For g < n, let b1, . . . , bn−g be n− g independent vectors in R with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when n → +∞ concerning the property that such a random basis is reduced in the sense of Lenstra, Lenstra & Lovász. The proof passes by the study of t...

A systematic method is presented for constructing effective Hamiltonians for general phonon-related structural transitions. The key feature is the application of group theoretical methods to identify the subspace in which the effective Hamiltonian acts and construct for it localized basis vectors, which are the analogue of electronic Wannier functions. The results of the symmetry analysis for t...

of n linearly independent vectors b1, . . . ,bn ∈ Rm in m-dimensional Euclidean space. For computational purposes, the lattice vectors b1, . . . ,bn are often assumed to have integer (or rational) entries, so that the lattice can be represented by an integer matrix B = [b1, . . . ,bn] ∈ Zm×n (called basis) having the generating vectors as columns. Using matrix notation, lattice points in L(B) c...

Let $L$ be a lattice in $ZZ^n$ of dimension $m$. We prove that there exist integer constants $D$ and $M$ which are basis-independent such that the total degree of any Graver element of $L$ is not greater than $m(n-m+1)MD$. The case $M=1$ occurs precisely when $L$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. As a corollary, we show t...

The purpose of this paper is to construct a weak hyper semi-quantale as a generalization of the concept of semi-quantale and used it as an appropriate hyperlattice-theoretic basis to formulate new lattice-valued topological theories. Based on such weak hyper semi-quantale, we aim to construct the notion of a weak hypervalued-topology as a generalized form of the so-called lattice-valued t...

The Lenstra-Lenstra-Lovász (LLL) algorithm is the most practical lattice reduction algorithm in digital communications. In this paper, several variants of the LLL algorithm with either lower theoretic complexity or fixed-complexity implementation are proposed and/or analyzed. Firstly, the O(n log n) theoretic average complexity of the standard LLL algorithm under the model of i.i.d. complex nor...

We compare Schnorr's algorithm for semi block 2k-reduction of lattice bases with Koy's primal-dual reduction for blocksize 2k. Koy's algorithm guarantees within the same time bound under known proofs better approximations of the shortest lattice vector. Under reasonable heuristics both algorithms are equally strong and much better than proven in worst-case. We combine primal-dual reduction with...

We consider the problem of finding the closest lattice point to a vector in n-dimensional Euclidean space when each component of the vector is available at a distinct node in a network. Our objectives are (i) minimize the communication cost and (ii) obtain the error probability. The approximate closest lattice point considered here is the one obtained using the nearest-plane (Babai) algorithm. ...

We study ”nearly orthogonal” lattice bases, or bases where the angle between any basis vector and the linear subspace spanned by the other basis vectors is greater than π 3 radians. We show that a nearly orthogonal lattice basis always contains a shortest lattice vector. Moreover, if the lengths of the basis vectors are “nearly equal”, then the basis is the unique nearly orthogonal lattice basi...

Abstract We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding physical stability properties dynamics its local low-rank nature. To address these aspects, we propose a...