# Tagged Questions

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
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### Unimodular matrix to increase the minimum eigenvalue

Given a positive definite matrix $P$, I would like to find a unimodular matrix $U$ so that $U P U^T$ raises the minimum eigenvalue as much as possible. How can one find such a matrix $U$?
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### Finding the odometer function of an abelian sandpile

As a computer scientist and "armchair" mathematician, I'm trying to replicate the images found here of Abelian sandpiles on a square lattice, where the initial configuration is $n$ chips on a single ...
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Let $\mathbb{A} = \widehat{\mathbb{Z}} \otimes \mathbb{Q} \times \mathbb{R}$ be the adeles over $\mathbb{Q}$. In Deligne's article "Formes modulaires et representations de GL(2)" he states without ...
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### $\Delta$ vs. $\delta$ in Lattice Theory

I'm learning about lattices and I'd like to confirm the difference between the density $\Delta$ and another density $\delta$. I would greatly appreciate if someone could confirm, correct, or even ...
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### lattice symmetries portrayed in animation

Below is an animated gif by Dave Whyte: What is the orbifold/fibrifold notation for the symmetries of the lattice depicted below? (a la /The Symmetries of Things/? Are there related theta function ...
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### Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
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### When does a simplex have an interior lattice point?

Given $r$ vectors $v_1, \dots, v_r$ in $\mathbb{Z}^n$, is there an easy way (in terms of the entries of the $v_i$) to determine if there is a point of $\mathbb{Z}^n$ in the interior of the simplex ...
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### Automorphisms of a lattice and changing to a nicer $\mathbb{Z}$-base

Suppose I have an integral lattice $L$ with an arbitrary $\mathbb{Z}$-base, equipped with a positive-definite nondegenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$, and an isometry $\nu$ ...
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### Term for intersection of lattice and convex region?

Is there a special term or convenient phrase for the restriction of a convex region to points of a lattice? This is motivated by wanting to talk about the feasible points of a discrete problem. I'd ...
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### Probability distribution of a self avoiding walk

Preliminary: Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks ...
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### Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$

I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring ...
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### Estimate for the co-volume of discs centered at lattice points in the plane?

Suppose I have a unimodular lattice $\Lambda = A \mathbb{Z^2}$ ($A\in SL(2,\mathbb{R})$) in the plane. I place a disc of fixed radius, $r$, around each point of $\Lambda$, so that I have a union of (...
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### The number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
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### Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...