Derivative Of Polynomial Research Articles

Polynomials with polar derivatives

In this paper, we present an integral inequality for the polar derivative of polynomials. Our theorem includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation

In the papers dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. Finally, we show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. Some numerical examples are provided to demonstrate the efficiency of the method.

On derivatives of polynomials over finite fields through integration

In this article, using rather elementary technique and the derived formula that relates the coefficients of a polynomial over a finite field and its derivative, we deduce many interesting results related to derivatives of Boolean functions and derivatives of mappings over finite fields. For instance, we easily identify several infinite classes of polynomials which cannot possess linear structures. The same technique can be applied for deducing a nontrivial upper bound on the degree of so-called planar mappings.

On partial derivatives of multivariate Bernstein polynomials

It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus.

Asymptotic Hilbert polynomial and a bound for Waldschmidt constants

In the paper we give an upper bound for the Waldschmidt constants of the wide class of ideals. This generalizes the result obtained by Dumnicki, Harbourne, Szemberg and Tutaj-Gasinska, Adv. Math. 2014. Our bound is given by a root of a suitable derivative of a certain polynomial associated with the asymptotic Hilbert polynomial.

Numerical algorithm for the variable-order Caputo fractional functional differential equation

While several high-order methods have been extensively developed for fixed-order fractional differential equations (FDEs), there are no such methods for variable-order FDEs. In this paper, we propose an accurate and robust approach to approximate the solution of functional Dirichlet boundary value problem with a type of variable-order Caputo fractional derivative. The proposed method is principally based on the shifted Chebyshev polynomials as basis functions and the matrix representation of variable-order fractional derivative of such polynomials. The underline variable-order FDE is then reduced to a system of algebraic equations, which greatly simplifies the solution process. Through numerical results, we confirm that the proposed scheme is very efficient and accurate for handling such problem.

Short-term power load forecasting with ordinary differential equation substitutions of polynomial networks

The purpose of the short-term electricity demand forecasting is to forecast in advance the system load, represented by the sum of all consumers load at the same time. Power load forecasting is important for an economically efficient operation and effective control of power systems and enables to plan the load of generating units. A precise load forecasting is required to avoid high generation costs and the spinning reserve capacity. Under-prediction of the demands leads to an insufficient reserve capacity preparation and can threaten the system stability, on the other hand, over-prediction leads to an unnecessarily large reserve that leads to high cost preparations. Differential polynomial neural network is a new neural network type, which decomposes and solves the selective general partial differential equation, which can model a searched function on the bases of observed data samples. It produces an output sum combination of convergent series of selected relative polynomial derivative terms, which can substitute for an ordinary differential equation solution to describe and forecast real data time-series. Partial derivative terms of several time-point variables substitute for the time derivatives of the converted general ordinary differential equation. The operating principles of the proposed method differ significantly from other conventional neural network techniques.

On the derivatives of unimodular polynomials

Let be the open unit disk of the complex plane; its boundary, the unit circle of the complex plane, is denoted by . Let denote the set of all algebraic polynomials of degree at most with complex coefficients. For , let The class is often called the collection of all (complex) unimodular polynomials of degree . Given a sequence of positive numbers tending to , we say that a sequence of polynomials is -ultraflat if Although we do not know, in general, whether or not -ultraflat sequences of polynomials exist for each fixed , we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences of either conjugate, or plain, or skew reciprocal unimodular polynomials such that with is a -ultraflat sequence of polynomials. Bibliography: 18 titles.

Approximation by simple partial fractions with constraints on the poles. II

It is shown that if a compact set not separating the plane lies in the union of the bounded components of the complement of another compact set , then the simple partial fractions (the logarithmic derivatives of polynomials) with poles in are dense in the space of functions that are continuous on and analytic in its interior. It is also shown that if a compact set with connected complement lies in the complement of the closure of a doubly connected domain with bounded connected components of the boundary and , then the differences of the simple partial fractions such that has its poles in and has its poles in are dense in the space . Bibliography: 9 titles.

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann-Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz's method. Examples show that the proposed approach is more accurate than recent methods available in the literature.

A New Sum-of-Squares Design Framework for Robust Control of Polynomial Fuzzy Systems With Uncertainties

This paper presents a new sum-of-squares (SOS, for brevity) design framework for robust control of polynomial fuzzy systems with uncertainties. Two kinds of robust stabilization conditions are derived in terms of SOS. One is global SOS robust stabilization conditions that guarantee the global and asymptotical stability of polynomial fuzzy control systems. The other is semiglobal SOS robust stabilization conditions. The latter is available for very complicated systems that are difficult to guarantee the global and asymptotical stability of polynomial fuzzy control systems. The main feature of all the SOS robust stabilization conditions derived in this paper are to be expressed as nonconvex formulations with respect to polynomial Lyapunov function parameters and polynomial feedback gains. Since a typical transformation from nonconvex SOS design conditions to convex SOS design conditions often results in some conservative issues, the new design framework presented in this paper gives key ideas to avoid the conservative issues. The first key idea is that we directly solve nonconvex SOS design conditions without applying the typical transformation. The second key idea is that we bring a so-called copositivity concept. These ideas provide some advantages in addition to relaxations. To solve our SOS robust stabilization conditions efficiently, we introduce a gradient algorithm formulated as a minimizing optimization problem of the upper bound of the time derivative of an SOS polynomial that can be regarded as a candidate of polynomial Lyapunov functions. Three design examples are provided to illustrate the validity and applicability of the proposed design framework. The examples demonstrate advantages of our new SOS design framework for the existing linear matrix inequality approaches and the existing convex SOS approach.

Parameter Derivatives of the Jacobi Polynomials with Three Variables on the Simplex

In this paper, an attempt has been made to derive parameter derivatives of Jacobi polynomials with three variables on the simplex. They are obtained via parameter derivatives of the classical Jacobi polynomials P n ( α,β ) ( x ) with respect to their parameters. The study is motivated by the expansions of parameter derivatives which are obtained for the classical Jacobi polynomials over the interval (−1, 1), Jacobi polynomials with two variables on the triangle and some other families of orthogonal polynomials in two variables.

The Zeros of Polar Derivative of Polynomials with Respect to Real Number

The Zeros of Polar Derivative of Polynomials with Respect to Real Number

On the $s^{th}$ derivative of a polynomial

For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.

Constructing general partial differential equations using polynomial and neural networks

Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.

Higher order derivatives of approximation polynomials on R $\mathbb{R}$

Leviatan has investigated the behavior of higher order derivatives of approximation polynomials of a differentiable function f on $[-1,1]$ . Especially, when $P_{n}$ is the best approximation of f, he estimates the differences $\|f^{(k)}-P_{n}^{(k)}\|_{L_{\infty}([-1,1])}$ , $k=0,1,2,\ldots $ . In this paper, we give the analogies for them with respect to the differentiable functions on $\mathbb{R}$ .

Computing the topology of an arrangement of parametric or implicit algebraic curves in the Lagrange basis

The study of arrangements of geometric objects is a problem of great interest in combinatorial and computational geometry, and in CAGD, with applications in computer-aided manufacturing, geometric modeling, robotics, NC-machining, molecular modeling, etc. [2, 3] We present a new algebraic method for computing the topology of the arrangement of a finite family of planar algebraic curves. The curves are given either by their implicit equations or by polynomial parametrizations. As far as we know, none of the previous works on this problem include the treatment of parametrized curves. The input of the method is assumed to be given as Lagrange data, i.e., by the values of the polynomials defining the curves, at a suitable set of nodes. The output is a graph representing critical points of the curves, all the intersection points between curves, and segments joining some of these points isotopic to the arcs of the curves. The method follows a sweep-line algorithm. To find critical points, intersections and self-intersections we use Bezout matrices. We take advantage of the existence of a basis of the nullspace of a Bezout matrix that has a nice and useful structure (see [4]). We use algebra by values techniques, which means that all computations (Bezout matrices, polynomial derivatives, etc.) are done in the Lagrange basis [5, 1]. This approach reduces the computational cost of manipulating high degree polynomials, and avoids conversion between different basis, which is known to be a numerically unstable procedure. To compute the roots of the determinant of a Bezout matrix we apply linearization, i.e., we use a straightforward procedure to build (in the Lagrange basis) a companion matrix pencil whose generalized eigenvalues are the roots of the aforementioned determinant. The algorithm has been tested using Maple 17, showing that it is accurate and quite efficient.

Effect of flint corn processing method and roughage level on finishing performance of Nellore-based cattle1

This study was conducted to evaluate the effects of flint corn processing method (CPM) and level of NDF from roughage (rNDF) on performance, carcass characteristics, and starch utilization by finishing Nellore-based cattle fed high-concentrate, flint corn-based diets. In this study, 112 Nellore type bulls (initial BW 384.07 ± 29.53 kg and 24-36 mo of age) were individually fed using Calan gates or individual pens. The animals were used in a randomized complete block design in a 2 × 4 factorial arrangement with 2 CPM, high-moisture flint corn (HMC) or finely ground dry flint corn (FGC), with 1 of 4 levels of rNDF, 3, 8, 13, and 18% (DM basis), using sugarcane silage (SS) as roughage. Bulls were adapted to the finishing diet over a 21-d period and fed for a total of 81 d. Fecal starch (FS) concentration was determined on d 46 and 74 of the feeding period. There was a quadratic effect of rNDF on final BW ( < 0.01) and ADG ( = 0.01). Optimal concentrations of rNDF were estimated using the first derivative of second order polynomials, indicating that final BW and ADG were maximized with 13.3 and 13.0% rNDF, respectively. An interaction was observed between CPM and rNDF ( = 0.05) for DMI, with peak DMI occurring at 11.3 and 13.7% rNDF with FGC and HMC, respectively. Cattle fed HMC had 13.9% greater G:F ( < 0.01) compared with those fed FGC (0.172 vs. 0.151, respectively). There were quadratic effects of rNDF on HCW ( = 0.04) and ME intake ( < 0.01); heaviest carcass weights were estimated, in both cases, to be achieved with 12.8% rNDF. A quadratic effect of rNDF for renal, pelvic, and inguinal fat weight ( = 0.04) was observed, with a peak estimated to occur at 12.6% rNDF. An interaction between CPM and rNDF also was observed for FS ( < 0.05). Bulls fed FGC with 3% rNDF had greater FS content, and FS linearly decreased as concentration of rNDF increased. For bulls fed HMC, FS was 3.0% of DM and was unaffected by rNDF in the diet. Lower FS from bulls fed HMC suggests that availability of starch from flint corn was greater than that of FGC. For Nellore-based cattle fed a flint corn-based diet containing SS and 8% whole lint cottonseed, performance was optimized with 12.8% rNDF. In the absence of cottonseed addition to diets, optimal performance would be expected with about 14.5% rNDF.

Systems of derivatives of polynomials, related to Chebyshev polynomials

Properties of the derivatives of polynomials of a complex variable that are related to Chebyshev polynomials and conditions of expansion of analytic functions in circle into series by them are investigated. Examples of such expansions are presented.

Inequalities for the Fractional Derivatives of Trigonometric Polynomials in Spaces with Integral Metrics

We establish necessary and sufficient conditions for the validity of Bernstein-type inequalities for the fractional derivatives of trigonometric polynomials of several variables in spaces with integral metrics. The problem of sharpness of these inequalities is investigated.