In this paper we study a representation of a fuzzy subgroup $mu$ of a group $G$, as a product of indecomposable fuzzy subgroups called the components of $mu$.  This representation is unique up to the number of components and their isomorphic copies. In the crisp group theory, this is a well-known Theorem attributed to Remak, Krull, and Schmidt. We consider the lattice of fuzzy subgroups and som...

Exercise 1 (Maximal compact subgroups of G). A lattice in Qp is a finitelygenerated Zp-submodule of Qp that generates Qp as vector space. In particular, it’s free of rank n. Note that G acts transitively on the set of lattices in Qp . (i) Show that K = StabG(Zp ). (ii) Suppose that K ′ is a compact subgroup of G. Show that K ′ stabilises a lattice. (Hint: show that the K ′-orbit of Zp is finite...

We study the algorithmic complexity of lattice problems based on the sieving technique due to Ajtai, Kumar, and Sivakumar [AKS01]. Given a k-dimensional subspace M ⊆ Rn and a full rank integer lattice L ⊆ Qn, the subspace avoiding problem SAP, defined by Blömer and Naewe [BN07], is to find a shortest vector in L \ M. We first give a 2O(n+k log k) time algorithm to solve the subspace avoiding pr...

Suppose G is a hyperbolic group whose boundary ∂∞G has topological dimension k. If ∂∞G is quasi-symmetrically homeomorphic to an Ahlfors kregular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group Isom(Hk+1) of hyperbolic (k+1)-space.

Abstract. Let L and M be two finite lattices. The ideal J(L,M) is a monomial ideal in a specific polynomial ring and whose minimal monomial generators correspond to lattice homomorphisms ϕ: L→M. This ideal is called the ideal of lattice homomorphism. In this paper, we study J(L,M) in the case that L is the product of two lattices L_1 and L_2 and M is the chain [2]. We first characterize the set...

In this note we classify all triples (a, b, i) such that there is a convex lattice polygon P with area a, and b respectively i lattice points on the boundary respectively in the interior. The crucial lemma for the classification is the necessity of b ≤ 2 i+ 7. We sketch three proofs of this fact: the original one by Scott [12], an elementary one, and one using algebraic geometry. As a refinemen...

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu [T. Lee, L. Liu, Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces, J. Comput. Phys. 229(2010) 8045-8063], is extended to simulate multiphase flows in porous media. A wetting boundary treatment is proposed for t...

Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz-ers, are all defined from the binary operation [x, y] = x −1 y −1 xy. Each of these notions, except centralizers of elements, may also be defined in terms of the commutator of normal subgroups. The commutator [M, N] (where M and N are normal subgrou...