— Recently, Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph G. In this paper, we develop a general Riemann-Roch Theory for sub-lattices of the root lattice An by following the work of Baker and Norine, and establish connections between the Riemann-Roch theory and th...

A theoretical analysis of Coulomb systems on lattices in general dimensions is presented. The thermodynamics is developed using Debye–Hückel theory with ion-pairing and dipole–ion solvation, specific calculations being performed for three-dimensional lattices. As for continuum electrolytes, low-density results for simple cubic ~sc!, body-centered cubic ~bcc!, and face-centered cubic ~fcc! latti...

In this paper we extend some aspects of the theory of 'supersolvable lattices' [3] to a more general class of finite lattices which includes the upper-semimodular lattices. In particular, all conjectures made in [33 concerning upper-semimodular lattices will be proved. For instance, we will prove that if L is finite upper-semimodular and if L' denotes L with any set of 'levels' removed, then th...

Lattices are discrete mathematical objects with widespread applications to integer programs as well as modern cryptography. A fundamental problem in both domains is the Closest Vector Problem (popularly known as CVP). It is well-known that CVP can be easily solved in lattices that have an orthogonal basis if the orthogonal basis is specified. This motivates the orthogonality decision problem: v...

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating func...

We extend some results of Brewer and Heinzer about integral domains with certain comaximal factorization properties for ideals, to the more general setup multiplicative lattices.

Left-modularity [2] is a more general concept than modularity in lattice theory. In this paper, we give a characterization of left-modular elements and demonstrate two formulae for the characteristic polynomial of a lattice with such an element, one of which generalizes Stanley’s Partial Factorization Theorem. Both formulae provide us with inductive proofs for Blass and Sagan’s Total Factorizat...

Coincidence Site Lattices (CSLs) are a well established tool in the theory of grain boundaries. For several lattices up to dimension d = 4, the CSLs are known explicitly as well as their indices and multiplicity functions. Many of them share a particular property: their multiplicity functions are multiplicative. We show how multiplicativity is connected to certain decompositions of CSLs and the...

We determine a general formula to compute the number of saturated chains in Dyck lattices, and we apply it to find the number of saturated chains of length 2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2 and 3) and the cardinality of the underlying poset.

Abstract We count primitive lattices of rank d inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect certain parameters such lattices. These include, for example, the subspace that a lattice spans, namely its projection Grassmannian; homothety class and equivalence modulo rescaling rotation, often referred shape. add prior work Schmidt by allowing sets in spaces are general ...