Description of aspheric surfaces

2019 ◽  
Vol 8 (3-4) ◽  
pp. 267-278
Author(s):  
Rainer Schuhmann
Keyword(s):  
Power Series ◽  
Orthogonal Polynomials ◽  
Aspheric Surfaces ◽  
Iso Standard ◽  
Toric Surfaces ◽  
Invariant Surfaces ◽  
Rotationally Invariant

Abstract Aspheric surfaces, in particular rotationally invariant surfaces, can be described according to the ISO standard 10110 Part 12 as sagitta functions of the surface coordinates. Usually, such functions are standardized as a combination of conic terms and power series or orthogonal polynomials. Similar functions are applied for surface forms, which are not rotationally invariant as cylindric and toric surfaces. In the following, different forms of describing aspheric surfaces as given in the standard as well as other forms will be presented and compared in an overview, and their special features will be discussed.

Padé approximation and orthogonal polynomials

1974 ◽  
Vol 10 (2) ◽  
pp. 263-270 ◽  
Author(s):  
G.D. Allen ◽  
C.K. Chui ◽  
W.R. Madych ◽  
F.J. Narcowich ◽  
P.W. Smith
Keyword(s):  
Power Series ◽  
Variational Method ◽  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Quadrature Formulas ◽  
The Past ◽  
Gaussian Quadrature Formulas ◽  
Poles And Zeros ◽  
Formal Power ◽  
Determinant Theory

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

The Orthogonal Polynomials of Power Series Probability Distributions and Their Uses

Biometrika ◽  
1966 ◽  
Vol 53 (1/2) ◽  
pp. 121
Author(s):  
D. F. I van Heerden ◽  
H. T. Gonin
Keyword(s):  
Power Series ◽  
Orthogonal Polynomials ◽  
Probability Distributions

scholarly journals Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis

2020 ◽  
Vol 10 (02) ◽  
pp. 2050003
Author(s):  
Diego Dominici
Keyword(s):  
Power Series ◽  
Asymptotic Analysis ◽  
Orthogonal Polynomials ◽  
Asymptotic Expansions ◽  
Linear Functional ◽  
Series Expansions ◽  
Recurrence Coefficients ◽  
Variable Formula ◽  
Polynomial Sequences ◽  
Power Series Expansions

We study the three-term recurrence coefficients [Formula: see text] of polynomial sequences orthogonal with respect to a perturbed linear functional depending on a variable [Formula: see text] We obtain power series expansions in [Formula: see text] and asymptotic expansions as [Formula: see text] We use our results to settle some conjectures proposed by Walter Van Assche and collaborators.

scholarly journals Gottlieb Polynomials and Their q-Extensions

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1499
Author(s):  
Esra ErkuŞ-Duman ◽  
Junesang Choi
Keyword(s):  
Power Series ◽  
Integral Representation ◽  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal ◽  
Finite Power

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan.

scholarly journals A combinatorial analysis of Severi degrees

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Fu Liu
Keyword(s):  
Power Series ◽  
Special Function ◽  
Joint Work ◽  
Toric Surfaces ◽  
Height Functions ◽  
Combinatorial Objects ◽  
Multivariate Function ◽  
Related Family ◽  
Combinatorial Formulas ◽  
International Audience

International audience Based on results by Brugallé and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees Nd,δ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic versionof a special function associated to long-edge graphs which appeared in Fomin-Mikhalkin’s formula, and conjecturedit to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we considera special multivariate function associated to long-edge graphs that generalizes their function. The main result of thispaper is that the multivariate function we define is always linear.The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of Qd,δ and a bound δ for the threshold of polynomiality ofNd,δ.Next, in joint work with Osserman, we apply thelinearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the Göttsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series B1(q) and B2(q) appearing in the Göttsche-Yau-Zaslow formula.The proof of our linearity result is completely combinatorial. We defineτ-graphs which generalize long-edge graphs,and a closely related family of combinatorial objects we call (τ,n)-words. By introducing height functions and aconcept of irreducibility, we describe ways to decompose certain families of (τ,n)-words into irreducible words,which leads to the desired results.

Modeling of Spray Deposition and Robot Motion Optimization for a Gas Turbine Application

2000 ◽  
Author(s):  
A. Hansbo ◽  
P.E. Nylén
Keyword(s):  
Coating Thickness ◽  
Spray Deposition ◽  
Engineering Practice ◽  
Programming System ◽  
Trial And Error ◽  
Invariant Surfaces ◽  
Motion Optimization ◽  
Matlab Implementation ◽  
Plasma Sprayed ◽  
Rotationally Invariant

Abstract The complexity of many components being coated in the aircraft industry today makes the traditional trial and error approach to obtain uniform coatings inadequate. To reduce programming time and further increase process accuracy a more systematic approach to develop robot trajectories is needed. In earlier work, a mathematical model was developed to predict coating thickness for thermal spray deposition on rotating objects with rotationally invariant surfaces. The model allows for varying spray distance and spray direction but is simple enough to give very short simulation times. An iterative method for robot feed optimization to obtain uniform coatings was also proposed. Currently, the use of the model in engineering practice is being evaluated. A MATLAB implementation of the model has been integrated with a commercial off-line programming system, giving a powerful and efficient tool to predict and optimize coating thickness. Simulations and experimental verifications are presented for two zirconia plasma sprayed parts.

scholarly journals On Fractional q-Extensions of Some q-Orthogonal Polynomials

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 97
Author(s):  
P. Njionou Sadjang ◽  
S. Mboutngam
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Series Method ◽  
Power Series Method ◽  
Gaussian Type

In this paper, we introduce a fractional q-extension of the q-differential operator Dq−1 and prove some of its main properties. Next, fractional q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties of the newly-defined functions are given. Finally, a fractional q-difference equation of Gaussian type is introduced and solved by means of the power series method.

Sign in / Sign up

Export Citation Format

Share Document