Consider the following relaxation of the Hadwiger Conjecture: For each t there exists Nt such that every graph with no Kt minor admits a vertex partition into dαt + βe parts, such that each component of the subgraph induced by each part has at most Nt vertices. The Hadwiger Conjecture corresponds to the case α = 1, β = −1 and Nt = 1. Kawarabayashi and Mohar [K. Kawarabayashi, B. Mohar, A relaxe...

For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis ...

We generalist Gabidulin codes to the case of infinite fields, eventually with characteristic zero. For this purpose, we consider an abstract field extension and any automorphism in the Galois group. We derive some conditions on the automorphism to be able to have a proper notion of rank metric which is in coherence with linearized polynomials. Under these conditions, we generalize Gabidulin cod...

where 3 is the centre of 51 and cl denotes closure in the norm topology. When 5t is a von Neumann algebra this result is already known (see [7; Lemma 9, p. 255], where the method requires only trivial modification to deal with the case of a left ideal), and the proof in that situation involves the restriction of an operator in M to a subspace on which it can be inverted. The proof given here fo...

The problem of the holomorphic contractibility Teichmüller spaces T(0,n) punctured spheres (n>4) arose in 1970s connection with solving algebraic equations Banach algebras. Recently it was solved by author. In present paper, we give a refined proof for all T(0,n), n>4 and provide two independent proofs low-dimensional spaces, which has intrinsic interest.