Polynomials In Variables Research Articles

A new direct second-order reliability analysis method

• A new study of the second-order reliability method (SORM) is presented. • All the information from the fitted quadratic polynomial model of the limit state function is used. • The cumulant generating function of the fitted quadratic polynomial surface in normalized standard variables is derived analytically. • The saddlepoint approximation is utilized for reliability analysis. • The proposed method is more accurate than the conventional SORM. In the second-order reliability method, the limit state function in arbitrarily distributed random variables is approximated by a quadratic polynomial of standard normal variables. The fitted quadratic polynomial is then used to calculate the probability of failure of the limit state. However, a closed-form solution for the probability of failure of a general quadratic polynomial surface is not available. As such, in this paper, a new second-order reliability method for reliability analysis is presented using saddlepoint approximation. The second-order approximation of the limit state function is obtained by the second-order Taylor series expansion at the most probable point. The cumulant generating function of the fitted quadratic polynomial of standard normal variables is derived analytically. The saddlepoint approximation is utilized to generate the probability density function, cumulative distribution function and probability of failure of the limit state. Finally, three numerical examples are used to compare the performance of the proposed second-order reliability with that of the first-order reliability method, the conventional second-order reliability method, and Monte Carlo simulation. The comparisons show that the proposed second-order reliability method gives accurate, convergent, and computationally efficient estimates of the probability of failure.

A characterization of Nikolskii–Besov classes via integration by parts

In this note we give a characterization of Nikolskii–Besov classes of functions of fractional smoothness (see [1–3]) by means of a nonlinear integration by parts formula in the form of a certain nonlinear inequality. This characterization is motivated by the recent papers [4–6] on distributions of polynomials in Gaussian random variables, where it has been shown that the distribution densities of nonconstant polynomials in Gaussian random variables belong to Nikolskii–Besov classes. Our main result is a generalization of the classical description of the class BV of functions of bounded variation in terms of integration by parts.

A Note on Lawton’s Theorem

AbstractWe prove Lawton’s conjecture about the upper bound on themeasure of the set on the unit circle on which a complex polynomial with a bounded number of coefficients takes small values. Namely, we prove that Lawton’s bound holds for polynomials that are not necessarily monic. We also provide an analogous bound for polynomials in several variables. Finally, we investigate the dependence of the bound on the multiplicity of zeros for polynomials in one variable.

Lower bounds for polynomials of many variables

A polynomial P(ξ) = P(ξ1,..., ξ n ) is said to be almost hypoelliptic if all its derivatives D ν P(ξ) can be estimated from above by P(ξ) (see [16]). By a theorem of Seidenberg-Tarski it follows that for each polynomial P(ξ) satisfying the condition P(ξ) > 0 for all ξ ∈ R n , there exist numbers σ > 0 and T ∈ R1 such that P(ξ) ≥ σ(1 + |ξ|) T for all ξ ∈ R n . The greatest of numbers T satisfying this condition, denoted by ST(P), is called Seidenberg-Tarski number of polynomial P. It is known that if, in addition, P ∈ I n , that is, |P(ξ)| → ∞ as |ξ| → ∞, then T = T(P) > 0. In this paper, for a class of almost hypoelliptic polynomials of n (≥ 2) variables we find a sufficient condition for ST(P) ≥ 1. Moreover, in the case n = 2, we prove that ST(P) ≥ 1 for any almost hypoelliptic polynomial P ∈ I2.

On the Properties of q-Bernstein-type Polynomials

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified $q$-Bernoulli numbers and $q$-Euler numbers applying $p$-adic $q$-integral representation on $\mathbb {Z}_p$ and $p$-adic fermionic $q$-invariant integral on $\mathbb {Z}_p$, respectively, to the inverse of $q$-Bernstein polynomials.

Primeness property for graded central polynomials of verbally prime algebras

Let F be an infinite field. The primeness property for central polynomials of Mn(F) was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for Mn(F) and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider Mn(R), where R admits a regular grading, with a grading such that Mn(F) is a homogeneous subalgebra and provide sufficient conditions – satisfied by Mn(E) with the trivial grading – to prove that Mn(R) has the primeness property if Mn(F) does. We also prove that the algebras Ma,b(E) satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.

A superintegrable discrete harmonic oscillator based on bivariate Charlier polynomials

A simple discrete model of the two-dimensional isotropic harmonic oscillator is presented. It is superintegrable with su(2) as its symmetry algebra. It is constructed with the help of the algebraic properties of the bivariate Charlier polynomials. This adds to the other discrete superintegrable models of the oscillator based on Krawtchouk and Meixner orthogonal polynomials in several variables.

Vieta’s formulae for regular polynomials of a quaternionic variable

Abstract Vieta’s classical formulae explicitly determine the coefficients of a polynomial p ∈ 𝔽[x] in terms of the roots of p, where 𝔽 is any commutative ring. In this paper, Vieta’s formulae are obtained for slice-regular polynomials over the noncommutative algebra of quaternions, by an argument which essentially relies on induction, without invoking quasideterminants or noncommutative symmetric functions.

Development and validation study of a non-alcoholic fatty liver disease risk scoring model among adults in China.

Non-alcoholic fatty liver disease (NAFLD) is one of the most common liver diseases in China. It is usually asymptomatic and transabdominal ultrasound (USS) is the usual means for diagnosis, but it may not be feasible to have USS screening of the whole population. To develop a risk scoring model for predicting the presence of NAFLD using parameters that can be easily obtain in clinical settings. A retrospective study on the data of 672 adults who had general health check including a transabdominal ultrasound. Fractional polynomial and multivariable logistic regressions of sociodemographic and biochemical variables on NAFLD were used to identify the predictors. A risk score was assigned to each predictor using the scaled standardized β-coefficient to create a risk prediction algorithm. The accuracy for NAFLD detection by each cut-off score in the risk algorithm was evaluated. The prevalence of NAFLD in our study population was 33.0% (222/672). Six significant factors were selected in the final prediction model. The areas under the curve (AUC) was 0.82 (95% CI: 0.78-0.85). The optimal cut-off score, based on the ROC was 35, with a sensitivity of 76.58% (95% CI: 70.44-81.98%) and specificity of 74.89% (95% CI: 70.62-78.83%). A NAFLD risk scoring model can be used to identify asymptomatic Chinese people who are at risk of NAFLD for further USS investigation.

FPGA-based implementation of online selective harmonic elimination PWM for voltage source inverter

ABSTRACTThis paper presents a field programmable gate array (FPGA)-based real-time selective harmonic elimination pulse width modulation (SHE-PWM) scheme for voltage source inverter (VSI). The proposed algorithm can generate the switching angles in real time through the approximation of the angles trajectories by simple variable coefficients-based polynomial equations. The proposed algorithm is first presented, analysed, and then its accuracy is checked through angle error analysis. Afterwards, an FPGA-based implementation is used and an induction motor drive is experimentally tested to validate the simulation results. Experimental results showed a good agreement with simulation and proved the validity as well as precision of the proposed algorithm. All intended selected harmonics are eliminated and simultaneously the fundamental is controlled. The output voltage and current present an excellent frequency spectrum with low harmonic distortion and power loss values.

Wiener–Hermite polynomial expansion for multivariate Gaussian probability measures

This paper introduces a new, transformation-free, generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation are discussed. New analytical formulae are proposed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Besides, the variance formula of the generalized PCE contains additional terms due to statistical dependence among Gaussian variables. The additional terms vanish when the Gaussian variables are statistically independent, reverting the generalized PCE to the classical PCE. Numerical examples illustrate the generalized PCE approximation in estimating the statistical properties of various output variables.

Linear equations over multiplicative groups, recurrences, and mixing III

Given an algebraic$\mathbf{Z}^{d}$-action corresponding to a prime ideal of a Laurent ring of polynomials in several variables, we show how to find the smallest order$n+1$of non-mixing. It is known that this is determined by the non-mixing sets of size$n+1$, and we show how to find these in an effective way. When the underlying characteristic is positive and$n\geq 2$, we prove that there are at most finitely many classes under a natural equivalence relation. We work out two examples, the first with five classes and the second with 134 classes.

New Hamiltonians for loop quantum cosmology with arbitrary spin representations

In loop quantum cosmology, one has to make a choice of SU(2) irreducible representation in which to compute holonomies and regularize the curvature of the connection. The systematic choice made in the literature is to work in the fundamental representation, and very little is known about the physics associated with higher spin labels. This constitutes an ambiguity of which the understanding, we believe, is fundamental for connecting loop quantum cosmology to full theories of quantum gravity like loop quantum gravity, its spin foam formulation, or cosmological group field theory. We take a step in this direction by providing here a new closed formula for the Hamiltonian of flat Friedmann-Lema\^{\i}tre-Robertson-Walker models regularized in a representation of arbitrary spin. This expression is furthermore polynomial in the basic variables which correspond to well-defined operators in the quantum theory, takes into account the so-called inverse-volume corrections, and treats in a unified way two different regularization schemes for the curvature. After studying the effective classical dynamics corresponding to single and multiple-spin Hamiltonians, we study the behavior of the critical density when the number of representations is increased and the stability of the difference equations in the quantum theory.

Connection coefficients for classical orthogonal polynomials of several variables

Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied. For the Jacobi polynomials on the simplex, it is shown that the connection coefficients can be expressed in terms of Tratnik's multivariable Racah polynomials and their weights. This gives, in particular, a new interpretation of the hidden duality between the variables and the degree indices of the Racah polynomials, which lies at the heart of their bispectral properties. These techniques also lead to explicit formulas for connection coefficients of Hahn and Krawtchouk polynomials of several variables, as well as for orthogonal polynomials on balls and spheres.

Value-sharing and uniqueness problems for non-Archimedean differential polynomials in several variables

We consider differential polynomials of Fermat–Waring type, const-ructed using polynomials of Yi’s type for meromorphic functions in a non-Archimedean field. Similarly to the Hayman Conjecture, we prove that the considered differential polynomials assume all values. We established also a uniqueness theorem for these differential polynomials.

Symmetric powers of symmetric bilinear forms, homogeneous orthogonal polynomials on the sphere and an application to compact Hyperkähler manifolds

The Beauville–Fujiki relation for a compact Hyperkähler manifold [Formula: see text] of dimension [Formula: see text] allows to equip the symmetric power [Formula: see text] with a symmetric bilinear form induced by the Beauville–Bogomolov form. We study some of its properties and compare it to the form given by the Poincaré pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere [Formula: see text], or equivalently, over [Formula: see text] with a Gaussian kernel.

An Averaged Chowla and Elliott Conjecture Along Independent Polynomials

We generalize a result of Matomaki, Radziwill, and Tao, by proving an averaged version of a conjecture of Chowla and a conjecture of Elliott regarding correlations of the Liouville function, or more general bounded multiplicative functions, with shifts given by independent polynomials in several variables. A new feature is that we recast the problem in ergodic terms and use a multiple ergodic theorem to prove it; its hypothesis is verified using recent results by Matomaki and Radziwill on mean values of multiplicative functions on typical short intervals. We deduce several consequences about patterns that can be found on the range of various arithmetic sequences along shifts of independent polynomials.

A multidimensional discontinuous Galerkin modeling framework for overland flow and channel routing

In this paper, we present the development and application of a new multidimensional, unstructured-mesh model for simulating coupled overland/open-channel flows in the kinematic wave approximation regime. The modeling approach makes use of discontinuous Galerkin (DG) finite element spatial discretizations of variable polynomial degree p, paired with explicit Runge–Kutta time steppers, and is supported by advancements made to an automatic mesh generation tool, Admesh +, that is used to construct constrained triangulations for channel routing. The developed modeling framework is applied to a set of four test cases, where numerical results are found to compare well with known analytic solutions, experimental data and results from another well-established (structured, finite difference) model within the area of application. The numerical results obtained demonstrate the accuracy and robustness of the developed modeling framework and highlight the potential benefits of using p (polynomial) refinement for hydrological simulations.

Numerical testing of quantitative phase-field models with different polynomials for isothermal solidification in binary alloys

Quantitative phase-field models have been developed as feasible computational tools for solving the free-boundary problem in solidification processes. These models are constructed with some polynomials of the phase-field variable that describe variations of the physical quantities inside the diffuse interface. The accuracy of the simulation depends on the choice of the polynomials and such dependence is indispensable for high-performance computing and valuable for extending the range of applications of the model to several physical systems. However, little is known about the dependence of the accuracy on the choice of the polynomials. In this study, numerical testing is carried out for quantitative phase-field models with extensive sets of polynomials (24 different models) for isothermal solidification in binary alloys. It is demonstrated in two-dimensional simulations of dendritic growth that a specific set of polynomials must be employed to achieve high accuracy in the models with double-well and double-obstacle potentials. Both types of model with the best set of polynomials yield almost the same numerical accuracy.

An efficient numerical progressive diagonalization scheme for the quantum Rabi model revisited

An efficient numerical progressive diagonalization scheme for the quantum Rabi model is revisited. The advantage of the scheme lies in the fact that the quantum Rabi model can be solved almost exactly by using the scheme that only involves a finite set of one variable polynomial equations. The scheme is especially efficient for a specified eigenstate of the model, for example, the ground state. Some low-lying level energies of the model for several sets of parameters are calculated, of which one set of the results is compared to that obtained from the Braak’s exact solution proposed recently. It is shown that the derivative of the entanglement measure defined in terms of the reduced von Neumann entropy with respect to the coupling parameter does reach the maximum near the critical point deduced from the classical limit of the Dicke model, which may provide a probe of the critical point of the crossover in finite quantum many-body systems, such as that in the quantum Rabi model.