arXiv research

Abstracts a theorem for non-smooth maps in infinite dimensions.

We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity…

Paper proves circle packings converge to Riemann mapping for Jordan domains.

The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.

In this note, we show that for any harmonic map into a non-compact symmetric space one can find naturally a "dual" harmonic map into a compact symmetric space which can be constructed from the same basic data (called "potentials" in the loop group formalism). Locally also the inverse/converse duality theorem holds.

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.

We show that the exponential map of the Bochner connection on the restricted holomorphic tangent bundle of a complex manifold admitting the positive-definite Bergman metric coincides with the inverse of Bergman's representative map. We also present a generalization of the Lu theorem, as an application.

The Reeb space of a smooth map whose codimension is minus is the space defined as the space of all connected components of inverse images. For generic maps such as Morse functions and their higher dimensional versions, they are polyhedra whose dimensions are equal to those of the target manifolds and which have simplic…

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism $f: M \to\mathbb{R}^n$ is bijective if and only if $H_{n-1}(M)=0$ and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometr…

This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending t…

The paper constructs metrics on spheres with families of minimal hypersurfaces.

Global inverse function theorem proved easily using Riemannian geometry.

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…

New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.

We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connecti…

The paper proves a generalized inverse function theorem for curved $L_\infty$ spaces.

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$--gerbe $Λ_{Z}$, th…

New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.

Discrete conformal maps on surfaces with vertex decorations are studied.

We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes $P$-adic solenoids, as well as the universal hyperbolic solenoid of Teichmüller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove…

This paper provides a full controlled version of algebraic $K$-theory. This includes a rich array of assembly maps; the controlled assembly isomorphism theorem identifying the controlled group with homology; and the stability theorem describing the behavior of the inverse limit as the control parameter goes to 0. There…

Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.

We provide a direct proof of Cramér's theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in $M$. Furthermore, we reveal the geometric obstructions one runs into …

The Cartan-Kähler theorem is extended to Lie algebroids.

In this paper we establish the basic tools to develop the "Calculus" associated with group-valued continuously Pansu differentiable mappings. We develop the technical machinery on which all of our results rely. In particular, the linearization of addends appearing in the Baker-Campbell-Hausdorff formula is one of the m…

The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.

The paper constructs a multi-valued inverse of quasiregular maps and develops pull-back theory for differential forms.

Proves existence of proper solutions for inverse mean curvature flow.

Study inverse problems with measure samples, improving estimator calibration and recovery.

Inverse function theorem and homotopy description for L-infinity bundles.

Here are two of our main results: Theorem 1. Let X be a normal space with dim X=n and m\geq n+1. Then the space C*(X,R^m) of all bounded maps from X into R^m equipped with the uniform convergence topology contains a dense G_δ-subset consisting of maps g such that \bar{g(X)}\capΠ^d is at most (n+d-m)-dimensional for eve…

A new VAE approach solves inverse problems without explicit inverse mapping.

Recently the author has introduced cobordism-like modules induced from generic maps whose codimensions are negative. They are generalizations of cobordism modules of manifolds. They have been introduced in generalizing the following theorem shown by Hiratuka and Saeki in 2013--14; for a generic map whose codimension is…

Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homot…

LazyDINO efficiently solves high-dimensional Bayesian inverse problems with fast and scalable solutions.

The paper studies global invertibility of maps on Finsler manifolds.

New approach uses distributed persistence for stable, parallelizable topological analysis of large point clouds.

3D metrics get scalar curvature bounds via IMCF.

New retraction on symplectic Stiefel manifold with closed-form inverse.

Neural network implementation of Brenier's polar factorization for vector fields.

Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map ever…

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomo…

We prove that structured vector bundles whose holonomies lie in GL(N,C), SO(N,C), or Sp(2N,C) have structured inverses. This generalizes a theorem of Simons and Sullivan.

Paper proves stability for recovering connections from holonomy traces.

New model for simulating and inferring from inverse problems.

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.

Framework combines machine learning and inverse methods to quantify uncertainties in model parameters.

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limi…