In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lovász) base reduction algorithm [20], cryptographic...

In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lovász) base reduction algorithm [20], cryptographic...

Sufficient conditions are given for groups of finite Morley rank having non-trivial torsion-free nilpotent normal subgroups to have linear representations with small kernels. In particular, centreless connected soluble groups of finite Morley rank with torsion-free Fitting subgroups have faithful linear representations. On the way, using a notion of definable weight space, we prove that certain...

In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds ho...

Fig. 1. Hyperspectral imagers resolve scenes at high spatial and spectral resolutions. We propose a novel architecture called KRISM that optically implements two operators: a spatially-coded spectrometer, and a spectrally-coded spatial imager. By iterating between the two, we can acquire a low rank approximation of the hyperspectral scene in a light efficient manner with very few measurements. ...

The operation V → V ω is a fundamental operation over finitary languages leading to ω-languages. Since the set Σ of infinite words over a finite alphabet Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers of finitary languages naturally arises and has been posed by Niwinski [Niw90], Simonnet [Sim92] and Staiger [Sta97a]. It has been recently...

A well-known result by P. Cameron provides us with a construction of the free group of rank 2א0 within the automorphism group of the rationals. We show that the full versatility of doubly transitive automorphism groups is not necessary by extending Cameron’s construction to a larger class of permutation groups and generalize his result by constructing pathological (permutations of unbounded sup...