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We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using partial fraction decomposition, the polynomial part of rational solutions is obtained by computing polynomial solutions of a non-homogeneous scalar integro-differential equation with a polynomial right-hand side. The paper is illustrated by examples where the computations are done with our Maple implementation. [ABSTRACT FROM AUTHOR]

This paper provides a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. This allows us to establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why classical algorithms then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of our algorithm. The algorithm is implemented in Maple and examples are given in Saade (2018). [ABSTRACT FROM AUTHOR]

In this paper, a symbolic, algorithmic procedure to compute an immersion that recasts a polynomial system into a linear one up to an output injection is proposed. Such a technique is based on computing, through algebraic geometry methods, the set of all the embeddings of the system and on matching the coefficients of these polynomials with the ones of the embeddings of a linear system up to an output injection. The given algorithm is then relaxed to compute an immersion that recasts a polynomial system into a form that is linear up to a finite order and an output injection and to compute an approximation of the immersion. [ABSTRACT FROM AUTHOR]

This paper proposes new iterative algorithms for the unknown input and state recovery from the system outputs using an approximate inverse of the strictly proper linear time-invariant (LTI) multivariable system. One of the unique advantages from previous system inverse algorithms is that the output differentiation is not required. The approximate system inverse is stable due to the systematic optimal design of a dummy feedthrough D matrix in the state-space model via the feedback stabilization. The optimal design procedure avoids trial and error to identify such a D matrix which saves tremendous amount of efforts. From the derived and proved convergence criteria, such an optimal D matrix also guarantees the convergence of algorithms. Illustrative examples show significant improvement of the reference input signal tracking by the algorithms and optimal D design over non-iterative counterparts on controllable or stabilizable LTI systems, respectively. Case studies of two Boeing-767 aircraft aerodynamic models further demonstrate the capability of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithm for implicitizing tensor product surfaces of bi-degree ( m , n ) with no base points, assuming that there are no moving planes of bi-degree ( m − 1 , n − 1 ) following the surface. The algorithm is based on some structural results: (1) There are exactly 2 n linearly independent moving planes of bi-degree ( m , n − 1 ) following the surface; (2) mn linearly independent moving quadrics of bi-degree ( m − 1 , n − 1 ) following the surface can be constructed from the 2 n linearly independent moving planes; (3) The mn linearly independent moving quadrics form a compact determinant of order mn which exactly gives the implicit equation of the rational surface. Complexity analysis and experimental results show that the new algorithm is significantly more efficient than the previous methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n th Weyl algebra, the polynomial n th shift algebra, and Z n -graded polynomials in the n th q _ -Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is Z n -graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in major computer algebra systems on nontrivial examples. [ABSTRACT FROM AUTHOR]

Classification arises in a wide range of applications. A variety of statistical tools have been developed for learning classification rules from data. Understanding of their relative merits and comparisons help users to choose a proper method in practice. This paper focuses on theoretical comparison of model-based classification methods in statistics with algorithmic methods in machine learning in terms of the error rate. Extending Efron’s comparison of logistic regression with linear discriminant analysis (LDA) under the normal setting, we contrast such algorithmic methods as the support vector machine (SVM) and boosting with the LDA and logistic regression and study their relative efficiencies in reducing the error rate based on the limiting behavior of the classification boundary of each method. We show that algorithmic methods are generally less effective than model-based methods in the normal setting. In particular, loss of efficiency in error rate is typically about 33% to 60% for the SVM and 50% to 80% for boosting when compared to the LDA. However, a smooth variant of the SVM is shown to be even more efficient than logistic regression. In addition to the theoretical study, we present results from numerical experiments under various settings for comparisons of finite-sample performance and robustness to mislabeling and model misspecification. [ABSTRACT FROM AUTHOR]

• The proposed simple algorithm is very different from conventional numerical methods. • A family of boundary shape functions in this algorithm plays the most important role of the accuracy. • Accurate and stable results are acquired for large noises. In this paper, we recover unknown space–time dependent forces imposed on the vibrating Euler–Bernoulli beams under different boundary conditions, including simply supported, clamped-hinged, cantilevered, and two-end fixed conditions. The data overspecified to recover the external force are the final time displacement and the right-side strain or right-side moment of the beam. We develop a family of boundary shape functions, which automatically satisfy the initial conditions, final time condition, and homogeneous boundary conditions for each type of beam. When the solution is obtained using the method of superimposing boundary shape functions and solving a small-scale linear system to satisfy an extra right-side boundary condition, the unknown force can be recovered through back-substitution of the solution into the Euler–Bernoulli beam equation. The accuracy and robustness of the proposed methods are confirmed by comparing the recovered results of seven examples to the exact forces, even though considerable noise is present in the overspecified data. [ABSTRACT FROM AUTHOR]

• We propose a novel numerical algorithm for solving the Cahn-Hilliard-Navier-Stokes (CHNS) equations. • The scheme is second-order and linear while preserving the energy dissipation law. • The proposed scheme only requires solving linear systems, and the solution existence and uniqueness are guaranteed. • The proposed scheme obeys an energy dissipation law in the original variables. In this paper, we consider the Cahn-Hilliard equation coupled with the incompressible Navier-Stokes equation, usually known as the Cahn-Hilliard-Navier-Stokes (CHNS) system. The CHNS system has been widely embraced to investigate the dynamics of a binary fluid mixture. By utilizing the modified leap-frog time-marching method, we propose a novel numerical algorithm for solving the CHNS system in an efficient and accurate manner. This newly proposed scheme has several advantages. First of all, the proposed scheme is linear in time and space, such that only a linear algebraic system needs to be solved at each time-marching step, making it extremely efficient. Also, the existence and uniqueness of numerical solutions are guaranteed for any time step size. In addition, the scheme is unconditionally energy stable with second-order accuracy in time and spectral accuracy in space, such that relatively large temporal and spatial mesh sizes can be used to obtain reliable numerical solutions. The rigorous proofs for the unconditional energy stable property and solution existence and uniqueness are given. Furthermore, we present several numerical examples to test the proposed numerical algorithm and illustrate its accuracy and efficiency. The differences of coarsening dynamics between the Cahn-Hilliard equation and the Cahn-Hilliard-Navier-Stokes equations have been investigated as well. [ABSTRACT FROM AUTHOR]