A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice Γ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diffraction measure is periodic, with the dual lattice Γ ∗ as lattice of periods. This state...

We give an abstract account of resource-bounded reducibilities as exemplified by the polynomially timeor logarithmically space-bounded reducibilities of Turing, truth-table, and many-one type. We introduce a small set of axioms that are satisfied for most of the specific resourcebounded reducibilities appearing in the literature. Some of the axioms are of a more algebraic nature, such as the re...

We establish the vanishing for non-trivial unitary representations of the bounded cohomology of SLd up to degree d− 1. It holds more generally for uniformly bounded representations on superreflexive spaces. The same results are obtained for lattices. We also prove that the real bounded cohomology of any lattice is invariant in the same range.

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind-MacNeille completion of P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of clos...

We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula.

Let L be a bounded distributive lattice. In this paper we focus on those functions f : L → L which can be expressed in the language of bounded lattices using variables and constants, the so-called “weighted” lattice polynomial functions. Clearly, such functions must be nondecreasing in each variable, but the converse does not hold in general. Thus it is natural to ask which nondecreasing functi...

We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a unive...

A central problem in the algorithmic study of lattices is the closest vector problem: given a lattice L represented by some basis, and a target point y, nd the lattice point closest to y. Bounded Distance Decoding is a variant of this problem in which the target is guaranteed to be close to the lattice, relative to the minimum distance 1(L) of the lattice. Speci cally, in the -Bounded Distance ...

We consider the problem of counting the number of lattice vectors of a given length and prove several results regarding its computational complexity. We show that the problem is ♯Pcomplete resolving an open problem. Furthermore, we show that the problem is at least as hard as integer factorization even for lattices of bounded rank or lattices generated by vectors of bounded norm. Next, we discu...

The lattice structure of bounded Coulomb systems is explored using a simple zero-temperature slab model. For slabs above a certain size in6nite-volume behavior (i.e., a bcc lattice) occurs. Below this size surface e6'ects dominate and the lattice type depends on the size, usually taking on an fcclike symmetry. This state is similar to the lattice observed in recent computer simulations. These r...