J. W. Neuberger

SNOWFLAKE HARMONICS AND COMPUTER GRAPHICS: NUMERICAL COMPUTATION OF SPECTRA ON FRACTAL DRUMS

In this work, we study the steady-states vibrations of the “Koch snowflake drum”, both numerically and by means of computer graphics. In particular, we approximate the smallest 50 eigenvalues (or “frequencies” of the drum), along with the corresponding eigenfunctions (called “snowflake harmonics”) of the Dirichlet Laplacian on the Koch snowflake domain. We describe the numerical methods used in the computations, and we display graphical representations of a selected set of the eigenfunctions (as well as of their gradients). In the case of the first harmonic, the graphical results agree with mathematically derived results (by Lapidus and Pang) concerning gradient behavior (“blow up” or “infinite stress” of the membrane) on the boundary and suggest further conjectures regarding the higher eigenfunctions. According to earlier work by the physicist Sapoval and his collaborators, this research may help better understand the formation and “stabilization” of fractal structures (e.g., coastlines, trees and blood vessels) in nature.

Norm of symmetric product compared with norm of tensor product

Suppose each of mm,n and k is a positive integerA is a (real-valued) symmetric n-linear function on Em and B is a symmetric k-linear function on Em . An inner product norm on the vector space of all (n+k)-linear functions on Em is defined. The tensor and symmetric products of A and B are denoted, respectively, by A⊗B and AċB.lt is shown that and that this result is best possible.

Minimal surfaces and Sobolev gradients

We treat the problem of computing triangle-based piecewise linear approximations to parametric minimal surfaces in

Energy minimization using Sobolev gradients: application to phase separation and ordering

{\bf R}^3

Sobolev Gradients and the Ginzburg--Landau Functional

. More specifically, given a triangulation T of the unit square

Tensor products and successive approximations for partial differential equations

\Omega

Analyticity and quasi-analyticity for one-parameter semigroups

and a function

Linear Extensions of Nonlinear Semigroups

f_0

Quasi-analyticity and semigroups

from the nodes of T into

Newton flow of the Riemann zeta function: separatrices control the appearance of zeros

{\bf R}^3