Construction Of Polynomials Research Articles

Construction of self-reciprocal normal polynomials over finite fields of even characteristic

In this paper, a computationally simple and explicit construction of some sequences of normal polynomials and self-reciprocal normal polynomials over finite fields of even characteristic are presented.

Two-Sided Derivatives for Regular Expressions and for Hairpin Expressions

The aim of this paper is to design a polynomial construction of a finite recognizer for hairpin completions of regular languages. This is achieved by considering completions as new expression operators and by applying derivation techniques to the associated extended expressions called hairpin expressions. More precisely, we extend partial derivation of regular expressions to two-sided partial derivation of hairpin expressions and we show how to deduce a recognizer for a hairpin expression from its two-sided derived term automaton, providing an alternative proof of the fact that hairpin completions of regular languages are linear context-free.

Basis of symmetric polynomials for many-boson light-front wave functions.

We provide an algorithm for the construction of orthonormal multivariate polynomials that are symmetric with respect to the interchange of any two coordinates on the unit hypercube and are constrained to the hyperplane where the sum of the coordinates is one. These polynomials form a basis for the expansion of bosonic light-front momentum-space wave functions, as functions of longitudinal momentum, where momentum conservation guarantees that the fractions are on the interval [0,1] and sum to one. This generalizes earlier work on three-boson wave functions to wave functions for arbitrarily many identical bosons. A simple application in two-dimensional ϕ(4) theory illustrates the use of these polynomials.

On constructing complete permutation polynomials over finite fields of even characteristic

In this paper, a construction of complete permutation polynomials over finite fields of even characteristic proposed by Tu et al. recently is generalized in a recursive manner. Besides, several classes of complete permutation polynomials are derived by computing compositional inverses of known ones.

Reconstruction of score sets

Abstract The score set of a tournament is defined as the set of its different outdegrees. In 1978 Reid [15] published the conjecture that for any set of nonnegative integers D there exists a tournament T whose degree set is D. Reid proved the conjecture for tournaments containing n = 1, 2, and 3 vertices. In 1986 Hager [4] published a constructive proof of the conjecture for n = 4 and 5 vertices. In 1989 Yao [18] presented an arithmetical proof of the conjecture, but general polynomial construction algorithm is not known. In [6] we described polynomial time algorithms which reconstruct the score sets containing only elements less than 7. In [5] we improved this bound to 9. In this paper we present and analyze new algorithms Hole-Map, Hole-Pairs, Hole-Max, Hole-Shift, Fill-All, Prefix-Deletion, and using them improve the above bound to 12, giving a constructive partial proof of Reid’s conjecture.

Constructing permutations and complete permutations over finite fields via subfield-valued polynomials

We describe a recursive construction of permutation and complete permutation polynomials over a finite field Fpn by using Fpk-valued polynomials for several same or different factors k of n. As a result, we obtain some specific permutation polynomials which unify and generalize several previous constructions.

Probabilistic collocation method for strongly nonlinear problems: 2. Transform by displacement

AbstractThe probabilistic collocation method (PCM) is widely used for uncertainty quantification and sensitivity analysis. In paper 1 of this series, we demonstrated that the PCM may provide inaccurate results when the relation between the random input parameter and the model response is strongly nonlinear, and presented a location‐based transformed PCM (xTPCM) to address this issue, relying on the transform between response and location. However, the xTPCM is only applicable for one‐dimensional problems, and two or three‐dimensional problems in homogeneous media. In this paper, we propose a displacement‐based transformed PCM (dTPCM), which is valid in two or three‐dimensional problems in heterogeneous media. In the PCM, we first select collocation points and run model/simulator to obtain response, and then approximate the response by polynomial construction. Whereas, in the dTPCM, we apply motion analysis to transform the response to displacement. That is, the response field is now represented by the displacement field. Next, we approximate the displacement instead of the response by polynomial, since the displacement is more linear to the input parameter than the response. Finally, we randomly generate a sufficient number of displacement samples and transform them back to obtain response samples to estimate statistical properties. Through multiphase flow and solute transport examples, we demonstrate that the dTPCM provides much more accurate statistics than does the PCM, and requires considerably less computer time than does the Monte Carlo (MC) method.

A fast algorithm for the construction of integrity bases associated to symmetry-adapted polynomial representations: application to tetrahedral $$\mathrm {XY_4}$$ XY 4 molecules

Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques, for the construction of symmetry-adapted polynomials in the symmetry coordinates of a molecular system, because it is based on a finer description of the mathematical structure of the latter polynomials. The present article extends its use to the construction of polynomial bases which span possibly, non-totally symmetric irreducible representations of a molecular symmetry group. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the algorithm presented in this article, which is faster than algorithms based on projection operators only. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry-adapted polynomial bases are constructed alongside in the same fashion. The method is illustrated in detail on $$\mathrm {XY_4}$$ type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the supplemental material associated with this paper.

ALGUNOS ASPECTOS DE LA TEORÍA DE ONDÍCULAS

El objetivo de este trabajo es analizar brevemente, algunas estrategias que han sido desarrolladas para la construcción del polinomio trigonométrico m0 bajo ciertas condiciones. Asimismo, mostraremos la relación que existe entre la función escala ϕ, la Ondícula ψ y el polinomio trigonométrico m0. Por ello, el Análisis Multiresolución (AMR) juega un rol vital para la construcción de bases ortonormales de Ondículas. Dentro de las nuevas técnicas usadas, una de las más importantes es la teoría de Frames que tiene una relación directa con el análisisde Ondículas.

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.

Towards ℛ-matrix construction of Khovanov-Rozansky polynomials I. Primary T-deformation of HOMFLY

We elaborate on the simple alternative from arXiv:1308.5759 to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction consists of 2 steps: first, with every link diagram with m vertices one associates an m-dimensional hypercube with certain q-graded vector spaces, associated to its 2^m vertices. A generating function for q-dimensions of these spaces is what we suggest to call the primary T-deformation of HOMFLY polynomial -- because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R-matrices. The second step is a certain minimization of residues of this new polynomial with respect to T+1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube -- this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of associated complex, just in the original Khovanov's construction at N=2. This second step is still somewhat sophisticated -- though incomparably simpler than its conventional matrix-factorization counterpart. In this paper we concentrate on the first step, and provide just a mnemonic treatment of the second step. Still, this is enough to demonstrate that all the currently known examples of KR polynomials in the fundamental representation can be easily reproduced in this new approach. As additional bonus we get a simple description of the DGR relation between KR polynomials and superpolynomials and demonstrate that the difference between reduced and unreduced cases, which looks essential at KR level, practically disappears after transition to superpolynomials. However, a careful derivation of all these results from cohomologies of cut-and-join morphisms remains for further studies.

An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials

In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.

DESIGN OF DYADIC-INTEGER-COEFFICIENTS BASED BI-ORTHOGONAL WAVELET FILTERS FOR IMAGE SUPER-RESOLUTION USING SUB-PIXEL IMAGE REGISTRATION

This paper presents image super-resolution scheme based on subpixel image registration by the design of a specific class of dyadicinteger-coefficient based wavelet filters derived from the construction of a half-band polynomial. First, the integer-coefficient based halfband polynomial is designed by the splitting approach. Next, this designed half-band polynomial is factorized and assigned specific number of vanishing moments and roots to obtain the dyadic-integer coefficients low-pass analysis and synthesis filters. The possibility of these dyadic-integer coefficients based wavelet filters is explored in the field of image super-resolution using sub-pixel image registration. The two-resolution frames are registered at a specific shift from one another to restore the resolution lost by CCD array of camera. The discrete wavelet transform (DWT) obtained from the designed coefficients is applied on these two low-resolution images to obtain the high resolution image. The developed approach is validated by comparing the quality metrics with existing filter banks.

Construction Techniques of Generator Polynomials of BCH Codes

LetA0 ⊂ A1 ⊂ · · · ⊂ At−1 ⊂ At be a chain of unitary commutative rings (eachAi is constructed by the direct product of suitable Galois rings with multiplicative groupAi of units) andK0 ⊂ K1 ⊂ · · · ⊂ Kt−1 ⊂ Kt be the corresponding chain of unitary commutative rings (each Ki is constructed by the direct product of corresponding residue fields of given Galois rings, with multiplicative groups K∗ i of units), where t is a non negative integer. In this work presents three different types of constructions of generator polynomials of sequences of BCH codes having entries from Ai and K∗ i for each i, where 0 ≤ i ≤ t.

Constructing permutation polynomials from piecewise permutations

We present a construction of permutation polynomials over finite fields by using some piecewise permutations. Based on a matrix approach and an interpolation approach, several classes of piecewise permutation polynomials are obtained.

Permutation polynomials over finite fields involving [formula omitted

Let p be a prime and q=ps. For integer a≥0, let Sa=x+xq+⋯+xqa−1∈Fp[x]. We present three constructions of permutation polynomials of Fqe involving Sa which generalize several recent results. When q is even, the third construction produces a large class of Dembowski–Ostrom permutation polynomials. We also discuss an interesting linear algebraic problem arising from the third construction.

Filtering of MS/MS data for peptide identification.

BackgroundThe identification of proteins based on analysis of tandem mass spectrometry (MS/MS) data is a valuable tool that is not fully realized because of the difficulty in carrying out automated analysis of large numbers of spectra. MS/MS spectra consist of peaks that represent each peptide fragment, usually b and y ions, with experimentally determined mass to charge ratios. Whether the strategy employed is database matching or De Novo sequencing, a major obstacle is distinguishing signal from noise. Improved ability to distinguish signal peaks of low intensity from background noise increases the likelihood of correctly identifying the peptide, as valuable information is preserved while extraneous information is not left to mislead.ResultsThis paper introduces an automated noise filtering method based on the construction of orthogonal polynomials. By subdividing the spectrum into a variable number (3 to 11) of bins, peaks that are considered "noise" are identified at a local level. Using a De Novo sequencing algorithm that we are developing, this filtering method was applied to a published dataset of more than 3000 mass spectra and an original dataset of more than 300 spectra. The samples were peptides from purified known proteins; therefore, the solutions could be compared to the correct sequences and the peaks corresponding to b, y and other fragments of significance could be identified. The same procedure was applied using two other published filtering methods. The ratios of the number of significant peaks that were preserved relative to the total number of peaks in each spectrum were determined. In the event that filtering out too many or too few signal peaks can lead to inaccuracy in sequence determination, the percentage of amino acid residues in the correct positions relative to the total number of amino acid residues in the correct sequence was also calculated for each sequence determined.ConclusionsThe results show that an orthogonal polynomial-based method of distinguishing signal peaks from background in mass spectra preserves a greater portion of signal peaks than compared methods, improving accuracy in sequence determination.

The action of the orthogonal group on planar vectors: invariants, covariants and syzygies

The construction of invariant and covariant polynomials from the x, y components of n planar vectors under the SO(2) and O(2) orthogonal groups is addressed. Molien functions determined under the SO(2) symmetry group are used as a guide to propose integrity bases for the algebra of invariants and the modules of covariants. The Molien functions that describe the structure of the algebra of invariants and the free modules of (m)-covariants, m ⩽ n − 1, are written as a ratio of a numerator in λ with positive coefficients over a (1 − λ2)2n − 1 denominator. This form of single rational function is standard in invariant theory and has a clear symbolic interpretation. However, its usefulness is lost for the non-free modules of (m)-covariants, m ⩾ n, due to negative coefficients in the numerator. We propose for these non-free modules a new representation of the Molien function as a sum of n rational functions with positive coefficients in the numerators and different numbers of terms in the denominators. This non-standard form is symbolically interpreted in terms of a generalized integrity basis. Integrity bases are explicitly given for n = 2, 3, 4 planar vectors and m ranging from 0 to 5. The integrity bases obtained under the SO(2) symmetry group are subsequently extended to the O(2) group.

Semifields from skew polynomial rings

Abstract Skew polynomial rings are used to construct finite semifields, following from a construction of Ore and Jacobson of associative division algebras. Johnson and Jha [10] constructed the so-called cyclic semifields, obtained using irreducible semilinear transformations. In this work we show that these two constructions in fact lead to isotopic semifields, show how the skew polynomial construction can be used to calculate the nuclei more easily, and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in [13] and implicitly by Dempwolff in [2].

Adaptive Construction of Surrogates for the Bayesian Solution of Inverse Problems

The Bayesian approach to inverse problems typically relies on posterior sampling approaches, such as Markov chain Monte Carlo, for which the generation of each sample requires one or more evaluations of the parameter-to-observable map or forward model. When these evaluations are computationally intensive, approximations of the forward model are essential to accelerating sample-based inference. Yet the construction of globally accurate approximations for nonlinear forward models can be computationally prohibitive and in fact unnecessary, as the posterior distribution typically concentrates on a small fraction of the support of the prior distribution. We present a new approach that uses stochastic optimization to construct polynomial approximations over a sequence of measures adaptively determined from the data, eventually concentrating on the posterior distribution. The approach yields substantial gains in efficiency and accuracy over prior-based surrogates, as demonstrated via application to inverse problems in partial differential equations.