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1. White-box Machine learning approaches to identify governing equations for overall dynamics of manufacturing systems: A case study on distillation column

   Subramanian, Renganathan; Moar, Raghav Rajesh; Singh, Shweta (March 2021, Machine learning with applications)

   null (Ed.)

   Dynamical equations form the basis of design for manufacturing processes and control systems; however, identifying governing equations using a mechanistic approach is tedious. Recently, Machine learning (ML) has shown promise to identify the governing dynamical equations for physical systems faster. This possibility of rapid identification of governing equations provides an exciting opportunity for advancing dynamical systems modeling. However, applicability of the ML approach in identifying governing mechanisms for the dynamics of complex systems relevant to manufacturing has not been tested. We test and compare the efficacy of two white-box ML approaches (SINDy and SymReg) for predicting dynamics and structure of dynamical equations for overall dynamics in a distillation column. Results demonstrate that a combination of ML approaches should be used to identify a full range of equations. In terms of physical law, few terms were interpretable as related to Fick’s law of diffusion and Henry’s law in SINDy, whereas SymReg identified energy balance as driving dynamics. 

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2. Constructing linear systems with particular kinds of solution sets

   Smith, J.; Lee, I.; Zandieh, M.; Andrews-Larson, C. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S.; & Higgins, A. (Ed.)

   Systems of equations is a core topic in linear algebra courses. Solving systems with no or infinitely many solutions tends to be less intuitive for students. In this study, we examined two students’ reasoning about the relationship between the structure of a system of linear equations and its solution set, particularly when creating systems with a certain number of equations and unknowns. Using data from a paired teaching experiment, we found that both students favored the notion of parallel planes, both geometrically and numerically, in the case of a system having no solution or infinitely many solutions. We also found that algebraic or numerical approaches were used as the main way of developing systems with a unique solution, especially in systems with more than two equations and two unknowns. In particular, one student gravitated toward geometric approaches and the other toward algebraic and numerical approaches. 

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3. Constructing linear systems with particular kinds of solution sets.

   Smith, J.; Lee, I.; Zandieh, M.; & Andrews-Larson, C. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S.; Higgins, A. (Ed.)

   Systems of equations is a core topic in linear algebra courses. Solving systems with no or infinitely many solutions tends to be less intuitive for students. In this study, we examined two students’ reasoning about the relationship between the structure of a system of linear equations and its solution set, particularly when creating systems with a certain number of equations and unknowns. Using data from a paired teaching experiment, we found that both students favored the notion of parallel planes, both geometrically and numerically, in the case of a system having no solution or infinitely many solutions. We also found that algebraic or numerical approaches were used as the main way of developing systems with a unique solution, especially in systems with more than two equations and two unknowns. In particular, one student gravitated toward geometric approaches and the other toward algebraic and numerical approaches. 

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4. Constructing linear systems with particular kinds of solution sets

   Smith, J. (January 2022, 24th conference on research in undergraduate mathematics education)

   S. S. Karunakaran; A. Higgins (Ed.)

   Systems of equations is a core topic in linear algebra courses. Solving systems with no or infinitely many solutions tends to be less intuitive for students. In this study, we examined two students’ reasoning about the relationship between the structure of a system of linear equations and its solution set, particularly when creating systems with a certain number of equations and unknowns. Using data from a paired teaching experiment, we found that both students favored the notion of parallel planes, both geometrically and numerically, in the case of a system having no solution or infinitely many solutions. We also found that algebraic or numerical approaches were used as the main way of developing systems with a unique solution, especially in systems with more than two equations and two unknowns. In particular, one student gravitated toward geometric approaches and the other toward algebraic and numerical approaches. 

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5. A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations

   https://doi.org/10.1007/978-3-030-69637-5_20  

   Hegde, Chinmay; Keinert, Fritz; Weber, Eric S. (February 2021, Applied and numerical harmonic analysis)

   The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a modified Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e., the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We prove that the modified algorithm converges under no additional assumptions on the equations. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also demonstrate that in the case of an inconsistent system of equations, the modified relaxed Kaczmarz algorithm converges to a weighted least squares solution as the relaxation parameter approaches 0. 

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