arXiv research

The paper links set cuspidality to function regularity and flatness.

The paper explores different smooth map notions on convex sets and their relationships.

In this note we prove that the level-set flow of the topologist's sine curve is a smooth closed curve. In previous work it was shown by the second author that under level-set flow, a locally-connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we…

The paper extends properties of smooth functions to closed sets and maps.

We prove that the space of smooth initial data and the set of smooth solutions of the Liouville equation are homeomorphic.

Smooth dec initial data sets may not extend to smooth spacetimes.

By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb R^d$ is smooth ($\mathcal C^\infty$) if and only if it is arc-smooth, i.e., $f\circ c$ is smooth for every smooth curve $c : \mathbb R \to U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on s…

MARINA-P improves non-smooth federated optimization with adaptive stepsizes.

Proves smoothness of conical singularities in mean curvature flow.

Smooths metrics with nonnegative scalar curvature near singular sets.

Determines higher smooth surgery structure sets of complex projective spaces.

Smooth maps bound Betti numbers of zero sets.

Simplicial sets deformation retract onto transverse simplices.

Smooth structures on diffeological spaces and sheaves, resolving conjectures.

We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…

Smooth approximation of integral cycles mod 2 in Riemannian manifolds.

Optimal smooth subspaces approximate large data sets efficiently.

Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.

Study shows area-minimizing submanifolds are mostly rough, not smooth.

New algorithm tackles smooth online learning with optimal regret.

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…

Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.

Proves rigidity for specific initial data sets under the dominant energy condition.

Diffusion models improve creativity by smoothing the score function, leading to interpolated data.

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $\mathbb{R}^{n+1}$ starting from any $n$-dimensional $(\varepsilon,R)$-Reifenberg flat set with $\varepsilon$ sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and …

Extends first-order flexes of surfaces to second-order flexes.

Smooth submetries between curved spaces are smooth.

The paper develops bounds and regularity for minimal boundaries in non-smooth spaces with Ricci curvature.

We present some methods to construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or non-symplectic cyclic isotropy point sets. All such actions are not compatible with any symplectic form.

The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of level sets such that the set of all vertices coincides with the set of all connected components of level sets including singular points. Reeb graphs are fundamental and important in the algebraic and diff…

Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$, with section $Σ\subset M$ and generalized Weyl group $W$. We show that restriction to $Σ$ is a surjective map from the set of smooth $G$-invariant tensors on $M$ onto the set of smooth $W$-invariant tensors on $Σ$. Moreover, we show that every s…

New algorithm solves federated minimax optimization problems.

We calculate the smooth structure set of $S^p \times S^q$, $S(p, q)$, for $p, q \geq 2$ and $p+q \geq 5$. As a consequence we show that in general $S(4j-1, 4k)$ cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of forgetful map $F:…

We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by smooth functions with no critical points. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as …

Generalizes smoothness conditions for optimization methods.

Proves higher regularity for anisotropic inverse mean curvature flow.

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some…

Estimates smooth functions and their derivatives from noisy data.

The paper defines singular evolutoids and uses them to derive an integral equality.

Derives smooth homogeneous structures for low-rank tensors.

Decision tree learning heuristics fail even in smoothed analysis for complex targets.

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure …

The paper explores the structure of Reeb spaces for smooth functions on manifolds.

Entropy data replaces classical charts for smooth manifolds.

This research optimizes Andrews plots for better visual clarity in high-dimensional data.

Smooth manifolds have functions with exactly two critical values.

Proposes SCD-split for CP to balance interpretability and efficiency.

Smooth solutions found for a specific type of Yamabe problem.