Polynomial Extensions Research Articles

Propagation of traveling wave solution of the strain wave equation in microcrystalline materials

Abstract This study focuses on the propagation behavior of traveling wave solution in microcrystalline materials using the polynomial complete discriminant system method. By establishing a complete discriminant system, we systematically analyze the formation and evolution process of traveling wave solution in microcrystalline materials. Specifically, we apply the cubic polynomial extension to the strain wave equation to obtain more accurate analytical solutions. Additionally, two-dimensional, three-dimensional, and contour plots are generated to visually illustrate the characteristics of the obtained solutions, facilitating a more intuitive understanding of their physical significance. These findings not only help reveal the propagation mechanism of traveling wave solution but also provide a theoretical foundation for the application of microcrystalline materials.

On the ascent of atomicity to monoid algebras

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid M and an integral domain R are both atomic implies that the monoid algebra R[M] of M over R is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.

Optimization-aided construction of multivariate Chebyshev polynomials

This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all mononials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial x12x22x3 on the euclidean ball and for the monomial x12x2x3 on the simplex.

Several Symmetric Identities of the Generalized Degenerate Fubini Polynomials by the Fermionic p-Adic Integral on Zp

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials.

Probabilistic degenerate Bernoulli and degenerate Euler polynomials

ABSTRACT Recently, many authors have studied degenerate Bernoulli and degenerate Euler polynomials. Let Y be a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of degenerate Bernoulli and degenerate Euler polynomials, namely the probabilistic degenerate Bernoulli polynomials associated with Y and the probabilistic degenerate Euler polynomials associated with Y . Also, we intoduce the probabilistic degenerate r -Stirling numbers of the second associated with Y and the probabilistic degenerate two variable Fubini polynomials associated with Y . We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As special cases of Y , we treat the gamma random variable with parameters α , β > 0 , the Poisson random variable with parameter α > 0 , and the Bernoulli random variable with probability of success p .

Data transfer within a finite cell remeshing approach applied to large deformation problems

AbstractThe present work is a comparative study of different data transfer techniques in the context of the finite cell method (FCM) in combination with remeshing for hyperelastic problems undergoing large deformations. The FCM is an immersed-boundary method that uses Cartesian grids for the discretization so as to avoid the generation of boundary conforming meshes. To overcome problems with heavily distorted meshes at large deformation states, we apply a remeshing procedure. During the remeshing, the data containing the deformation history has to be transferred between the meshes. In the present study, different methods are considered and compared: radial basis functions without and with polynomial extension, inverse distance weighting, and $$\textit{L}_\text {2}$$ L 2 -projection applying the shape functions used in the FCM for the trial and test functions.

Simulations and analysis of underwater acoustic wave propagation model based on Helmholtz equation

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L 2 and L ∞ norms.

CONSTRUCTION FOR q-HYPERGEOMETRIC BERNOULLI POLYNOMIALS OF A COMPLEX VARIABLE WITH APPLICATIONS COMPUTER MODELING

Two new extensions of the familiar Bernoulli polynomials are considered by using q-sine, q-cosine, q-hypergeometric and q-exponential functions. We call q-sine and q-cosine hypergeometric Bernoulli polynomials. Then, diverse formulas and properties for these polynomials, such as summation formulas, addition formulas, q-derivative properties, q-integral representations and some correlations are derived. Also, q-sine and q-cosine hypergeometric Bernoulli polynomials with two parameters are introduced and some relations and identities are investigated. Furthermore, some computational values are given by tables, and the beautiful zeros representations of the q-sine hypergeometric Bernoulli polynomials and q-cosine hypergeometric Bernoulli polynomials are showed by the figures

Polynomial extension of some symmetric partition regular structures

The study of symmetric structures is a new trend in combinatorial number theory. Recently in [9], Di Nasso proved symmetrized versions of some classical theorems like Hindman's theorem, Van der Waerden's theorem and Deuber's theorem. This opens the question of which other classical theorems can be symmetrized, as well as if other symmetric operations allow for such generalizations. Here we give positive answers to both questions, by showing that symmetrization of the polynomial extension of Van der Waerden's and Deuber's Theorem is possible.

On the Two-Variable Analogue Matrix of Bessel Polynomials and Their Properties

In this paper, we explore a study focused on a two-variable extension of matrix Bessel polynomials. We initiate the discussion by introducing the matrix Bessel polynomials involving two variables and derive specific differential formulas and recurrence relations associated with them. Additionally, we present a segment detailing integral formulas for the extended matrix Bessel polynomials. Lastly, we introduce the Laplace–Carson transform for the two-variable matrix Bessel polynomial analogue.

A New Family of Appell-Type Changhee Polynomials with Geometric Applications

Recently, Appell-type polynomials have been investigated and applied in several ways. In this paper, we consider a new extension of Appell-type Changhee polynomials. We introduce two-variable generalized Appell-type λ-Changhee polynomials (2VGATλCHP). The generating function, series representations, and summation identities related to these polynomials are explored. Further, certain symmetry identities involving two-variable generalized Appell-type λ-Changhee polynomials are established. Finally, Mathematica was used to examine the zero distributions of two-variable truncated-exponential Appell-type Changhee polynomials.

The Properties of Topological Manifolds of Simplicial Polynomials

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition.

Polynomial Extension of the Stronger Central Sets Theorem

In [F81] Furstenberg introduced the notion of central set and established his famous Central Sets Theorem. Since then, several improved versions of Furstenberg's result have been found. The strongest generalization has been published by De, Hindman and Strauss in [DHS08], whilst a polynomial extension by Bergelson, Johnson and Moreira appeared in [BJM17].
 In this article, we will establish a polynomial extension of the stronger version of the central sets theorem, and we will discuss properties of the families of sets that this result leads to consider.

A Combinatorial Interpretation of the Padovan Generalized Polynomial Sequence

We investigate a combinatorial interpretation of the Padovan polynomial sequence, also addressing its polynomial extensions. We thus include the Tridovan polynomial sequence, Tetradovan polynomial sequences, leading up to the Z-dovan polynomial generalization.

Rings with the annihilator condition and their extensions

Let R be an associative unital ring and let α be an endomorphism of R. In this article, we study rings having the annihilator condition (a.c.) and some related rings. Also, many polynomial extensions of R having the condition (a.c.) are stud-ied. In particular, α-skew quasi-Armendariz rings and α-(sps) quasi-Armendariz rings. We prove that this class of rings have always the property (a.c.) on the left but not necessary on the right. Furthermore, we investigate the transmission of the property (a.c) from a ring R to R[x, α] and R[[x, α]].

On Extension of Multilinear Operators and Homogeneous Polynomials in Vector Lattices

On Extension of Multilinear Operators and Homogeneous Polynomials in Vector Lattices

Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

AbstractA formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second‐order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional‐type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state‐deletion and state‐addition occur.

Spectra Reconstruction for Human Facial Color from RGB Images via Clusters in 3D Uniform CIELab* and Its Subordinate Color Space.

Previous research has demonstrated the potential to reconstruct human facial skin spectra based on the responses of RGB cameras to achieve high-fidelity color reproduction of human facial skin in various industrial applications. Nonetheless, the level of precision is still expected to improve. Inspired by the asymmetricity of human facial skin color in the CIELab* color space, we propose a practical framework, HPCAPR, for skin facial reflectance reconstruction based on calibrated datasets which reconstruct the facial spectra in subsets derived from clustering techniques in several spectrometric and colorimetric spaces, i.e., the spectral reflectance space, Principal Component (PC) space, CIELab*, and its three 2D subordinate color spaces, La*, Lb*, and ab*. The spectra reconstruction algorithm is optimized by combining state-of-art algorithms and thoroughly scanning the parameters. The results show that the hybrid of PCA and RGB polynomial regression algorithm with 3PCs plus 1st-order polynomial extension gives the best results. The performance can be improved substantially by operating the spectral reconstruction framework within the subset classified in the La* color subspace. Comparing with not conducting the clustering technique, it attains values of 25.2% and 57.1% for the median and maximum errors for the best cluster, respectively; for the worst, the maximum error was reduced by 42.2%.

Left Nil Zero Semicommutative rings

This paper introduces a class of rings called left nil zero semicommutative rings ( LNZS rings ), wherein a ring R is said to be LNZS if the left annihilator of every nilpotent element of R is an ideal of R. It is observed that reduced rings are LNZS but not the other way around. So, this paper provides some conditions for an LNZS ring to be reduced and among other results, it is proved that R is reduced if and only if the ring of upper triangular matrices over R is LNZS. Furthermore, it is shown that the polynomial ring over an LNZS may not be LNZS and so is the case of the skew polynomial over an LNZS ring. Therefore, this paper investigates the LNZS property over the polynomial extension and skew polynomial extension of an LNZS ring with some additional conditions.

Extension operators for trimmed spline spaces

We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree p with k continuous derivatives. The construction is based on polynomial extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem.