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 can I find the number of the shortest paths between two points on a 2D lattice grid?
How do you find the number of the shortest distances between two points on a grid where you can only move one unit up, down, left, or right? Is there a formula for this?
Eg. The shortest path between ...
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integer lattice points on a sphere
Suppose we have a sphere centered at the origin of $\mathbb{R^{n}}$ with radius $r$. Are there known theorems that state the number of integer lattice points that lie on the sphere? It seems like this ...
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Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$
Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
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Elementary proof that if $A$ is $m \times n$ matrix map from $\mathbb{Z}^m$ then the map is surjective iff the gcd of det of minors is 1.
I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by $A$, an $m \times n$ matrix the map is surjective iff the gcd*strong ...
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Which internal angles can a lattice polygon have?
I am wondering if for a lattice polygon an internal angle can take any value? If no which ones not and why?
I guess there will be some restrictions due to the discrete nature of the grid but I am ...
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A book useful to learn lattices (discrete groups)
Does anyone know a good book about lattices (as subgroups of a vector space $V$)?
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Index of a sublattice in a lattice and a homomorphism between them
I am asked to show that if $\phi_A$ is the homomorphism from $\mathbb{Z}^k \rightarrow \mathbb{Z}^k$ given by $\phi_A(x)=xA$ then the index of $\phi(\mathbb{Z}^k)$ in $\mathbb{Z}^k$ is finite if ...
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topology on n-colorings of a lattice?
Consider the collection of all $n$-colorings of $\mathbb{Z^{d}}$ (i.e. the collection of all ways to color each lattice point one of $n$ colors). What are some non-trivial ways to define a topology on ...
