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The Maximum Likelihood Decoding Problem (MLD) is known to be NP-hard and its complexity is strictly related to the security of some post-quantum cryptosystems, that is, the so-called code-based primitives. Analogously, the Multivariate Quadratic System Problem (MQ) is NP-hard and its complexity is necessary for the security of the so-called multivariate-based primitives. In this paper we present a closed formula for a polynomial-time reduction from any instance of MLD to an instance of MQ, and viceversa. We also show a polynomial-time isomorphism between MQ and MLD, thus demonstrating the direct link between the two post-quantum cryptographic families., Accepted for publication in Annali di Matematica Pura ed Applicata (1923 -)

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection with Jordan algebras., Comments are very welcome!

A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves connection tensors by a Galerkin method in the augmented subspace spanned by the new and old bases. This is followed by rank truncation within a specified error tolerance. The memory requirements are linear in the order of the tensor and linear in the maximal mode dimension. The integrator is robust to small singular values of matricizations of the connection tensors. Up to the rank truncation error, which is controlled by the given error tolerance, the integrator preserves norm and energy for Schro"\dinger equations, and it dissipates the energy in gradient systems. Numerical experiments with a basic quantum spin system illustrate the behavior of the proposed algorithm.

We consider a partial differential equation - ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model, the bulk flow of human drivers is represented by a scalar conservation law, while each autonomous vehicle is described by an ordinary differential equation. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. Next, we propose a control strategy for the speeds of the autonomous vehicles to minimize the average fuel consumption of the entire traffic flow. Existence of solutions for the optimal control problem is proved, and we numerically show that a reduction in average fuel consumption is possible with an AV acting as a moving bottleneck.

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual cone certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. As a result, rational WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. As an additional algorithmic application, we present an almost entirely numerical hybrid algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence., Comment: Preprint accepted in SIAM Journal on Optimization. Major revision compared to v1

The goal of this paper is to discover modern soliton solutions to long and short-wave interaction system by procedures called extended rational sine-cosine and rational sinh-cosh methods. We assume that the equation has a hypothetical soliton solutions. By reorganizing the resulting equations, we obtain a system of equations. Using Maple software, we get unknown coefficients in the system and writing them in the original equation, we obtain new solition solutions of the equation. The results show that the soliton solutions generated by the method for the long and short-wave interaction system are bright, kink type, bright periodic and dark solutions. We provided 3-D figures to illustrate the solutions. Computational results indicate that the method employed in this paper is superior than some other methods used in the literature to solve the same system equations.

The main objective of this article is to study the Wiener index of zero-divisor graph of the ring of integer modulo n . We present a constructed method to calculate the Wiener index of zero-divisor graph of Z n for any positive integer n .

Currently, DNA strand displacement is often used to build neural networks or solve logical problems. While there are few studies on the use of DNA strand displacement to solve the higher order equations. In this paper, the catalysis, degradation, annihilation and adjusted reaction modules are built through DNA strand displacement. The chemical reaction networks of the corresponding higher order equations and simultaneous equations are established through these modules, and these chemical reaction networks can be used to build analog circuits to solve binary primary simultaneous equations and binary quadratic simultaneous equations. Finally, through Visual DSD software verification, this design can realize the solution of binary primary simultaneous equations and binary quadratic simultaneous equations, which provides a reference for DNA computation in the future.

This paper deals with the extended-dissipativity-based adaptive event-triggered (AET) control problem for stochastic polynomial fuzzy singular systems (SPFSSs). The purpose of this work is to design a state-feedback controller through the AET mechanism,which not only makes the closed-loop system admissible and extended dissipative,but also saves the network resources. Firstly,with a suitable Lyapunov-Krasovskii functional and an integral inequality in stochastic setting,an AET control criterion of SPFSSs considering asynchronous premise is established to ensure the mean square admissibility and extended dissipativity of the close-loop singular system. Additionally,a simple lemma is employed to approximate the non-convex sum-of-squares (SOS) conditions with the convex SOS conditions due to the difficulty in solving the non-convex SOS design conditions. Then both the desired feedback controller gain and event-triggered weighting polynomial matrix are co-designed based on the derived criterion. Finally,simulation examples are provided to illustrate the effectiveness of the proposed approaches.

In this article, a numerical technique based on polynomials is proposed for the solution of one and two-dimensional time-fractional Burgers equation. First, the given problem is reduced to time discrete form using θ -weighted scheme. Then, with the help of Lucas and Fibonacci polynomials the given PDEs transformed to system of algebraic equations which is easy to solve. The proposed algorithm is validated by solving some numerical examples. Despite this, convergence analysis of the scheme is briefly discussed and verified numerically. The main objective of this paper is to show that polynomial based method is convenient for 1D and 2D nonlinear time-fractional partial differential equations (TFPDEs). Efficiency and performance of the proposed technique are examined by calculating L 2 and L ∞ error norms. Obtained accurate results confirm applicability and efficiency of the method.

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension of the Davis--Kahan method, takes advantage of the dual formulation of differential operators along with the Prager--Synge technique and provides greatly improved accuracy of the estimate, especially for the finite element approximations of eigenfunctions. Numerical examples of eigenvalue problems of matrices and the Laplace operators over convex and non-convex domains illustrate the efficiency of the proposed algorithms., 39 pages, 12 tables, 9 figures

This paper studies an infinite horizon optimal control problem for discrete-time linear systems and quadratic criteria, both with random parameters which are independent and identically distributed with respect to time. A classical approach is to solve an algebraic Riccati equation that involves mathematical expectations and requires certain statistical information of the parameters. In this paper, we propose an online iterative algorithm in the spirit of Q-learning for the situation where only one random sample of parameters emerges at each time step. The first theorem proves the equivalence of three properties: the convergence of the learning sequence, the well-posedness of the control problem, and the solvability of the algebraic Riccati equation. The second theorem shows that the adaptive feedback control in terms of the learning sequence stabilizes the system as long as the control problem is well-posed. Numerical examples are presented to illustrate our results., Comment: 24 pages, 3 figures

This paper presents a new sampled-data-based event-triggered pointwise security controller by pointwise measurements for partial differential equation (PDE) systems under stochastic cyber-attacks. First, according to the Takagi--Sugeno fuzzy model, the considered nonlinear system is reconstructed and an event-triggered pointwise fuzzy controller is proposed with pointwise measurements. Moreover, networked control systems (NCSs) are introduced to improve the transmission convenience; however, two deception cyber-attacks with different characteristics are brought into PDE systems for the first time. In addition, based on novel Lyapunov-Krasovskii functionals (LKFs), some relaxed conditions to assure system's exponential stability are established. Finally, the effectiveness and practicability of the designed controller are demonstrated by numerical and application examples.

We improve truncated differential attacks on t-branch contracting Feistel ciphers with a domain size of Nt. Based on new truncated differentials, a generic distinguisher for t2 + t − 2 rounds using O(Nt−1) data and time is obtained. In addition, we obtain a key-recovery attack on t2 + 1 rounds with Õ(Nt−2) data and Õ(Nt−1) time. Compared to previous results by Guo et al. (ToSC 2016), our attacks cover more rounds with a lower data-complexity. Applications of the generic truncated differential to concrete ciphers include full-round attacks on some instances of GMiMC-crf, and the best-known key-recovery attack on 17 rounds of the Chinese block cipher standard SM4. In addition, we propose an automated search method for truncated differentials using SMT, which is effective even for trails with probability below the probability of the truncated differential for a random permutation.

Singularly perturbed systems have found widespread applications in practice. The existing results on singularly perturbed systems mainly focused on the Lyapunov asymptotic stability, which are unable to deal with the cases that the system states cannot exceed a given threshold during a fixed time interval. This paper addresses the finite-time fuzzy control issue for discrete-time nonlinear singularly perturbed systems with input constraints. The aim is to guarantee the boundedness of the states of singularly perturbed systems during a finite-time interval. Based on the matrix inequality technique, some conditions are established to guarantee the finite-time boundedness of the fuzzy singularly perturbed systems, where the singularly perturbed parameter is independent so as to avoid the ill-conditioned problem caused by the small singularly perturbed parameter. The gains of the finite-time fuzzy controller can be obtained by solving some singularly perturbed parameter independent linear matrix inequalities. Finally, the proposed finite-time fuzzy controller design approach for nonlinear singularly perturbed systems is illustrated via the Van der Pol circuit.

This article presents an efficient method to solve of Eikonal equation by the new differential quadrature method. The purpose of this article is to obtain the accuracy and performance method for the Eikonal equation. We introduce a novel grid points to reduce errors and compared them on different issues to accelerate convergence. The accuracy and stability of numerical results for two and three-dimensional of the Eikonal equation are presented. In these numerical examples, we investigate the stability of the proposed method by adding random noises. The new differential quadrature method is a capable method for the some numerical examples and we evaluate the accuracy and performance of our proposed method.

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These algorithms extend the classical trace test in numerical algebraic geometry. Our results rely on both the analysis of the structure of sparse resultants as well as an extension of Esterov’s results on monodromy groups of sparse systems.

In this paper, we investigate the complexity of the central path of semidefinite optimization through the lens of real algebraic geometry. To that end, we propose an algorithm to compute real univariate representations describing the central path and its limit point, where the limit point is described by taking the limit of central solutions, as bounded points in the field of algebraic Puiseux series. As a result, we derive an upper bound $2^{O(m+n^2)}$ on the degree of the Zariski closure of the central path, when $\mu$ is sufficiently small, and for the complexity of describing the limit point, where $m$ and $n$ denote the number of affine constraints and size of the symmetric matrix, respectively. Furthermore, by the application of the quantifier elimination to the real univariate representations, we provide a lower bound $1/\gamma$, with $\gamma =2^{O(m+n^2)}$, on the convergence rate of the central path.

The Convex Hull Membership (CHM) tests whether \( p \in conv(S) \) , where p and the n points of S lie in \( \mathbb { R}^m \) . CHM finds applications in Linear Programming, Computational Geometry, and Machine Learning. The Triangle Algorithm (TA), previously developed, in \( O(1/\varepsilon ^2) \) iterations computes \( p^{\prime } \in conv(S) \) , either an \( \varepsilon \) - approximate solution , or a witness certifying \( p \not\in conv(S) \) . We first prove the equivalence of exact and approximate versions of CHM and Spherical -CHM, where \( p=0 \) and \( \Vert v\Vert =1 \) for each v in S . If for some \( M \ge 1 \) every non-witness with \( \Vert p^{\prime }\Vert \gt \varepsilon \) admits \( v \in S \) satisfying \( \Vert p^{\prime } - v\Vert \ge \sqrt {1+\varepsilon /M} \) , we prove the number of iterations improves to \( O(M/\varepsilon) \) and \( M \le 1/\varepsilon \) always holds. Equivalence of CHM and Spherical-CHM implies Minimum Enclosing Ball (MEB) algorithms can be modified to solve CHM. However, we prove \( (1+ \varepsilon) \) -approximation in MEB is \( \Omega (\sqrt {\varepsilon }) \) -approximation in Spherical-CHM. Thus, even \( O(1/\varepsilon) \) iteration MEB algorithms are not superior to Spherical-TA. Similar weakness is proved for MEB core sets. Spherical-TA also results a variant of the All Vertex Triangle Algorithm (AVTA) for computing all vertices of \( conv(S) \) . Substantial computations on distinct problems demonstrate that TA and Spherical-TA generally achieve superior efficiency over algorithms such as Frank–Wolfe, MEB, and LP-Solver.

In this paper, a new computationally efficient approach to solve fractional differential equations with Atangana–Baleanu operator is introduced. Controlled Picard’s method is employed for solving a class of fractional differential equations with order 0 α 1 . The proposed approach can cover wide range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for nonlinear differential equations. Two models of fractional Riccati equation are solved to validate and illustrate the accuracy of the new approach. Figures has been used to construct the results obtained from the presented approach. It is shown that the proposed method is efficient, credible, and easy to implement for various related problems in science and engineering.

The main target of our research is to achieve the solution to a variable-order fractional mobile-immobile advection-dispersion equation within the Atangana-Baleanu (AB) fractional operator. This equation is the best tool for modeling dynamical systems. The presented scheme is based on the collocation method and Lagrange polynomials. The given approach uses derivative operational matrices of integer and variable orders. The operational matrix of variable-order derivative in the AB sense is computed based on Lagrange polynomials in the presented study. The key theory of this approach is to convert the corresponding problem to a system of algebraic equations. The solution to the problem under study can be computed by utilizing the mentioned system and the collocation points. The aim of this paper is to show that this technique is easily used to solve fractional partial differential equations with excellent accuracy in results. To demonstrate the accuracy and efficiency of the suggested method, some numerical examples are provided.

A fundamental theory of deterministic linear-quadratic (LQ) control is the equivalent relationship between control problems, two-point boundary value problems and Riccati equations. In this paper, we extend the equivalence to a general time-inconsistent deterministic LQ problem, where the inconsistency arises from non-exponential discount functions. By studying the solvability of the Riccati equation, we show the existence and uniqueness of the linear equilibrium for the time-inconsistent LQ problem.

We describe the relations among the ℓ \ell -torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the ℓ \ell -torsion conjecture for ℓ \ell -groups and the other two conjectures for nilpotent groups.

In this paper, we introduce new kinds of graphs based on equality algebras. First of all, by using the meet operation we define the notion of zero divisors on equality algebra and study related properties. Then we introduce a meet graph on equality algebra by using zero divisors. In addition, we investigate some conditions for a meet graph to be a complete, connected and a star graph. Also, by using the notion of filters in equality algebras, we define an equivalence relation on equality algebras and then by using the equivalence classes we introduce two kinds of graphs on them. Finally, conditions for a graph based on these classes to be connected, or bipartite, or complete, or planar or an outer-planar graph are provided.

A finite difference algorithm based on the integral Laguerre transform in time for solving a three-dimensional one-way wave equation is proposed. This allows achieving high accuracy of calculation results. In contrast to the Fourier method, the approach does not need to solve systems of linear algebraic equations with indefinite matrices. To filter the unstable components of a wave field, Richardson extrapolation or spline approximation can be used. However, these methods impose additional limitations on the integration step in depth. This problem can be solved if the filtering is performed not in the direction of extrapolation of the wave field, but in a horizontal plane. This approach called for fast methods of converting the Laguerre series coefficients into the Fourier series coefficients and vice versa. The high stability of the new algorithm allows calculations with a large depth step without loss of accuracy and, in combination with Marchuk-Strang splitting, this can significantly reduce the calculation time. Computational experiments are performed. The results have shown that this algorithm is highly accurate and efficient in solving the problems of seismic migration.

In this article, based on polynomial differential inclusions, we propose a heuristic iterative approach for estimating the domains of attraction for nonpolynomial systems. First, we use the fuzzy model to construct a polynomial differential inclusion for the nonpolynomial system, which can be equivalently written as a time-invariant uncertain polynomial system. Then, beginning with an initial inner estimation, we present an iterative approach to enlarge this initial inner estimation by calculating common Lyapunov-like functions. Furthermore, the domains of attraction are estimated by combining this iterative approach with heuristic construction of differential inclusions. In the end, our heuristic iterative approach is implemented with linear semidefinite programming and then tested on some nonpolynomial examples with comparisons to the existing methods in the literature.

This article proposes a distributed optimization design to compute continuous-time algebraic Riccati inequalities (ARIs), where the information of matrices is distributed among agents. We propose a design procedure to tackle the nonlinearity, the inequality, and the coupled information structure of ARI; then, we design a distributed algorithm based on an optimization approach and analyze its convergence properties. The proposed algorithm is able to verify whether ARI is feasible in a distributed way and converges to a solution if ARI is feasible for any initial condition.

We propose an algorithm to compute global conformal parameterizations of high-genus meshes, which is based on an implementation of holomorphic quadratic differentials. First, we design a novel diffusion method which is capable of computing a pole-free discrete harmonic measured foliation. Second, we propose a definition for discrete holomorphic quadratic differential which consists of a horizontal and a vertical harmonic measured foliation. Third, we present a practical algorithm to approximate the discrete natural coordinates for a holomorphic quadratic differential, which represents a flat metric with cones conformal to the original metric, i.e., a parameterization. Finally, we apply the discrete natural coordinates for parameterization of high genus meshes. Our parameterization method is global conformal and simple to implement. The advantage of our method over the approach based on holomorphic differential one-forms is that ours has a larger space of parameterizations. We demonstrate our approach with hundreds of configurations on dozens of meshes to show its robustness on conformal parameterization.

Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use (Formula presented.) functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the (Formula presented.) function action over vectors for two-dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential-related matrix function that we express using Kronecker products of one-dimensional matrices. We exploit state-of-the-art implementations of (Formula presented.) functions to compute this auxiliary function's action and then recover the original (Formula presented.) action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation.

Model order reduction is a technique that is used to construct low-order approximations of large-scale dynamical systems. In this paper, we investigate a balancing based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. To this aim, we propose a new algebraic observability Gramian for the system based on Hilbert space adjoint theory. We then show the proposed Gramians satisfy a particular type of generalized Lyapunov equations and we investigate their connections to energy functionals, namely, the controllability and observability. This allows us to find the states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order systems. Finally, based on $\mathcal H_2$ energy considerations, we, furthermore, derive error bounds, depending on the neglected singular values. The efficiency of the proposed method is demonstrated by means of two semi-discretized partial differential equations and is compared with the existing model reduction techniques in the literature., 26 pages

In this paper we investigate non-linear matroid optimization problems with polynomial objective functions where the monomials satisfy certain monotonicity properties. Indeed, we study problems where the set of non-linear monomials consists of all non-linear monomials that can be built from a given subset of the variables. Linearizing all non-linear monomials we study the respective polytope. We present a complete description of this polytope. Apart from linearization constraints one needs appropriately strengthened rank inequalities. The separation problem for these inequalities reduces to a submodular function minimization problem. These polyhedral results give rise to a new hierarchy for the solution of matroid optimization problems with polynomial objectives. This hierarchy allows to strengthen the relaxations of arbitrary linearized combinatorial optimization problems with polynomial objective functions and matroidal substructures. Finally, we give suggestions for future work.

In this paper, fuzzy linear singular differential equations under so-called granular differentiability are investigated in which the coefficients and initial conditions are as fuzzy numbers. Some new notions such as fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors, rank, index and fuzzy Jordan canonical form of a fuzzy matrix are introduced. Moreover, we compare the proposed method with other ones based on other derivatives, using some examples.

In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular Quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed.

In this paper we obtain further improvement of index bounds for character sums of polynomials over finite fields. We present some examples, which show that our new bound is an improved bound compared to both the Weil bound and the index bound given by Wan and Wang. As an application, we count the number of all the solutions of some algebraic curves by using our result., Comment: 9 pages, to appear in DCC

This paper deals with estimation of interval controls for interval Linear Time Invariant (LTI) plants using pole placement technique. Generally, the parameters associated with LTI plant system are assumed to be crisp values, but due to the occurrence of error in experimentation, measurement or due to maintenance induced errors, the parameters are imprecise. As such, the LTI problem reduces to interval LTI. The pole placement problem is then reduced to interval algebraic equation viz. interval Diophantine equation using proper transfer function. Further, the interval Diophantine equation is transformed to Interval System of Linear Equations (ISLE) viz. interval Sylvester matrix equation. Initially, using interval arithmetic an interval algebraic approach has been applied to solve the ISLE for computing non-negative or non-positive controls. Then, another approach using Artificial Neural Network (ANN) is proposed for computing the interval controls. Further, algorithms based on sign function and ANN procedures have been discussed. Finally, the efficiency of the proposed algorithms has been verified using different order interval plants.

By employing the properties of centered realizations, we devise a modified version of the dual Riccati equation approach to optimal $\mathcal {H}_{2}$ control for differential-algebraic systems that is guaranteed to be more computationally efficient and numerically accurate than other methods from literature. Moreover, we show that if the optimal controller has an improper transfer function matrix, then all controllers which ensure a finite $\mathcal {H}_{2}$ norm will share this property.

We present a method to compute polynomial conservation laws for systems of partial differential equations ( PDE s). The method only relies on linear algebraic computations and is complete, in the sense it can find a basis for all polynomial fluxes that yield conservation laws, up to a specified order of derivatives and degree. We compare our method to state-of-the-art algorithms based on the direct approach on a few PDE systems drawn from mathematical physics.

The present work is to solve the nonlinear singular models using the framework of the stochastic computing approaches. The purpose of these investigations is not only focused to solve the singular models, but the solution of these models will be presented to the extended form of the delayed, prediction and pantograph differential models. The Gudermannian function is designed using the neural networks optimized through the global scheme "genetic algorithms (GA)", local method "sequential quadratic programming (SQP)" and the hybridization of GA-SQP. The comparison of the singular equations will be presented with the exact solutions along with the extended form of delayed, prediction and pantograph based on these singular models. Moreover, the neuron analysis will be provided to authenticate the efficiency and complexity of the designed approach. For the correctness and effectiveness of the proposed approach, the plots of absolute error will be drawn for the singular delayed, prediction and pantograph differential models. For the reliability and stability of the proposed method, the statistical performances "Theil inequality coefficient", "variance account for" and "mean absolute deviation'' are observed for multiple executions to solve singular delayed, prediction and pantograph differential models.

In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap(.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples.

Constrained solutions are required in some practical problems. This paper studies the problem of different constrained solutions for a class of multivariate quadratic matrix equations, which has rarely been studied before. First, the quadratic matrix equations are transformed into linear matrix equations by Newton's method. Second, the different constrained solutions of the linear matrix equations are solved by the modified conjugate gradient method, and then the different constrained solutions of the multivariate quadratic matrix equations are obtained. Finally, the convergence of the algorithm is proved. The algorithm can be well performed on the computers, and the effectiveness of the method is verified by numerical examples.

The synthesis of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs is studied. An approach based on the State-dependent Riccati Equation (SDRE) is presented for 2 and ∞ control problems. Depending on the nonlinearity and the dimension of the resulting problem, offline, online, and hybrid offline-online alternatives to the SDRE synthesis are proposed. The hybrid offline-online SDRE method reduces to the sequential solution of Lyapunov equations, effectively enabling the computation of suboptimal feedback controls for two-dimensional PDEs. Numerical tests for the Sine-Gordon, degenerate Zeldovich, and viscous Burgers’ PDEs are presented, providing a thorough experimental assessment of the proposed methodology.

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations. To fulfill the input requirement of Algorithm I, we develop a numerical GCD-finder containing a successive singular value updating and an iterative GCD refinement as the main engine of Algorithm II that calculates the multiplicity structure and the initial root approximation. While limitations of our algorithm exist in identifying the multiplicity structure in certain situations, the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are established. According to our computational experiments and error analysis, a polynomial being ill-conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved, and its multiple roots can be computed with high accuracy.

Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to other established matrix equation solvers, and investigate the applicability and performance in large-scale sparse examples., 19 pages, 2 figures, 5 tables

The solution of a system of simultaneous linear equations is a fundamental problem in numerical linear algebra and is a basic ingredient of many scientific simulations. Examples are scientific or engineering problems modeled by ordinary or partial differential equations. The numerical solution is often based on discretization methods leading to a system of linear equations.

Assessing non-negativity of multivariate polynomials over the reals, through the computation of {\em certificates of non-negativity}, is a topical issue in polynomial optimization. This is usually tackled through the computation of {\em sums-of-squares decompositions} which rely on efficient numerical solvers for semi-definite programming. This method faces two difficulties. The first one is that the certificates obtained this way are {\em approximate} and then non-exact. The second one is due to the fact that not all non-negative polynomials are sums-of-squares. In this paper, we build on previous works by Parrilo, Nie, Demmel and Sturmfels who introduced certificates of non-negativity modulo {\em gradient ideals}. We prove that, actually, such certificates can be obtained {\em exactly}, over the rationals if the polynomial under consideration has rational coefficients and we provide {\em exact} algorithms to compute them. We analyze the bit complexity of these algorithms and deduce bit size bounds of such certificates., Comment: 24 pages, 2 tables

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods, especially the augmentation method, successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle., Comment: A preliminary version of this paper is to appear in Proceedings of the 44th International Symposium on Symbolic and Algebraic Computation (ISSAC 2019), Beijing, China, July 2019

This paper investigates minimal solutions of fuzzy relation inequalities with addition-min composition. It first shows the conditions that an element is a minimal solution of the inequalities, and presents the conditions that the inequalities have a unique minimal solution. It then proves that every solution of the inequalities has a minimal one and proposes an algorithm to searching for a minimal solution with computational complexity O (n2) where n is the number of unknown variables of the inequalities. This paper finally describes all minimal solutions of the inequalities.

Algorithm 982: Explicit solutions of triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients provides an explicit solution for an homogeneous system, and a brief description of how to compute a solution for the inhomogeneous case. The method described is not directly useful if the coefficient matrix is singular. This remark explains more completely how to compute the solution for the inhomogeneous case and for the singular coefficient matrix case.

In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.