Piecewise Analytic Curve Research Articles

Zeros of Faber Polynomials for Joukowski Airfoils

Let K be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for K are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized counting measures of the Faber polynomials for Joukowski airfoils; it is never equal to the potential-theoretic equilibrium measure of K. This implies that these airfoils admit an electrostatic skeleton and also explains an interesting class of examples of Ullman (Mich Math J 13:417–423, 1966) related to Chebyshev quadrature.

Uniform Convergence of Some Extremal Polynomials in Domain with Corners on the Boundary

The aim of this paper is to investigate approximation properties of some extremal polynomials in , space. We are interested in finding approximation rate of extremal polynomials to Riemann function in and -norms on domains bounded by piecewise analytic curve.

On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial Fn(z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)]n. Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂Ω, we obtain asymptotic formulas for Fn that yield fine results on the limiting distribution of the zeros of Faber polynomials.

Vision guided navigation for a nonholonomic mobile robot

Theoretical and analytical aspects of the visual servoing problem have not received much attention. Furthermore, the problem of estimation from the vision measurements has been considered separately from the design of the control strategies. Instead of addressing the pose estimation and control problems separately, we attempt to characterize the types of control tasks which can be achieved using only quantities directly measurable in the image, bypassing the pose estimation phase. We consider the task of navigation for a nonholonomic ground mobile base tracking an arbitrarily shaped continuous ground curve. This tracking problem is formulated as one of controlling the shape of the curve in the image plane. We study the controllability of the system characterizing the dynamics of the image curve, and show that the shape of the image curve is controllable only up to its "linear" curvature parameters. We present stabilizing control laws for tracking piecewise analytic curves, and propose to track arbitrary curves by approximating them by piecewise "linear" curvature curves. Simulation results are given for these control schemes. Observability of the curve dynamics by using direct measurements from vision sensors as the outputs is studied and an extended Kalman filter is proposed to dynamically estimate the image quantities needed for feedback control from the actual noisy images.

Asymptotic distribution of the zeros of Faber polynomials

AbstractUsing potential theoretic methods and exploiting the connection with eigenvalues of Toeplitz matrices, we investigate the limiting behaviour of zeros of Faber polynomials generated by a Laurent series. Our results build upon fundamental work of J. L. Ullman. For example, we show that if E is a compact set with simply connected complement and connected interior whose boundary is either (i) not a piecewise analytic curve or (ii) a piecewise analytic curve but with a singularity other than an outward cusp, then the equilibrium distribution for E is a limit measure of the sequence of normalized zero counting measures for the Faber polynomials associated with E.

HARMONIC ANALYSIS IN A MULTIPLY CONNECTED DOMAIN. II

The purpose of this paper is to construct harmonic analysis in a multiply connected plane domain Ω+ bounded by n piecewise-analytic curves. Part I (§§1-3) appeared in the preceding issue of this journal.

HARMONIC ANALYSIS IN A MULTIPLY-CONNECTED DOMAIN. I

The purpose of this paper is to construct harmonic analysis in a multiply connected plane domain Ω+ bounded by n piecewise-analytic curves. Part I (§§1-3) appeared in the preceding issue of this journal.

Stress Functions for Plates Bounded by Piecewise Analytic Curves

Stress Functions for Plates Bounded by Piecewise Analytic Curves

On Maximum Pressure Rise with Turbulent Boundary Layer

Stress Functions for Plates Bounded by Piecewise Analytic Curves M I N G M. CHEN AND P. T. Hsu 240 R B. S. Stratford published ' 2 the results of an analytical and experimental investigation concerning the maximum attainable pressure rise in the presence of incompressible turbulent boundary layer. The examination of his experimental data reveals that it is also in essential agreement with some of our work, in which the allowable turbulent boundary-layer pressure rise was used as a criterion for the prescription of cascade blade velocity distribution. A comparison between the analytical results of reference 3 (computed, however, for the diffussion coefficient DT = —0.017) and the Stratford experimental data is shown in Figs. 1,2, and 3. As may be seen from Fig. 1, the agreement of the analytically predicted surface velocity distribution is not too good initially, but, for ( U/UQ) < 0.65, the data are very close to the results of our calculation method. This initial discrepancy stems probably from the fact that the turbulent boundary layer can withstand some suddenly applied pressure rise (such as produced by shocks, for example) without separation, while our calculation scheme did not allow for such happenings. In this respect, the analytical method suggested by Stratford is somewhat better in this region (see Fig. 4 of reference 2). I t should be noted, however, that, from the practical standpoint, this difference in the velocity distribution in the initial region is rather unimportant, since it occurs for only very small surface length (1.0 < x/x0 < 1.15) and, thus, should not affect substantially the aerodynamic design. Turning now to Fig. 2, the predicted behavior of the momentum thickness (6/do) as a function of the pressure coefficient Cv