The MacLane’s class of analytic functions is the class of nonconstant analytic functions in the unit disk that have asymptotic values at a dense subset of the unit circle. In this paper, we define a subclass of consisting of those functions that have asymptotic values at a dense subset of the unit circle reached along rectifiable asymptotic paths. We also show that the class is a proper subclas...

We prove that a normal operator on a separable Hubert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made HilbertSchmidt. In 1909 Hermann Weyl proved [3] that each bounded Hermitian operator on a separable Hubert space can be written as the sum of a diagonal operator and a co...

A physically natural potential energy for simple closed curves in R3 is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist and are regular. Finally, the number of ...

We show that if $M$ is a sub-Riemannian manifold and $N$ Carnot group such the nilpotentization of at almost every point isomorphic to $N$, then there are subsets positive measure embed into by bilipschitz maps. Furthermore, countably $N$--rectifiable, i.e., all except for null set can be covered many

Let C be a rectifiable Jordan curve of the finite z plane. We shall say that a function f(z) belongs to the class Lip (C, j , a) if ƒ (z) is regular in the limited region bounded by C (which we shall call the interior of C), if f{z) is continuous in the corresponding closed region, and if the jth. derivative of f(z) is also continuous in this closed region and satisfies a Lipschitz condition wi...

I prove that the solutions of the phase-field equations, on a subsequence, converge to a weak solution of the Mullins-Sekerka problem with kinetic undercooling. The method is based on energy estimates, a monotonicity formula, and the equipartition of the energy at each time. I also show that for almost all t, the limiting interface is ( d 1)-rectifiable with a square-integrable mean-curvature v...