Set Of Polynomials Research Articles

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

Shape-Constrained Regression Using Sum of Squares Polynomials

Shape-Constrained Regression Using Algebraic Techniques How can one fit a multivariate polynomial to data points in such a way that this polynomial is guaranteed to have certain shape constraints, such as convexity or monotonicity in some of its variables? In “Shape-constrained regression using sum-of-squares polynomials,” M. Curmei and G. Hall propose a hierarchy of semidefinite programs to address this problem. They show that polynomial functions that are optimal to any fixed level of our hierarchy form a consistent estimator of the underlying shape-constrained function, which generates the data points. As a by-product of the proof, they establish that sum-of-squares-convex polynomials are dense in the set of polynomials that are convex over an arbitrary box. They further demonstrate the performance of the regressor for the problem of computing optimal transport maps in a color transfer task and that of estimating the optimal value function of a conic program. A real-time application of the latter problem to inventory management contract negotiation is presented.

On the Ding and Helleseth's 7th open problem about optimal ternary cyclic codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. The first two and the 6th problems were completely solved, and the third and last three problems were partially solved. In this paper, we focus on the 7th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer and then present a counterexample. Furthermore, we construct two new classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound by studying some special polynomials over finite fields.

Clustering in the Lazard method for Cylindrical Algebraic Decomposition

Cylindrical Algebraic Decomposition (CAD) is an algorithm originally proposed by Collins in [4]. Given a set of multivariate polynomials, CAD decomposes the multidimensional real space into connected subsets called cells, within which those polynomials are sign-invariant.

Conditional GrÖbner Basis: GrÖbner Basis Detection with Parameters

For a given finite polynomial set, finding a monomial order such that the given set is already a GrÖbner basis for the ideal generated by the given set with respect to the found monomial order is called GrÖbner basis detection (GBD) problem and there is also its simpler version, called structural GrÖbner basis detection (SGBD) problem. In this short communication, we give algorithms to solve these problems for polynomials with parameters on their coefficients.

Uniqueness theorems for weighted harmonic functions in the upper half-plane

We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin.

Eighth-Kind Chebyshev Polynomials Collocation Algorithm for the Nonlinear Time-Fractional Generalized Kawahara Equation

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method’s applicability and precision are demonstrated.

A study on fractional tumor-immune interaction model related to lung cancer via generalized Laguerre polynomials

BackgroundCancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies.MethodsIn this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data.ResultsWe propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results.ConclusionThe findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems.

Dynamical cancellation of polynomials

AbstractExtending the work of Bell, Matsuzawa and Satriano, we consider a finite set of polynomials over a number field and give a necessary and sufficient condition for the existence of an and a finite set such that for any , we have the cancellation result: If and are maps in such that , then in fact . Moreover, the conditions we give for this cancellation result to hold can be checked by a finite number of computations from the given set of polynomials.

Intersecting families of graphs of functions over a finite field

Let U be a set of polynomials of degree at most k over Fq, the finite field of q elements. Assume that U is an intersecting family, that is, the graphs of any two of the polynomials in U share a common point. Adriaensen proved that the size of U is at most qk with equality if and only if U is the set of all polynomials of degree at most k passing through a common point. In this manuscript, using a different, polynomial approach, we prove a stability version of this result, that is, the same conclusion holds if |U| > qk − qk − 1. We prove a stronger result when k = 2. For our purposes, we also prove the following results. If the set of directions determined by the graph of f is contained in an additive subgroup of Fq, then the graph of f is a line. If the set of directions determined by at least q - sqrt{q}/2 affine points is contained in the set of squares/non-squares plus the common point of either the vertical or the horizontal lines, then up to an affinity the point set is contained in the graph of some polynomial of the form αxpk.

Caputo’s fractional discrete-time stability connection for stabilizing controllers

Abstract This work aims to give a method to connect a set of polynomials having all of their zeros inside the stability zone for fractional difference systems with Caputo’s fractional discrete operator. Due to the complexity of the stability zone, it is necessary to use a set that describes explicitly the stability zone for fractional-order difference systems, in order to build a polynomial family with zeros belonging to the described zone. Such a construction of the polynomial family will be based on the connection of their zeros. Moreover, the applicability is shown with the design of a robust stabilizing controller, which is illustrated by stabilizing the fractional discrete Duffing oscillator.

On the zeros of certain polynomials and entire functions

Entire functions whose coefficients are polynomials having real and negative roots are built from Touchard polynomials. The particular set of polynomials is shown to have purely complex roots, where we show a connection of these polynomials with certain approximations of the Riemann’s zeta function. Also a certain class of Fourier transforms is shown to have only real roots.

Fractal transformations of recursive walks in the Julia fern

One of the most studied fractals corresponds to the Julia Set and one of its elements can be obtained with the following recursive formula in the complex plane: Zi+1 = Zi2 - 1, the result depends on the chosen starting complex number. We can define the Julia set of a complex variable polynomial as the boundary of the set of points that escape to infinity when iterating this polynomial. This means that the orbit of an element of the Julia set does not escape infinity.

Hungry Lotka–Volterra lattice under nonzero boundaries, block‐Hankel determinant solution, and biorthogonal polynomials

AbstractIn this paper, we first extend the hungry Lotka–Volterra lattice to a case of nonzero boundary conditions and present its corresponding exact solution expressed in terms of a block‐Hankel determinant. Then, we establish a connection between this hungry Lotka–Volterra lattice under nonzero boundary conditions and a set of biorthogonal polynomials. It turns out that the hungry Lotka–Volterra lattice under nonzero boundary conditions possesses a Lax pair expressed in terms of the biorthogonal polynomials. Moreover, we consider two special cases of the hungry Lotka–Volterra lattice. For the case , it reduces to the Lotka–Volterra lattice under nonzero boundary condition, which has been discussed in literature. We also present the result for in detail, which extends a known result to a case of nonzero boundary functions. All these results are obtained by virtue of Hirota's bilinear method and determinant techniques.

Numerical method for fractional Advection–Dispersion equation using shifted Vieta–Lucas polynomials

In the pursuit of creating more precise and flexible mathematical models for complex physical phenomena, this study constructs a unique fractional model for the Advection–Dispersion equation. The Advection–Dispersion equation is a fundamental mathematical tool for analyzing fluid dynamics and mass transfer processes, but traditional integer-order models often fail to accurately capture anomalous transport behaviors. To overcome this limitation, we employ a non-singular non-integer operator based on the Heydari–Hosseininia notion. This operator allows us to transform the classical advection–dispersion equation into a more robust and flexible fractional model, which can better represent a variety of transport phenomena across different scales and mediums.The numerical approximation of the proposed fractional Advection–Dispersion equation model is achieved using a set of specific polynomials known as shifted Vieta–Lucas polynomials. Aided by their unique properties, the shifted Vieta–Lucas polynomials enable us to transform the original differential system into a more computationally tractable algebraic system via a non-integer derivative matrix. Furthermore, we undertake a detailed analysis of convergence and truncation error associated with the shifted Vieta–Lucas polynomials, underpinning the validity and stability of our numerical method. To illustrate the efficiency and robustness of the derived method, we provide several example problems which are solved using our method. The corresponding solutions are extensively presented with the aid of figures and tables, demonstrating the method’s remarkable performance in solving the fractional Advection–Dispersion equation model. The proposed method shows promising potential for further application and development in solving other fractional differential equations and related scientific problems.

Recurrence and Eigenfunction Methods for Non-Trivial Models of Discrete Binary Choice.

Understanding how systems relax to equilibrium is a core theme of statistical physics, especially in economics, where systems are known to be subject to extrinsic noise not included in simple agent-based models. In models of binary choice-ones not much more complicated than Kirman's model of ant recruitment-such relaxation dynamics become difficult to determine analytically and require solving a three-term recurrence relation in the eigendecomposition of the stochastic process. In this paper, we derive a concise closed-form solution to this linear three-term recurrence relation. Its solution has traditionally relied on cumbersome continued fractions, and we instead employ a linear algebraic approach that leverages the properties of lower-triangular and tridiagonal matrices to express the terms in the recurrence relation using a finite set of orthogonal polynomials. We pay special attention to the power series coefficients of Heun functions, which are also important in fields such as quantum mechanics and general relativity, as well as the binary choice models studied here. We then apply the solution to find equations describing the relaxation to steady-state behavior in social choice models through eigendecomposition. This application showcases the potential of our solution as an off-the-shelf solution to the recurrence that has not previously been reported, allowing for the easy identification of the eigenspectra of one-dimensional, one-step, continuous-time Markov processes.

A phase-field Lattice Boltzmann method for liquid-vapor phase change problems based on conservative Allen-Cahn equation and adaptive treegrid

A phase-field Lattice Boltzmann method (LBM) based on Allen-Cahn (A-C) equation with a modified adaptive mesh refinement (AMR) method is presented to simulate the liquid-vapor phase change problem. Three Lattice Boltzmann equations are solved: one for the conservative Allen-Cahn equation, one for the incompressible Navier-Stokes equations, and one for the energy equation. These three equations are written in the form of multiple-relaxation-time, together with the equilibrium and forcing terms using the set of Hermite polynomials. The adaptive mesh refinement technique with quadtree grid data structure is implemented to capture the liquid-vapor interface in simulations, in which the mesh can be locally refined by subdividing blocks identified by user-defined criterion. The Lax-Wendroff finite-difference scheme is adopted to calculate the whole computational domain continuously under nonuniform grids. The present method is validated by the classical Stefan problem and then performed to simulate the film boiling heat transfer. The numerical experiments verify the ability of the proposed method to capture the correct evolution of the interface, guarantee the numerical accuracy and remarkably save the computational cost.

The separating variety for matrix semi-invariants

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V, and let k[V]G be the corresponding algebra of invariant polynomial functions. A separating set S⊆k[V]G is a set of polynomials with the property that for all v,w∈V, if there exists f∈k[V]G separating v and w, then there exists f∈S separating v and w.In this article we consider the action of G=SL2(C)×SL2(C) on the C-vector space M2,2n of n-tuples of 2×2 matrices by multiplication on the left and the right. Minimal generating sets Sn of C[M2,2n]G are known, and |Sn|=124(n4−6n3+23n2+6n). In recent work, Domokos [8] showed that for all n≥1, Sn is a minimal separating set by inclusion, i.e. that no proper subset of Sn is a separating set. This does not necessarily mean that Sn has minimum cardinality among all separating sets for C[M2,2n]G. Our main result shows that any separating set for C[M2,2n]G has cardinality ≥5n−9. In particular, there is no separating set of size dim⁡(C[M2n]G)=4n−6 for n≥4. Further, S4 has indeed minimum cardinality as a separating set, but for n≥5 there may exist a smaller separating set than Sn. We also consider the action of G=SLl(C) on Ml,n by left multiplication. In that case the algebra of invariants has a minimum generating set of size (nl) (the l×l minors of a generic matrix) and dimension ln−l2+1. We show that a separating set for C[Ml,n]G must have size at least (2l−2)n−2(l2−l). In particular, C[Ml,n]G does not contain a separating set of size dim⁡(C[Ml,n]G) for l≥3 and n≥l+2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem.

Monomiality and a New Family of Hermite Polynomials

The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms.

On critical renormalization of complex polynomials

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization.