Polynomial Research Articles

Fuzzy pogroupoids

In this paper we show that, for given a pogroupoid ( X, ·), the associated poset ( X, ⩽) is ( C 2 + 1)-free if and only if the relation ▷ μ is transitive for any fuzzy subset μ of X. Also we determine the set C( X, ·) of fuzzy subsets μ such that μ( x · y) = μ( y · x) for all x, y ∈ X. Furthermore, with a given finite poset ( X, ⩽) or the associated pogroupoid ( X, ·) we may then associate a polynomial whose coefficients count the number of congruence classes mod( X, ·) of fuzzy subsets of X along with another polynomial invariant of interest.

Oriented Matroids and Associated Valuations

It is possible to associate a valuation on the “orthant lattice” with each oriented matroid. In the case of uniform oriented matroids, it is not difficult to provide a characterization of the corresponding valuations. This is done here, thereby establishing a new characterization of the uniform oriented matroids themselves. Additionally, the connection between the valuations and the total polynomials associated with uniform oriented matroids is noted.

Essential covers of the cube by hyperplanes

A set L of linear polynomials in variables X 1 , X 2 , … , X n with real coefficients is said to be an essential cover of the cube { 0 , 1 } n if (E1) for each v ∈ { 0 , 1 } n , there is a p ∈ L such that p ( v ) = 0 ; (E2) no proper subset of L satisfies (E1), that is, for every p ∈ L , there is a v ∈ { 0 , 1 } n such that p alone takes the value 0 on v; (E3) every variable appears (in some monomial with non-zero coefficient) in some polynomial of L. Let e ( n ) be the size of the smallest essential cover of { 0 , 1 } n . In the present note we show that 1 2 ( 4 n + 1 + 1 ) ⩽ e ( n ) ⩽ n 2 + 1 .

The Complexity of Factors of Multivariate Polynomials

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework o...

On building polynomials

On building polynomials

A generalized trigonometric series function model for determining ionospheric delay*

Abstract A generalized trigonometric series function (GTSF) model, with an adjustable number of parameters, is proposed and analyzed to study ionosphere by using GPS, especially to provide ionospheric delay correction for single frequency GPS users. The preliminary results show that, in comparison with the trigonometric series function (TSF) model and the polynomial (POLY) model, the GTSF model can more precisely describe the ionospheric variation and more efficiently provide the ionospheric correction when GPS data are used to investigate or extract the earth's ionospheric total electron content. It is also shown that the GTSF model can further improve the precision and accuracy of modeling local ionospheric delays.

On multilinear oscillatory singular integrals with rough kernels

In this paper, for the multilinear oscillatory singular integral operators T A defined by T Af(x)= p.v. ∫ R n e iP(x,y) Ω(x−y) |x−y| n+m R m+1(A;x,y)f(y) dy, n⩾2, where P( x, y) is a nontrivial and real-valued polynomial defined on R n× R n , Ω(x) is homogeneous of degree zero on R n , A( x) has derivatives of order m in Λ ̇ β (0< β<1), R m+1 ( A; x, y) denotes the ( m+1)th remainder of the Taylor series of A at x expended about y, the author proves that if Ω∈L q(S n−1) for some q>1/(1− β), then for any p∈(1,∞), T A is bounded on L p( R n) . Meanwhile, the weighted L p -boundedness of T A is also given.

THE ABELIAN INTEGRALS OF A ONE-PARAMETER HAMILTONIAN SYSTEM UNDER POLYNOMIAL PERTURBATIONS

An upper bound Z(2,n)≤[n/2]+6[(n-1)/2]+5 is derived for the number of zeros of Abelian integrals Ĩ(h)=∮γ(h)g(x,y)dx-f(x,y)dy on the open interval (0,1/6), where γ(h) is an oval lying on the algebraic curve Hλ=(1/3)x3+(1/2)x2+λy3-y2=h, f(x,y), g(x,y) are polynomials of x and y, and n= max { deg f(x,y), deg g(x,y)}. The proof exploits the expansion of [Formula: see text] near λ=0.

Interpolation and approximation by polynomials

* Preface * Univariate Interpolation * Best Approximation * Numerical Integration * Peano's Theorem and Applications * Multivariate Interpolation * Splines * Bernstein Polynomials * Properties of the q-integers * References * Index *

A goodness-of-fit test for a polynomial errors-in-variables model

A goodness-of-fit test for a polynomial errors-in-variables model

Rapid quantification of carotenoids and fat in Atlantic Salmon (Salmo salar L.) by Raman spectroscopy and chemometrics.

Raman spectroscopy (785 nm excitation) was used to determine the overall carotenoid (astaxanthin and cantaxanthin) and fat content in 49 samples of ground muscle tissue from farmed Atlantic salmon (Salmo salar L.). Chemically determined contents ranged from 1.0 to 6.8 mg/kg carotenoids and 36 to 205 g/kg fat. In addition to the raw Raman spectra, three types of spectral preprocessing were evaluated: the first derivative, subtraction of the fitted fourth-order polynomial (POLY), and the intensity normalized versions of POLY (POLY-SNV). Further, variable selection based on significance testing by use of jack-knifing was performed on each spectral data set. Partial least-squares regression resulted in a root mean square error of prediction of 0.33 mg/kg (R = 0.97) for carotenoids for the variable selected versions of all the preprocessed spectral data sets. The fat content was best estimated by the variable selected POLYSNV, resulting in a root mean square error of prediction of 15.5 g/kg (R = 0.95). Both preprocessing and variable selection improved the regression models significantly. The results demonstrate that Raman spectroscopy is a suitable method for simultaneous, rapid, and nondestructive quantification of both pigments and fat in ground salmon muscle tissue.

The Fedoryuk Condition and the Łojasiewicz Exponent near a Fibre of a Polynomial

The Fedoryuk Condition and the Łojasiewicz Exponent near a Fibre of a Polynomial

On the Łojasiewicz Exponent near the Fibre of a Polynomial

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than $-1$ then they are attained at a curve meromorphic at infinity.

Diophantine equations E(x) = P(x) with E exponential, P polynomial

Diophantine equations E(x) = P(x) with E exponential, P polynomial

Arc-analyticity and polynomial arcs

Arc-analyticity and polynomial arcs

Kloosterman sum identities over [formula omitted

We introduce Kloosterman polynomials over F 2 m , and use these polynomials to prove three identities involving Kloosterman sums over F 2 m .

An algorithm to find a coordinate’s mate

In this paper an algorithm is given to decide if a given polynomial in two variables with coefficients in a finitely generated K-algebra is a coordinate and if so, find a mate for this polynomial.

Solution to a problem of S. Payne

A problem posed by S. Payne calls for determination of all linearized polynomials $f(x)\in \mathbb {F}_{2^n}[x]$ such that $f(x)$ and $f(x)/x$ are permutations of $\mathbb {F}_{2^n}$ and $\mathbb {F}_{2^n}^*$ respectively. We show that such polynomials are exactly of the form $f(x)=ax^{2^k}$ with $a\in \mathbb {F}_{2^n}^*$ and $(k,n)=1$. In fact, we solve a $q$-ary version of Payne’s problem.

On the use of orthogonal polynomials in the study of anisotropic plates

Fil: Nallim, Liz. Universidad Nacional de Salta. Facultad de Ingenieria; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Salta; Argentina

Necessary conditions for a polynomial to be chromatic

Necessary conditions for a polynomial to be chromatic