Search

It has been empirically observed that eigenfunctions of Laplace's equation $-\Delta \phi = \lambda \phi$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has some nontrivial implications in acoustics that has been put to real-life use (sound attenuation by noise-protective walls); this short paper describes the mathematical mechanism responsible for this and describes the quantitative strength of the phenomenon for some examples.

In this paper we present a theoretical foundation for a representation of a data set as a measure in a very large hierarchically parametrized family of positive measures, whose parameters can be computed explicitly (rather than estimated by optimization), and illustrate its applicability to a wide range of data types. The pre-processing step then consists of representing data sets as simple measures. The theoretical foundation consists of a dyadic product formula representation lemma, a visualization theorem. We also define an additive multiscale noise model which can be used to sample from dyadic measures and a more general multiplicative multiscale noise model which can be used to perturb continuous functions, Borel measures, and dyadic measures. The first two results are based on theorems. The representation uses the very simple concept of a dyadic tree, and hence is widely applicable, easily understood, and easily computed. Since the data sample is represented as a measure, subsequent analysis can exploit statistical and measure theoretic concepts and theories. Because the representation uses the very simple concept of a dyadic tree defined on the universe of a data set and the parameters are simply and explicitly computable and easily interpretable and visualizable, we hope that this approach will be broadly useful to mathematicians, statisticians, and computer scientists who are intrigued by or involved in data science including its mathematical foundations., Comment: submitted

We revise the Levy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains.

We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The main technical ingredient is a construction, for every continuum K, of a Borel probabilistic measure \mu with the property that on every ball B(x,r), with x in K, the measure is bounded by a universal constant multiple of r\exp(-g(x,r)), where g(x,r) > 0 is an explicit function. The continuum K is mean wiggly at exactly those points x in K where g(x, r) has a logarithmic growth to infinity as r goes to 0. The theory of mean wiggly continua leads, via the product formula for dimensions, to new estimates of the Hausdorff dimension for Cantor sets. We prove also that asymptotically flat sets are of Hausdorff dimension 1 and that asymptotically non-porous continua are of the maximal dimension. Another application of the theory is geometric Bowen's dichotomy for Topological Collet-Eckmann maps in rational dynamics. In particular, mean wiggly continua are dynamically natural as they occur as Julia sets of quadratic polynomials for parameters from a generic set on the boundary of the Mandelbrot set.

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with $\mathcal{C}^\alpha$ metric). These coordinates are bi-Lipschitz on embedded balls of the domain or manifold, with distortion constants that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient. These estimates hold in the non-smooth category, and are stable with respect to perturbations within this category. Finally, these coordinate systems are intrinsic and efficiently computable, and are of value in applications., Comment: 41 pages, 2 figures. V2 some minor corrections. V3 more minor corrections. V4 minor corrections and significant rewriting for better readability

We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is either zero or one.