We use difference sets to construct interesting sets of lines in complex space. Using (v, k, 1)-difference sets, we obtain k2−k+1 equiangular lines in Ck when k − 1 is a prime power. Using semiregular relative difference sets with parameters (k, n, k, λ) we construct sets of n + 1 mutually unbiased bases in Ck. We show how to construct these difference sets from commutative semifields and that ...

In this paper, we show that the eigenvectors associated with the zero eigenvalues of the Laplacian and signless Lapacian tensors of a k-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector associated with the zero eigenvalue of the Laplacian or signless Lapacian tensor have the same modulus. Moreover, under a cano...

Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T n Ru(G) for any maximal torus T of G [Bo, 10.6(4)]. Over more general k, an analogous such structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and an element a ∈ k − kp, so k′ := k(a1/p) is a degree-p purely inseparable extension of k. Note that k′ s :...

We are concerned with conditions under which a locally compact group G has a maximal compact normal subgroup K and whether or not G/K is a Lie group. If G has small compact normal subgroups K such that G/K is a Lie group, then G is pro-Lie. If in G there is a collection of closed normal subgroups {Ha} such that f~| Ha = e and G/Ha is a Lie group for each a, then G is a residual Lie group. We de...

Let SCRc = {σ ∈ 2n : K(σ) ≥ n + K(n) − c}, where K denotes prefix-free Kolmogorov complexity. These are the strings with essentially maximal prefix-free complexity. We prove that SCRc is not a Π1 set for sufficiently large c. This implies Solovay’s result that strings with maximal plain Kolmogorov complexity need not have maximal prefix-free Kolmogorov complexity, even up to a constant. We show...

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the of birational transformations projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out overall study centralizers elements infinite order in which leads to a classification embeddings $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well maximal non-torsion abelian subgroups ...

We exhibit a canonical connection between maximal (0, 1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. Mor...

Let G be a connected reductive linear algebraic group defined over an field k of characteristic not 2, θ ∈ Aut(G) an involutional k-automorphism of G and K = Gθ = {g ∈ G | θ(g) = g} the set of fixed points of θ. Denote the set of k-rational points of G by Gk. In this paper we shall classify the K-conjugacy classes of θ-stable maximal tori of G. This is shown to be independent of the characteris...

In recent years, soft sets have become popular in various fields. For this reason, many studies been carried out the field of algebra. study, intersection k-ideals are defined with help a semiring, and some algebraic structures examined. Moreover, quotient rings by k-semiring. Isomorphism theorems examined rings. Finally, properties investigated defining maximal k-ideals.