Symmetry results for variational energies on convex polygons

We prove that, in the class of convex polygons with a given number of sides, the regular $n$-gon is optimal for some shape optimization problems involving the torsional rigidity, the principal frequency of the Laplacian, or the logarithmic capacity.

The spectrum of the mean curvature operator

We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝ N (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

A Novel Technique for Resolver-to-Digital Conversion Using Principal Frequency Component S-Transform

A novel technique for resolver-to-digital conversion (RDC) using principal frequency component S-transform (PFCST) is proposed in this paper. First, the mode envelope of two output signals of the resolver is extracted by PFCST. The envelope extracted by PFCST maintains the same time resolution as the original signal because it performs time-frequency conversion for each sampling point. Then, the quadrant of the resolver is determined by the judgment rule formed by the polarity of the optimum nonzero region of the signals, and the quadrant information is used to correct the arctangent to obtain the accurate rotor position. Finally, the simulations prove that the maximum angle error of the resolver estimated by this method occurs at the quadrant junction but does not exceed one deg., and the experiments are used to verify the effectiveness of the proposed method.

On the monotonicity of the principal frequency of the p-Laplacian

For any fixed integer {D>1} we show that there exists {M\in[e^{-1},1]} such that for any open, bounded, convex domain {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M, the principal frequency of the p-Laplacian on Ω is an increasing function of p on {(1,\infty)} . Moreover, for any real number {s>M} there exists an open, bounded, convex domain {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary which has the maximum of the distance function to the boundary of Ω equal to s such that the principal frequency of the p-Laplacian is not a monotone function of {p\in(1,\infty)} .

Minimization problems for inhomogeneous Rayleigh quotients

In this paper, the minimization problem [Formula: see text] where [Formula: see text] is studied when [Formula: see text] ([Formula: see text]) is an open, bounded, convex domain with smooth boundary and [Formula: see text]. We show that [Formula: see text] is either zero, when the maximum of the distance function to the boundary of [Formula: see text] is greater than [Formula: see text], or it is a positive real number, when the maximum of the distance function to the boundary of [Formula: see text] belongs to the interval [Formula: see text]. In the latter case, we provide estimates for [Formula: see text] and show that for [Formula: see text] sufficiently large [Formula: see text] coincides with the principal frequency of the [Formula: see text]-Laplacian in [Formula: see text]. Some particular cases and related problems are also discussed.

Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving y′′′x=fx,y,y′

Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type (MRKT) methods for solving y′′′x=fx,y,y′ are derived in this paper. These methods are constructed which exactly integrate initial value problems whose solutions are linear combinations of the set functions eωx and e-ωx for exponentially fitted and sin⁡ωx and cos⁡ωx for trigonometrically fitted with ω∈R being the principal frequency of the problem and the frequency will be used to raise the accuracy of the methods. The new four-stage fifth-order exponentially fitted and trigonometrically fitted explicit MRKT methods are called EFMRKT5 and TFMRKT5, respectively, for solving initial value problems whose solutions involve exponential or trigonometric functions. The numerical results indicate that the new exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods are more efficient than existing methods in the literature.