This work considers the convergence rate of multi-agent systems with discrete-time single-integrator dynamics and undirected interaction topologies. In recent work it has been proven that in case of lattice interaction topologies the convergence rate can be bounded away from zero, independent of the network size, using asymmetric weightings that give the interaction graph a preferred communicat...

An insight on the lattice decoder for flat-fading multiple antenna wireless communications systems is presented in this paper. In particular, we show that by formulating the decoding problem as a bounded-error subset selection, the resultant decoder finds the nearest lattice point to the received signal vector such that the search is bounded inside a hypercube centered at the received vector. T...

In [5], a representation of bounded lattices within so-called standard topological contexts has been developed. Based on the theory of formal concept analysis [14] it includes Stone’s representation of Boolean algebras by totally disconnected compact spaces [10], Priestley’s representation of bounded distributive lattices by totally order disconnected compact spaces [7] as well as Urquhardt’s r...

We have already considered instances of the following type of problem: given a bounded subset Ω of Euclidean space R N , to determine #(Ω ∩ Z N), the number of integral points in Ω. It is clear however that there is no answer to the problem in this level of generality: an arbitrary Ω can have any number of lattice points whatsoever, including none at all. In [Gauss's Circle Problem], we counted...

Monotone bounded distributive lattice expansions (DLMs) are boundeddistributive lattices augmented by nitary operations that are isotone orantitone in each coordinate. Such algebras encompass most algebras withbounded distributive lattice reducts that arise from logic, and general-ize bounded distributive lattices with operators. We de ne the canonicalextension for DLMs and ...

We develop further basic tools in the theory of bounded continuous cohomology of locally compact groups; as such, this paper can be considered a sequel to [18], [39], and [11]. We apply these tools to establish a Milnor–Wood type inequality in a very general context and to prove a global rigidity result which was originally announced in [13] and [33] with a sketch of a proof using bounded cohom...

A bstmct-A new efficient algorithm for bounded-distance decoding of the Leech la t t ice is presented. The algori thm decodes correctly at least ulp to the guaranteed error-correction radius of the Leech lattice. The proposed decoder is based on projecting the points of the Leech lattice onto the codewords of the (6,3,4) quar te rnary code, the hezacode Ne. Project ion on the hexacode induces p...

Two different elementary approaches for deriving an explicit formula for the distribution of the range of a simple random walk on Z of length n are presented. Both of them rely on Hermann Weyl’s discrepancy norm, which equals the maximal partial sum of the elements of a sequence. By this the original combinatorial problem on Z can be turned into a known path-enumeration problem on a bounded lat...

and Applied Analysis 3 With respect to the pointwise order u ≤ v ⇐⇒ ( ∀x ∈ Ω : u (x) ≤ v (x) ) (11) the setNL(Ω) is a Dedekind complete lattice. In particular, the supremum and infimum of a setA ⊂ NL(Ω) is given by inf A = (I ∘ S) (φ) , (12) supA = (I ∘ S) (ψ) , (13) respectively, where φ : Ω ∋ x 󳨃→ inf{u(x) : u ∈ A} and ψ : Ω ∋ x 󳨃→ sup{u(x) : u ∈ A}. Furthermore, the lattice NL(Ω) is fully di...

If M is a finite complemented modular lattice with n atoms and D is a bounded distributive lattice, then the Priestley power M [D] is shown to be isomorphic to the poset of normal elements of Dn, thus solving a problem of E. T. Schmidt from 1974. It is shown that there exist a finite modular lattice A not having M4 as a sublattice and a finite modular lattice B such that A⊗B is not semimodular,...