Related papers: The Surface Signature and Rough Surfaces
We give some basic properties of strongly topologically transitive, supermixing, and hypermixing maps on general topological spaces. Then we present some other results for which our mappings need to be continuous.
This paper develops the angular and static-channel component of Geometric and Spectral Alignment for residual Jacobian chains. Starting from Cartan-coordinate rigidity and fitted effective-rank windows, we study how dominant singular…
The real world and its geographic objects are modeled and represented in different spatial databases. Each of these databases provides only a partial description (in space and time) of the geographic objects represented. Sometimes,…
We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…
Splines are central objects for the interpolation of discrete data via piecewise smooth paths. Their iterated-integral signature is an infinite collection of tensors which characterizes paths almost uniquely. We study truncations of this…
Geometry and topology constitute complementary descriptors of three-dimensional shape, yet existing benchmark datasets primarily capture geometric information while neglecting topological structure. This work addresses this limitation by…
This manuscript complements the Hirsch-Pugh-Shub (HPS) theory on persistence of normally hyperbolic laminations and the theorem of Robinson on the structural stability of diffeomorphisms that satisfy Axiom A and the strong transversality…
This article treats the question of fundamentality of the translates of a polyharmonic spline kernel (also known as a surface spline) in the space of continuous functions on a compact set $\Omega\subset \RR^d$ when the translates are…
We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded…
We use vielbein bundle's horizontal lift path integral formulation and gauge theory's holonomy map to compactly describe parallel transport and geodesic equations on a manifold. This is first applied to the geometry of general relativistic…
Surface reconstruction from point clouds is a fundamental challenge in computer graphics and medical imaging. In this paper, we explore the application of advanced neural network architectures for the accurate and efficient reconstruction…
We propose a novel approach for performing convolution of signals on curved surfaces and show its utility in a variety of geometric deep learning applications. Key to our construction is the notion of directional functions defined on the…
The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann…
We introduce a theoretical framework for computing transport coefficients for complex materials. As a first example, we resolve long-standing inconsistencies between experiment and theory pertaining to the conductivity and Hall mobility for…
We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a…
Many surface reconstruction methods incorporate normal integration, which is a process to obtain a depth map from surface gradients. In this process, the input may represent a surface with discontinuities, e.g., due to self-occlusion. To…
Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties…
We present Uni-Fusion, a universal continuous mapping framework for surfaces, surface properties (color, infrared, etc.) and more (latent features in CLIP embedding space, etc.). We propose the first universal implicit encoding model that…
In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce…
Sequential and temporal data arise in many fields of research, such as quantitative finance, medicine, or computer vision. A novel approach for sequential learning, called the signature method and rooted in rough path theory, is considered.…