Related papers: Discrete Geometric Singular Perturbation Theory
We show that there is generically non-uniqueness for the anisotropic Calder\'on problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show…
We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem…
Scattered disk objects (SDOs) are distant minor bodies that orbit the sun on highly eccentric orbits, frequently with perhelia near Neptune's orbit. Gravitational perturbations due to Neptune frequently lead to chaotic dynamics, with the…
Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely…
The renormalization group (RG) method is one of the singular perturbation methods which is used in search for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost)…
Much of the progress in the gravitational self-force problem has involved the use of singular perturbation techniques. Yet the formalism underlying these techniques is not widely known. I remedy this situation by explicating the foundations…
Special-generic-like maps or SGL maps are introduced by the author motivated by observing and investigating algebraic topological or differential topological properties of manifolds via nice smooth maps whose codimensions are negative. The…
We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example…
We derive, and discuss the properties of, a symplectic map for the dynamics of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a circular, coplanar orbit interior to the pericenter of the parabolic orbit. The map…
Continual learning systems operating in fixed-dimensional spaces face a fundamental geometric barrier: the flat manifold problem. When experience is represented as a linear trajectory in Euclidean space, the geodesic distance between…
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations…
In this text we consider discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets, and their continuous analogous defined on the self-similar sets themselves. We are interested in the spectral properties…
Multiscale models of materials, consisting of upscaling discrete simulations to continuum models, are unique in their capability to simulate complex materials behavior. The fundamental limitation in multiscale models is the presence of…
Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing…
The bifurcation theory of ordinary differential equations (ODEs), and its application to deterministic population models, are by now well established. In this article, we begin to develop a complementary theory for diffusion-like…
We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be…
Dissipative cognitive architectures maintain computation through continuous energy expenditure, where units that exhaust their energy are stochastically replaced with fresh random state. This creates a fundamental challenge: how can…
For a class of $(N+1)$-dimensional systems of differential delay equations with a cyclic and monotone negative feedback structure, we construct a two-dimensional invariant manifold, on which phase curves spiral outward towards a bounding…
This paper addresses the morphing of manifold-valued images based on the time discrete geodesic paths model of Berkels, Effland and Rumpf 2015. Although for our manifold-valued setting such an interpretation of the energy functional is not…
This article establishes the foundation for a new theory of invariant/integral manifolds for non-autonomous dynamical systems. Current rigorous support for dimensional reduction modelling of slow-fast systems is limited by the rare events…