It is shown that lattice codes can achieve capacity on the additive white Gaussian noise channel. More precisely, for any rate R less than capacity and > 0, there exists a lattice code with rate no less than R and average error probability upper-bounded by : These lattice codes include all points of the (translated) lattice within the spherical bounding region (not just the ones inside a thin s...

We present deterministic polynomially space bounded algorithms for the closest vector problem for all lp-norms, 1 < p < ∞, and all polyhedral norms, in particular for the l1norm and the l∞-norm. For all lp-norms with 1 < p < ∞ the running time of the algorithm is p · log2(r)n, where r is an upper bound on the size of the coefficients of the target vector and the lattice basis and n is the dimen...

The notation and terminology used here are introduced in the following papers: [18], [13], [17], [14], [19], [7], [1], [8], [6], [20], [3], [9], [2], [10], [15], [16], [5], [11], [4], and [12]. Let us observe that there exists a lattice which is finite. Let us mention that every lattice which is finite is also complete. Let L be a lattice and let D be a subset of the carrier of L. The functor D...

We study the interfaces’ time evolution in one-dimensional bistable extended dynamical systems with discrete time. The dynamics is governed by the competition between a local piece-wise affine bistable mapping and any couplings given by the convolution with a function of bounded variation. We prove the existence of travelling wave interfaces, namely fronts, and the uniqueness of the correspondi...

We describe the transfer operator approach to coupled map lattices (CML) in cases where the local map is expanding but has no Markov partition (e.g. a general tent map). The coupling is allowed to be non-local, but the total innuence of all sites j 6 = i on site i must be small. The main technical tool are lattice-size independent estimates of Lasota-Yorke type which show that the transfer (Per...

∈ , ∈ ∨ q ) – fuzzy ideal (prime) of bounded lattice is introduced and its properties are studied. Mathematics Subject Classifications: 06B10, 06D72

This paper proposes a new lattice filter structure that has the following properties. When the filter is Linear Time Invariant (LTI), it is equivalent to the celebrated Gray Markel Lattice. When the lattice parameters vary with time it sustains arbitrary rate of time variations without sacrificing a prescribed degree of stability, provided that the lattice coefficients are magnitude bounded in ...

A k-dimensional lattice simplex σ ⊆ Rd is the convex hull of k + 1 affinely independent integer points. General lattice polytopes are obtained by taking convex hulls of arbitrary finite subsets of Zd . A lattice simplex or polytope is called empty if it intersects the lattice Zd only at its vertices. (Such polytopes are studied also under the names elementary and latticefree.) In dimensions d >...

Pseudo-hoops are algebraic structures introduced in [1, 2] by B. Bosbach under the name of complementary semigroups. This is a formalization of the paper [4]. Following [4] we prove some properties of pseudo-hoops and we define the basic concepts of filter and normal filter. The lattice of normal filters is isomorphic with the lattice of congruences of a pseudo-hoop. We also study some importan...