Related papers: On Proper Polynomial Maps of $\mathbb{C}^2.$

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if they have equivalent independence polynomials. We extend…

Soient donn\'es deux graphes $\Gamma_1$, $\Gamma_2$ \`a $n$ sommets. Sont-ils isomorphes? S'ils le sont, l'ensemble des isomorphismes de $\Gamma_1$ \`a $\Gamma_2$ peut \^etre identifi\'e avec une classe $H \pi$ du groupe sym\'etrique sur…

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In…

We study Chebyshev constants and transfinite diameter on the graph of a polynomial mapping $f\colon\mathbb{C}^2\to\mathbb{C}^2$. We show that two transfinite diameters of a compact subset of the graph (i.e., defined with respect to two…

We introduce and study the problem \mpd, which asks for two planar graphs $G_1$ and $G_2$ whether $G_1$ can be embedded such that its dual is isomorphic to $G_2$. Our algorithmic main result is an NP-completeness proof for the general case…

We describe the topology of a general polynomial mapping $F=(f, g):X\to\Bbb C^2$, where $X$ is a complex plane or a complex sphere.

Let $F,E\subseteq \mathbb{R}^2$ be two self similar sets. First, assuming $F$ is generated by an IFS $\Phi$ with strong separation, we characterize the affine maps $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $g(F)\subseteq F$. Our…

Given a principal $G$-bundle $P \to M$ and two $C^1$ curves in $M$ with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on $P$. The…

A problem of completing a linear map on C*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some…

We derive a correspondence between the eigenvalues of the adjacency matrix $A$ and the signless Laplacian matrix $Q$ of a graph $G$ when $G$ is $(d_1,d_2)$-biregular by using the relation $A^2=(Q-d_1I)(Q-d_2I)$. This motivates asking when…

We claimed that there is a polynomial algorithm to test if two graphs are isomorphic. But the algorithm is wrong. It only tests if the adjacency matrices of two graphs have the same eigenvalues. There is a counterexample of two…

A frame is an overcomplete set that can represent vectors(signals) faithfully and stably. Two frames are equivalent if signals can be essentially represented in the same way, which means two frames differ by a permutation, sign change or…

In recent work [Nien et al. 2016], the authors enumerated a classification of quadratic maps of the plane according to their critical sets and images. It is straightforward to show that quadratic maps which are affinely map equivalent are…

Let $G$ be a graph embedded in a surface and let $\mathcal F$ be a set of even faces of $G$ (faces bounded by a cycle of even length). The resonance graph of $G$ with respect to $\mathcal F$, denoted by $R(G;\mathcal F)$, is a graph such…

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such…

Let $F=(F_1,F_2):C^2 ---> C^2 be a polynomial automorphism. It is well know that deg F_1 | deg F_2 or deg F_2 | deg F_1. On the other hand, if (d_1,d_2) \in (N\{0})^2 is such that d_1 | d_2 or d_2 | d_1, then one can construct a polynomial…

We show the existence of polynomial maps which have a regular bifurcation value, while over a neighbourhood of this value the fibres are connected and diffeomorphic.

In this article we study the bicanonical map $\phi_2$ of quadruple Galois canonical covers X of surfaces of minimal degree. We show that $\phi_2$ has diverse behavior and exhibit most of the complexities that are possible for a bicanonical…

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we…

We study proper holomorphic maps between type-$\mathrm{I}$ irreducible bounded symmetric domains. In particular, we obtain rigidity results for such maps under certain assumptions. More precisely, let $f:D^{\mathrm{I}}_{p,q}\to…