The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that H G is a normal subgroup, for amalgamated free products G ∗H (H ⋊ F ). Here F is a free group and H ⋊ F is an arbitrary semi-direct product. This includes free products G∗F and semi-...

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G H of graphs. As the main result of this paper, we prove that η(G1 G2) ≥ h √ l (1− o(1)) for any two g...

Let G,H be graphs and G?H represent a particular graph product of G H. We define im(G) to the largest t such that has Kt-immersion ask: given im(G)=t im(H)=r, how large is im(G?H)? Best possible lower bounds are provided when ? Cartesian or lexicographic product, conjecture offered for each direct strong products, along with some partial results.

The goal of this article is to prove the Sum Squares Conjecture for real polynomials $$r(z,\bar{z})$$ on $$\mathbb {C}^3$$ with diagonal coefficient matrix. This conjecture describes possible values rank $$r(z,\bar{z}) \left\Vert {z} \right\Vert ^2$$ under hypothesis that $$r(z,\bar{z})\left\Vert ^2=\left\Vert {h(z)} some holomorphic polynomial mapping h. Our approach connect problem degree est...

The goal of this work is to study homomorphism problems (from a computational point of view) on two superclasses of graphs: 2-edge-coloured graphs and signed graphs. On the one hand, we consider the H-Colouring problem when H is a 2-edge-coloured graph, and we show that a dichotomy theorem would imply the dichotomy conjecture of Feder and Vardi. On the other hand, we prove a dichotomy theorem f...

For graphs G and H, a homomorphism from G to H, or H-coloring of G, is an adjacency preserving map from the vertex set of G to the vertex set of H. Our concern in this paper is the maximum number of H-colorings admitted by an n-vertex, d-regular graph, for each H. Specifically, writing hom(G,H) for the number of H-colorings admitted by G, we conjecture that for any simple finite graph H (perhap...

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank> 1. This partly exposito...

In 1965, Fine and Wilf proved the following theorem: if (fn)n¿0 and (gn)n¿0 are periodic sequences of real numbers, of period lengths h and k, respectively, and fn = gn for 06 n¡h+ k − gcd(h; k), then fn = gn for all n¿ 0. Furthermore, the constant h + k − gcd(h; k) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where fn 6 gn inste...

A “double star” is a tree with two internal vertices. It known that the Gyárfás–Sumner conjecture holds for double stars, is, every star H $H$ , there function f ${f}_{H}$ such if G $G$ does not contain as an induced subgraph then ? ( ) ? ? $\chi (G)\le {f}_{H}(\omega (G))$ (where ,\omega $ are chromatic number and clique of ). Here we prove can be chosen to polynomial.

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...