In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this arti...

In this paper we investigate how the complexity of the shortest vector problem in a lattice Λ depends on the cycle structure of the additive group Z/Λ. We give a proof that the shortest vector problem is NP-complete in the max-norm for n-dimensional lattices Λ where Z/Λ has n−1 cycles. We also give experimental data that show that the LLL algorithm does not perform significantly better on latti...

In this article, we formalize in Mizar [5] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorit...

We give a new family of Lovász Local Lemmas (LLL), with applications. Shearer has given the most general condition under which the LLL holds, but the original condition of Lovász is simpler and more practical. Do we have to make a choice between practical and optimal? In this article we present a continuum of LLLs between the original and Shearer’s conditions. One of these, which we call Clique...

This study examined the convergent construct validity of a new performance-based assessment instrument called the Comprehensive High-Level Activity Mobility Predictor (CHAMP) as a measure of high-level mobility in servicemembers (SMs) with traumatic lower-limb loss (LLL). The study was completed by 118 SMs. Convergent construct validity of the CHAMP was established using the 6-minute walk test ...

In 1982, Arjen Lenstra, Hendrik Lenstra Jr. and László Lovász introduced an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLL-reduction. The precise definition involves the R-factor of the QR factorization of the basis matrix. In order to circumvent the use of rational/exact arithmetic with large bit-sizes, it is tempting to consi...

In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in...

Higher than the lowest Landau level contributions to magnetization and specific heat of superconductors are calculated using Ginzburg Landau equations approach. Corrections to the excitation spectrum around solution of these equations (treated perturbatively) are found. Due to symmetries of the problem leading to numerous cancellations the range of validity of the LLL approximation in mean fiel...

Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the anyonic nature of particle...