Polynomial Systems Research Articles

Input–output finite-time stabilisation for periodic piecewise polynomial systems with nonlinear actuator faults: an observer-based approach

This paper is concerned with the input–output finite-time (IO-FT) stabilisation problem for a class of continuous-time periodic piecewise polynomial systems (PPPSs) with immeasurable states and external disturbances via state estimation-based robust reliable controller. Firstly, to reflect the actuality, a parameter uncertainty that exhibits the random nature and the actuator faults with nonlinear character are considered in the addressed PPPSs and the control scheme, respectively. In detail, the randomness of uncertain parameters is portrayed by the stochastic variable and it is presumed to pursue the Bernoulli distributions. Secondly, to estimate the immeasurable states of PPPSs, a periodic piecewise polynomial observer is designed depending on the output of PPPSs. The main intent of this article is to devise a state estimation-based robust reliable controller to ascertain the IO-FT stabilisation of the PPPSs. Moreover, by bridging the Lyapunov stability theory, linear matrix inequality technique and IO-FT stability theory, the required IO-FT stabilisation conditions in the frame of linear matrix inequality are procured for the system under consideration. Eventually, simulation results of the addressed PPPSs are shown in line with the proposed analytical findings, revealing the competence, inherent capability and utility of the devised control protocol.

On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1, \dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1, \dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1, \dots,\kappa_\rho+\delta_\rho+1;x), $$ and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots, \delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1; \alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x), \qquad n\in\mathbb{Z}_+, $$ where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures. The importance of the orthogonality property was our main reason to concentrate our attention on polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$. The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties. We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

Integrability and dynamics of a simplified class B laser system.

A simplified class B laser system is a family of differential polynomial systems of degree two depending on the parameters a and b. Its rich dynamics has already been observed in 1980s, see Arecchi et al. [Opt. Commun. 51, 308-314 (1984)] and Politi et al. [Phys. Rev. A 33, 4055 (1986)], and still nowadays, it attracts the interest of the researchers. In this paper, we characterize its dynamics near infinity for all values of the parameters. When a=0, the partial integrability was already proved by Oppo and Politi [Z. Phys. B Con. Mat. 59, 111-115 (1985)]. Here, we prove that for a=0, it is completely integrable with two independent first integrals given by Liouvillian functions, and we present a complete study of its dynamics. When a≠0, we study its dynamics in the Poincaré ball B3, i.e., the interior of this ball is identified with R3 and its boundary the two-dimensional sphere S2 is identified with the infinity of R3.

New traveling wave solutions, phase portrait and chaotic pattern for the stochastic modified Korteweg–de Vries equation

This article mainly studies the new traveling wave solutions of the stochastic modified Korteweg–de Vries equation with multiplicative noise. The traveling wave solutions in the form of hyperbolic function, trigonometric function, rational function and Jacobi elliptic function are obtained by transforming the modified Korteweg–de Vries equation into an ordinary differential equation through traveling wave transformation, and combining it with the complete discriminant system of polynomials. This article fully applies visual analysis technology and provides graphical representations of the obtained solutions in multiple dimensions such as phase maps, sensitivity maps, 3D maps and 2D maps, providing an intuitive and convenient channel for people to further understand the physical characteristics of the solutions for the stochastic modified Korteweg–de Vries equation. This article provides a rich graphical analysis of the solutions to the stochastic modified Korteweg–de Vries equation from multiple perspectives, and provides a fast and efficient solution method for the exact solution of the equation. The research method in this paper can also be used to study nonlinear partial differential equation in other fields.

On functional observers for polynomial systems with nonlinear oscillations

AbstractIn this contribution we discuss the design of functional observers for polynomial systems. Our approach is based on a high gain design employing an embedded observer. The functional to be estimated is generated from the high gain observer's state. The restriction to polynomial systems allows the representation of the system's and the observer's dynamics as polynomial ideals. For the calculations we utilize methods from algebraic geometry. Our approach is illustrated on a chaotic system.

Multiplicative brownian motion stabilizes traveling wave solutions and dynamical behavior analysis of the stochastic Davey–Stewartson equations

This article investigates the traveling wave solutions of the stochastic Davey–Stewartson equations based on the theory of the complete discriminant system of quadratic polynomials. Firstly, the stochastic Davey–Stewartson equations are transformed into the nonlinear ordinary differential equations through traveling wave transformation. By using Maple software, some two-dimensional and three-dimensional phase portraits are drawn. Secondly, according to the classification of the roots of quadratic polynomial equation, new traveling wave solutions of the stochastic Davey–Stewartson equations are obtained by using the complete discriminant system, which include rational function solutions, triangle function solutions and the Jacobi elliptic function solutions. Finally, three-dimensional-surface plots and two-dimensional-shape plots of some obtained solutions are drawn by using Maple software.

Polynomial differential systems with hyperbolic limit cycles

One of the main problems of the qualitative theory of the planar differential equations is the study their limit cycles. In this paper given an algebraic curve of degree n we characterize the planar polynomial differential systems of degree greater or equal than n which admit the ovals of the algebraic curve as hyperbolic limit cycles.

A Novel Application of Polynomial Solvers in mmWave Analog Radio Beamforming

Beamforming is a signal processing technique where an array of antenna elements can be steered to transmit and receive radio signals in a specific direction. The usage of millimeter wave (mmWave) frequencies and multiple input multiple output (MIMO) beamforming are considered as the key innovations of 5 th Generation (5G) and beyond communication systems. The mmWave radio waves enable high capacity and directive communication, but suffer from many challenges such as rapid channel variation, blockage effects, atmospheric attenuations, etc. The technique initially performs beam alignment procedure, followed by data transfer in the aligned directions between the transmitter and the receiver [1]. Traditionally, beam alignment involves periodical and exhaustive beam sweeping at both transmitter and the receiver, which is a slow process causing extra communication overhead with MIMO and massive MIMO radio units. In applications such as beam tracking, angular velocity, beam steering etc. [2], beam alignment procedure is optimized by estimating the beam directions using first order polynomial approximations. Recent learning-based SOTA strategies [3] for fast mmWave beam alignment also require exploration over exhaustive beam pairs during the training procedure, causing overhead to learning strategies for higher antenna configurations. Therefore, our goal is to optimize the beam alignment cost functions e.g. , data rate, to reduce the beam sweeping overhead by applying polynomial approximations of its partial derivatives which can then be solved as a system of polynomial equations. Specifically, we aim to reduce the beam search space by estimating approximate beam directions using the polynomial solvers. Here, we assume both transmitter (TX) and receiver (RX) to be equipped with uniform linear array (ULA) configuration, each having only one degree of freedom (d.o.f.) with N t and N r antennas, respectively.

On the polynomial integrability of the critical systems for optimal eigenvalue gaps

This exploration consists of two parts. First, we will deduce a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent p ∈ (1, ∞) of the Lebesgue spaces concerned. These systems can be used to obtain the optimal lower or upper bounds for eigenvalue gaps of Sturm–Liouville operators and are equivalent to non-convex Hamiltonian systems of two degrees of freedom. Second, with appropriate choices of exponents p, the critical systems are polynomial systems in four dimensions. These systems will be investigated from two aspects. The first one is that by applying the canonical transformation and the Darboux polynomial, we obtain the necessary and sufficient conditions for polynomial integrability of these polynomial critical systems. As a special example, we conclude that the system with p = 2 is polynomial completely integrable in the sense of Liouville. The second is that the linear stability of isolated singular points is characterized. By performing the Poincaré cross section technique, we observe that the systems have very rich dynamical behaviors, including periodic trajectories, quasi-periodic trajectories, and chaos.

Introduction to Algebraic Geometry

Summary A classical algebraic geometry is study of zero points of system of multivariate polynomials [3], [7] and those zero points would be corresponding to points, lines, curves, surfaces in an affine space. In this article we give some basic definition of the area of affine algebraic geometry such as algebraic set, ideal of set of points, and those properties according to [4] in the Mizar system[5], [2]. We treat an affine space as the n-fold Cartesian product k n as the same manner appeared in [4]. Points in this space are identified as n-tuples of elements from the set k. The formalization of points, which are n-tuples of numbers, is described in terms of a mapping from n to k, where the domain n corresponds to the set n = {0, 1, . . ., n − 1}, and the target domain k is the same as the scalar ring or field of polynomials. The same approach has been applied when evaluating multivariate polynomials using n-tuples of numbers [10]. This formalization aims at providing basic notions of the field which enable to formalize geometric objects such as algebraic curves which is used e.g. in coding theory [11] as well as further formalization of the fields [8] in the Mizar system, including the theory of polynomials [6].

A sufficient condition for the super-linearization of polynomial systems

A sufficient condition for the super-linearization of polynomial systems

A message recovery attack on multivariate polynomial trapdoor function.

Cybersecurity guarantees the exchange of information through a public channel in a secure way. That is the data must be protected from unauthorized parties and transmitted to the intended parties with confidentiality and integrity. In this work, we mount an attack on a cryptosystem based on multivariate polynomial trapdoor function over the field of rational numbers Q. The developers claim that the security of their proposed scheme depends on the fact that a polynomial system consisting of 2n (where n is a natural number) equations and 3n unknowns constructed by using quasigroup string transformations, has infinitely many solutions and finding exact solution is not possible. We explain that the proposed trapdoor function is vulnerable to a Gröbner basis attack. Selected polynomials in the corresponding Gröbner basis can be used to recover the plaintext against a given ciphertext without the knowledge of the secret key.

Cohomology Rings of Toric Bundles and the Ring of Conditions

The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}$$\\end{document} can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley–Reisner algebras. In Sankaran and Uma (Comment Math Helv 78(3):540–554, 2003), Sankaran and Uma generalized the “Stanley–Reisner description” to the case of toric bundles, i.e., equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion–Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety G/B. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties G/B previously obtained by Kaveh (J Lie Theory 21(2):263–283, 2011).

On computing modified moments for half-range Hermite weights

AbstractIn this paper, we consider the computation of the modified moments for the system of Laguerre polynomials on the real semiaxis with the Hermite weight. These moments can be used for the computation of integrals with the Hermite weight on the real semiaxis via product rules. We propose a new computational method based on the construction of the null-space of a rectangular matrix derived from the three-term recurrence relation of the system of orthonormal Laguerre polynomials. It is shown that the proposed algorithm computes the modified moments with high relative accuracy and linear complexity. Numerical examples illustrate the effectiveness of the proposed method.

Time-Reversibility and Ivariants of Some 3-dim Systems

We study time-reversibility and invariants of the group of transformations $x\to x, \ y\to \alpha y, \ z \to \alpha ^{-1}z$ for three-dimensional polynomial systems with $0:1:-1$ resonant singular point at the origin. An algorithm to find the Zariski closure of the set of time-reversible systems in the space of parameters is proposed. The interconnection of time-reversibility and invariants of the group mentioned above is discussed.

Gröbner Basis Approach for Solving Fuzzy Complex System of Linear Equations

The foundation of Gröbner’s work is a potent approach for separating polynomial systems of equations. In this study, we solve complex fuzzy systems of linear equations utilizing Gröbner’s basic technique. First, a system of explicit polynomial equations is generated by applying fuzzy computations and fuzzy complex computations to the fundamental system. Taking into account the lexicographical order, Gröbner’s basis is then determined for the ideal given by a definite equivalent polynomial system. The Gröbnerbase that was obtained has a high triangular structure. Thus, the system solution may be readily implemented by proceeding. Hence, all of the system’s answers are readily attainable. Next, a criterion is established to identify whether or not a solution exists for the primary system. The advantage of the proposed approach is that it acquires all of the problem’s solutions simultaneously. Using this technique, a mathematical example and an application issue based on an electrical circuit are resolved. Comparing the generated findings to the known outcomes reveals a high level of congruence.

Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems

Recently Chen and Gao \cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over C, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over C by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables.Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over Fq improving on subsequent work by Chen, Gao and Yuan \citeChenGao2018. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.

Robust observer-based tracking control for polynomial uncertain systems via a homogeneous Lyapunov approach

In this paper, an observer-based tracking control problem is considered for a polynomial system with parameter uncertainties and external disturbances. Noting that some states are difficult to measure in practice, a polynomial state observer is employed to obtain the unknown states. Then a robust observer-based tracking controller is designed to force the states of the polynomial uncertain system to follow the ones of the reference model and satisfy H ∞ performance. By using a full-state-dependent homogeneous polynomial function, the solvability condition is derived for the observer-based tracking control strategy, which reduces conservatism. Sum of squares technique is utilised to calculate the corresponding observer and controller, effectively dealing with the calculation problem in polynomial systems. Simulation examples are given to illustrate the effectiveness of the presented control strategy.

Pattern-Multiplicative Average of Nonnegative Matrices Revisited: Eigenvalue Approximation Is the Best of Versatile Optimization Tools

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ1(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots over 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero.

Forecasting Value-at-Risk using functional volatility incorporating an exogenous effect

This paper proposes a novel extension of log and exponential GARCH models, where time-varying parameters are approximated by orthogonal polynomial systems. These expansions enable us to add and study the effects of market-wide and external international shocks on the volatility forecasts and provide a flexible mechanism to capture various dynamics of the parameters. We examine the performance of the new model in both theoretical and empirical analysis. We investigate the asymptotic properties of the quasi-maximum likelihood estimators under mild conditions. The small-sample behavior of the estimators is studied via Monte Carlo simulation. The performance of the proposed models, in terms of accuracy of both volatility estimation and Value-at-Risk forecasts, is assessed in an empirical study of a set of major stock market indices. The results support the proposed specifications with respect to the corresponding constant-parameters models and to other time-varying parameter models.