In this expression, R is a lattice vector between a pair of unit cells: R = ua + vb + wc ; u, v, and w are integers and the dot product k R = kau + kbv + kcw . (In two dimensions, R = ua + vb and k R = kau + kbv .) At this point, we need to clarify the meaning of the vector k and find a way to define the twoand three-dimensional Brillouin zones. To do this, let’s consider the general, oblique, ...

Asymptotic expressions for the optimal scaling factor and resulting minimum distortion , as a function of codebook size N, are found for xed-rate k-dimensional lattice vector quantization of generalized Gaussian sources with decay parameter 1. These expressions are derived by minimizing upper and lower bounds to distortion. It is shown that the optimal scaling factor a N decreases as (lnN) 1== ...

In the present paper we show how to speed up lattice parameter searches for Monte Carlo and quasi–Monte Carlo node sets. The classical measure for such parameter searches is the spectral test which is based on a calculation of the shortest nonzero vector in a lattice. Instead of the shortest vector we apply an approximation given by the LLL algorithm for lattice basis reduction. We empirically ...

We describe how a shortest vector of a 2-dimensional integral lattice corresponds to a best approximation of a unique rational number defined by the lattice. This rational number and its best approximations can be computed with the euclidean algorithm and its speedup by Schönhage (1971) from any basis of the lattice. The described correspondence allows, on the one hand, to reduce a basis of a 2...

For a left vector space V over a totally ordered division ring F, let Co(V ) denote the lattice of convex subsets of V . We prove that every lattice L can be embedded into Co(V ) for some left F-vector space V . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Ω) = {X ∩Ω | X ∈ Co(V )}, for some f...

We introduce algorithms for lattice basis reduction that are improvements of the famous L 3-algorithm. If a random L 3 {reduced lattice basis b1; : : : ; bn is given such that the vector of reduced Gram{ Schmidt coeecients (fi;jg 1 j < i n) is uniformly distributed in 0; 1) (n 2) , then the pruned enumeration nds with positive probability a shortest lattice vector. We demonstrate the power of t...

Lattice studies of hadronic matrix elements require matching between operators in full continuum QCD and those in the lattice theory being simulated. In the present article we will focus on an effective lattice theory which combines nonrelativistic (NRQCD) heavy quarks and clover light quarks. We wish to match the theories correct through O( p M , α p M , α ap). For heavy meson (e.g. B and B∗) ...

It is known that the generating vector of a rank-1 lattice rule can be constructed component-by-component to achieve strong tractability error bounds in both weighted Korobov spaces and weighted Sobolev spaces. Since the weights for these spaces are nonincreasing, the first few variables are in a sense more important than the rest. We thus propose to copy the points of a rank-1 lattice rule a n...

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. ...

Point-to-point vacuum correlation functions for spatially separated hadron currents are calculated in quenched lattice QCD on a 16 × 24 lattice with 6/g = 5.7. The lattice data are analyzed in terms of dispersion relations, which enable us to extract physical information from small distances where asymptotic freedom is apparent to large distances where the hadronic resonances dominate. In the p...