The component-by-component construction algorithm for rank-1 lattices can be formulated elegantly as a repeated matrix-vector product. As was shown in an earlier paper, this matrix-vector product can be done in time O(n log(n)) for n prime. Here we extend this result to general n using facts from algebra to obtain a construction cost of O(sn log(n)) for a rank-1 lattice in s dimensions with n p...

Abstract We reveal a divergent issue associated with the mean-field theory for Bose gases in optical lattices constructed by widely used straightforward decoupling of hopping term, where corresponding Hamiltonian generally assumes no lower energy bound once spatial dependence superfluid order parameter is taken into account. Via systematic functional integral approach, we solve this establishin...

We present general fixed point theorems for spaces that are equipped with a transitive relation. We apply them to prove corresponding theorems for ultrametric spaces, topological spaces, complete lattices, and ordered abelian groups and fields.

In this article, we investigate the set of γ-sortable elements, associated with a Coxeter group W and a Coxeter element γ ∈ W , under Bruhat order, and we denote this poset by Bγ . We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation ...

It is well known that some lattices in SO(n, 1) can be nontrivially deformed when included in SO(n+1, 1) (e.g., via bending on a totally geodesic hypersurface); this contrasts with the (super) rigidity of higher rank lattices. M. Kapovich recently gave the first examples of lattices in SO(3, 1) which are locally rigid in SO(4, 1) by considering closed hyperbolic 3-manifolds obtained by Dehn fil...

We introduce a composite optical lattice created by two mutually rotated square patterns and allowing observation of continuous transformation between incommensurate and completely periodic structures upon variation of the rotation angle θ. Such lattices acquire periodicity only for rotation angles cos θ = a/c, sin θ = b/c, set by Pythagorean triples of natural numbers (a, b, c). While linear e...

We investigate three closely related constructions of lattices from linear codes: the classical Construction D, Construction D, and the recently developed Construction A′. These constructions have been proven useful and result in efficient encoding and decoding algorithms for Barnes-Wall lattices. Here, we analyze their applications in a general setting. We show that Construction D produces a l...

We observe that the known fact that difference logic and hybrid logic with universal modality have the same expressive power on Kripke frames can be strengthened for a far wider class of general frames. This observation, together with a general completeness result, is used to show that lattices of difference logics and of hybrid logics are isomorphic.

In the context of a general lattice Γ in Rn and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the multiwavelets associated with a Multiresolution Analysis (MRA) of multiplicity d ≥ 1, and all the scaling functions. Moreover, we give several examples: in particular, we construct a single, MRA and C∞(Rn) wavelet, which ...