Section (1) is devoted to a discussion of the model-fitting problem, which finds its explicit solution in equation (1.13). In section (2) the maximum likelihood, (ML), estimates of the model parameters are investigated, and for the class of series considered shown to possess the same optimum properties as in the case of independent series. Next, the covariance matrix of the parameter estimates is expressed in terms of the spectral function of the generating process (eq. 3.7). The last section is concerned with certain working approximations to the ML statistics.

We give a detailed description of nonconstant harmonic functions and their level sets on the Sierpiński gasket. We introduce a parameter, called$eccentricity$, which classifies these functions up to affine transformations$h→ah+b$. We describe three (presumably) distinct measures that describe how the eccentricities are distributed in the limit as we subdivide the gasket into smaller copies (cells) and restrict the harmonic function to the small cells. One measure simply counts the number of small cells with eccentricity in a specified range. One counts the contribution to the total energy coming from those cells. And one counts just those cells that intersect a fixed generic level set. The last measure yields a formula for the box dimension of a generic level set. All three measures are defined by invariance equations with respect to the same iterated function system, but with different weights. We also give a construction for a rectifiable curve containing a given level set. We exhibit examples where the curve has infinite winding number with respect to some points.

Motivated by Gutkin curves arising in capillary floating and billiard ball problem, we introduce a new projection function for convex plane curves, and study the curves when the defined projection function is constant. These curves can be regarded as a generalization of curves with constant width. We give a necessary and sufficient condition for the existence of such curves, and provide an explicit expression for the radius of curvature. Properties of our curves and their relation to Gutkin curves are also discussed. Finally, we construct and exhibit some curves with constant projection function.

No abstract uploaded!

No abstract uploaded!