Related papers: Tangential approximation of analytic sets

Given a submanifold $M\subset \mathbf{R}^\nu$, a curve $\gamma:I\to M$ and tangent vectors $v$ along $\gamma$, we roll the tangent space along $\gamma$. In doing so, we get an imprint/trace of $\gamma$ on the tangent space, as well as an…

We introduce an equivalence relation on $W^{s,p}({\mathbb S}^N;{\mathbb S}^N)$ involving the topological degree, and we evaluate the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. In some special…

Assuming $V=L$, we construct a plane set $E$ of Hausdorff dimension $1$ whose every orthogonal projection onto straight lines through the origin has Hausdorff dimension $0$. This is a counterexample to J. M. Marstrand's seminal projection…

Valiant's Holant theorem is a powerful tool for algorithms and reductions for counting problems. It states that if two sets $\mathcal{F}$ and $\mathcal{G}$ of tensors (a.k.a. constraint functions or signatures) are related by a…

We extend Nguyen's characterization of Sobolev spaces $W^{1,p}$ to the setting of PI-metric measure spaces such that at a.e. point the tangent space (in the Gromov-Hausdorff sense) is unique and Euclidean with a fixed dimension. We also…

The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension…

In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…

Given $S= \{v_1, \dots, v_n\} \subset \mathbb{R} ^m$ and $p \in \mathbb{R} ^m$, testing if $p \in conv(S)$, the convex hull of $S$, is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean {\it…

In this paper it is shown that for locally trivial complex analytic morphisms between some reduced spaces the Relative Riemann-Hilbert Theorem still holds up to torsion, i.e. tame flat relative connections on torsion-free sheaves are in…

It is shown (Theorem A and its corollary) that if g is any nonconstant nonunivalent analytic function on a half-plane H and if D is either a half-plane or a smoothly bounded Jordan domain, then there is a function f on D for which f'(D)…

Let $n, m$ be positive integers, $n\geq m$. We make several remarks on the relationship between approximate differentiability of higher order and Morse-Sard properties. For instance, among other things we show that if a function…

Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. An $(s,t)$-Furstenberg set is a set $K \subset \mathbb{R}^{2}$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{L} \geq t$…

In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, $S(A)$ and $S(B)$, of two uniformly…

Let $f:{\mathbb B}^n \to {\mathbb B}^N$ be a holomorphic map. We study subgroups $\Gamma_f \subseteq {\rm Aut}({\mathbb B}^n)$ and $T_f \subseteq {\rm Aut}({\mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups.…

The connections between quasi-Assouad dimension and tangents are studied. We apply these results to the calculation of the quasi-Assouad dimension for a class of planar self-affine sets. We also show that sets with decreasing gaps have…

We prove that any analytic set in $\C^n$ with a unique tangent cone at infinity is an algebraic set. We prove that the degree of a complex algebraic set in $\C^n$, which is Lipschitz normally embedded at infinity, is equal to the degree of…

Two subsets $S$ and $T$ of $\mathbb{F}_2^n$ are \textit{affinely equivalent} if there is an affine automorphism of $\mathbb{F}_2^n$ taking $S$ to $T$. Given a basis of the affine span of $S$, we can construct a Venn diagram whose regions…

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…

Building upon previous works of Andr{\'e} and Chudnovsky, we prove a general result concerning the approximations of values at rational points a/b of any G-function F with rational Taylor coefficients by fractions of the form n/(B…

Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint…