Class Of Polynomials Research Articles

A New First Finite Class of Classical Orthogonal Polynomials Operational Matrices: An Application for Solving Fractional Differential Equations

In this paper, new operational matrices (OMs) of ordinary and fractional derivatives (FDs) of a first finite class of classical orthogonal polynomials (FFCOP) are introduced. Also, two algorithms are proposed for using the tau and collocation spectral methods (SPMs) to get new approximate solutions to the given fractional differential equations (FDEs). These algorithms convert the given FDEs subject to initial/boundary conditions (I/BCs) into linear or nonlinear systems of algebraic equations that can be solved using appropriate solvers. To demonstrate the robustness, efficiency, and accuracy of the proposed spectral solutions, several illustrative examples are presented. The obtained results show that the proposed algorithms exhibit higher accuracy compared to existing techniques in the literature. Furthermore, an error analysis is provided.

Lateral torsional buckling of unsymmetric beams by FEM with linear elements

As the exact solution of lateral torsional buckling of elastic prismatic beams is practically limited to the simple case of simply supported beams under equal end moments, other loading conditions and boundary conditions require more practical solutions to the problem. As the general solution of lateral torsional buckling of unsymmetric beams is a function of the sign of the bending moment along the axis of the beam, i.e. location of the shear center concerning the compression zone of the cross section, application of the current traditional empirical expressions such as coefficient of the moment, Cb, becomes highly inconsistent and unwarranted to capture the correct critical moment as acknowledged by the American Institute of Steel Construction, AISC. The finite element method, FEM, offers a feasible alternative to overcome these shortcomings. FEM is formulated in its simplest form of linear elements for the general case of lateral torsional buckling of unsymmetric cross sections. Finite element development shows that the characteristic equation is of the nonlinear quadratic eigenvalue problem type. Using the classical polynomial shape functions for beams, FEM proves to be extremely accurate and can overcome the high inconsistencies and discrepancies embedded in the application of the classical methods.

Some results on complete permutation polynomials and mutually orthogonal Latin squares

In this paper, we investigate some classes of complete permutation polynomials (CPPs) with the form (L1(x))t+L2(x) for some specific linearized polynomials L1(x) and L2(x) over finite fields. Some constructions of PPs and CPPs over finite fields using the AGW criterion are also proposed. We also obtain some constructions of sets of Mutually orthogonal Latin squares (MOLS) using permutation polynomials over finite fields.

Refined multiplicative tensor product of matrix factorizations

An algorithm for matrix factorization of polynomials was proposed in [11] and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible polynomials. In this paper, we improve this algorithm by refining the construction of one of its two main ingredients, namely the multiplicative tensor product ⊗˜ of matrix factorizations to obtain another different bifunctorial operation denoted by ⊗‾. We refer to ⊗‾ as the refined multiplicative tensor product of matrix factorizations. In fact, we observe that in the algorithm for matrix factorization of polynomials developed in [11], if we replace ⊗˜ by ⊗‾, we obtain better results on the class of summand-reducible polynomials in the sense that the refined algorithm produces matrix factors which are of smaller sizes.

Holomorphic Hörmander-type functional calculus on sectors and strips

In this paper, the abstract multiplier theorems for 0 0 -sectorial and 0 0 -strip type operators by Kriegler and Weis [Math. Z. 289 (2018), pp. 405–444] are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic Hörmander-type functions on sectors and strips are introduced, with even a finer scale of smoothness than the classical polynomial scale. Moreover, we establish alternative descriptions of these spaces involving Schwartz and “holomorphic Schwartz” functions. Finally, the abstract results are combined with a result by Carbonaro and Dragičević [Duke Math. J. 166 (2017), pp. 937–974] to obtain an improvement—with respect to the smoothness condition—of the known Hörmander-type multiplier theorem for general symmetric contraction semigroups.

The Reaction–Diffusion Models in Biomedicine: Highly Accurate Calculations via a Hybrid Matrix Collocation Algorithm

A hybrid efficient and highly accurate spectral matrix technique is adapted for numerical treatments of a class of two-pint boundary value problems (BVPs) with singularity and strong nonlinearity. The underlying model is a reaction-diffusion equation arising in the modeling of biomedical, chemical, and physical applications based on the assumptions of Michaelis–Menten kinetics for enzymatic reactions. The manuscript presents a highly computational spectral collocation algorithm for the model combined with the quasilinearization method (QLM) to make the proposed technique more efficient than the corresponding direct spectral collocation algorithm. A novel class of polynomials introduced by S.K. Chatterjea is used in the spectral method. A detailed proof of the convergence analysis of the Chatterjea polynomials (ChPs) is given in the L2 norm. Different numerical examples for substrate concentrations with all values of parameters are performed for the case of planar and spherical shapes of enzymes. For validation, these results are compared with those obtained via wavelet-based procedures and the Adomian decomposition scheme. To further improve the approximate solutions obtained by the QLM–ChPs method, the technique of error correction is introduced and applied based on the concept of residual error function. Overall, the presented results with exponential convergence indicate that the QLM–ChPs algorithm is simple and flexible enough to be applicable in solving many similar problems in science and engineering.

Some new results on permutation polynomials of the form [formula omitted] over [formula omitted

This paper investigates the permutation behavior of polynomials in the form b(xpm+ax+δ)s−ax over Fp2m by employing the Akbary–Ghioca–Wang criterion, where b∈Fpm⁎, a,δ∈Fp2m with aq+1=1. As a result, we present several classes of permutation polynomials of the form b(x2m+ax+δ)s+ax over F22m and three classes of permutation polynomials of the form b(xpm−x+δ)s+x over Fp2m with infinitely many odd primes p.

Characterization of the unit group of ℤ[G × C2n

Let [Formula: see text] be a group and [Formula: see text] be the cyclic group of order 2. We characterize the unit group [Formula: see text] of the integral group ring [Formula: see text] in terms of the unit group [Formula: see text], for every positive integer [Formula: see text]. As a consequence, for each [Formula: see text], we determine conditions on some class of matrices [Formula: see text]s, to be in GL[Formula: see text]. Using this we calculate the integer solutions of some specific class of symmetric polynomial [Formula: see text] with integer coefficients.

The Number of Zeros in a Disk of a Complex Polynomial with Coefficients Satisfying Various Monotonicity Conditions

Motivated by results on the location of the zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Eneström–Kakeya theorem and its recent generalizations), we impose similar conditions and give bounds on the number of zeros in certain regions. We do so by introducing a reversal in monotonicity conditions on the real and imaginary parts of the coefficients and also on their moduli. The conditions imposed are less restrictive than many of those in the current literature and hence apply to polynomials not covered by previous results. The results presented naturally apply to certain classes of lacunary polynomials. In particular, the results apply to certain polynomials with two gaps in their coefficients.

Some γ-positive polynomials arising from enumerations of the pseudo Schröder paths

We introduce a notion of pseudo Schröder paths, which are counted by the Schröder numbers. Using the symbolic method and the Lagrange inversion formula, we provide an alternative proof that the descent polynomials of separable permutations are γ-positive. We also present several classes of γ-positive polynomials via enumerating the pseudo Schröder paths according to several combinatorial statistics by applying Riordan arrays.

An Interesting Class of Non-Kac Random Polynomials

An Interesting Class of Non-Kac Random Polynomials

Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

EXPONENTIAL POLYNOMIALS AND STRATIFICATION IN THE THEORY OF ANALYTIC INEQUALITIES

This paper considers MEP - Mixed Exponential Polynomials as one class of real exponential polynomials. We introduce a method for proving the positivity of MEP inequalities over positive intervals using the Maclaurin series to approximate the exponential function precisely. Additionally, we discuss the relation between MEPs and stratified families of functions from [1] through two applications, referring to inequalities from papers [2] and [3].

Some New Families of Finite Orthogonal Polynomials in Two Variables

In this paper, we generalize the study of finite sequences of orthogonal polynomials from one to two variables. In doing so, twenty three new classes of bivariate finite orthogonal polynomials are presented, obtained from the product of a finite and an infinite family of univariate orthogonal polynomials. For these new classes of bivariate finite orthogonal polynomials, we present a bivariate weight function, the domain of orthogonality, the orthogonality relation, the recurrence relations, the second-order partial differential equations, the generating functions, as well as the parameter derivatives. The limit relations among these families are also presented in Labelle’s flavor.

Bounded Poincaré operators for twisted and BGG complexes

We construct bounded Poincaré operators for twisted complexes and BGG complexes with a wide class of function classes (e.g., Sobolev spaces) on bounded Lipschitz domains. These operators are derived from the de Rham versions using BGG diagrams and, for vanishing cohomology, satisfy the homotopy identity dP+Pd=I in degrees >0. The operators preserve polynomial classes if the de Rham versions do so. Nontrivial cohomology and the complex property P∘P=0 can be incorporated. We present applications to polynomial exact sequences.

Mixed-Type Hypergeometric Bernoulli–Gegenbauer Polynomials

In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials do not fulfill either Hanh or Appell conditions.

KRLS post-distorter with adaptive kernel width for visible light communications.

As a viable supplement to the fifth generation wireless communication, visible light communications (VLC) with affluent spectrum resources can cater to the ever-increasing high speed data transmission demand. However, the nonlinear characteristics of light emitting diode (LED) can distort the transmitted signal in the VLC link, which damages the communication quality. To mitigate the nonlinear impairments, a reproducing kernel Hilbert space post-distortion scheme is proposed in this paper, which is based on kernel recursive least squares (KRLS) with adaptive kernel width. In this kernel based method, the kernel width will affect the approximation ability of the model. Therefore, in the recursive process of KRLS, Gauss-Newton (GN) algorithm is adopted to update the kernel width. In addition, combined with the enhanced novelty criterion (ENC), the KRLS-GN post-distorter learns the sparse dictionary adaptively according to the input data, which is beneficial to complete the linearization under the limited memory budget constraints. The performance of the proposed KRLS-GN-ENC scheme is verified by simulations, and the results show that KRLS-GN-ENC can achieve a significant improvement over KRLS-ENC. Compared with the schemes based on classical polynomial filtering, KRLS-GN-ENC exhibits better nonlinear compensation performance and faster convergence speed.

Symmetry Adapted Bases for Trigonometric Optimization

We present an algorithm to compute the global minimum of a trigonometric polynomial, when it is invariant under the exponential action of a Weyl group. This is based on a common relaxation technique that leads to a semi-definite program (SDP). It is then shown how to exploit the invariance in order to reduce the number of variables of the SDP and to simplify its structure significantly. This approach complements the one that was proposed as a poster at the recent ISSAC 2022 conference [HMMR22] and later extended to [HMMR23]. In the previous work, we first used the invariance of the objective function to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. In the present work, we first apply the relaxation and then exploit symmetry. We show that the Weyl group action is induced by an orthogonal representation and describe its isotypic decomposition.

Fractional Polynomial Models as Special Cases of Bayesian Generalized Nonlinear Models

We propose a framework for fitting multivariable fractional polynomial models as special cases of Bayesian generalized nonlinear models, applying an adapted version of the genetically modified mode jumping Markov chain Monte Carlo algorithm. The universality of the Bayesian generalized nonlinear models allows us to employ a Bayesian version of fractional polynomials in any supervised learning task, including regression, classification, and time-to-event data analysis. We show through a simulation study that our novel approach performs similarly to the classical frequentist multivariable fractional polynomials approach in terms of variable selection, identification of the true functional forms, and prediction ability, while naturally providing, in contrast to its frequentist version, a coherent inference framework. Real-data examples provide further evidence in favor of our approach and show its flexibility.

Newfangled Linearization Formula of Certain Nonsymmetric Jacobi Polynomials: Numerical Treatment of Nonlinear Fisher’s Equation

This article is devoted to deriving a new linearization formula of a class for Jacobi polynomials that generalizes the third-kind Chebyshev polynomials class. In fact, this new linearization formula generalizes some existing ones in the literature. The derivation of this formula is based on employing a new moment formula of this class of polynomials and after that using suitable symbolic computation to reduce the resulting linearization coefficients into simplified forms that do not contain any hypergeometric functions or sums. The new formula is employed along with some other formulas and with the utilization of the spectral tau method to obtain numerical solutions to the nonlinear Fisher equation. The presented method is used to convert the equation governed by its underlying conditions into a nonlinear system of equations. The solution of the resulting system can be obtained through any suitable standard numerical scheme. To demonstrate the efficiency and usefulness of the proposed algorithm, some examples are shown, including comparisons with some existing techniques in the literature.