Two variations of a method for reducing controllers for systems described by partial diierential equations (PDEs) are compared. In the rst method, the controller reduction is accomplished by projection of a large scale nite element approximation of the PDE controller onto a low order basis that is computed using the proper orthogonal decomposition (POD). The second method involves computing a r...

Upper and lower bounds are derived for the decoding complexity of a general lattice L. The bounds are in terms of the dimension n and the coding gain of L, and are obtained based on a decoding algorithm which is an improved version of Kannan’s method. The latter is currently the fastest known method for the decoding of a general lattice. For the decoding of a point x, the proposed algorithm rec...

Vector perturbation (VP) precoding is a promising technique for multiuser communication systems operating in the downlink. In this work, we introduce a hybrid framework to improve the performance of lattice reduction (LR) aided (LRA) VP. Firstly, we perform a simple precoding using zero forcing (ZF) or successive interference cancellation (SIC) based on a reduced lattice basis. Since the signal...

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We show that for a prime p the smallest a with ap−1 6≡ 1 (mod p2) does not exceed (log p)463/252+o(1) which improves the previous bound O((log p)2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p)5/3+o(1).

A lattice in euclidean space which is an orthogonal sum of nontrivial sublattices is called decomposable. We present an algorithm to construct a lattice’s decomposition into indecomposable sublattices. Similar methods are used to prove a covering theorem for generating systems of lattices and to speed up variations of the LLL algorithm for the computation of lattice bases from large generating ...

Given d complex numbers z1, ..., zd, it is classical that linear dependencies λ1 z1+ ···+ λd zd=0 with λ1, ..., λd∈Z can be guessed using the LLL-algorithm. Similarly, given d formal power series f1, ..., fd ∈ C[[z]], algorithms for computing Padé-Hermite forms provide a way to guess relations P1 f1 + ···+ Pd fd = 0 with P1, ..., Pd ∈C[z]. Assuming that f1, ..., fd have a radius of convergence ...

The Lenstra–Lenstra–Lovász lattice basis reduction algorithm (called LLL or L3) is a fundamental tool in computational number theory and theoretical computer science, which can be viewed as an efficient algorithmic version of Hermite’s inequality on Hermite’s constant. Given an integer d-dimensional lattice basis with vectors of Euclidean norm less than B in an ndimensional space, the L3 algori...

We establish a link between some heuristic asymptotic formulas (due to Cohen and Lenstra) concerning the moments of the p–part of the class groups of quadratic fields and formulas giving the frequency of the values of the p–rank of these class groups. Furthermore we report on new results for 4–ranks of class groups of quadratic number fields.