Derivative Of Polynomial Research Articles

Linear sparse differential resultant formulas

Let P be a system of n linear nonhomogeneous ordinary differential polynomials in a set U of n−1 differential indeterminates. Differential resultant formulas are presented to eliminate the differential indeterminates in U from P. These formulas are determinants of coefficient matrices of appropriate sets of derivatives of the differential polynomials in P, or in a linear perturbation Pε of P. In particular, the formula ∂FRes(P) is the determinant of a matrix M(P) having no zero columns if the system P is “super essential”. As an application, if the system P is sparse generic, such formulas can be used to compute the differential resultant ∂Res(P) introduced by Li et al. (2011) [19].

Approximation of conjugate functions by trigonometric polynomials in weighted Orlicz spaces

We investigate the approximation of a conjugate function by the Fejer sums of the Fourier series of the conjugate function and obtain the estimate between the derivatives of the conjugate functions and the derivatives of the conjugate trigonometric polynomials in the weighted Orlicz spaces with Muckenhoupt weights. We prove inverse theorem of approximation theory for the derivatives conjugate functions in the weighted Orlicz spaces.

Approximation of functions in by trigonometric polynomials

We consider the Lebesgue space with variable exponent . It consists of measurable functions for which the integral exists. We establish an analogue of Jackson's first theorem in the case when the -periodic variable exponent satisfies the conditionUnder the additional assumption we also get an analogue of Jackson's second theorem. We establish an -analogue of Bernstein's estimate for the derivative of a trigonometric polynomial and use it to prove an inverse theorem for the analogues of the Lipschitz classes for . Thus we establish direct and inverse theorems of the theory of approximation by trigonometric polynomials in the classes . In the definition of the modulus of continuity of a function , we replace the ordinary shift by an averaged shift determined by Steklov's function .

Roots of complex polynomials and foci of real algebraic curves

We give a new proof of results of B. Z. Linfield presenting the roots of the derivative of a complex polynomial as the foci of a certain real algebraic curve in the complex plane \mathbb C .

Approximation by simple partial fractions with constraints on the poles

Under various constraints on a compact subset of the complex plane and a subset disjoint from , the problem of density in the space (the space of functions that are continuous on a compact set and analytic in its interior) of the set of simple partial fractions (logarithmic derivatives of polynomials) with poles in is studied. The present investigation also involves examining some properties of additive subgroups of a Hilbert space.Bibliography: 19 titles.

A tau method for nonlinear dynamical systems

In this work we present a new Tau method for the solution of nonlinear systems of differential equations which are linear in the derivative of highest order and polynomial in the remaining. We avoid the linearization of the problem by associating to it a nonlinear algebraic system and combine a forward substitution with the Tau method. We develop an adaptive step by step version of this alternative nonlinear tau method and we apply it to several nonlinear dynamical systems.

Vortical flow simulations by a high-order accurate unstructured hexahedral grid method

Vortical flows over a delta wing are numerically simulated by unstructured high-order accurate methods using flux reconstruction (FR) approach by Huynh. The order of accuracy of the scheme for a two-dimensional vortex convection is first tested on various types of grids and with some flux correction functions. Cares for calculating the derivative of the flux polynomial are shown. The flow over a delta wing is then simulated by the fourth order (polynomial degree 3, or p3) and third order (p2) FR as well as the structured MUSCL approach for the comparison. The p3 FR shows the highest suction peak due to the leading-edge vortices and the p2 results are closer to the p3 regardless of the smaller degrees of freedom than the case of MUSCL calculation.

Siebeck curves and two refinements of the

We prove two results which improve the well known Gauss-Lucas theorem by locating the roots of the derivative of a complex polynomial $f$ in sets smaller than the convex hull of the roots of $f$.

Sharp integral inequalities for fractional derivatives of trigonometric polynomials

We study sharp estimates of integral functionals for operators on the set Tn of real trigonometric polynomials fn of degree n≥1 in terms of the uniform norm ‖fn‖C2π of the polynomials and similar questions for algebraic polynomials on the unit circle of the complex plane. P. Erdös, A.P. Calderon, G. Klein, L.V. Taikov, and others investigated such inequalities. In this paper, we, in particular, show that the sharp inequality ‖Dαfn‖q≤nα‖cost‖q‖fn‖∞ holds on the set Tn for the Weyl fractional derivatives Dαfn of order α≥1 for 0≤q<∞. For q=∞ (α≥1), this fact was proved by Lizorkin (1965) [12]. For 1≤q<∞ and positive integer α, the inequality was proved by Taikov (1965) [23]; however, in this case, the inequality follows from results of an earlier paper by Calderon and Klein (1951) [6].

NDD Functions: Properties and Applications

A function whose numerator polynomial is the derivative of the denominator polynomial is designated as NDD function. Some properties of such functions are derived and their applications are given. The NDD functions resemble with the driving point impedance/admittance function.

Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations

The paper is concerned with one problem of function interpolation on a triangle. We consider a large class of interpolation conditions guaranteeing the smoothness of order m of the resulting piecewise polynomial function on the triangulated domain. It is known that, for smoothness m ≥ 1, the known upper estimates for the error of approximation of derivatives of order 2 and higher by derivatives of interpolation polynomials defined on a triangulation element contain the sine of the smallest angle in the denominator. As a result, the “smallest angle condition” must be imposed on the triangulation. It was shown earlier that the effect of the smallest angle could be weakened (which does not mean that it can be eliminated in all cases). The principal aim of this paper is to show that, for a large number of methods of choosing interpolation conditions, including traditional methods, the effect of the smallest angle of the triangle on the error of approximation of derivatives of a function by derivatives of the interpolation polynomial is essential for some derivatives of order 2 and higher for m ≥ 1. In the case m = 0, the effect of the middle (largest) angle is important. As a consequence, the results on the unimprovability of the upper estimates obtained earlier are strengthened.

Sharp Markov brothers type inequality in the spaces L p and L 1 on a closed interval

We study an inequality between the Lp-mean of the (n − 1)th derivative of an algebraic polynomial of degree n ≥ 2 and the L1-mean of this polynomial on a closed interval. Sharp constants and extremal polynomials are written for all p ∈ [0,∞].

Symbolic differentiation of factorized polynomials with repeated roots and the identification of their loci

The mathematical apparatus of algebraic combinatorics is excellently suited to the investigation of symbolic differentiation of analytic functions. In particular, the relationship between the algebraic roots of polynomials and those of their derivatives remains an important topic with wide applications to various branches of pure and applied mathematics. In this paper, a methodology inspired by combinatorial theory is employed to derive analytic expressions for the k-th derivative q(k) of factorized polynomial functions with repeated roots \((x^{\xi} - a_{1})^{\alpha_{1}} (x^{\xi} - a_{2})^{\alpha_{2}} \cdots (x^{\xi} - a_{n})^{\alpha_{n}}\), where ξ∈ℝ∗, ai, αi∈ℝ and i=1,…,n. It is shown that these derivatives are generating functions for classes of integer sequences whose properties are employed to develop a binary tree algorithm that is suitable for the symbolic evaluation of q(k). Compared to the application of Faa di Bruno’s famous differentiation formula for composite functions and to other existing methods for symbolic differentiation, the algorithm is superior because it does not involve finding integer partitions, which is an NP-complete problem. Mathematical identities that relate this topic to other branches of mathematics (e.g. to statistics via the multinomial distribution and multinomial coefficients) are derived and, in addition, a method for identifying the loci of the roots of polynomial derivatives is outlined. The practical significance of these contributions lies in their applicability to various areas of engineering and physics.

On the convergence of Kergin and Hakopian interpolants at Leja sequences for the disk

We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk D of a sufficiently smooth function f in a neighbourhood of D converge uniformly to f on D. Moreover, when f∈C∞(D), all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of f.

Bernstein-type inequalities

It is shown that a Bernstein-type inequality always implies its Szegő-variant, and several corollaries are derived. Then, it is proven that the original Bernstein inequality on derivatives of trigonometric polynomials implies both Videnskii’s inequality (which estimates the derivative of trigonometric polynomials on a subinterval of the period), as well as its half-integer variant. The methods for these two results are then combined to derive the general sharp form of Videnskii’s inequality on symmetric E⊂[−π,π] sets. The sharp Bernstein factor turns out to be 2π times the equilibrium density of the set ΓE={eit|t∈E} on the unit circle C1 that corresponds to E when we identify C1 by R/(mod2π).

Some Inequalities for the Polar Derivative of Polynomials in Complex Domain

Let p(z) be a polynomial of degree n. In this paper we prove results con- cerning maximum modulus of the polar derivative of p(z) with restricted zeros. Our results refine and generalize certain well-known polynomial inequalities.

On approximation by trigonometric polynomials in Orlicz spaces

Let T denote the interval . In this paper, some theorems for approximation by trigonometric polynomials in the Orlicz space are proved. Under certain conditions, we prove the existence of derivatives of functions belonging to the Orlicz space and investigate the approximation properties of the derivatives of trigonometric polynomials. Note that the estimate obtained between the derivatives of functions and the derivatives of trigonometric polynomials depends on a sequence of best approximations in the Orlicz space .

Discriminants, Symmetrized Graph Monomials, and Sums Of Squares

In 1878, motivated by the requirements of the invariant theory of binary forms, J. J. Sylvester constructed, for every graph with possible multiple edges but without loops, its symmetrized graph monomial, which is a polynomial in the vertex labels of the original graph. We pose the question for which graphs this polynomial is nonnegative or a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on the discriminant of the derivative of a univariate polynomial and by an interesting example of P. and A. Lax of a graph with four edges whose symmetrized graph monomial is nonnegative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.

On the Derivative of a Polynomial

Certain refinements and generalizations of some well known inequalities concerning the polynomials and their derivatives are obtained.

A new parallel polynomial division by a separable polynomial via hermite interpolation with applications, pp. 770-781.

A new parallel division of polynomials by a common separable divisor over a perfect field is presented and this is done by expressing the remainders as derivatives of a unique polynomial. In order to get this result, a novel variant expression of the classical Lagrange–Sylvester Hermite interpolating polynomial has been utilised, although any known variant may be used. The above findings are utilized to obtain a number of new identities involving polynomial derivatives, including a closed formula for the semi–simple part of the Jordan decomposition of a matrix.