Related papers: Random Surfaces and Higher Algebra
In this paper, we propose a novel formulation to extend CNNs to two-dimensional (2D) manifolds using orthogonal basis functions, called Zernike polynomials. In many areas, geometric features play a key role in understanding scientific…
We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path…
In these notes we discuss tools and concepts that emerge when studying high-dimensional random landscapes, i.e., random functions on high-dimensional spaces. As an illustrative example, we consider an inference problem in two forms:…
The failure of roughness parameters to predict surface properties stems from their inherent scale-dependence; in other words, the measured value depends on the way it was measured. Here we take advantage of this scale-dependence to develop…
We consider the geometric random (GR) graph on the $d-$dimensional torus with the $L_\sigma$ distance measure ($1 \leq \sigma \leq \infty$). Our main result is an exact characterization of the probability that a particular labeled cycle…
The critical dynamics of conformal field theories on random surfaces is investigated beyond the previously studied dynamics of the overall area and the genus. It is found that the evolution of the order parameter in physical time performs a…
This paper identifies a structural property of data distributions that enables deep neural networks to learn hierarchically. We define the "staircase" property for functions over the Boolean hypercube, which posits that high-order Fourier…
This thesis deals with automorphisms of real algebraic surfaces, which are polynomial transformations with a polynomial inverse. The main concern is whether their restriction to the real locus reflects all the richness of the complex…
In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the…
Passive random walker dynamics is introduced on a growing surface. The walker is designed to drift upward or downward and then follow specific topological features, such as hill tops or valley bottoms, of the fluctuating surface. The…
We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface.
Stochastic evolution equations describing the dynamics of systems under the influence of both deterministic and stochastic forces are prevalent in all fields of science. Yet, identifying these systems from sparse-in-time observations…
In the last decade, the concept of path signature has achieved significant success in data science applications. It offers a powerful set of features that effectively capture and describe the characteristics of paths or sequential data.…
A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. In this paper we consider a randomizing model for STSs and generalizations to branched covers of other simple translation…
We study the random growth of surfaces from within the perspective of a single column, namely, the fluctuation of the column height around the mean value, y(t)= h(t)-< h(t)>, which is depicted as being subordinated to a standard…
In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
Surface growth, by association or dissociation of material on the boundaries of a body, is ubiquitous in both natural and engineering systems. It is the fundamental mechanism by which biological materials grow, starting from the level of a…
We demonstrate two examples of stochastic processes whose lifts to geometric rough paths require a renormalisation procedure to obtain convergence in rough path topologies. Our first example involves a physical Brownian motion subject to a…
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave…