such that f(g(a)) = a for all a ∈ L. Projective lattices are characterized in [3] by four conditions. This talk will discuss two of them that are of current interest. If g in (2) is only required to be order-preserving, it is called an isotone section of the epimorphism (1). We will characterize which lattices L have an isotope section for every epimorphism (1). We will use this to characterize...

In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sph...

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, Z[x]/〈f〉 (here f is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the no...

The magnetic field probability P (B) is calculated from Ginzburg-Landau theory for various lattices of vortex lines in type-II superconductors: Ideal triangular lattices, lattices with various shear strains and with a super lattice of vacancies, and lattices of short vortices in films whose magnetic field “mushrooms” near the surface.

Boris Venkov passed away on November 10, 2011, just 5 days before his 77th birthday. His death overshadowed the conference “Diophantine methods, lattices, and arithmetic theory of quadratic forms” November 13-18, 2011, at the BIRS in Banff (Canada), where his important contributions to the theory of lattices, modular forms and spherical designs played a central role. This article gives a short ...

The concept and existence of sphere-boundachieving and capacity-achieving lattices has been explained on the AWGN channels by Forney [4]. In this paper we focus on regular LDPC lattices [7]. We introduce and investigate an ensemble of LDPC lattices with known properties. It is shown that these lattices are sphere-bound-achieving and capacityachieving.

We find out for which t shells of selfdual lattices and of theirs shadows are spherical t-designs. The method uses theta series of lattices, which are modular forms. We analyse fully cubic and Witt lattices, as well as all selfdual lattices of rank at most 24.

Conjecture 1.1 is out of reach for nonarithmetic lattices in O.n; 1/ and SU.n; 1/, since we do not understand the structure of such lattices. However, all known constructions of nonarithmetic lattices lead to noncoherent groups: See the author, Potyagailo and Vinberg [28] for the case of Gromov–Piatetsky–Shapiro construction; the same argument proves noncoherence of nonarithmetic reflection lat...