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Molecules have various computational representations, including numerical descriptors, strings, graphs, point clouds, and surfaces. Each representation method enables the application of various machine learning methodologies from linear…
Euler diagrams are a tool for the graphical representation of set relations. Due to their simple way of visualizing elements in the sets by geometric containment, they are easily readable by an inexperienced reader. Euler diagrams where the…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
Controlled ordinary differential equations driven by continuous bounded variation curves can be considered a continuous time analogue of recurrent neural networks for the construction of expressive features of the input curves. We ask up to…
Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Liu et al., Nature:473(7346):167-173,…
In this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we…
Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally…
Vector displacements expressed in spherical coordinates are proposed. They correspond to electromagnetic fields in vacuum that globally rotate about an axis and display many circular patterns on the surface of a sphere. The fields basically…
Preserving geometric structure is important in learning. We propose a unified class of geometry-aware architectures that interleave geometric updates between layers, where both projection layers and intrinsic exponential map updates arise…
We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix…
In this paper, we study uniform ensemble controllability (UEC) of linear ensemble systems defined in an infinite-dimensional space through finite-dimensional settings. Specifically, with the help of the Stone-Weierstrass theorem for…
Developing on a recent work on localized bubbles of ordinary relativistic fluids, we study the comparatively richer leading order surface physics of relativistic superfluids, coupled to an arbitrary stationary background metric and gauge…
A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be…
We study time- and parameter-dependent ordinary differential equations in the geometric setting of vector fields and their flows. Various degrees of regularities in state are considered, including Lipschitz, finitely diferentiable, smooth,…
We consider the bilinear Schroedinger equation on a bounded one-dimensional domain and we provide explicit times such that the global exact controllability is verified. In addition, we show how to construct controls for the global…
Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies controllability by taking in consideration the eigenvalues of an associated derivation D. When the state…
We survey results of recent activity towards studying controllability and accessibility issues for equations of dynamics of incompressible fluids controlled by low-dimensional or, degenerate, forcing. New results concerning controllability…
Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space…
Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions of exactly marginal operators. We show that the curvature of such conformal manifolds can be computed…
The successful application of Quantum Optimal Control (QOC) over the past decades unlocked the possibility of directing the dynamics of quantum systems. Nevertheless, solutions obtained from QOC algorithms are usually highly irregular,…