Sum Of Polynomials Research Articles

Extraction of Several Harmonics from Trigonometric Polynomials. Fejer-Type Inequalities

Given a trigonometric polynomial $${T_n}(t) = \sum\nolimits_{k = 1}^n {{\tau _k}\left( t \right),{\tau _k}\left( t \right): = {a_k}\cos kt + {b_k}\sin kt}$$ we consider the problem of extracting the sum of harmonics $$\sum \tau_{\mu_s}(t)$$ prescribed orders µs by the method of amplitude and phase transformations. Such transformations map the polynomials Tn(t) into similar ones using two simple operations: the multiplication by a real constant X and the shift by a real phase λ, i.e., Tn(t) → XTn(t — λ). We represent the sum of harmonics as a sum of such polynomials and then use this representation to obtain sharp Fejer-type estimates.

Computation of Sum of Squares Polynomials from Data Points

We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition ...

Fuzzy-based Dissipative Consensus for Multi-Agent Systems with Markov Switching Topologies

In this paper, the dissipativity-based consensus problem for polynomial fuzzy multi-agent systems is considered. First, a novel fuzzy modeling method is proposed to describe the error dynamics. By establishing the changing topologies by Markov process, a new consensus protocols are designed. By polynomial Lyapunov function and sum of squares, sufficient condition is given to assure even-square consensus with dissipative performance. Finally, an illustrative example is employed to verify the proposed dissipativity-based even-square consensus design schemes.

Sum of squares approach for ground vehicle lateral control under tire saturation forces

This work presents the stability analysis and design of a lateral controller for a nonlinear ground vehicle applying the concept of polynomial sum of squares relaxations. The system is approximated by a polynomial vector field that describes the lateral vehicle dynamics. The resulting polynomial system falls on a class of non-affine in input system, which makes the control synthesis more involved. This issue is circumvented by an input-affine approximation, simplifying the stability analysis and the design procedure of a polynomial state-feedback controller able to enlarge the region of attraction (RoA). We also compare the estimated region of attraction with the standard LQR optimal controller.

A Polynomial Sieve and Sums of Deligne Type

AbstractLet $f\in \mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in \mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we present a new bound for $N(f,F,B):=|\{\textbf{x}\in \mathbb{Z}^{n+1}:\max _{0\leq i\leq n}|x_{i}|\leq B,\exists t\in \mathbb{Z}\textrm{ such that}\ f(t)=F(\textbf{x})\}|.$ To do this, we introduce a generalization of the power sieve [ 14, 28] and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.

On the two-term exponential sums and character sums of polynomials

Abstract The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and acquire several accurate asymptotic formulas for them.

On the Construction of Converging Hierarchies for Polynomial Optimization Based on Certificates of Global Positivity

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellensätze from the late 20th century (e.g., due to Stengle, Putinar, or Schmüdgen) that certify positivity of a polynomial on an arbitrary closed basic semialgebraic set. In this paper, we show that such hierarchies could in fact be designed from much more limited Positivstellensätze dating back to the early 20th century that only certify positivity of a polynomial globally. More precisely, we show that any inner approximation to the cone of positive homogeneous polynomials that is arbitrarily tight can be turned into a converging hierarchy of lower bounds for general polynomial minimization problems with compact feasible sets. This in particular leads to a semidefinite programming–based hierarchy that relies solely on Artin’s solution to Hilbert’s 17th problem. We also use a classical result from Pólya on global positivity of even forms to construct an “optimization-free” converging hierarchy for general polynomial minimization problems with compact feasible sets. This hierarchy requires only polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming–based and second-order cone programming–based hierarchies for polynomial minimization problems that rely on the recently introduced concepts of diagonally dominant sum of squares and scaled diagonally dominant sum of squares polynomials. We remark that the scope of this paper is theoretical at this stage, as our hierarchies—though they involve at most two sum of squares constraints or only elementary arithmetic at each level—require the use of bisection and increase the number of variables (respectively, the degree) of the problem by the number of inequality constraints plus three (respectively, by a factor of two).

Some Extended Gibonacci Polynomial Sums with Dividends

Some Extended Gibonacci Polynomial Sums with Dividends

Ordinary and degenerate Euler numbers and polynomials

In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on mathbb{Z}_{p}. Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

Facial reduction for exact polynomial sum of squares decomposition

Facial reduction for exact polynomial sum of squares decomposition

Time evolution of the electric field using the rapid expansion method with pseudospectral evaluation of spatial derivatives — Part 1

We have developed a method for modeling transient electromagnetic data acquired on land or at sea. The 3D 3C time response of the electric wavefield is simulated by considering the spatial and temporal responses separately and using numerical methods best suited for each. We have used the rapid expansion method to develop the 3C electric wavefield time response from the spatial responses found using a pseudospectral method. The results are free of numerical dispersion and accurate to the Nyquist frequency in time and space. The resulting diffusive field is a weighted sum of Chebyshev polynomials that exhibit a wave-like character and are very sensitive to small perturbations in the medium. The method is intrinsically parallel, which leads to computational efficiency. Numerical results are compared favorably with the analytic response and 1D methods even though the computations are innately 3D.

Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

English

English

One kind of character sum modulo a prime p and its recurrence formula

The aim of this paper is to use an analytic method and the properties of the classical Gauss sums to research the computational problem of one kind of character sum of polynomials modulo an odd prime p and obtain several meaningful third- and fourth-order linear recurrence formulae for them.

On the finite reciprocal sums of Fibonacci and Lucas polynomials

In this note, we consider the finite reciprocal sums of Fibonacci and Lucas polynomials and derive some identities involving these sums.

Power sums of arithmetic progressions and Bernoulli polynomials

Power sums of arithmetic progressions and Bernoulli polynomials

Nikolskii Inequalities for Trigonometric Polynomials in Different Metrics

S. M. Nikolsky’s inequalities for trigonometric polynomials in different metrics are well-known. We generalize the inequalities for the sums of trigonometric polynomials.

Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) = 1\}$. The results are first stated for the case $f(x_1,\dots,x_d) = x_1^2+\dots+x_d^2,$ which is of particular interest.

Almost sure limit theorems on Wiener chaos: the non-central case

In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Ito integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].

ELASTICITY ANISOTROPY OF COMPOSITES

The article presents a solution to a problem on the establishment of mathematical relations between elastic constants of anisotropic fiber composites in principal anisotropy directions. A stress function in the form of a polynomial sum was used to solve the fourth-order differential equation in partial derivatives for an orthotropic body, thus allowing determining elastic constants of fiber composites with different directions of reinforcing fibers by means of mathematical calculations. It was established that all anisotropic materials could be conditionally divided into two groups. One group includes composites in which elastic constants take an intermediate extreme value at 60° upon change of the reinforcing fibers position from 0 to 90°. The other group includes composites with two extreme values — at 0 and 90°. This new knowledge obtained for the first time shall be used to provide rationale for the method of composite strength assessment by means of calculations depending on the position of reinforcing fibers, which was previously possible only upon execution of experimental studies.