Related papers: Injectivity of differentiable maps R^2 --> R^2 at …

Let X:R2\Dr->R2 be a differentiable (but not necessarily C1) vector field, where r>0 and Dr={z\in R2:|z|\le r}. If for some e>0 and for all p\in R2\Dr, no eigenvalue of D_p X belongs to (-e,0]\cup {z\in\C:\mathcal{R}(z)\ge 0}, then (a)For…

(a) Let X=(f,g) be a differentiable map in the plane (not necessarily C^1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DX(p) when p varies in R^2. If, for some \epsilon>0, the set Spec(X) is disjoint of…

One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for one-dimensional piecewise affine maps with two intervals. In this paper we prove that for injective maps, it is…

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…

Let $X$ be a topological space, $U$ -- opened subset of $X$. We will say that point $x \in \partial U$ is {\it accessible} from $U$ if there exists continuous injective mapping $\phi : I \to \Cl D$ such that $\phi(1)=x$, $\phi([0,1))…

Let $R$ be any ring containing a non-tivial idempotent element $e$. Let $\Im: R\rightarrow R$ be a mapping such that $\Im(ab)=\Im(b)a+b\Im(a)$ for all $a,b\in R$. In this note, our aim is to show that under some suitable restrictions…

We prove the projective plane $\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by…

This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps $F(x,y)$ defined on a finite-dimensional Euclidean space. There are no hypothesis on the continuity of the partial…

It is shown that there is a computable conformal map of the unit disk onto a domain $D$ that has a computable extension to the closure of the unit disk even though the boundary of $D$ is not effectively locally connected. The proof encodes…

We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional…

Rademacher theorem asserts that Lipschitz continuous functions between Euclidean spaces are differentiable almost everywhere. In this work we extend this result to set-valued maps using an adequate notion of set-valued differentiability…

A notion of differentiability is being proposed for maps between Wasserstein spaces of order 2 of smooth, connected and complete Riemannian manifolds. Due to the nature of the tangent space construction on Wasserstein spaces, we only give a…

This paper presents sufficient graph-theoretic conditions for injectivity of collections of differentiable functions on rectangular subsets of R^n. The results have implications for the possibility of multiple fixed points of maps and…

Establishing that a demand mapping is injective is core first step for a variety of methodologies. When a version of the law of demand holds, global injectivity can be checked by seeing whether the demand mapping is constant over any line…

Extension dimension is characterized in terms of $\omega$-maps. We apply this result to prove that extension dimension is preserved by refinable maps between metrizable spaces. It is also shown that refinable maps preserve some…

Let $X:U-->R^2$ be a differentiable vector field. Set $Spc(X)={eigenvalues of DX(z) : z\in U}$. This $X$ is called Hurwitz if $Spc(X)\subset{z\in C:\Re(z)<0}$. Suppose that $X$ is Hurwitz and $U\subset R^2$ is the complement of a compact…

Expanding maps with indifferent fixed points, a.k.a. intermittent maps, are popular models in nonlinear dynamics and infinite ergodic theory. We present a simple proof of the exactness of a wide class of expanding maps of [0,1], with…

Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most…

In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the…

In a recent paper, it was shown that the problem of existence of a continuous map $X \to Y$ extending a given map $A \to Y$ defined on a subspace $A \subseteq X$ is undecidable, even for $Y$ an even-dimensional sphere. In the present paper,…