scholarly journals Quasi-Spherical and Multi-Quasi-Spherical Polynomial Quaternionic Equations: Introduction of the Notions and Some Examples

2010 ◽  
Vol 21 (2) ◽  
pp. 407-416 ◽  
Author(s):  
Dmytro Mierzejewski
Keyword(s):  
Spherical Polynomial

Spark plug reliability improvement with regard to ceramic insulator form factor

2020 ◽  
pp. 095440702094662
Author(s):  
Bernard Fryskowski
Keyword(s):  
Form Factor ◽  
Surface Resistance ◽  
Design Method ◽  
Three Dimensional ◽  
Fuel Mixture ◽  
Spark Plug ◽  
Ceramic Insulators ◽  
Spherical Polynomial ◽  
Novel Concept ◽  
High Degree

Ceramic insulators for spark plugs are important components responsible for the dielectric barrier to generate high-voltage pulses required to ignite the air–fuel mixture and for providing mechanical support to the center electrode. To ensure a high degree of reliability, many manufacturers apply corrugation patterns to the glazed upper part of an insulator to prevent flashover and parasitic discharges as well as to reduce the leakage current. The corrugation pattern design based mostly on simple cylindrical and conical shapes has changed very little over the years. This gives rise to the question whether an application of more sophisticated curves such as spherical, polynomial, or exponential can improve the surface resistance of a ribbed spark plug insulator. Quantitative analysis based on form factor and leakage distance calculations was used as a design method to optimize the shape of the spark plug insulator and its pollution performance. Furthermore, a novel concept of concave insulator corrugation patterns formed by quadratic and exponential functions was proposed and discussed as an alternative solution suitable for practical application. It was found that insulator ribs completed with supplementary concave corrugation ensure a longer leakage distance than conventional patterns. According to the results of calculations and measurements performed on three-dimensional printed samples, it was stated that the novel concave corrugation patterns can significantly increase the surface resistance of spark plug insulators.

REGULARIZED LEAST SQUARE REGRESSION WITH SPHERICAL POLYNOMIAL KERNELS

2009 ◽  
Vol 07 (06) ◽  
pp. 781-801 ◽  
Author(s):  
LUOQING LI
Keyword(s):  
Theoretical Analysis ◽  
Integral Operators ◽  
Excess Risk ◽  
Least Square ◽  
Least Square Regression ◽  
Learning Rates ◽  
Explicit Bounds ◽  
Spherical Polynomial ◽  
Polynomial Kernels

This article considers regularized least square regression on the sphere. It develops a theoretical analysis of the generalization performances of regularized least square regression algorithm with spherical polynomial kernels. The explicit bounds are derived for the excess risk error. The learning rates depend on the eigenvalues of spherical polynomial integral operators and on the dimension of spherical polynomial spaces.

Approximation of functions on the sphere on a Sobolev space with a Gaussian measure in the probabilistic case setting

2014 ◽  
Vol 12 (05) ◽  
pp. 1461012 ◽  
Author(s):  
Heping Wang ◽  
Xuebo Zhai
Keyword(s):  
Sobolev Space ◽  
Gaussian Measure ◽  
Linear Operators ◽  
Partial Summation ◽  
Approximation Of Functions ◽  
Worst Case ◽  
Average Case ◽  
The Best Approximation ◽  
Optimal Linear ◽  
Spherical Polynomial

In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the Lq space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the Lq space for 2 < q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the Lq space for 1 ≤ q < ∞. Also, in the Lq space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.

scholarly journals Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere

2015 ◽  
Vol 766 ◽  
pp. 468-498 ◽  
Author(s):  
D. J. Ivers ◽  
A. Jackson ◽  
D. Winch
Keyword(s):  
Incompressible Flows ◽  
Dimensional Space ◽  
Bounded Operator ◽  
Neumann Boundary ◽  
Analytic Solutions ◽  
Careful Consideration ◽  
Finite Dimensional Space ◽  
Rotating Boundary ◽  
Spherical Polynomial ◽  
Polynomial Flows

AbstractWe consider incompressible flows in the rapid-rotation limit of small Rossby number and vanishing Ekman number, in a bounded volume with a rigid impenetrable rotating boundary. Physically the flows are inviscid, almost rigid rotations. We interpret the Coriolis force, modified by a pressure gradient, as a linear operator acting on smooth inviscid incompressible flows in the volume. The eigenfunctions of the Coriolis operator $\boldsymbol{{\mathcal{C}}}$ so defined are the inertial modes (including any Rossby modes) and geostrophic modes of the rotating volume. We show $\boldsymbol{{\mathcal{C}}}$ is a bounded operator and that $-\text{i}\boldsymbol{{\mathcal{C}}}$ is symmetric, so that the Coriolis modes of different frequencies are orthogonal. We prove that the space of incompressible polynomial flows of degree $N$ or less in a sphere is invariant under $\boldsymbol{{\mathcal{C}}}$. The symmetry of $-\text{i}\boldsymbol{{\mathcal{C}}}$ thus implies the Coriolis operator is non-defective on the finite-dimensional space of spherical polynomial flows. This enables us to enumerate the Coriolis modes, and to establish their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spherical polynomial flows under $\boldsymbol{{\mathcal{C}}}$ and completeness, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson’s equation with a Neumann boundary condition and polynomial data, in a sphere is a polynomial. We also enumerate the Coriolis modes in a sphere, with careful consideration of the geostrophic modes, directly from the known analytic solutions.

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