In CRYPTO 2010, Aguilar et al. proposed a somewhat homomorphic encryption scheme, i.e. an encryption scheme allowing to compute a limited amount of sums and products over encrypted data, with a security reduction from LWE over general lattices. General lattices (as opposed to ideal lattices) do not have an inherent multiplicative structure but, using a tensorial product, Aguilar et al. managed ...

This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps de…ned on spaces of random variables, possibly with the so-called Fatou property. We show …rst that the celebrated Namioka-Klee theorem for linear, positive functionals holds also for convex monotone maps on Frechet lattices. It is well-known among the specialists th...

Solitons are localized nonlinear waves and occur in many branches of physics. Their properties have provided a deep and fundamental understanding of complex nonlinear systems. In recent years, there has been considerable interest in studying solitons in systems with periodic potentials or lattices, in particular, those that can be generated in nonlinear optical materials [1]–[5]. In periodic la...

The Shortest Vector Problem (SVP) is the most famous and widely studied computational problem on lattices. Given a lattice L (typically represented by a basis), SVP asks to find the shortest nonzero vector in L. The problem can be defined with respect to any norm, but the Euclidean norm is the most common (see the entry lattice for a definition). A variant of SVP (commonly studied in computatio...

Atom trapping and cooling using optical lattices is reviewed. A general theory and an example of a one-dimensional lattice is discussed. Some experimental achievements in the field are reviewed. PACS numbers: 32.80.Pj, 32.80.Qk Typeset using REVTEX

Let K be a locally compact field and G a simple AT-group, G = G(K). A discrete subgroup T of G is called a lattice if G/F carries a finite G-invariant measure. It is a uniform (or cocompact) lattice if G/T is compact and nonuniform otherwise. When the jRf-rank of G is greater than one, Margulis [Ma, Z] proved that T is arithmetic, establishing the conjecture of Selberg and PiatetskiShapiro. Thi...

We introduce a new type of dcpo-completion of posets, called D-completion. For any poset P , the D-completion exists, and P and its Dcompletion have the isomorphic Scott closed set lattices. This completion is idempotent. A poset P is continuous (algebraic) if and only if its D-completion is continuous(algebraic). Using the D-completion, we construct the local dcpocompletion of posets, that rev...

The paper proposes a general approach of interaction between players or attributes. It generalizes the notion of interaction defined for players modeled by games, by considering functions defined on distributive lattices. A general definition of the interaction transform is provided, as well as the construction of operators establishing transforms between games, their Möbius transforms and thei...

We investigate generalized phase transitions of type localization delocalization from one to several Sinai-Bowen-Ruelle invariant measures in finite networks of chaotic elements (coupled map lattices) with general graphs of connections in the limit of weak coupling.

Concept lattices are systems of conceptual clusters, called formal concepts, which are partially ordered by the subconcept/superconcept relationship. Concept lattices are basic structures used in formal concept analysis. In general, a concept lattice may contain overlapping clusters and need not be a tree. On the other hand, tree-like classification schemes are appealing and are produced by sev...