Erratum: Occupation Ratio of Large Atoms under Fermi–Dirac Statistics on Integer Lattices

A SHORT NOTE ON ATOMS AND COATOMS IN SUBGROUP LATTICES OF GROUPS

In this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. It is proved that an abelian group of finite exponent is strongly coatomic.

Void Statistics in Large Galaxy Redshift Surveys: Does Halo Occupation of Field Galaxies Depend on Environment?

We use measurements of the projected galaxy correlation function wp(rp) and galaxy void statistics to test whether the galaxy content of halos of fixed mass is systematically different in low density environments. We present new measurements of the void probability function (VPF) and underdensity probability function (UPF) from Data Release Four of the Sloan Digital Sky Survey (SDSS), as well a...

Completely Empty Pyramids on Integer Lattices

A Note on Integer Factorization Using Lattices

We revisit Schnorr’s lattice-based integer factorization algorithm, now with an effective point of view. We present effective versions of Theorem 2 of [11], as well as new properties of the Prime Number Lattice bases of Schnorr and Adleman.

Integer Optimization and Lattices

• Lattices. We will see basic properties of lattices, followed by Minkowski’s Theorem which guarantees that any symmetric convex set with volume bigger than 2 must have an non-zero integer point. We will show an application of Minkowski’s theorem to Diophantine approximations. Then we will introduce the powerful concept of Lattice basis reduction which finds an almost orthogonal basis for a lat...