Construction Of Polynomials Research Articles

On Polynomial General Helices in n-Dimensional Euclidean Space $$\mathbb {R}^{n}$$ R n

In this work, we study the so-called polynomial general helices in an arbitrary dimensional Euclidean space. First, we give a method to construct helices from polynomial curves in n-dimensional Euclidean space \(\mathbb {R}^n\), and another method to construct polynomial general helices in \(\mathbb {R}^n\) from polynomial general helices in \(\mathbb {R}^{n+1}\) or \(\mathbb {R}^{n+2}\). Then, we proceed with a method to construct rational helices from polynomial general helices.

Exceptional Laguerre Polynomials

AbstractThe aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials, we associate them with two partitions. We find that the use of partitions is an elegant way to express these polynomials and we restate some of their known properties in terms of partitions. We discuss the asymptotic behavior of the regular zeros and the exceptional zeros of exceptional Laguerre polynomials as the degree tends to infinity.

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions. Nonnegativity is enforced by requiring the polynomials to be sums of squares, a condition which is then formulated as a semidefinite program (SDP) that can be solved computationally. Although such computations are subject to numerical error, we demonstrate two ways to obtain rigorous results: using interval arithmetic to control the error of an approximate SDP solution, and finding exact analytical solutions to relatively small SDPs. Previous formulations are extended to allow for bounds depending analytically on parametric variables. These methods are illustrated using the Lorenz equations, a system with three state variables $(x,y,z)$ and three parameters $(\beta,\sigma,r)$. Bounds are reported for infinite-time averages of all eighteen moments $x^ly^mz^n$ up to quartic degree that are symmetric under $(x,y)\mapsto(-x,-y)$. These bounds apply to all solutions regardless of stability, including chaotic trajectories, periodic orbits, and equilibrium points. The analytical approach yields two novel bounds that are sharp: the mean of $z^3$ can be no larger than its value of $(r-1)^3$ at the nonzero equilibria, and the mean of $xy^3$ must be nonnegative. The interval arithmetic approach is applied at the standard chaotic parameters to bound eleven average moments that all appear to be maximized on the shortest periodic orbit. Our best upper bound on each such average exceeds its value on the maximizing orbit by less than 1%. Many bounds reported here are much tighter than would be possible without computer assistance.

Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz.

On the construction of irreducible polynomials over finite fields via odd prime degree endomorphisms of elliptic curves

In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.

Construction of Double Grothendieck Polynomials of Classical Types using IdCoxeter Algebras

We construct double Grothendieck polynomials of classical types which are essentially equivalent to but simpler than the polynomials defined by A.~N. Kirillov in arXiv:1504.01469 and identify them with the polynomials defined by T.~Ikeda and H.~Naruse in Adv. Math. (2013) for the case of maximal Grassmannian permutations. We also give geometric interpretation of them in terms of algebraic localization map and give explicit combinatorial formulas.

Kempe’s Universality Theorem for Rational Space Curves

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 9/2 d - 6c + 1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicity construction of the linkage.

On moduli space of symmetric orthogonal matrices and exclusive Racah matrixS¯for representation R = [3,1] with multiplicities

Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of arXiv:1612.00422, we obtain the exclusive Racah matrix $\bar S$ for the currently-front-line case of representation $R=[3,1]$ with non-trivial multiplicities, where it is actually operator valued, i.e. depends on the choice of basises in the intertwiner spaces. Effective field theory for arborescent knots in this case possesses gauge invariance, which is not yet properly described and understood. Because of this lack of knowledge a big part (about a half) of $\bar S$ needs to be reconstructed from orthogonality conditions. Therefore we discuss the abundance of symmetric orthogonal matrices, to which $\bar S$ belong, and explain that dimension of their moduli space is also about a half of that for the ordinary orthogonal matrices. Thus the knowledge approximately matches the freedom and this explains why the method can work -- with some limited addition of educated guesses. A similar calculation for $R=[r,1]$ for $r>3$ should also be doable.

A construction of primitive polynomials over finite fields

In this work, a new construction based on companion matrices of primitive polynomials is provided. Given two primitive polynomials over the finite fields and , we construct a ring isomorphism that transforms the companion matrix of the primitive polynomial over into a matrix with elements in whose characteristic polynomial is another primitive polynomial over .

Several classes of permutation trinomials from Niho exponents

Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field $\mathbb{F}_{2^n}$, where $n$ is a positive even integer, we focus on the construction of permutation trinomials over $\mathbb{F}_{2^n}$ from Niho exponents. As a consequence, several new classes of permutation trinomials over $\mathbb{F}_{2^n}$ are constructed from Niho exponents based on some subtle manipulation of solving equations with low degrees over finite fields.

On certain Diophantine equations of the form z2 = f(x)2 ± g(y)2

In this note we study the existence of integer solutions of the Diophantine equationz2=f(x)2±g(y)2 for certain polynomials f,g∈Z[x] of degree ≥3. In particular, for given k∈N we prove that for all a,b∈Z satisfying the condition a2+b2≠0, the above Diophantine equation, with f(x)=xk(x+a), g(x)=xk(x+b) and any choice of the sign, has infinitely many solutions in integers x,y,z such that f(x)g(y)≠0. Moreover, we prove that for f(x)=x3 and g(x)=x(x+1)(x+2) the system of Diophantine equationsz12=f(x)2+g(y)2,z22=f(x)2+g(y+1)2 has infinitely many solutions in positive integers x,y,z1,z2 with gcd⁡(x,y)=1. Similar result is proved for the systemz12=f(x)2+g(y)2,z22=f(x+1)2+g(y)2. We also present some experimental results concerning the construction of polynomials with rational coefficients for which the Diophantine equation z2=f(x)2±g(y)2 has infinitely many solutions in integers.

On the Number of Integer Recurrence Relations

<p>This paper presents the number of k-stage integer recurrence relations (IRR) over the ring Z<sub>2</sub> which generates sequences of maximum possible period (2<sup>k</sup>-1)2<sup>e-1</sup> for e<span style="text-decoration: underline;">></span>1. This number corresponds to the primitive polynomials mod 2 which satisfy the condition proposed by Brent and is2<sup>(e-2)k+1</sup>(2<sup>k-1</sup>-1) for <span> e<span style="text-decoration: underline;">></span>3</span>. This number is same as measured by Dai but arrived at with a different condition for maximum period. Our way of counting gives an explicit method for construction of such polynomials. Furthermore, this paper also presents the number of different sequences corresponding to such IRRs of maximum period.</p>

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these — the eigenvalue density of the real eigenvalues — gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.

Matrix approach to hypercomplex Appell polynomials

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy–Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras.

Algebraic soft decoding of Reed–Solomon codes with improved progressive interpolation

The algebraic soft decoding (ASD) algorithm for Reed–Solomon (RS) codes can correct errors beyond the half distance bound with a polynomial time complexity. However, the decoding complexity remains high due to the computationally expensive interpolation that is an iterative polynomial construction process. By performing the interpolation progressively, the progressive ASD (PASD) algorithm can adapt the decoding computation to the need, leveraging the average complexity of multiple decoding events. But the complexity reduction is realised at the expense of system memory, since the intermediate interpolation information needs to be memorised. Addressing this challenge, this paper proposes an improved PASD (I-PASD) algorithm that can alleviate the memory requirement and further reduce the decoding complexity. A condition on expanding the set of interpolated polynomials will be introduced, which excepts the need of performing iterative updates for the newly introduced polynomial. Further incorporating the re-encoding transform, the I-PASD algorithm can reduce the decoding complexity over the PASD algorithm by a factor of 1/3 and its memory requirement is at most half of the PASD algorithm. The complexity and memory requirement will be theoretically analysed and validated by numerical results. Finally, we will confirm that the complexity and memory reductions are realised with preserving the error-correction capability of the ASD algorithm.

Algorithm 960

The design and implementation of a Matlab object-oriented software library for working with polynomials is presented. The construction and evaluation of polynomials in Bernstein form are motivated and justified. Efficient constructions for the coefficients of a polynomial in Bernstein form when the polynomial is not given with this representation are provided. The presented adaptive evaluation algorithm uses the VS (Volk and Schumaker) algorithm, the de Casteljau algorithm, and a compensated VS algorithm. In addition, we have completed the library with other algorithms to perform other usual operations with polynomials in Bernstein form.

Finding minimal polynomials of algebraic numbers of the form tan2(π/n) using Tschirnhausen’s transform

Solutions of two problems are proposed based on the Tschirnhausen transform. The first problem is connected with the construction of minimal polynomials of the numbers of the form tan2(π/n) by means of the Tschirnhausen transform for all natural n > 2. The second problem consists in finding the exact roots of the equation x3 − 7x − 7 = 0. A solution of the problem is obtained from the fact that the roots of the equation produce the cyclotomic field Q7. Examples of construction of minimal polynomials are provided.

Orthogonal polynomials, repeated measures, and SPSS

ABSTRACTWe discuss the construction of discrete orthonormal polynomials, using MAPLE procedures. We also study two important applications of these polynomials in statistics: in multiple linear regression and in repeated measures analysis. In particular, it is argued that the tests given by SPSS for linear and other trends in a within-subject factor are inefficient. Examples are given, including two (from psychology and medicine, respectively) which involve repeated measures and SPSS. Extensive tables of discrete orthornormal polynomials are given in the Appendix.

Kinetic feedback design for polynomial systems

New computational methods are proposed in this paper to construct polynomial feedback controllers for the stabilization of polynomial systems with linear input structure around a positive equilibrium point. Using the theory of chemical reaction networks (CRNs) and previous results on dynamical equivalence, a complex balanced or weakly reversible zero deficiency closed loop realization is achieved by computing the gain matrix of a polynomial feedback using optimization. It is shown that the feedback resulting in a complex balanced closed loop system having a prescribed equilibrium point can be computed using linear programming (LP). The robust version of the problem, when a convex set of polynomial systems is given over which a stabilizing controller is searched for, is also solvable with an LP solver. The feedback computation for rendering a polynomial system to deficiency zero weakly reversible form can be solved in the mixed integer linear programming (MILP) framework. It is also shown that involving new monomials (complexes) into the feedback does not improve the solvability of the problems. The proposed methods and tools are illustrated on simple examples, including stabilizing an open chemical reaction network.

On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan

In this paper we deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the $Q$-transform employed previously by Varshamov and Meyn for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper we construct infinite sequences of irreducible polynomials of non-decreasing degree starting from any irreducible polynomial.