Related papers: Injectivity of non-singular planar maps with one c…

We establish a new characterization for a conformal mapping of the unit disk $\mathbb{D}$ to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets…

In the field of the Jacobian conjecture it is well-known after Druzkowski that from a polynomial "cubic-homogeneous" mapping we can build a higher-dimensional "cubic-linear" mapping and the other way round, so that one of them is invertible…

We show that extending an embedding of a graph $\Gamma$ in a surface to an embedding of a Hamiltonian supergraph can be blocked by certain planar subgraphs but, for some subdivisions of $\Gamma$, Hamiltonian extensions must exist.

Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the…

We call a (not necessarily planar) embedding of a graph $G$ in the plane \emph{sequential} if its vertices lie in $\mathbb Z^2$ and the line segments between adjacent vertices contain no interior integer points. In this note, we prove (i) a…

We give an example of a local normal domain $R$ such that the map of Grothendieck groups $\G(R) \to \G(\hat R)$ is not injective. We also raise some questions about the kernel of that map.

A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of…

We prove that Mean Value mapping between convex quadrilaterals is injective, affirmatively proving a conjecture stated in M. S. Floater and J. Kosinka, On the injectivity of Wachspress and mean value mappings between convex polygons, Adv.…

We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding.

The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right half-plane mapping or a normalized vertical strip mapping is convex in the direction of the real axis.…

Given a strictly convex domain $\Omega$ in $\R^2$, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we…

We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse.

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be…

Let C be a general curve. Mukai asked whether for every stable rank 2 vector bundle E on C with det(E) = K, the multiplication map Sym^2 H^0(C,E) --> H^0(C, Sym^2 E) is injective. We observe that this holds when h^0(E) < 7 .

One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.

Let $k$ be an algebraically closed field of characteristic zero. Let $H:k^2\to k^2$ be a polynomial mapping such that the Jacobian $\text{Jac}\,H$ is a non-zero constant. In this note we prove, that if there is a line $l \subset k^2$ such…

In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a certain sub-domain. In our approach…

Let ${\mathcal B}_u$ be the Springer fiber over a nilpotent endomorphism $u\in End(\mathbb{C}^n)$. Let $J(u)$ be the Jordan form of $u$ regarded as a partition of $n$. The irreducible components of ${\mathcal B}_u$ are all of the same…

We show that for finite groups the Loday assembly map with coefficients in finite fields is in general not injective.

In this note we provide two special examples of non-injective polynomial maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ with non-vanishing Jacobian: the first one is surjective, the second one has non-dense image.