Related papers: Cubic sublattices
We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev-polynomials. If the dimension $d$ of the lattice is a power of two, i.e. $d=2^m, m \in \mathbb{N}$, the resulting lattice is an…
Many challenging Graver bases computations, like for multi-way tables in statistics, have a highly symmetric problem structure that is not exploited so far computationally. In this paper we present a Graver basis algorithm for sublattices…
A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational…
Replacing the continuous space by a cubic lattice we find a deformation of the Poincar\'e algebra. A deformation of the relativistic mass operator is shown to be a Casimir of the algebra. The real structure is preserved.
In this paper, we study the discrete cubic nonlinear Schroedinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In…
We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in $\mathbb{R}^d$ and its edges connect any two polytopes that can be obtained from one another by either…
In this note, we discuss Hassett maximal cubic fourfolds and construct an explicit irreducible component of maximal dimension sixteen of the locus $\mathcal{Z}$ of Hassett maximal cubic fourfolds. We utilize algebraic and arithmetic methods…
The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an…
We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized…
Dilworth's theorem. Every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice $L$. We want: Every finite distributive lattice $D$ can be represented as the congruence lattice of a nice finite…
We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially…
Generalized Fibonacci cube $Q_{d}(f)$, introduced by Ili\'{c}, Klav\v{z}ar and Rho, is the graph obtained from the $d$-hypercube $Q_{d}$ by removing all vertices that contain $f$ as a substring. The smallest integer $d$ such that $Q_{d}(f)$…
Usually $Z_2$ topological insulators are protected by time reversal symmetry. Here, we present a new type of $Z_2$ topological insulators in a cubic lattice which is protected by a novel hidden symmetry, while time reversal symmetry is…
In this paper we study sequences of lattices which are, up to similarity, projections of $\mathbb{Z}^{n+1}$ onto a hyperplane $\bm{v}^{\perp}$, with $\bm{v} \in \mathbb{Z}^{n+1}$ and converge to a target lattice $\Lambda$ which is…
A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological…
We study $b$-property of a sublattice (or an order ideal) $F$ of a vector lattice $E$. In particular, $b$-property of $E$ in $E^\delta$, the Dedekind completion of $E$, $b$-property of $E$ in $E^u$, the universal completion of $E$, and…
We prove that a variety of Novikov algebras has a distributive lattice of subvarieties if and only if the lattice of its subvarieties defined by identities of degree three is distributive, thus answering, in the case of Novikov algebras, a…
We obtain an explicit formula to enumerate closed random walks on a cubic lattice with a specified length and 3D algebraic area. The 3D algebraic area is defined as the sum of algebraic areas obtained from the walk's projection onto the…
A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf…
In this article, we study subloci of solvable curves in $\mathcal{M}_g$ which are contained in either a K3-surface or a quadric or a cubic surface. We give a bound on the dimension of such subloci. In the case of complete intersection genus…