arXiv research

A dimension allowing in particular to state necessary and sufficient conditions of the Morse-Sard Theorem for real valued functions is introduced.

The paper proves Sard's theorem for polynomial maps in infinite dimensions.

In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.

We study Thom Transversality Theorem using a point of view, suggested by Gromov, which allows to avoid the use of Sard Theorem and gives finer informations on the structure of the set of non-transverse maps.

Extends transversality to supergeometry, proving stability and genericity.

Analytic proof for minimal rank Sard conjecture.

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known …

In this paper, we study Lipschitz-Fredholm vector fields on Bounded-Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded-Fréchet-Finsler manifold endowed with a strengthened connection $\mathcal{K}$ and if $ξ$ is a smooth Lipschitz-Fr…

New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.

Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the ca…

Proves Sard conjecture for specific distributions, controlling divergence of vector fields.

Necessary and sufficient condition is given for a set $A\subset R^1$ to be a subset of the critical values set for a $C^k$ function $f:R^m \to R^1$.

We introduce a dynamical-systems approach for the study of the Sard problem in sub-Riemannian Carnot groups. We show that singular curves can be obtained by concatenating trajectories of suitable dynamical systems. As an applications, we positively answer the Sard problem in some classes of Carnot groups.

We prove generic regularity and Uhlenbeck-type compactification theorems for the moduli spaces of PU(2)-monopoles. Generic regularity is NOT obtained in the usual way (by applying Sard theorem to a smooth parameterized moduli space), since the parameterized moduli space can be a priori singular. We explain why, using t…

The paper bounds abnormal and Goh-abnormal sets for metabelian Lie groups with polarizations.

Study Sard problem in step 2 and filiform Carnot groups.

New bounds on geodesic dimension and curvature exponent in Carnot groups.

Let $n, m, k$ be positive integers with $k=n-m+1$. We establish an abstract Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous result of De Pascale's for Sobolev $W^{k,p}_{\textrm{loc}}(\mathbb{R}^n, \mathbb{R}^m)$ functions with $p>n$ and, on the other hand, also the following new result: i…

We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the general…

SARD improves adversarial robustness in two-stage L2D systems.

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence o…

We prove that every function $f:\mathbb{R}^n\to \mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$ is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential $\partial_{…

SARD improves deep learning clinical prediction performance.

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--…

Study finds isolated SL submanifolds on non-Kähler Calabi-Yau threefolds.

The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple graphs for which every unit sphere is a d-sphere, the zero locus of (f-c) is a …

Improved semialgebraic choices with linear complexity.

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 …

Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard conjecture states that that set should be a subset of the so-called Martinet surface o…

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…

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal …

{\bf Construction.} For a dominating polynomial mapping {$F: K^n\to K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {\it bundle} $G_F \subset T^{*}K^n$. We restrict $G_F$ over the critical points $Sing(F)$ of $F$ in $F^{-1}(0)$ and partiti…

For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite …

The sigma-point filters, such as the UKF, which exploit numerical quadrature to obtain an additional order of accuracy in the moment transformation step, are popular alternatives to the ubiquitous EKF. The classical quadrature rules used in the sigma-point filters are motivated via polynomial approximation of the integ…

The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate local symmetries. The whole package provides a functional analytic framework to de…

New sub-Riemannian structures fail synthetic curvature bounds.

Global inverse function theorem proved easily using Riemannian geometry.

A new comparison theorem for geometric spaces.

Paper develops formulas and theorems in Hermitian geometry.

The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.

Revises a theorem by Thurston, finding a counter-example and a weaker version.

Proofs for Moon's theorem and its generalization.

We consider the task of automated theorem proving, a key AI task. Deep learning has shown promise for training theorem provers, but there are limited human-written theorems and proofs available for supervised learning. To address this limitation, we propose to learn a neural generator that automatically synthesizes the…

Analyzes Saito vanishing theorem using $L^{2}$ methods.

Investigates proving geometric theorems over complex and real numbers using tilings.

Extends symplectic reduction and theorem to Lie algebroids.

Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.

Proves Thurston's bounded image theorem for Haken manifolds.