We examine the relationship between the prices paid by households and their shopping patterns measured in terms of shopping frequency and the range of stores visited. We use the TNS data which allows us to control for household heterogeneity. The main contribution of the paper is that we find...

Chapter 1. Gauss lattice theory -- Chapter 2. Reduction principle of Ajtai -- Chapter 3. Learning with errors -- Chapter 4. LWE cryptosystem -- Chapter 5. Cyclic lattice and Ideal lattice -- Chapter 6. Fully Homomorphic Encryption -- Chapter 7. General NTRU cryptosystem.

We provide a new approach to study phase transition for Ising models on closed Cayley trees. It is shown that there may exist more than one limiting measure for an Ising model on an infinite Cayley tree in both ferromagnetic and antiferromagnetic cases. The exact critical values of the...