Polynomials In Variables Research Articles

Three term relations for multivariate Uvarov orthogonal polynomials

Three term relations for orthogonal polynomials in several variables associated to a moment linear functional obtained by a Uvarov modification of a given moment functional are studied. Existence of Uvarov orthogonal polynomials is analyzed, stating conditions to ensure it. The matrices of the three term relations of the Uvarov orthogonal polynomials are explicitly given in terms of the matrices of the three term relations satisfied by the original family. Two examples are presented in order to show that the results are valid for positive definite linear functionals and also for some quasi definite linear functionals which are not positive definite.

Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms

As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure are applied. Second, completions and other results on solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed. Third, the characterization of the constraints T1≤T≤T2 for the linear operator T, only in terms of quadratic forms, is deduced. Here, T1, T,and T2 are bounded linear operators. Concrete spaces, operators, and functionals are involved in our corollaries or examples.

Two variable Freud orthogonal polynomials and matrix Painlevé-type difference equations

We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyse relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the coefficients of the structure relations satisfied by these bivariate semiclassical orthogonal polynomials, also a matrix differential-difference equation for the bivariate orthogonal polynomials is deduced. The extension of the Painlevé equation for the coefficients of the three term relations to the bivariate case and a two dimensional version of the Langmuir lattice are obtained.

VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV

AbstractWe prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$ . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$ -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

Dimension-Free Entanglement Detection in Multipartite Werner States

Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a complete characterization of their entanglement that is independent of the underlying local Hilbert space: for every entangled Werner state there exists a dimension-free entanglement witness. The construction of such a witness is formulated as an optimization problem. To solve it, two semidefinite programming hierarchies are introduced. The first one is derived using real algebraic geometry applied to positive polynomials in the entries of a Gram matrix, and is complete in the sense that for every entangled Werner state it converges to a witness. The second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables, and is a relaxation that involves smaller semidefinite constraints.

On the zeros of a quaternionic polynomial with restricted coefficients

In this paper, we are concerned with the problem of locating the zeros of regular polynomials of a quaternionic variable with restricted quaternionic coefficients. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results generalize some recently proved results about the distribution of zeros of a quaternionic polynomial.

Polynomial-based model for estimating instantaneous fuel consumption

The current steady-state fuel consumption models are not optimal due to their large estimation error, complex structure, excess model coefficients, and relatively low predicting accuracy of fuel consumption. Based on experimental data of a passenger car’s fuel consumption, this study analyzes advantages and disadvantages of existing models, proposes a unified expression of the transient variable polynomial fuel-consumption model, introduces the Bayesian information criterion (BIC) for model selection, and establishes an instantaneous fuel-consumption model with simple structure and high accuracy. The simulation results show that, compared with the BIT-TFCM models, the root mean square error (RMSE) and the mean absolute percentage error (MAPE) of the proposed model are 6.27% and 6.43% smaller, respectively, on average. The proposed model has the advantages of simple structure, high precision, less influence from working conditions, stable performance, and easy prediction of the fuel consumption of future working conditions, hence providing a foundation for economic path planning.

Irreducibility of integer-valued polynomials in several variables

Let \(\underline{S}\) be an arbitrary subset of \(R^n\) where R is a domain with the field of fractions \(\mathbb {K}\). Denote the ring of polynomials in n variables over \(\mathbb {K}\) by \(\mathbb {K}[\underline{x}] \). The ring of integer-valued polynomials over \(\underline{S}\), denoted by Int\((\underline{S},R)\), is defined as the set of the polynomials of \(\mathbb {K}[\underline{x}] \), which maps \(\underline{S}\) to R. In this article, we study the irreducibility of the polynomials of Int\((\underline{S},R)\) for the first time in the case when R is a Unique Factorization Domain. We also show that our results remain valid when R is a Dedekind domain or, sometimes, any domain.

Deflection and stress analysis of piezoelectric laminated composite plate under variable polynomial transverse loading

Among many smart materials, piezoelectric materials have emerged as the most studied ones for practical applications. They owe their success to several factors, including low price, high bandwidth, availability in various formats, and ease of handling and implementation. The present study focused on the performance of piezoelectric laminated composite plate under various electromechanical loading conditions by utilizing the first-order shear deformation theory with the Newton–Raphson residual and iteration with Gauss integration point in Ansys. For the first time, the effects of electrical loading, circuit arrangement, voltage variation, and polynomial variable transverse loading are studied over piezoelectric composite plate (PCP). The effects of plate aspect ratio, thickness ratio, boundary conditions, ply orientations, nature of loading conditions, and voltage variation are presented. The study also utilized open and close circuit arrangements as sensors and actuators to gauge the performance of PCP in the form of static bending analysis. The maximum OC (open circuit) output voltage is generated with the N4 type of loading compared with Ni (i = 1–4); on the other hand, the OC output voltage is minimum with N5. The combined effect of external load and voltage presented in the study will be useful for analyzing the deflection variation, and it can further be implemented in reducing deflection or vibration. It is noted that, with a higher piezoelectric to laminate thickness (t/h) ratio, the maximum OC output voltage is observed. In addition, the rate of voltage generation observed is the highest under the N4 loading condition and the lowest under the N5 type of load.

Maximal functions and multiplier theorem for Fourier orthogonal series

Under the assumption that orthogonal polynomials of several variables admit an addition formula, we can define a convolution structure and use it to study the Fourier orthogonal expansions on a space of homogeneous type. We define a maximal function via the convolution structure induced by the addition formula and use it to establish a Marcinkiewicz multiplier theorem. For the space of homogeneous type defined by a family of weight functions on conic domains, we show that the maximal function is bounded by the Hardy-Littlewood maximal function so that the multiplier theorem holds on conic domains.

Summing Hecke eigenvalues over polynomials

In this paper we estimate sums of the form \(\sum _{n\le X}|a_{{\text {Sym}}^m \pi }(|f(n)|)|\), for symmetric power lifts of automorphic representations \(\pi \) attached to holomorphic forms and polynomials \(f(x)\in {\mathbb {Z}}[x]\) of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when \(\deg (f)\le 4\). Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves.

Composite sets of polynomials of several complex variables

Composite sets of polynomials of several complex variables

Saturation of Newton polytopes of type A and D cluster variables

We study Newton polytopes for cluster variables in cluster algebras \(\mathcal{A}(\Sigma)\) of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the cluster variables of the initial seed \(\Sigma\). The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if \(\Sigma\) has principal coefficients or boundary frozen variables, then all cluster variable Newton polytopes are saturated. We also characterize when these Newton polytopes are empty; that is, when they have no non-vertex lattice points.Mathematics Subject Classifications: 13F60, 52B20Keywords: Cluster algebras, Newton polytopes, snake graphs

Optimal Number of Required Points for Evaluation - Interpolation Technique in Complex Basis

According to polynomial evaluation-interpolation technique, first of all, the set of required points, in which interpolation will be executed, must be defined in evaluation part. Taking into account that working in complex domain provides as with the advantage of conjugate properties, the number of evaluations at these points could be reduced. This paper presents the appropriate form of required points, aiming to an optimal reduction of their amount. The appropriate form of points for two variable polynomials depends on the degree of each variable. As a repercussion the number of evaluations could be reduced even up to the half.

Distributions of Polynomials in Gaussian Random Variables under Constraints on the Powers of Variables

Distributions of Polynomials in Gaussian Random Variables under Constraints on the Powers of Variables

Limits of the Tristram–Levine signature function

We show that under a precise condition on the single variable Alexander polynomial, the limit at one of the Tristram–Levine signature of a link is determined by the linking matrix.

Norms on complex matrices induced by complete homogeneous symmetric polynomials

We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension-independent tracial inequalities. Their potential merits further investigation.

Longitudinal shear performance of lightweight steel-UHPC composite connections based on large-diameter high strength friction-grip bolts

In order to improve the construction efficiency and reduce the self-weight of composite bridges, prefabricated thin UHPC (ultra-high performance concrete) decks can be connected to steel girders by high-strength, large-diameter friction grip through-bolts. This paper presents a pilot experimental study into the shear performances of such composite connections. Thirty push-out tests were carried out to observe the influences of bolt diameter, slab thickness, bolt strength, pretension force and random asymmetric clearance of the bolt within the slab and joist. The observed failure modes include bolt fracture, pronounced permanent flexure of the bolts due to extensive yield, and crushing/splitting of the UHPC. 36% and 25% increases in diameter (from 22 to 30 mm) and tensile resistance (from 795 to 995 MPa) in 55 mm thick slabs led to 43% and 11% improvement respectively in connection shear resistance. Moreover, a bolt pretension force increase from 150 to 200 kN and a slab decrease in thickness from 55 to 35 mm respectively changed the failure mode from bolt bending to bolt fracture and reduced the shear resistance by 15%. Random asymmetric clearances of the bolts between 0.17 and 5.43 mm led to 88%-100% reduction of connection stiffness at the first slip stage and decreased shear resistance by up to 11%. Regression analysis shows that connection shear strength, stiffness and ductility are mixed polynomial and exponential functions of the key connection variables.

How many roots of a system of random Laurent polynomials are real?

Abstract We say that a zero of a Laurent polynomial that lies on the unit circle with centre is real. We also say that a Laurent polynomial that is real on this circle is real. In contrast with ordinary polynomials, it is known that for random real Laurent polynomials of increasing degree the average proportion of real roots tends to rather than to . We show that this phenomenon of the asymptotically nonvanishing proportion of real roots also holds for systems of Laurent polynomials of several variables. The corresponding asymptotic formula is obtained in terms of the mixed volumes of certain convex compact sets determining the growth of the system of polynomials. Bibliography: 11 titles.

Attacking Jacobian Problem Using Resultant Theory

This paper introduces a relation between resultant and the Jacobian determinant by generalizing Sakkalis theorem from two polynomials in two variables to the case of (n) polynomials in (n) variables. This leads us to study the results of the type: , and use this relation to attack the Jacobian problem. The last section shows our contribution to proving the conjecture.