Incomplete Polynomials Research Articles

When D-companion matrix meets incomplete polynomials

In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we obtain a majorization relationship between the zeros of convex combinations of incomplete polynomials and the original polynomial. Moreover, we prove that the set of zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. Location of zeros of convex combinations of incomplete polynomials is determined.

Pluripotential theory associated with general bodies

Pluripotential theory associated with a convex body $P\subset (\mathbb R^+)^d$ is a recently developing field of study. In this paper, motivated by work of J. J. Callaghan [Ann. Polon. Math. 90 (2007), 21–35] on $\theta $-incomplete polynomials in $\mathb

Impact of Initial Titanium Magnetite Ore Material Composition on Fluxed Pellet Metallurgical Properties

Research work was carried out on the test units with the conditions close to the industrial ones to study the material composition impact of the concentrates produced from various types of the Kachkanar titanium magnetite ores on the fluxed pellet metallurgical properties. In order to study the pellet properties with a small number of experiments the simplex-lattice design method was used. Data was quoted on variation of key quality parameters of pellets from particular concentrate types and their mixture at certain horizons throughout the bed height with regard to a conveyer-type roast machine. An expression was obtained for the weighted average strength of commercial pellets. According to the data obtained in the course of the experiments with certain concentrate types and their well-balanced mixtures the expressions were introduced as incomplete cubic polynomials describing the key quality parameter variations of the pellets from optional composition mixtures. The work results are of certain interest for the process engineers and specialists dealing with iron ore raw material preparation, as they allow forecasting of high quality fluxed pellet production from various material composition concentrates.

Some new formulas of complete and incomplete degenerate Bell polynomials

The aim of this paper is to study the complete and incomplete degenerate Bell polynomials, which are degenerate versions of the complete and incomplete Bell polynomials, and to derive some properties and identities for those polynomials. In particular, we introduce some new polynomials associated with the incomplete degenerate Bell polynomials. In fact, they are the coefficients of the reciprocal of the power series given by 1 plus the one appearing as the exponent of the generating function of the complete degenerate Bell polynomials.

Some identities for degenerate complete and incomplete r-Bell polynomials

In this paper, we study degenerate complete and incomplete r-Bell polynomials. They are generalizations of the recently introduced degenerate r-Bell polynomials and degenerate analogues for the complete and incomplete r-Bell polynomials. We investigate some properties and identities for these polynomials. In particular, we give explicit formulas for the degenerate complete and incomplete r-Bell polynomials.

Reverse Markov- and Bernstein-type inequalities for incomplete polynomials

Let Pn,k be the set of all algebraic polynomials, with real coefficients, of degree at most n+k having at least n+1 zeros at 0. Let ‖f‖A≔supx∈A|f(x)|for real-valued functions f defined on a set A⊂R. Let Vab(f)≔∫ab|f′(x)|dxdenote the total variation of a continuously differentiable function f on an interval [a,b]. We prove that there are absolute constants c1>0 and c2>0 such that c1nk≤minP∈Pn,k‖P′‖[0,1]V01(P)≤minP∈Pn,k‖P′‖[0,1]|P(1)|≤c2nk+1for all integers n≥1 and k≥1. We also prove that there are absolute constants c1>0 and c2>0 such that c1nk1∕2≤minP∈Pn,k‖P′(x)1−x2‖[0,1]V01(P)≤minP∈Pn,k‖P′(x)1−x2‖[0,1]|P(1)|≤c2nk+11∕2for all integers n≥1 and k≥1.

The location of critical points of polynomials

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials.

On r-Central Incomplete and Complete Bell Polynomials

Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well.

Generating Functions for Orthogonal Polynomials of A2, C2 and G2

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

Generalized incomplete poly-Bernoulli polynomials and generalized incomplete poly-Cauchy polynomials

By using the restricted and associated Stirling numbers of the first kind, we define the generalized restricted and associated poly-Cauchy polynomials. By using the restricted and associated Stirling numbers of the second kind, we define the generalized restricted and associated poly-Bernoulli polynomials. These polynomials are generalizations of original poly-Cauchy polynomials and original poly-Bernoulli polynomials, respectively. We also study their characteristic and combinatorial properties.

INCOMPLETE BESSEL POLYNOMIALS: A NEW CLASS OF SPECIAL POLYNOMIALS FOR ELECTROMAGNETICS

A new class of incomplete Bessel polynomials is introduced, and its application to the solution of electromagnetic problems regarding transient wave radiation phenomena in truncated spherical structures is discussed. The general definition and main analytical properties of said special functions are provided. The definition is such that the interrelationships between the incomplete polynomials parallel, as far as it is feasible, those for canonical Bessel polynomials.

Symmetric global partition polynomials for reproducing kernel elements

The reproducing kernel element method is a numerical technique that combines finite element and meshless methods to construct shape functions of arbitrary order and continuity, yet retains the Kronecker- $$\delta $$ ? property. Central to constructing these shape functions is the construction of global partition polynomials on an element. This paper shows that asymmetric interpolations may arise due to such things as changes in the local to global node numbering and that may adversely affect the interpolation capability of the method. This issue arises due to the use of incomplete polynomials that are subsequently non-affine invariant. This paper lays out the new framework for generating general, symmetric, truly minimal and complete affine invariant global partition polynomials for triangular and tetrahedral elements. Optimal convergence rates were observed in the solution of Kirchhoff plate problems with rectangular domains.

On Multivariate Incomplete Polynomials on Starlike Domains

Multivariate incomplete polynomials are considered on compact 0-symmetric starlike domains. Problems of density and quantitative approximation properties of such polynomials are investigated. It is shown that density holds for a certain class of starlike domains which includes both convex and some nonconvex domains. On the other hand, a family of nonconvex starlike domains is also found for which density fails. In addition, it is also shown that on 0-symmetric convex bodies in \(\mathbb{R}^{d}\), continuous functions can be approximated by θ-incomplete polynomials with the rate O(ω2(n−1/(d+3))). Moreover, if the convex body is the intersection of simplexes with vertex at the origin, then this order improves to \(O (\omega_{2}(f,1/\sqrt{n}) )\).

On Higher Order Pyramidal Finite Elements

AbstractIn this paper we first prove a theorem on the nonexistence of pyramidal polynomial basis functions. Then we present a new symmetric composite pyramidal finite element which yields a better convergence than the nonsymmetric one. It has fourteen degrees of freedom and its basis functions are incomplete piecewise triquadratic polynomials. The space of ansatz functions contains all quadratic functions on each of four subtetrahedra that form a given pyramidal element.

Minimal Multi-Convex Projections Onto Subspaces of Incomplete Algebraic Polynomials

Let X = (C N [0, 1], ‖·‖), where N ≥ 3 and let V be a linear subspace of Π N , where Π N denotes the space of algebraic polynomials of degree less than or equal to N. Denote by 𝒫 S = 𝒫 S (X, V) = {P: X → V | P-linear and bounded P| V = id V , PS ⊂ S}, where S denotes a cone of multi-convex functions. In [25, 26], the multi-convex projections were defined and it was shown the explicite formula for projection with minimal norm in 𝒫 S for V = Π N . In this article we present a generalization of these results in the case of V being certain, proper subspaces of Π N .

A generalization of the Gauss-Lucas theorem

Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.

A Green's function for θ-incomplete polynomials

Let $K$ be any subset of $ \mathbb C^N $. We define a pluricomplex Green's function $V_{K,\theta} $ for $ \theta $-incomplete polynomials. We establish properties of $V_{K,\theta} $ analogous to those of the weighted pluricomplex Green's function. When $K

On the value of the max-norm of the orthogonal projector onto splines with multiple knots

The supremum over all knot sequences of the max-norm of the orthogonal spline projector is studied with respect to the order k of the splines and their smoothness. It is first bounded from below in terms of the max-norm of the orthogonal projector onto a space of incomplete polynomials. Then, for continuous and for differentiable splines, its order of growth is shown to be k .

Incomplete locator polynomials and redundant iterations reduction in the Welch–Berlekamp algorithm

A concept named 'incomplete locator polynomial' is introduced as a replacement for the traditional approach. Theorems for finding the incomplete locators were established when the Welch-Berlekamp algorithm (1986) was employed for decoding Reed-Solomon codes, and an incomplete iteration strategy of decoding Reed-Solomon codes, based on finding incomplete locator polynomials is presented, by which the redundant iterations in the Welch-Berlekamp algorithm are reduced significantly.

The role of the endpoint in weighted polynomial approximation with varying weights

For a weight functionw: [a, b]→(0, ∞), we consider weighted polynomials of the formw n Pn where the degree ofP n is at mostn. The class of functions that can be approximated with such polynomials depends on the behavior of the densityv(t) of the extremal measure associated withw. We show that every approximable function must vanish at the endpointa ifv(t) behaves like (t−a) β ast→a with β>−1/2. We also present an analogous result for internal points. Our results solve some open problems posed by V. Totik and disprove a conjecture of G.G. Lorentz on incomplete polynomials.