The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G we prove that n 2 + ∗1 − 1 rb(G) n− ∗0 + 1, where ∗0 and ∗1 denote the independence number and the edge independence number, respectively, of th...

Let k be a field, G˜ connected reductive k-group, and Γ finite group. In previous work, the authors defined what it means for k-group G to parascopic (G˜,Γ). Roughly, this is simultaneous generalization of several settings. For example, could act on G˜, part group Γ-fixed points in G˜. Or an endoscopic group, pseudo-Levi subgroup, or isogenous image If such both are k-quasisplit, then we constr...

Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that TrF/E(g(x), h(x)) = δg,h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char(E) 6= 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char(E) = 2, then F ...

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group $$G^\vee $$ , and Poisson–Lie $$G^*$$ respectively. The main result of this paper is the following relation between these objects: integral cone defined by cluster structure Berenstein–Kazhdan potential on double Bruhat cell $$G^{\vee ; w_0, e} \subset G^\vee isomorp...

A k-block in a graph G is a maximal set of at least k vertices no two of which can be separated in G by fewer than k other vertices. The block number (G) of G is the largest integer k such that G has a k-block. We investigate how interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a k-block, or which...

The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which have different colours. For a cubic polyhedral (i.e. 3-connected plane) graph G we prove that n 2 + α∗ 1 − 1 ≤ rb(G) ≤ n− α∗ 0 + 1 , where α∗ 0 and α ∗ 1 denote the independence number and the edge independ...

We develop an analytic approach to the Radon transform f̂(ζ) = ∫ τ⊂ζ f(τ), where f(τ) is a function on the affine Grassmann manifold G(n, k) of k-dimensional planes in Rn, and ζ is a k′-dimensional plane in the similar manifold G(n, k′), k′ > k. For f ∈ Lp(G(n, k)), we prove that this transform is finite almost everywhere on G(n, k′) if and only if 1 ≤ p < (n− k)/(k′− k), and obtain explicit inv...

The paper contains results that characterize the Donkin–Koppinen filtration of coordinate superalgebra K[G] general linear supergroup G = GL(m|n) by its subsupermodules CΓ OΓ(K[G]). Here, supermodule is largest subsupermodule whose composition factors are irreducible supermodules highest weight ⋋, where ⋋ belongs to a finitely-generated ideal Γ poset X(T)+ dominant weights G. A decomposition as...

Let V be 2n-dimensional vector space over a field K equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let K0 ⊆ K be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, . . . , 2n}, there exist natural representations of the groups G ∼= U(2n,K/K0) and H = G∩SL(V ) ∼= SU(2n,K/K0) on the k-th exterior power of V . With the aid of linear...

Let g ∈ L2(R) be a compactly supported function, whose integertranslates {Tkg}k∈Z form a partition of unity. We prove that for certain translationand modulation parameters, such a function g generates a Gabor frame, with a (non-canonical) dual generated by a finite linear combination h of the functions {Tkg}k∈Z; the coefficients in the linear combination are given explicitly. Thus, h has compac...