Set Of Polynomials Research Articles

Cotangent spaces and separating re-embeddings

Given an affine algebra [Formula: see text], where [Formula: see text] is a polynomial ring over a field [Formula: see text] and [Formula: see text] is an ideal in [Formula: see text], we study re-embeddings of the affine scheme [Formula: see text], i.e. presentations [Formula: see text] such that [Formula: see text] is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials [Formula: see text] in the ideal [Formula: see text] which are coherently separating in the sense that they are of the form [Formula: see text] with an indeterminate [Formula: see text] which divides neither a term in the support of [Formula: see text] nor in the support of [Formula: see text] for [Formula: see text]. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of [Formula: see text]. The dimension of the cotangent space of [Formula: see text] at a [Formula: see text]-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of [Formula: see text] and found an optimal re-embedding.

Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian

In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension N≥3. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.

Dynamic modeling and orbit maneuvering response analysis for a three-axis attitude stabilized large scale flexible spacecraft installed with hinged solar arrays

An analytical dynamic model for a three-axis attitude stabilized large scale flexible spacecraft (LSFS) installed with hinged solar arrays is established. The natural property of the LSFS is investigated by simplifying it as a rigid central body hinged with two sets of multi-panel structures in this paper. The Gram-Schmidt process is used to construct a set of characteristic orthogonal polynomials as the displacement field of the solar panel. Lagrange multipliers are introduced to describe the constraints at hinges. For the model of LSFS, the Rayleigh-Ritz method is employed to obtain analytical global modes which are explicitly functions of space coordinates and effective for the modeling of lower-order discrete dynamic equations and design of active vibration controllers for the system. Moreover, the method is applied to investigate the parametric influence on natural characteristics for the LSFS. The paper develops a novel method to obtain rigid-flexible coupling global modes and construct the low-dimensional dynamical model for LSFS and study the rigid-flexible coupling dynamic response of in-orbit LSFS with complicated external excitation.

On Properties of a Regular Simplex Inscribed into a Ball

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$.

The solvability cardinal of the class of polynomials

Let G be an Abelian group, and let {{mathbb {C}}}^G denote the set of complex valued functions defined on G. A map D: {{mathbb {C}}}^G rightarrow {{mathbb {C}}}^G is a difference operator, if there are complex numbers a_i and elements b_i in G(i=1,ldots , n) such that (Df)(x)=sum _{i=1}^n a_i f(x+b_i) for every fin {{mathbb {C}}}^G and xin G. By a system of difference equations we mean a set of equations { D_i f=g_i : iin I}, where I is an arbitrary set of indices, D_i is a difference operator and g_i in {{mathbb {C}}}^G is a given function for every iin I, and f is the unknown function. The solvability cardinal mathrm{sc} ,({{mathcal {F}}}) of a class of functions {{mathcal {F}}} subset {{mathbb {C}}}^G is the smallest cardinal number kappa with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality <kappa has a solution in {{mathcal {F}}}, then S itself has a solution in {{mathcal {F}}}. The behaviour of mathrm{sc} ,({{mathcal {F}}}) is rather erratic, even for classes of functions defined on {{mathbb {R}}}. For example, mathrm{sc} ,({{mathbb {C}}}[x])=3, but mathrm{sc} ,({mathcal {TP}}) =omega _1, where {mathcal {TP}} is the set of trigonometric polynomials; mathrm{sc} ,({{mathbb {C}}}^{{mathbb {R}}})=omega , but mathrm{sc} ,({mathcal {DF}}) =(2^omega )^+, where {mathcal {DF}} is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on {{{mathbb {R}}}}^n, on normed linear spaces and, in general, on topological Abelian groups. Let {{mathcal {P}}}_G denote the class of polynomials defined on the group G. After presenting some general estimates we prove that mathrm{sc} ,({{mathbb {C}}}[x_1 ,ldots ,x_n ])=omega if 2le n<infty , and mathrm{sc} ,({{mathcal {P}}}_X)=omega _1 if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that mathrm{sc} ,({{mathcal {P}}}_G)=3 if r_0 (G)le 1, mathrm{sc} ,({{mathcal {P}}}_G)=omega if 2le r_0 (G)<infty , and mathrm{sc} ,({{mathcal {P}}}_G)ge omega _1 if r_0 (G) is infinite, where r_0 (G) denotes the torsion free rank of G. The solvability of systems of difference equations is closely connected to the existence of projections of function classes commuting with translations (see Theorem 7.1). As an application we construct a projection from {{mathbb {C}}}^{{{{mathbb {R}}}}^n} onto {{mathbb {C}}}[x_1 ,ldots ,x_n ] commuting with translations by vectors having rational coordinates (Theorem 7.4).

Parameterization of robust multi-objective PID-based automatic voltage regulators: Generalized Hurwitz approach

This paper presents robust design of multi-objective PID controllers for automatic voltage regulators (AVRs). The set of stabilizing PID controllers is computed analytically using Routh-Hurwitz criterion (RHC). Minimal damping factors and coefficients are guaranteed by pole-placement in convex regions located in the open left half plane (LHP) and bounded by the damping isoclines. Optimal α-pole placement PID is derived where all gains are computed explicitly. Further, frequency domain specifications specifically gain and phase margins (GM, PM) are considered. Enforcing pole-clustering, GM and PM requirements are reduced to solve Hurwitz stability of either a real or complex characteristic polynomial which are tackled by generalizing RHC. Control basins (CBs) realizing any design objective other than stability are traced in kp-ki parameter plane, and hence different design objectives can be met simultaneously if their corresponding basins intersect. Model parametric uncertainties are captured by a set of principle polynomials derived by Polyak’s corollary to guarantee robust stability and performance simultaneously. Computation and comparative simulation results are given to confirm the effectiveness and simplicity of the proposed design approach.

On inherited properties and Julia sets of exceptional Jacobi polynomials

Some inherited properties of exceptional Jacobi polynomials are derived and as an application it is shown that similarly to the standard case, the equilibrium measure of Julia sets of exceptional Jacobi polynomials tends to the equilibrium measure of the interval of orthogonality in weak-star sense.

Solving nonlinear differential equations with differentiable quantum circuits

We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle.

Sets of range uniqueness for multivariate polynomials and linear functions with rank k

Let be a set of functions with a common domain X and a common range Y. A set is called a set of range uniqueness (SRU) for , if for all , Let be the set of all real polynomials in n variables of degree at most k and let be the set of all linear functions with rank k. We show that there are SRU's for of cardinality , but there are no such SRU's of size or less. Moreover, we show that there are SRU's for of size but there are no such SRU's of smaller size.

Inversion and Symmetries of the Star Transform

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along “star-shaped” trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

Sobolev Orthogonal Polynomials on the Sierpinski Gasket

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ( $$SG$$ ), which is a fractal set that can be viewed as a limit of a sequence of finite graphs. These orthogonal polynomials arise through the Gram–Schmidt orthogonalisation process applied on the set of monomials on $$SG$$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $$L^2$$ , $$L^\infty $$ , and Sobolev norms, and study their asymptotic behavior. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.

Pseudo-Lucas Functions of Fractional Degree and Applications

In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case.

Eigenvalue paths arising from matrix paths

It is known (see e.g. [2], [4], [5], [6]) that continuous variations in the entries of a complex square matrix induce continuous variations in its eigenvalues. If such a variation arises from one real parameter α∈[0,1], then the eigenvalues follow continuous paths in the complex plane as α shifts from 0 to 1. The intent here is to study the nature of these eigenpaths, including their behavior under small perturbations of the matrix variations, as well as the resulting eigenpairings of the matrices that occur at α=0 and α=1. We also give analogs of our results in the setting of monic polynomials.

A three-dimensional phase-field lattice Boltzmann method for incompressible two-components flows

In this paper, a lattice Boltzmann model for the coupled Allen–Cahn–Navier–Stokes equations in three dimensions is presented. Two equations are solved: one for the fluid velocity and one for the order parameter. Both are written within the general multiple-relaxation-time framework, where all the equilibrium and forcing terms are described by using the full set of Hermite polynomials. The resultant practical implementation is compact. The gradient of the order parameter can be computed by the non-local finite differences or the local central moments. The latter suffers from grid-scale oscillations. The very good accuracy properties are demonstrated against nine well-consolidated benchmark tests. Specifically, two groups of tests are tackled. In the former, the velocity field is superimposed. Hence, only the equation for the evolution of the order parameter is solved. These numerical experiments demonstrate the ability of the proposed scheme to capture the correct evolution of the interface. In the latter, two immiscible fluids are considered and the two equations are solved. Simulations of the vertical penetration of a wedge-shaped body, two- and three-dimensional Rayleigh–Taylor instability prove that two-fluids systems can be successfully simulated by our approach.

Low-Degree Approximation of Random Polynomials

We prove that with “high probability” a random Kostlan polynomial in n+1 many variables and of degree d can be approximated by a polynomial of “low degree” without changing the topology of its zero set on the sphere mathbb {S}^n. The dependence between the “low degree” of the approximation and the “high probability” is quantitative: for example, with overwhelming probability, the zero set of a Kostlan polynomial of degree d is isotopic to the zero set of a polynomial of degree O(sqrt{d log d}). The proof is based on a probabilistic study of the size of C^1-stable neighborhoods of Kostlan polynomials. As a corollary, we prove that certain topological types (e.g., curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials.

APPROXIMATE SOLUTIONS FOR THE KOLMOGOROV-WIENER FILTER WEIGHT FUNCTION FOR CONTINUOUS FRACTIONAL GAUSSIAN NOISE

Context. We consider the Kolmogorov-Wiener filter for forecasting of telecommunication traffic in the framework of a continuous fractional Gaussian noise model.&#x0D; Objective. The aim of the work is to obtain the filter weight function as an approximate solution of the corresponding WienerHopf integral equation. Also the aim of the work is to show the convergence of the proposed method of solution of the corresponding equation.&#x0D; Method. The Wiener-Hopf integral equation for the filter weight function is a Fredholm integral equation of the first kind. We use the truncated polynomial expansion method in order to obtain an approximate solution of the corresponding equation. A set of Chebyshev polynomials of the first kind is used.&#x0D; Results. We obtained approximate solutions for the Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise. The solutions are obtained in the approximations of different number of polynomials; the results are obtained up to the nineteen-polynomial approximation. It is shown that the proposed method is convergent for the problem under consideration, i.e. the accuracy of the coincidence of the left-hand and right-hand sides of the integral equation increases with the number of polynomials. Such convergence takes place due to the fact that the correlation function of continuous fractional Gaussian noise, which is the kernel of the corresponding integral equation, is a positively-defined function.&#x0D; Conclusions. The Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise is obtained as an approximate solution of the corresponding Fredholm integral equation of the first kind. The proposed truncated polynomial expansion method is convergent for the problem under consideration. As is known, one of the simplest telecommunication traffic models is the model of continuous fractional Gaussian noise, so the results of the paper may be useful for telecommunication traffic forecast.

A Monoparametric Family of Piecewise Linear Systems to Generate Scroll Attractors via Path-Connected Set of Polynomials

In this work, we present a monoparametric family of piecewise linear systems to generate multiscroll attractors through a polynomial family defined by path curves that connect to the roots. The idea is to define path curves where the roots of a polynomial can take values by determining an initial and a final polynomial. As a consequence, structural stability and bifurcation of the system can be obtained. Structural stability is obtained by preserving the same stability of the initial and final polynomials. However, the system bifurcates by changing the stability of the final polynomial with respect to the initial polynomial. The aim is achieved by the design of a piecewise linear controller that is applied to affine linear systems. Our results are mathematically proved and numerical examples are also provided to illustrate the approach.

Ideals of the Multiview Variety.

The multiview variety of an arrangement of cameras is the Zariski closure of the images of world points in the cameras. The prime vanishing ideal of this complex projective variety is called the multiview ideal. We show that the bifocal and trifocal polynomials from the cameras generate the multiview ideal when the foci are distinct. In the computer vision literature, many sets of (determinantal) polynomials have been proposed to describe the multiview variety. We establish precise algebraic relationships between the multiview ideal and these various ideals. When the camera foci are noncoplanar, we prove that the ideal of bifocal polynomials saturate to give the multiview ideal. Finally, we prove that all the ideals we consider coincide when dehomogenized, to cut out the space of finite images.

Total disconnectedness of Julia sets of random quadratic polynomials

AbstractFor a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

Reducing conservatism in robust stability analysis of fractional-order-polytopic systems

This paper studies the robust stability of the fractional-order (FO) LTI systems with polytopic uncertainty. Generally, the characteristic polynomial of the system dynamic matrix is not an affine function of the uncertain parameters. Consequently, the robust stability of the uncertain system cannot be evaluated by well-known approaches including LMIs or exposed edges theorem. Here, an over-parameterization technique is developed to convert the main characteristic polynomial into a set of local over-parameterized characteristic polynomials (LOPCPs). It is proved that the robust stability of LOPCPs implies the robust stability of the uncertain system. Then, an algorithm is proposed to explore system’s robust stability through investigating the robust stability of these LOPCPs based on the exposed edges idea. For the sake of feasibility comparison, extensive examples are elaborated that reveal the superiority of the proposed algorithm.