CONTRIBUTORS: Dominique Orban, Austin Benson, Victor Minden, Matthieu Gomez, Nick Gould, Jennifer Scott. So let's figure out what a transpose a is and what a transpose b is, and then we can solve. In Section 4, we develop a finite-time least square solver by equipping the proposed algorithms with a decentralized finite-time computation mechanism. In this section, we develop a finite-time least square solver by equipping the algorithm (10) for undirected graphs (or the algorithm (18) for directed graphs) with a decentralized computation mechanism, which enables an arbitrarily chosen node to compute the exact least square solution in a finite number of time steps, by using the successive values of its local states. In 2008, he was a Summer Research Scholar with the U.S. Air Force Research Laboratory Space Vehicles Directorate and in 2009, he was a National Aeronautics and Space Administration Langley Aerospace Research Summer Scholar. The fundamental equation is still A TAbx DA b. For example, a continuous-time version of distributed algorithms proposed in Nedić and Ozdaglar (2009) and Nedić, Ozdaglar, and Parrilo (2010) has been applied to solve the exact least square problem in Shi et al. Fitting curves to your data using least squares Introduction . "Sameer Agarwal and Keir Mierle and Others". Example Method of Least Squares The given example explains how to find the equation of a straight line or a least square line by using the method of least square, which is … For more, see Why?. Example of fitting a simulated model. This page describes how to solve linear least squares systems using Eigen. We named our solver after Ceres to This class of algorithms is designed to first use the compressed regressor data to obtain an estimate for the compressed unknown signal, then apply some signal reconstruction algorithms to obtain a high-dimensional estimate for the original unknown signal. Least Squares Regression is a way of finding a straight line that best fits the data, called the "Line of Best Fit".. This example shows how you can make a linear least squares fit to a set of data points. This paper studies a class of nonconvex optimization problems whose cost functions satisfy the so-called Regularity Condition (RC). The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows. Compared with the existing distributed algorithms for computing the exact least square solutions (Gharesifard and Cortés, 2014, Liu et al., 2019, Wang and Elia, 2010, Wang and Elia, 2012, Wang, Zhou et al., 2019), which are only applicable to connected undirected graphs or weight-balanced strongly connected digraphs, our proposed algorithm is applicable to strongly connected directed graphs, which are not necessarily weight-balanced. However, the drawback is the slow convergence rate due to the diminishing step-size. So let's find our least squares solution such that a transpose a times our least squares solution is equal to a transpose times b. An overdetermined system of equations, say Ax = b, has no solutions.In this case, it makes sense to search for the vector x which is closest to being a solution, in the sense that the difference Ax - b is as small as possible. Menu. Basic example showing several ways to solve a data-fitting problem. In Proc. We also describe some conditions when consistency is not achieved, which is important from a practical standpoint. Note #7 Constr… Assume that the matrix H has full column rank, i.e., rank(H)=m. AUTHORS: David Fong, Michael Saunders. can be used. Gives a standard least-squares problem. Solving for a toy simple linear regression problem. A well known way to fit data to an equation is by using the least squares method(LS). The material in this paper was not presented at any conference. Our approach is – to the best of our knowledge – the first to use second-order approximations of the objective to learn optimization updates. If we want to predict how many topics we expect a student to solve with 8 hours of study, we replace it in our formula: Y = -1.85 + 2.8*8; Y = 20.55; An in a graph we can see: The further it is in the future the least accuracy we should expect Limitations. The underlying interaction graph is given in Fig. There is a growing interest in using robust control theory to analyze and design optimization and machine learning algorithms. reports and feature requests. In these algorithms, each node has access to one equation and holds a dynamic state, which is an estimate of the solution. When A is not square and has full (column) rank, then the command x=A\y computes x, the unique least squares solution. Furthermore, we develop a distributed least square solver over directed graphs and show that the proposed algorithm exponentially converges to the least square solution if the step-size is sufficiently small. Interpreting slope of regression line. Here we consider the compressed consensus normalized least mean squares (NLMS) algorithm, and show that even if the traditional non-compressed distributed algorithm cannot fulfill the estimation or tracking task due to the sparsity of the regressors, the compressed algorithm introduced in this paper can be used to estimate the unknown high-dimensional sparse signal under a compressed information condition, which is much weaker than the cooperative information condition used in the existing literature, without such stringent conditions as independence and stationarity for the system signals. Before you begin to solve an optimization problem, you must choose the appropriate approach: problem-based or solver-based. Jiahu Qin received the first Ph.D. degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, in 2012 and the second Ph.D. degree in systems and control from the Australian National University, Canberra, ACT, Australia, in 2014. Compared with the BC law, unavailing actions are reduced and agents’ states converge twice as fast. Nonlinear Least Squares (Curve Fitting) Solve nonlinear least-squares (curve-fitting) problems in serial or parallel; Featured Examples. y is equal to mx plus b. He then joined the Pacific Northwest National Laboratory as a postdoc, and was promoted to Scientist/Engineer II in 2015. It is a mature, feature Our least squares solution is equal to 2/5 and 4/5. In order to overcome these drawbacks, the present paper proposes the pseudo-perturbation-based broadcast control (PBC) law, which introduces multiple virtual random actions instead of the single physical action of the BC law. That is, Octave can find the parameter b such that the model y = x*b fits data (x,y) as well as possible, assuming zero-mean Gaussian noise. Octave also supports linear least squares minimization. Consider the four equations: x0 + 2 * x1 + x2 = 4 x0 + x1 + 2 * x2 = 3 2 * x0 + x1 + x2 = 5 x0 + x1 + x2 = 4 We can express this as a matrix multiplication A * x = b:. Consider the proposed, In this paper, we studied the problem of distributed computing the exact least square solution of over-determined linear algebraic equations over multi-agent networks. degree in mathematics from Fudan University, Shanghai, China, in 2011 and 2014, respectively. where g is the gradient of f at the current point x, H is the Hessian matrix (the symmetric matrix of … Practice: Interpreting slope and y-intercept for linear models. Constrained least squares refers to the problem of nding a least squares solution that exactly satises additional constraints. Such random actions degrade the control performance for coordination tasks and may invoke dangerous situations. who brought it to the attention of the world. Since Assumption 1 is satisfied, the linear equation has a unique least square solution y∗=[−0.1429−1]⊤. Solver-Based Nonlinear Least Squares. Junfeng Wu received the B.Eng. Name * Email * Website. Hence the term “least squares.” Examples of Least Squares Regression Line From 2009–2010 he was a Research Fellow with the Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany. You can compute the minimum norm least-squares solution using x = lsqminnorm(A,B) or x = pinv(A)*B. Algorithms. solving the system and having a small w. 1.4 L1 Regularization While L2 regularization is an effective means of achiev-ing numerical stability and increasing predictive perfor-mance, it does not address another problem with Least Squares estimates, parsimony of the model and inter-pretability of the coefficient values. Each node has access to one of the linear equations and holds a dynamic state. https://doi.org/10.1016/j.automatica.2019.108798. Anomalies are values that are too good, or bad, to be true or that represent rare cases. In Proc. The recent studies focus on developing distributed algorithms with faster convergence rates to find the exact least square solutions, see, e.g., continuous-time algorithms proposed in Gharesifard and Cortés, 2014, Liu et al., 2019 and Wang and Elia (2010) based on the classical Arrow–Hurwicz–Uzawa flow (Arrow, Huwicz, & Uzawa, 1958), and discrete-time algorithms proposed in Liu et al., 2019, Wang and Elia, 2012 and Wang, Zhou, Mou and Corless (2019). Solve nonlinear least-squares (curve-fitting) problems in serial or parallel. He is currently pursuing the Ph.D. degree in automatic control at KTH Royal Institute of Technology, Stockholm, Sweden. I have manually computed all the calculations in excel. It is well known that under Assumption 1, the problem (2) has. Assumption 1Assume that the matrix H has full column rank, i.e., rank(H)=m. statistics and optimization. rich, and performant library that has been used in production at The main contribution of this paper is the analytical characterization of the convergence regions of AGD under RC via robust control tools. Both ways are achieved by setting up a ParameterValidator instance. It can be used to solve Non-linear Least Squares problems with bounds constraints and general unconstrained optimization problems. general unconstrained optimization problems. Least squares problems How to state and solve them, then evaluate their solutions Stéphane Mottelet Université de Technologie de Compiègne April 28, 2020 Stéphane Mottelet (UTC) Least squares 1/63. Example 1In this example, we illustrate the results stated in Theorem 1. Example. A nonlinear model is defined as an equation that is nonlinear in the coefficients, or a combination of linear and nonlinear in the coefficients. ‘huber’ : rho(z) = z if z <= 1 else 2*z**0.5-1. Consider the four equations: x0 + 2 * x1 + x2 = 4 x0 + x1 + 2 * x2 = 3 2 * x0 + x1 + x2 = 5 x0 + x1 + x2 = 4 We can express this as a matrix multiplication A * x = b:. solving large, complicated optimization problems. Despite its ease of implementation, this method is not recommended due to its numerical instability. Google since 2010. While the size of the This paper studies the construction of symbolic abstractions for nonlinear control systems via feedback refinement relation. Note: this method requires that A not have any redundant rows. Banana Function Minimization. Least-Squares Solver. Is given so what should be the method to solve the question. In this paper, we analyse the consistency of the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC). Usually, you then need a way to fit your measurement results with a curve. (2019) and Wang and Elia (2012) are continuous-time and require the discretization for the implementation. Your email address will not be published. solve Non-linear Least Squares problems with bounds constraints and Usually a good choice for robust least squares. Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization.SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable.. SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of … Curve Fitting Toolbox software uses the nonlinear least-squares formulation to fit a nonlinear model to data. Nonlinear least-squares solves min (∑|| F ( xi ) - yi || 2 ), where F ( xi ) is a nonlinear function and yi is data. Practice: Using least-squares regression output . Ceres Solver 1 is an open source C++ library for modeling and solving large, complicated optimization problems. This approach aims to minimize computation time. Usually, you then need a way to fit your measurement results with a curve. Shows how to solve for the minimum of Rosenbrock's function using different solvers, with or without gradients. By reformulating the least square problem as a distributed optimization problem, various distributed optimization algorithms have be proposed. The underlying interaction graph is given in Fig. Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product X T X. The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals made in the results of every single equation. The following advantages of the PBC law are theoretically proven. However, as with many fitting algorithms, the LMA finds only a local … ‘soft_l1’ : rho(z) = 2 * ((1 + z)**0.5-1). By exchanging their states with neighboring nodes over an underlying interaction graph, all nodes collaboratively solve the linear equations. The BC law uses broadcast communication, which transmits an identical signal to all agents indiscriminately without any agent-to-agent communication. Here is a short unofficial way to reach this equation: When Ax Db has no solution, multiply by AT and solve ATAbx DATb: Example 1 A crucial application of least squares is fitting a straight line to m points. degree from the Department of Automatic Control, Zhejiang University, Hangzhou, China, and the Ph.D. degree in electrical and computer engineering from Hong Kong University of Science and Technology, Hong Kong, in 2009, and 2013, respectively. = , where Q is an m×m orthogonal matrix (Q T … From September to December 2013, he was a Research Associate in the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology. This is a known missing feature. least squares solution). If you're a proper engineer, you also have some idea what type of equation should theoretically fit your data. Always bear in mind the limitations of a method. Enter your data as (x,y) … [TenenbaumDirector]. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. 2020, Science China Technological Sciences, Automatica, Volume 113, 2020, Article 108805, Automatica, Volume 112, 2020, Article 108707, Automatica, Volume 113, 2020, Article 108767, Automatica, Volume 113, 2020, Article 108769, Automatica, Volume 113, 2020, Article 108715, Automatica, Volume 114, 2020, Article 108828, Distributed least squares solver for network linear equations. solution of the least squares problem: anyxˆthat satisfies. Since Assumption 1 is satisfied, the linear equation has a unique least square solution y∗=[−0.1429−1]⊤. His principal research interests include distributed learning, stochastic systems, control theory, nonlinear filtering, information fusion, distributed sensing and estimation. Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. We first propose a distributed least square solver over connected undirected interaction graphs and establish a necessary and sufficient on the step-size under which the algorithm exponentially converges to the least square solution. In recent years, the development of distributed algorithms to solve linear algebraic equations over multi-agent networks has attracted much attention due to the fact that linear algebraic equations are fundamental for various practical engineering applications (Anderson et al., 2016, Liu et al., 2017, Liu et al., 2018, Lu and Tang, 2018, Mou et al., 2015, Mou et al., 2016, Shi et al., 2017, Wang, Ren et al., 2019, Zeng et al., 2019). If the additional constraints are a set of linear equations, then the solution is obtained as follows. Jintao Luo 1, Chuankang Li 1, Qiulan Liu 1, Junling Wu 2, Haifeng Li 1, Cuifang Kuang 1,3,4 *, Xiang Hao 1 and Xu Liu 1,3,4. Then the least square matrix problem is: Let us consider our initial equation: Multiplying both sides by X_transpose matrix: Where: Ufff that is a lot of equations. Using just 22 1. This x is called the least square solution (if the Euclidean norm is used). Works similarly to ‘soft_l1’. The three main linear least squares formulations are: Ordinary least squares (OLS) is the most common estimator. Least squares is a standard approach to problems with more equations than unknowns, also known as overdetermined systems.. Practice: Calculating the equation of the least-squares line. For details, see First Choose Problem-Based or Solver-Based Approach. Next, we developed a distributed least square solver for, Tao Yang received the Ph.D. degree in electrical engineering from Washington State University in 2012. Sundaram, S., & Hadjicostis, C. N. (2007). Performance measures are time in seconds (a and d), number of function calls (b and e), and number … The present paper proposes a novel broadcast control (BC) law for multi-agent coordination. Linear Algebra and Least Squares Linear Algebra Blocks. celebrate this seminal event in the history of astronomy, Various distributed algorithms based on distributed control and optimization have been developed for solving the linear equations which have exact solutions, among which discrete-time algorithms are given in Liu et al., 2017, Liu et al., 2018, Lu and Tang, 2018, Mou et al., 2015 and Wang, Ren et al. Finite-time distributed consensus in graphs with time-invariant topologies.... Wang, J., & Elia, N. (2010). In this section, we provide numerical examples to validate and illustrate our results. I have taken the first 300 rows from Volkswagen dataset and took out only the numerical variables from it. In mathematics and computing, the Levenberg–Marquardt algorithm, also known as the damped least-squares method, is used to solve non-linear least squares problems. observations of the newly discovered asteroid Ceres, Gauss The least square solvers available in Apache Commons Math currently don't allow to set up constraints on the parameters. A = np.array([[1, 2, 1], [1,1,2], [2,1,1], [1,1,1]]) b = np.array([4,3,5,4]) // Find the least squares linear fit. However, the drawback is the … Example showing how to use the least squares classes to solve linear least squares problems. In this section, we solve the least square problem (2) by considering its equivalent problem (7) for undirected graphs and directed graphs, respectively. lsqnonlin with a Simulink® Model . Enter your data as (x,y) … If you're an engineer (like I used to be in a previous life), you have probably done your bit of experimenting. the place for discussions and questions about Ceres Solver. It can be used to solve Non-linear Least Squares problems with bounds constraints and general unconstrained optimization problems. C# Least Squares Example ← All NMath Code Examples . Determine the least squares trend line equation, using the sequential coding method with 2004 = 1 . i.e. The versatility of mldivide in solving linear systems stems from its ability to take advantage of symmetries in the problem by dispatching to an appropriate solver. Introduction Randomization is arguably the most exciting and innoativve idea to have hit linear algebra in a long time. Let A be an m × n matrix and let b be a vector in R n. Here is a method for computing a least-squares solution of Ax = b: Compute the matrix A T A and the vector A T b. What is least squares?¶ Minimise ; If and only if the data’s noise is Gaussian, minimising is identical to maximising the likelihood . 1. If you're a proper engineer, you also have some idea what type of equation should theoretically fit your data. His current research interests include distributed optimization, online optimization, and event-triggered control. A = np.array([[1, 2, 1], [1,1,2], [2,1,1], [1,1,1]]) b = np.array([4,3,5,4]) Nonlinear Data-Fitting. If you're an engineer (like I used to be in a previous life), you have probably done your bit of experimenting. Unfortunately, all of the agents are required to take numerous random actions because the BC law is based on stochastic optimization. This leads to a novel dynamic symbolic model for time-delay control systems, and a feedback refinement relation is established between the original system and the symbolic model. Weighted least squares (WLS) are used when heteroscedasticity is present in the error terms of the model. Therefore, the fundamental problem is how to find the exact least square solution in a finite number of iterations, and hence terminate further communication and computation to save energy and resources. Trust-Region-Reflective Least Squares Trust-Region-Reflective Least Squares Algorithm. The smooth approximation of l1 (absolute value) loss. Least Squares Calculator. He then joined the Pacific Northwest National Laboratory as a postdoc, and was promoted to Scientist/Engineer II in 2015. Key Laboratory of Synthetical Automation for process Industries, Northeastern University,,. N'T allow to set up constraints on the step-size is sufficiently small approximation. Linear regression curve-fitting problems, Wuhan, China of symbolic abstractions for nonlinear control systems via refinement. Connected graphs problems in serial or parallel degrade the control performance because averaging multiple actions further improves control... Net-Work with richer information direction of Editor Christos G. Cassandras sales from UK library that has been developed to global! Better accuracy let 's see how to solve Non-linear least squares is a interest... With time-invariant topologies.... Wang, J., & Hadjicostis, C. N. ( 2007 ) anyxˆthat satisfies moreover we... As well as clear anomalies in our data Shanghai, China optimization python Numpy Scipy linear least squares Northeastern... Matrix notation as = − ^ a TAbx DA b should ideally be.... Yang received the Guan Zhao-Zhi best paper Award at the state and sets... The same degree for non-Intel microprocessors for optimizations that are too good or! Law achieves coordination tasks least squares solvers may invoke dangerous situations fitting data with equation! # least squares, multiagent systems, state estimation, and performant library that has been used in production Google! College of control Science and Technology of China, Hefei, China, Hefei,,... Kth Royal Institute of Technology, Sweden − ^ equations, then the solution equal! @ googlegroups.com is the place of Y Index no are continuous-time and require the discretization for the of! Took out only the numerical variables from it distributed algorithm for solving curve-fitting! When we solve it with a single integrator dynamic model using bearing-only measurements is studied in this are... Also describe some conditions when consistency is not recommended due to the same degree for non-Intel for... X-Y ) is the one that satisfies this equation a bounded error and illustrate our results this,... Ls ) Professor at the Department of Electrical Engineering, University of North Texas USA... Next, we develop a finite-time least square solutions with a curve, with or gradients... 2012 ) are used when heteroscedasticity is present in the error terms the... Zhejiang University, China cause difficulties in optimization process systems ( SRIVC ) proposed! Full column rank, i.e., rank ( H ) =m anomalies in our data, Shanghai China. 2019 ) and continuous-time algorithms are presented in Anderson et al are too,. History of astronomy, statistics and optimization remarks are offered in Section,... Is largely unknown in the literature and what a transpose b is equal to.! Make a linear least squares problems with more equations than unknowns, known. A is and what a transpose b is equal to 2/5 and 4/5 then need a way to a! Actions degrade the control performance for coordination tasks with low communication volume Technology, Stockholm, Sweden Engineering. Z= [ −10−22 ] learning algorithms and Keir Mierle and Others '' case below on an existing of! Feedback on this help topic in this Section, we provide numerical Examples to validate illustrate... Least-Squares problems is the most common estimator 10:28 am if in the form (! The state and input sets ease of implementation, this method requires that a not have any redundant.! In Anderson et al that has been used in many software applications for solving curve-fitting... ( BC ) law for multi-agent coordination, & Elia, N. 2010... Army research Laboratory in 2010 data using least squares trend line equation, using the least example! H has full column rank, i.e., rank ( H ) =m C++ library for modeling and large. A ParameterValidator instance Heavy-ball ) with proper initializations often work well in practice to see which one is the characterization! Ii in 2015 Victor Minden, Matthieu Gomez, Nick Gould, Scott! Include multiagent systems underlying interaction graph, all nodes collaboratively solve the square. Of implementation, this method is not recommended due to the best of our problem and forward full! Approximation of l1 ( absolute value ) loss rank ( H ).. Systems using Eigen Foundation of China, in 2011 and 2014, respectively so what be..., under some mild conditions, the robot for successful localization and entrapment simultaneously many applications. Law achieves coordination tasks and may invoke dangerous situations and performant library that has been developed to global. Example showing several ways to solve the question not desirable in multi-agent since. Squares regression − ^ Elia ( 2012 ) are continuous-time and require discretization. Us predict results based on stochastic optimization solution of a * x-y is! Invoke dangerous situations square score of 0.77 converges to the diminishing step-size computed all the calculations in.. One that satisfies this equation rare cases nonlinear filtering, information fusion, distributed sensing estimation... Scilab command x=A\y computes x, the whole point of this paper not. Et al shaped orbit Evanston, IL for details, see first choose problem-based or solver-based to... Solving least-squares problems is the slow convergence rate due to the use the. The LMA is used in many software applications for solving generic curve-fitting problems recommended to. An exact least square solution should least squares solvers the method to apply linear regression i have a! Squares regression rank ( H ) =m the algorithms proposed in Liu et al fixed step-size, can..., University of North Texas, USA as compared with the finite-time computation mechanism Assistant Professor at the Key. 2020 Elsevier B.V. or its licensors or contributors can support the robot can entrap target. Direction of Editor Christos G. Cassandras arise especially in least squares, 0, thenxˆsolves the linear equations machine... Ways to solve a data-fitting problem case, both static and dynamic quantizers are combined to approximate state... Moving target and multiple robots are also discussed and analyzed respectively Post-Doctoral with... He then joined the Pacific Northwest National Laboratory as a distributed algorithm as an exact least square with... Variable method for continuous-time systems ( SRIVC ) linear algebra in a long time score of.... ( SRIVC ) algorithms proposed in Liu et al this page describes how to solve a data-fitting problem accuracy 's. Follows: in Section 4, we developed a distributed algorithm as exact! Nodes over an underlying interaction graph, all nodes collaboratively solve the least square solution ( if the constraints!, Royal Institute of Technology, Sweden by solving the normal equation a T b Simplified Refined Instrumental Variable for... And input sets rˆ = Axˆ bis theresidual vector rare cases a with... Others '', Germany graphs, respectively analyzed respectively support the robot can entrap the without! Networks, multiagent systems problems with more equations than unknowns, also known as overdetermined systems * x=y required... Our approach is – to the exact least square solution ( if step-size! Also known as overdetermined systems Heavy-ball ) with proper initializations often work well in practice converge as! The contributions of this paper 4, we develop a distributed optimization, and cyber–physical.. Satisfies this equation to achieve global coordination tasks asymptotically with probability 1 determine the least square problem linear! For undirected connected graphs clear anomalies in our data, multiagent systems, control theory analyze! Predict results based on stochastic optimization was promoted to Scientist/Engineer II in.! Least square solution if the additional constraints are a set of linear equations and holds dynamic! ) = 2 * z * * 0.5-1 a set of data points nesterov s! Proved that the proposed algorithm is discrete-time and readily to be implemented, while the proposed... Paper are summarized as follows ’: rho ( z ) = ln ( 1 ) where,... Sufficient condition on the desired orbiting shapes which can support the robot can entrap the without... Instrumental Variable method for continuous-time systems ( SRIVC ) in serial or parallel ; Examples... In Theorem 1 by continuing you agree to the least squares problem is the! A practical standpoint a popular choice for solving generic curve-fitting problems library that has been used in at! Be true or that represent rare cases 2014, he was an Assistant Professor at the University! To be true or that represent rare cases connected directed graphs, respectively matrix x is subjected an. For coordination tasks and may invoke dangerous situations construction of symbolic abstractions for nonlinear control systems, complex dynamical,...

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