNDARY CONDITIONS OF THE STRESS FUNCTION WITH UNKNOWN COEFFICIENTS OF AN ELASTIC HALF-PLANE WITH CIRCULAR CAVITIES." Scientific works/Elmi eserler 2 (April 2, 1996): 56–62. http://dx.doi.org/10.58225/sw.2022.2.56-62.

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Abstract. In this paper, discusses the solution of the first fundamental boundary value problem for an elastic half-space with a circular space.In the article, the definition of an analytical function with an unknown coefficient given in the inner contour is not related to the separations by the contour variable of the boundary condition, but to a system of infinite linear algebraic equations obtained in the process of orthogonalization of this boundary condition.The coefficients of a system of algebraic equations are expressed in terms of integrals along a circular contour, that is, it is obtained from comparing the coefficients of the same overhead forces satisfying the contour conditions.In order for the resulting infinite system of algebraic linear equations to have a solution, they must be regular.The rapid attenuation (approaching zero) of the coefficients that make up the system of algebraic linear equations as a result of the calculations carried out confirms this condition.The problem under consideration is a special case of the classical Bussinex problem. The coefficients of the system of algebraic equations are expressed by integrals over a circular contour. Keywords: boundary conditions, system of algebraic equations, stress, integrall, regularization condition of a system of algebraic equations

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32

Mavoungou, J. P., and C. Nkuimi-Jugnia. "On a characterization of the lattice of subsystems of a transition system." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–4. http://dx.doi.org/10.1155/ijmms/2006/82318.

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It was first proved by Birkhoff and Frink, and the result now belongs to the folklore, that any algebraic lattice is up to isomorphism the lattice of subuniverses of a universal algebra. A study of subsystems of a transition system yields a new algebraic concept, that of a strongly algebraic lattice. We give here a representation theorem to the manner of Birkhoff and Frink of such lattices.

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33

Diop, S., and Y. Aït-Amirat. "On a differential algebraic system structure theory." IFAC Proceedings Volumes 46, no. 2 (2013): 290–95. http://dx.doi.org/10.3182/20130204-3-fr-2033.00204.

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34

Ivaschenko, E. A., and V. G. Skobelev. "PRESENTATION OF CRYPTOSYSTEMS VIA POLYBASIC ALGEBRAIC SYSTEM." Prikladnaya diskretnaya matematika, no. 2 (December 1, 2008): 33–38. http://dx.doi.org/10.17223/20710410/2/8.

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35

Jing, YANG, TAN Wenhui, and WEI Zhouchao. "Invariant Algebraic Surfaces of the Vallis System." 应用数学和力学 43, no. 1 (2022): 84–93. http://dx.doi.org/10.21656/1000-0887.420112.

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36

Kahramanlı, Ş., and N. Allahverdi. "Algebraic Approach to Transformations on Hypercube System." Mathematical and Computational Applications 1, no. 1 (June 1, 1996): 50–59. http://dx.doi.org/10.3390/mca1010050.

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37

Krutitskii, P. A., and V. V. Kolybasova. "Numerical method for an integral-algebraic system." Differential Equations 53, no. 10 (October 2017): 1364–71. http://dx.doi.org/10.1134/s0012266117100135.

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38

Sabadni, Irene, Michael V. Shapiro, and Daniele C. Struppa. "Algebraic analysis of the moisil—theodorescu system." Complex Variables, Theory and Application: An International Journal 40, no. 4 (February 2000): 333–57. http://dx.doi.org/10.1080/17476930008815227.

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39

Llibre, Jaume, and Xiang Zhang. "Invariant algebraic surfaces of the Rikitake system." Journal of Physics A: Mathematical and General 33, no. 42 (October 11, 2000): 7613–35. http://dx.doi.org/10.1088/0305-4470/33/42/310.

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40

Berdjag, D., C. Christophe, and V. Cocquempot. "AN ALGEBRAIC METHOD FOR NONLINEAR SYSTEM DECOMPOSITION." IFAC Proceedings Volumes 39, no. 13 (2006): 48–53. http://dx.doi.org/10.3182/20060829-4-cn-2909.00007.

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41

Nett, C. "Algebraic aspects of linear control system stability." IEEE Transactions on Automatic Control 31, no. 10 (October 1986): 941–49. http://dx.doi.org/10.1109/tac.1986.1104137.

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42

Rachmaputri, Gantina. "Algebraic structure of the anti-causal system." Journal of Physics: Conference Series 890 (September 2017): 012122. http://dx.doi.org/10.1088/1742-6596/890/1/012122.

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43

Hall, E. B., and G. L. Wise. "An algebraic aspect of linear system theory." IEEE Transactions on Circuits and Systems 37, no. 5 (May 1990): 651–53. http://dx.doi.org/10.1109/31.55011.

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44

Llibre, Jaume, and Xiang Zhang. "Invariant algebraic surfaces of the Lorenz system." Journal of Mathematical Physics 43, no. 3 (March 2002): 1622–45. http://dx.doi.org/10.1063/1.1435078.

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45

DENG, XIJUN, and AIYONG CHEN. "INVARIANT ALGEBRAIC SURFACES OF THE CHEN SYSTEM." International Journal of Bifurcation and Chaos 21, no. 06 (June 2011): 1645–51. http://dx.doi.org/10.1142/s0218127411029331.

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In this paper, enlightened by the idea of the weight of a polynomial introduced by Swinnerton-Dyer [2002], we find all the invariant algebraic surfaces of the Chen system x′ = a(y - x), y′ = (c - a)x + cy - xz, z′ = xy - bz.

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46

Vuk, Martin. "Algebraic integrability of the confluent Neumann system." Journal of Physics A: Mathematical and Theoretical 41, no. 39 (August 29, 2008): 395201. http://dx.doi.org/10.1088/1751-8113/41/39/395201.

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47

Dengyuan, C., and Z. Hangwei. "Lie algebraic structure for the AKNS system." Journal of Physics A: Mathematical and General 24, no. 2 (January 21, 1991): 377–83. http://dx.doi.org/10.1088/0305-4470/24/2/010.

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48

Kokenovna, Zhukabayeva Tamara. "Introduction of Cryptosystems by Polybasic Algebraic System." Procedia - Social and Behavioral Sciences 46 (2012): 5816–23. http://dx.doi.org/10.1016/j.sbspro.2012.06.521.

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49

Chen, Dengyuan. "Lie algebraic structure of akns matrix system." Acta Mathematicae Applicatae Sinica 9, no. 3 (July 1993): 213–22. http://dx.doi.org/10.1007/bf02032916.

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50

Tam, Honwah, and Yufeng Zhang. "A new algebraic system and its applications." Chaos, Solitons & Fractals 23, no. 1 (January 2005): 151–57. http://dx.doi.org/10.1016/j.chaos.2004.04.003.

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