Admissible Polynomials Research Articles

Chebyshev admissible meshes and Lebesgue constants of complex polynomial projections

We construct admissible polynomial meshes on piecewise polynomial or trigonometric curves of the complex plane, by mapping univariate Chebyshev points. Such meshes can be used for polynomial least-squares, for the extraction of Fekete-like and Leja-like interpolation sets, and also for the evaluation of their Lebesgue constants.

Sunset similarity solution for a receding hydraulic fracture

This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used to show that a receding hydraulic fracture has a linear aperture asymptote$\hat {w}\sim \hat {s}$in the fracture tip, where$\hat {s}$is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers$N\ge 2$and values of the radius decay exponent$\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree$N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with$\gamma =1/2$called the sunset solution. For the reverse time$t^{\prime }$measured from closure, the sunset solution is characterized by$w\sim t^{\prime }$and$R\sim t^{\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as$t^{\prime }\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.

Toward super‐approximation in positive characteristic

In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $\Omega$ is a finite symmetric subset of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $\Gamma$ be the group generated by $\Omega$. Suppose the Zariski-closure of $\Gamma$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of ${\rm Ad}(\Gamma)$ is $\mathbb{F}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\rm Cay}(\pi_{f(x)}(\Gamma),\pi_{f(x)}(\Omega))$ as $f$ ranges in the set of $c_0$-admissible polynomials is a family of expanders, where $\pi_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.

Sparse bounds for discrete singular Radon transforms

We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:\mathbb{Z}^d\to \mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.

Adomian Decomposition Method applied to anisotropic thick plates in bending

The aim of this report is to apply the Adomian Decomposition Method (ADM) to anisotropic rectangular moderately thick plates under linear bending using a first-order shear deformation theory. Given the plate's homogeneity, the transverse displacement and rotations degrees-of-freedom are globally interpolated by kinematically admissible polynomials functions. The Rayleigh-Ritz Method is employed to solve the equilibrium equations. As proposed by the original Adomian Decomposition Method, the degrees-of-freedom are expanded into an infinite series, but the differential operator is additively decomposed following constitutive hierarchy. The initial solution refers to a lower lower constitutive symmetry in the operator decomposition and is enhanced by non-isotropic ones. The recursive system requirements in order to have convergence are presented and they depend only on the anisotropic index. The results obtained by the methodology are discussed and compared to those found in the literature. Solutions for different boundary conditions types as well as different thickness under uniform loading are presented so as to provide benchmark solutions. Furthermore, analysis of the anticlastic curvature and the effects of the transverse shear at corners with soft simply-supported edges, largely ignored by the literature, are shown and discussed.

Polynomial fitting and interpolation on circular sections

We construct Weakly Admissible polynomial Meshes (WAMs) on circular sections, such as symmetric and asymmetric circular sectors, circular segments, zones, lenses and lunes. The construction resorts to recent results on subperiodic trigonometric interpolation. The paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions.

Probabilistic spherical Marcinkiewicz–Zygmund inequalities

Recently, norm equivalences between spherical polynomials and their sample values at scattered sites have been proved. These so-called Marcinkiewicz–Zygmund inequalities involve a parameter that characterizes the density of the sampling set and they are applicable to all polynomials whose degree does not exceed an upper bound that is determined by the density parameter. We show that if one is satisfied by norm equivalences that hold with prescribed probability only, then the upper bound for the degree of the admissible polynomials can be enlarged significantly and that then, moreover, there exist fixed sampling sets which work for polynomials of all degrees.

Vibrations of skewed cantilevered triangular, trapezoidal and parallelogram Mindlin plates with considering corner stress singularities

Based on the Mindlin shear deformation plate theory, a method is presented for determining natural frequencies of skewed cantilevered triangular, trapezoidal and parallelogram plates using the Ritz method, considering the effects of stress singularities at the clamped re-entrant corner. The admissible displacement functions include polynomials and corner functions. The admissible polynomials form a mathematically complete set and guarantee the solution convergent to the exact frequencies when sufficient terms are used. The corner functions properly account for the singularities of moments and shear forces at the re-entrant corner and accelerate the convergence of the solution. Detailed convergence studies are carried out for plates of various shapes to elucidate the positive effects of corner functions on the accuracy of the solution. The results obtained herein are compared with those obtained by other investigators to demonstrate the validity and accuracy of the solution. Copyright © 2005 John Wiley & Sons, Ltd.

On the Low Velocity Impact Response of Laminated Composite Plates using the P-Version Ritz Method

Low energy impact in composite laminates is often a crucial and destructive loading condition since it leads to significant internal damage, undetectable by visual inspection. Low velocity impact upon a laminated plate imposes a complex stress state mainly due to the structural heterogeneity resulting by the ply orientation of the constituent laminae and the contact boundary conditions, which lead to a loading path that varies with the impact energy and the properties of the impactor-impacted plate system. The present work deals with the development of a numerical scheme for the calculation of the dynamic response of any type of laminated composite plates under low-velocity impact. The governing non-linear, second order differential equations are derived using p-Ritz admissible polynomial functions and the elastoplastic version of the Hertzian contact law. The dynamic response of fully clamped, cross ply and angle ply composite plates are investigated.

Polynomial controllers for linear systems

In this paper, we study the problem of controlling a linear system with polynomial controls. We establish upper bounds of admissible polynomial controls in two special cases: systems with distinct eigenvalues and systems with all real eigenvalues. For certain other classes we show that the existence of such control of a minimum possible degree is equivalent to questions about the existence of multiple zeros for certain classes of entire functions.

Parsimonious modelling of link flexibility in a robotic system

Using a Hamiltonean approach for a flexible robot arm allows the derivation of a complete distributed parameter model, i.e. including equations of motion and boundary conditions. Then a modal expansion based on a finite set of low degree admissible polynomials results into a reduced order lumped parameter model. This approximated model has been validated by comparison with theoretical and experimental results.

Admissible sequential polynomial estimators for stochastic processes

The problem of sequential estimation for stochastic processes whose likelihood functions are of exponential type is considered. In sequential estimation plans satisfying such regularity conditions under which the Wald identities hold the form of admissible polynomial estimators under quadratic loss function is determined.

The open case for simple alternative rings

2. Shirshov's result. We now state a convenient version of one of Shirshov's theorems. Let X be any set (of 'indeterminates'). Let Nx(F) and Sx(F) be, respectively, the free nonassociative and free associative algebras over the field F and on the free generating set X. There is an obvious F-homomorphism a from Nx(F) onto Sx(F): in the standard representation az deletes all parentheses. Say p CNx(F) is admissible provided pa;50. Now suppose R is any (nonassociative) algebra over F. Say R satisfies the admissible polynomial identity (p.i.) p, provided p ENx(F) is admissible, and for every F-homomorphism 3 of Nx(F) into R, po3=O. We canl now state