Let r1, . . . , rs be non-zero integers satisfying r1 + · · ·+ rs = 0. Let G ' Z/k1Z⊕ · · · ⊕ Z/knZ be a finite abelian group with ki|ki−1 (2 ≤ i ≤ n), and suppose that (ri, k1) = 1 (1 ≤ i ≤ s). Let Dr(G) denote the maximal cardinality of a set A ⊆ G which contains no non-trivial solution of r1x1 + · · · + rsxs = 0 with xi ∈ A (1 ≤ i ≤ s). We prove that Dr(G) |G|/ns−2. We also apply this result...

We consider polytopes in Rn that are generated by N vectors in Rn whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random ±1 polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur distance between them is essentially of a maximal order n. This result is a...

Let I be a finite or infinite index set, let X be a topological space, and let Yi, φNi i∈I be a family of FC-spaces. For each i ∈ I, letAi : X → 2i be a set-valued mapping. Some new existence theorems of maximal elements for a set-valued mapping and a family of set-valued mappings involving a better admissible set-valuedmapping are established under noncompact setting of FC-spaces. Our results ...

We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras gl(∞), sl(∞), so(∞), and sp(∞). For simple finite–dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin.

Let P be a Poisson homogeneous G-space. In [Dr2], Drinfeld shows that corresponding to each p ∈ P , there is a maximal isotropic Lie subalgebra lp of the Lie algebra d, the double Lie algebra of the tangent Lie bialgebra (g, g∗) of G. Moreover, for g ∈ G, the two Lie algebras lp and lgp are related by lgp = Adg lp via the Adjoint action of G on d. In particular, they are isomorphic as Lie algeb...

The following theorem is proved. The collection of lines of an n-uniform projective Hjelmslev plane is maximal when considered as a collectiion of mutually intersecting sets of equal cardinality.

There is well-established evidence that decision makers consistently fail to consider all available options. Instead, they restrict attention to only a subset of alternatives and then undertake a more detailed analysis of this reduced set. This systematic lack of consideration of available options can lead to a “more is less” effect, where excess of options can be welfare-reducing for a decisio...

Let G be a finite group. We let m(G) and σ(G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper we determine the structure of G/Φ(G) when G is a finite soluble group with m(G) ≤ 2σ(G).

We consider voting rules that are based on the majority graph. Such rules typically output large sets of winners. Our goal is to investigate a general method which leads to refinements of such rules. In particular, we use the idea of parallel universes, where each universe is connected with a permutation over alternatives. The permutation allows us to construct resolute voting rules (i.e. rules...

We study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group using an extension of the method due to Borel and Prasad.