相关论文: Construction of some perfect integral lattices wit…
There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on $D_4$-quartic fields with…
A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The goal of this paper is to initiate a systematic study of the geometry…
The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese $3$-fold projections. More precisely, for any integer $d\ge 4$ and any $d$-th root $e$ of…
In this paper, we characterize the congruences of an arbitrary i--lattice, investigate the structure of the lattice they form and how it relates to the structure of the lattice of lattice congruences, then, for an arbitrary non--zero…
$ $Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators $d(\Gamma)$ of a lattice $\Gamma$ in a high rank simple Lie group $H$ grows sub-linearly with $v = \mu(H / \Gamma)$, the co-volume of $\Gamma$ in $H$. We prove…
We construct explicit rate-one, full-diversity, geometrically dense matrix lattices with large, non-vanishing determinants (NVD) for four transmit antenna multiple-input single-output (MISO) space-time (ST) applications. The constructions…
In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.
We prove that every lattice with more than one element has a proper congruence-preserving extension.
Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of…
There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor…
Given $d\geq 2$, we show that the number of approximates $\frac{1}{q}\mathbf{p}\in \mathbb{Q}^d$ of $\mathbf{x}\in\mathbb{R}^d$ satisfying $|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{1}{d}}$ with denominator $1\leq q < T$ decays to the…
We are entering an era where a number of large-scale lattice simulations of four-dimensional supersymmetric theories are under way. Moreover, proposals for how to approach such studies continue to progress. One particular line of research…
Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$-element set has a four-element generating set. In 1983, L. Z\'adori gave a new proof of this fact with a particularly elegant…
We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.
We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let $\phi$: Con K $\to$ D be a {∨, 0}-homomorphism, where Conc K denotes the {∨, 0}-semilattice of all finitely generated…
A main goal in lattice theory is the construction of dense lattices. Most of the remarkable dense lattices in small dimensions have an additional symmetry, they are modular, i.e. similar to their dual lattice. Extremal lattices are densest…
We give a new existence proof for the rank 2^d even lattices usually called the Barnes-Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs…
We show how to construct SO(4) invariant functions for strongly correlated Fermi systems on lattices of finite sizes. We illustrate the method on the case of the 1D Hubbard chain with four and with six sites.
A lattice Delaunay polytope D is called perfect if it has the property that there is a unique circumscribing ellipsoid with interior free of lattice points, and with the surface containing only those lattice points that are the vertices of…
For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $\sqrt{s}\cdot L$ is isometric to a sublattice of $\mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$…