arXiv research

Proves a theorem for normal distributions on manifolds with boundary.

The generalization of Frobenius' theorem to foliations with singularities is usually attributed to Stefan and Sussmann, for their simultaneous discovery around 1973. However, their result is often referred to without caring much on the precise statement, as some sort of magic spell. This may be explained by the fact th…

The paper establishes a correspondence between normal distributions and neat foliations on manifolds with boundary.

Using the definition of entropy of a family of increasing distances on a compact metric set given in [10] we introduce a notion of Finsler entropy for smooth distributions and Stefan-Sussmann foliations. This concept generalizes most of classical topological entropy on a compact Riemannian manifold : the entropy of a f…

This is a survey concerning the relationship between Lie Groupoids (and their morphisms) and singular foliations in the sense of Sussmann-Stefan (considered from a purely geometrical point of view). We focus on the interaction between the algebraic and differentiable structures underlying Lie groupoids, and between gro…

Desingularizes singular foliations with a locally compact groupoid.

Survey and extend work on singular foliations in diffeology.

Introduces Carrollian Lie algebroids to handle singular Carrollian geometries.

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then t…

The paper describes fitting submanifolds to data using Sussmann's orbit theorem.

Proposes neural networks for solving complex free boundary problems.

New method solves supercooled Stefan problem, proving minimal solutions are physical.

Anisotropic obstacle problems and Stefan problem studied with evolving surfaces.

A map between manifolds which matches up families of complete vector fields is a fiber bundle mapping on each orbit of those vector fields.

Central bank optimizes bailout cash injection to limit defaults.

We consider a family of $C^1$ vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincaré inequality.

We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves.

Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the classical Stefan-problem in one space dimension, where the paths of the moving in…

Probabilistic proof of smooth boundaries in optimal stopping problems.

This paper is a short version of some joint work with Stefan Haller. It describes the structure of "smooth manifold with corners" on the space of possibly broken instantons and on the completion of unstable manifolds of a generic smooth vector field. The result is stated in Theorem 1.4.

The paper characterizes stochastic incompleteness in Riemannian manifolds.

Study proves existence and uniqueness for differential equations with non-Lipschitz coefficients.

We discuss smooth nonlinear control systems with symmetry. For a free and proper action of the symmetry group, the reduction of symmetry gives rise to a reduced smooth nonlinear control system. If the action of the symmetry group is only proper, the reduced nonlinear control system need not be smooth. Using the smooth …

We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this …

The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting…

These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of local transformation groups, based on the Stefan-Sussman theory on the integrab…

Paper tackles circularity issues in machine learning predictions.

Let $X$ be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed $L$-Lipschitz curve $γ:S^1\rightarrow X$ may be extended to an $L$-Lipschitz map defined on the hemisphere $f:H^2\rightarrow X$. This implies that $X$ satisfies a quadratic isoperimetri…

This note contains two remarks about the application of the d-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace t…

AI models solved the Kaczmarz algorithm's worst-case complexity.

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions;…

Let $(X^{m+1}, g)$ be an $(m+1)$-dimensional globally hyperbolic spacetime with Cauchy surface $M^m$, and let $\widetilde M^m$ be the universal cover of the Cauchy surface. Let $\mathcal N_{X}$ be the contact manifold of all future directed unparameterized light rays in $X$ that we identify with the spherical cotangent…

We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds $\mathcal P$ and $\mathcal Q$ of a manifold $\mathcal M$ endowed with a distribution $\mathcal D\subset T\M$. We give a different proof, that holds in a more general context, of a result by Bismut (Larg…

Solves financial and non-financial problems using heat potentials.

The paper introduces controllable principal connections and estimates distances between bundles and spaces.

In this paper, we will be concerned with the explicit classification of closed, oriented, simply-connected spin manifolds in dimension eight with vanishing cohomology in the odd dimensions. The study of such manifolds was begun by Stefan Müller. In order to understand the structure of these manifolds, we will analyze t…

Solves Plateau-Douglas problem for singular configurations in general metric spaces.

JuliaConnectoR integrates Julia functions into R for deep learning.

The paper analyzes McKean-Vlasov equations with hitting times, proving global solvability.

This survey presents a review of state-of-the-art deep neural network architectures, algorithms, and systems in vision and speech applications. Recent advances in deep artificial neural network algorithms and architectures have spurred rapid innovation and development of intelligent vision and speech systems. With avai…

Survey of large language models in financial prediction and trading.

Synthesizes machine learning applications in reliability and safety.

The book contains a collection of works on Riemann-Cartan and metric-affine manifolds provided with nonlinear connection structure and on generalized Finsler-Lagrange and Cartan-Hamilton geometries and Clifford structures modelled on such manifolds. The choice of material presented has evolved from various applications…

The paper refutes standard asset pricing models and introduces new theories.

This paper examines anomalies and frauds in blockchain networks and proposes detection techniques.

Software and hardware co-design and optimization of HPC systems has become intolerably complex, ad-hoc, time consuming and error prone due to enormous number of available design and optimization choices, complex interactions between all software and hardware components, and multiple strict requirements placed on perfor…

FinTech uses data science and AI to transform finance.

The ever-growing big data and emerging artificial intelligence (AI) demand the use of machine learning (ML) and deep learning (DL) methods. Cybersecurity also benefits from ML and DL methods for various types of applications. These methods however are susceptible to security attacks. The adversaries can exploit the tra…