# 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.

10 views

### Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle

I'm doing some research on the Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle and I was wondering if anyone knew why we consider the integer lattice points within ...
90 views

### Count the number of elements of ring [closed]

1/ How to count the number of elements of $\mathbb{Z}[i]/(1+2i)^n$? 2/ How to write $\mathbb{Z}[i]/(1+2i)^n$ as direct sum of cyclic groups (in view of the structure theorem of finite abelian ...
66 views

### Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?

It is a standard fact that if $M$ is a nonsingular $n\times n$ integer matrix, the index of the $\mathbb{Z}$-span of its columns as an abelian group in $\mathbb{Z}^n$ is $|\det M|$. What happens if we ...
15 views

### How do you take the discrete Fourier transform (DFT) of a parallelogram or a Bravais lattice in general?

I'm working on implementing a method that extracts the corresponding wallpaper group given a gray-scale image/pattern. But to do so, I need to take the DFT of a unit cell in the image which, in the ...
29 views

32 views

31 views

46 views

### Short Primitive Vectors in a Lattice in $\mathbb{Z}^2$

Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$. I want to find an order-of-magnitude bound on the shortest ...
31 views

### How can I visualize ideals on a ring of integers of imaginary quadratic fields?

If I were to visualize the ideal $(2, 3+3i)$ of $Z[i]$ on the complex plane, I would find a gcd of $2$ and $3+3i$ (for example $1+i$) and the ideal $(2, 3+3i)$ is identical to $(1+i)Z[i]$, which forms ...
551 views

### Proving Binomial Identities Using Bijections To Lattice Paths

Let $n$ be a positive integer. How can I derive a bijection to show that the following equality holds? $2\displaystyle\sum\limits_{j=0}^{n-1} \binom{n-1+j}{j} = \binom{2n}{n}$ In class, we've been ...
14 views

### Complete toric varieties with given codimension of the singular locus

Let $N\cong \mathbb{Z}^n$ be a lattice and $\Delta\subseteq N_\mathbb{R}$ be a fan such that $X=X(\Delta)$ is a complete and simplicial (i.e. $\mathbb{Q}-$factorial) toric variety of dimension $n$ ...
80 views

### Lattice generated by vectors orthogonal to an integer vector

Given a non-zero vector $\boldsymbol{v}$ composed of integers, imagine the set of all non-zero integer vectors $\boldsymbol{u}$, such that $\boldsymbol{u} \cdot \boldsymbol{v} = 0$, i.e., the integer ...
231 views

### how to find the number of integer coordinates in the interior of triangle

How to find the number of integer coordinates in the interior of the triangle with vertices(0,0) (0,21) (21,0).
1k views

### Pick's Theorem on a triangular (or hex) grid

Pick's theorem says that given a square grid consisting of all points in the plane with integer coordinates, and a polygon without holes and non selt-intersecting whose vertices are grid points, its ...
42 views

### Area of the lattice generated from $(n, n\sqrt{2} \mod 1)$

I plotted $\Big\{ (n, n \sqrt{2} \, \mathrm{mod} \,1) \;\Big| -50 \leq n \leq 50 \Big\}$ and even though the $n \sqrt{2}$ is a line, the pattern that emerges is a lattice. What is the basis of this ...
95 views

### How many edges, faces, cells in a $2\times 2 \times 2 \times 2$ hyper cubic lattice?

If I have a $2\times 2\times 2\times 2$ hyper cubic lattice, how many corners, edges, faces, and cells will it be composed of? E.g. the 4D analogue of figure below. Assume the faces within the figure ...
272 views

### Number of self-avoiding rook walks in a rectangular grid

I was wondering how many self-avoiding rook walks there are on an $m×n$ grid. A self-avoiding rook walk is a path from the bottom left corner to the top right corner of the grid, composed only of ...
36 views

### Creating n-dimensional lattices from lower dimensional parts

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
66 views

### Farthest point on parallelogram lattice

On points arranged in a parallelogram lattice, like on the image in this Wikipedia article, how to calculate the maximal distance any point on the plane may have to its closest point from the lattice. ...