Parabolic Polynomials Research Articles

On the family of cubic parabolic polynomials

<p style='text-indent:20px;'>For a sequence <inline-formula><tex-math id="M1">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> of complex numbers we consider the cubic parabolic polynomials <inline-formula><tex-math id="M2">\begin{document}$ f_n(z) = z^3+a_n z^2+z $\end{document}</tex-math></inline-formula> and the sequence <inline-formula><tex-math id="M3">\begin{document}$ (F_n) $\end{document}</tex-math></inline-formula> of iterates <inline-formula><tex-math id="M4">\begin{document}$ F_n = f_n\circ\dots\circ f_1 $\end{document}</tex-math></inline-formula>. The Fatou set <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{F}_0 $\end{document}</tex-math></inline-formula> is the set of all <inline-formula><tex-math id="M6">\begin{document}$ z\in\hat{\mathbb{C}} $\end{document}</tex-math></inline-formula> such that the sequence <inline-formula><tex-math id="M7">\begin{document}$ (F_n) $\end{document}</tex-math></inline-formula> is normal. The complement of the Fatou set is called the Julia set and denoted by <inline-formula><tex-math id="M8">\begin{document}$ \mathcal{J}_0 $\end{document}</tex-math></inline-formula>. The aim of this paper is to study some properties of <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{J}_0 $\end{document}</tex-math></inline-formula>. As a particular case, when the sequence <inline-formula><tex-math id="M10">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> is constant, <inline-formula><tex-math id="M11">\begin{document}$ a_n = a $\end{document}</tex-math></inline-formula>, then the iteration <inline-formula><tex-math id="M12">\begin{document}$ F_n $\end{document}</tex-math></inline-formula> becomes the classical iteration <inline-formula><tex-math id="M13">\begin{document}$ f^n $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M14">\begin{document}$ f(z) = z^3+a z^2+z $\end{document}</tex-math></inline-formula>. The connectedness locus, <inline-formula><tex-math id="M15">\begin{document}$ M $\end{document}</tex-math></inline-formula>, is the set of all <inline-formula><tex-math id="M16">\begin{document}$ a\in\mathbb{C} $\end{document}</tex-math></inline-formula> such that the Julia set is connected. In this paper we investigate some symmetric properties of <inline-formula><tex-math id="M17">\begin{document}$ M $\end{document}</tex-math></inline-formula> as well.</p>

Multiscale Polynomial-Based High-Order Central High Resolution Schemes

Polynomial-based high order central high resolution schemes with semi-discrete forms are integrated with multiresolution-based adapted cells/grids. To preserve the positivity condition on non-uniform cells/grids, the corresponding formulations are studied, redesigned or developed. Two general approaches can be used for polynomial-based reconstructions: (a) direct interpolation by a polynomial, (b) proper combination of different polynomials to construct a new polynomial with desired features. Based on these approaches, three polynomial-based reconstructions are considered: (i) parabolic polynomials interpolating average solutions of three successive cells; (ii) piece-wise parabolic methods (PPMs) obtained with two different local stencils; (iii) central-WENO schemes [based on the results of approach (i)]. For the first approach, the corresponding features, stability conditions, formulations and nonlinear limiters are studied and updated. For the second approach, for more localized stencils, new independent variables (e.g., first and second spatial derivatives) are introduced by adding new conservation laws. Two PPM-based central schemes are formulated and a new limiter and a new updating procedure are introduced. For the third approach, the average-interpolating parabolic polynomial [in approach (i)] is used in the framework of the central-WENO formulation. Third and fourth order formulations are provided on non-uniform grids/cells. Finally some numerical examples are presented to verify the results and to assess effectiveness and robustness of the three approaches.

Real Analyticity of Hausdorff Dimension Function of Disconnected Julia Sets of Parabolic Polynomials fa,b(z) = z(1 – az – bz<sup>2</sup>)

Abstract not available
 GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 119-125

Determination of Shearing Properties for Tubular Pinewood under Torsion

In this study, tubular pinewood (Pinus sylvestris L.) specimens are tested and shear strain measurements are performed by applying torsion in z direction in the consideration of light weight aircraft engineering. The objective of this paper is to contribute and generate the nonlinear material model in terms of shear modulus presented with power functions under the consideration of nonlinear behavior of wood under torque. Strain gauge measurements are performed for the maximum shear stresses which develop on the tubular specimen, along the radial r(rin, rout), circumferential Φ(Φin, Φout) and z directions, in a point-wise manner. The data is gathered and examined for the determination of the local variations of empirical shear modulus functions on transversely isotropic surfaces of the specimens. The coordinate dependent shear modulus functions of GzΦ(r), GzΦ(Φ), GzΦ(z) are derived for GzΦ(r, Φ, z)as the function of r, Φ and z, respectively, by analyzing the gathered data. It is proposed to represent the shear modulus functions, GzΦ(Φ) and GzΦ(z) with the parabolic polynomials, and, to represent the shear modulus function GzΦ(r) with a linear equation.

Isomorphisms of Hecke modules and parabolic Kazhdan–Lusztig polynomials

We define and study some Hecke modules which generalize the ones defined by Deodhar in [10] and we find isomorphisms between them. As consequences, we obtain equalities between parabolic Kazhdan–Lusztig polynomials of different Coxeter groups. In particular, we show that any parabolic Kazhdan–Lusztig polynomial equals a parabolic polynomial of a maximal quotient, and we find equalities between the parabolic polynomials of quasi-minuscule quotients and those of certain maximal quotients of the corresponding affine Weyl groups.

Real analyticity of Hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$

Real analyticity of Hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$

A Direct Digital Frequency Synthesizer Based on the Quasi-Linear Interpolation Method

This paper introduces a novel direct digital frequency synthesizer (DDFS) with an architecture based on the quasi-linear interpolation method (QLIP). The QLIP method is a hybrid polynomial interpolation in which the first quarter of a cosine function is approximated by two sets of linear and parabolic polynomials. The section of the cosine function that is closer to its peak is interpolated by parabolic polynomials, due to its resemblance to a parabola. The rest of the function, which is closer to where it approaches zero, is interpolated by linear polynomials. The paper describes the proposed interpolation method and its VLSI implementation. The performance of the proposed implementations is compared to several state-of-the-art DDFS designs.

Asymptotic formulas for parabolic cylindrical functions and Hermite polynomials for large argument values

Expansions in the elgenfunctions of the Sturm-Liouville problem and perturbation expansions are applied to obtain asymptotic formulas for parabolic cylindrical functions and Hermite polynomials for large n. The formulas are compared with previously published formulas and, in particular, a numerical comparison is made for one Hermite polynomial.

A new kind of splines and their use for fast ray-tracing in reflective cavities

In this paper we are presenting a new kind of splines that are very effective in ray-tracing applications. They are designed in such a way to enable the fast and efficient computation of line-spline intersections (line representing the light ray, and spline representing the reflective cavity). These splines are piecewise parabolic polynomials, but with additional degrees of freedom. Polynomial sections of the spline can be rotated to a certain angle (each section has its own angle of rotation), enabling thus the continuity of the first derivative.

Efficiency of annular finite elements for flexural vibrations of thick disks

The efficiency is evaluated of various semianalytical annular finite elements in predicting the vibrational modes for disks where the effect of transverse shear and rotary inertia becomes important, with reference both to the minimum number of degrees of freedom and to the minimum number of operations required to achieve a certain approximation for the exact natural frequency value. From this point of view, a new element, in which normal displacements are interpolated by cubic polynomials and rotations by parabolic polynomials along the radius, is compared with other elements described in the literature.