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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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 ...
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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 ...
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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 ...
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Lattice Reduction in Mathematica

I have some trouble understanding the concept of lattice reduction. As I understand, an integer lattice $$\{ A k : k \in \mathbb{Z}^n \} \subset \mathbb{Z}^n $$ is defined by a regular matrix $A \in ...
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Reference Request For Hermite normal form of non full row rank matrix

Could someone recommend me some references which discuss the problem of the reduction of a matrix which is not full row rank into its Hermite normal form?
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Topology of network from adjacency matrix : honeycomb?

In a percolative problem, I have noticed that all of the nodes of my system are connected to 3 other nodes. I started drawing a bit and realized that this could look like a honeycomb lattice. The ...
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Lower bounds on lattice points on a convex curve

I was just reading this paper on the number of integral points on a convex curve of arc length l. The paper begins: In 1926, Jarnik [4] proved that a strictly convex arc y = f(x) of length l ...
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existence of a lattice rectangle in a $13 \times 13$ grid

Problem: Prove that if 53 points are chosen from a $13\times 13$ grid then there will necessarily exist a rectangle whose vertices are among the 53 points chosen. My try: I am guessing we have to ...
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What are the dual polyhedra of the face-centered cubic lattice?

For a given lattice $L$ we could define the set of points closest to one point more than any other. $$ \{ x \} = \min_{\ell \in L} \|x - \ell \| \in \mathbb{R}^3$$ This generalizes the "fractional ...
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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 ...
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A question on finitely generated modules over Z in matrix form

in my class on module theory I have been given this problem on finitely generated $ \mathbb{Z} $ modules (Abelian groups) stating the following: We define the vectors $ v_1 = (1,0,-1) $ $ v_2 = ...
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Lenstra's integer programming algorithm: Finding a lattice point “near the center”

Preliminaries: As part of Lenstra's algorithm for integer programming (see here, page 4) we compute a linear transformation $\tau$ and a point $z \in \mathbb{R}^n$ which meet certain conditions (step ...
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Order of the group associated to a quotient of a lattice

Let $A=[A_{ij}]$ be a $n\times n$ symmetric positive definite integer-valued matrix. Define elements of $\mathbb{Z}^n$ $v_i=[A_{i1},A_{i2},...,A_{in}]$ where I am treating them as row vectors. ...
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Properties/Significance of the Radical (Square-free kernel) of an integer

The radical, or square-free kernel, of an integer is the product of its distinct prime factors. So that if $n=\prod_{i=1}^{s} p_i^{\alpha_i}$ is a prime decomposition of $n$ then the radical of $n$ ...
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Question in Lattice Reduction Using LLL Algorithm for d>n

Suppose we have a Lattice $L$ in $R^n$ generated by a basis $L(b_1,...,b_d)$, where $b_i$ is a column vector with $n$ elements, assume $d>n$, could we reduce the basis $L(b_1,...,b_d)$ to new ...
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Number of lattice points in homotetic image

Let $\Lambda$ be a lattice in $\mathbb R^n$, with covolume $\Gamma$. Moreover, let $S$ be a bounded (Lebesgue-)measurable subset of $\mathbb R^n$ and for each $\alpha > 0$ define $\alpha S := ...
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How can I translate this affine hyperplane?

Let $\mu\in\Bbb Z^d$. Suppose that $H=\{x\in\Bbb Z^d:\langle\mu,x\rangle=1\}$ is nonempty and fix $v\in H$. Now, let $\{a_1,\dotsc,a_{d-1}\}$ be a basis for the kernel of the map $\langle ...
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Lattice points with next-largest norm

In a 2D integer grid, the points in increasing distance from the origin are: $(0,0)$ $(\pm1,0)$ and $(0,\pm1)$ $(\pm1,\pm1)$ etc By symmetry we need only consider one-eighth of the lattice, $x\ge0$ ...
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Basis for the intersection of two integer lattices

If $B_1$ and $B_2$ are the bases of two integer lattices $L_1$ and $L_2$, i.e. $L_1=\{B_1n:n\in\mathbb Z^d\}$ and $L_2=\{B_2n:n\in\mathbb Z^d\}$, is there an easy way to determine a basis for ...
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Expected time to walk in Manhattan

Suppose you are at coordinate $(m,n)$, and wish to go to $(0,0)$ using unit steps, either down or left. However, at each point, there is a traffic light, which switches from allowing horizontal to ...
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Number of integer distance points on a circle.

Here is a paper claiming that given a circle with an integer diameter $D$, one can determine the maximum number of points on the circle with the integer distance property (meaning that any pair of two ...
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Real dimension of $\mathbb{Z}^d \otimes \mathbb{R}$

I have following, probably really trivial question. Lets take $\mathbb{Z}^d \otimes \mathbb{R}$. Consider this as vector space over $R$ by defining $\mu (v\otimes \lambda) = v \otimes (\lambda \mu)$ ...
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Inverse Gauss circle problem

The Gauss circle problem focuses on the amount of lattice points of a square lattice that can fit inside a circle with radius r. http://i.stack.imgur.com/y4Yf2.png But say I want to fit N lattice ...
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What is an ideal lattice?

Can anyone describe in layman's terms what an ideal lattice is? I've seen them mentioned in many places, but haven't found a good definition of what exactly they are, nor any good terms to know where ...
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Index of a sublattice

Assume that $\Lambda \subset \mathbb{Z}^2$ is an integer lattice of rank $2$ and define for each integer $k$, $$\Lambda(k):=\Big\{(x_1,x_2) \in \Lambda: k\mid (x_1,x_2)\Big\}.$$ What is the index of ...
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An inequality related to lattice points 'around' a circle

Take a circle of radius $r$ with centre at the origin such that $r^2=N_1^2+N_2^2$ for $N_1,N_2\in\mathbb{N}$. Consider a lattice coordinate $(a,b)$ such that $a\in(-r,-2)$ and define $b$ to be the ...
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Interpreting infinite integer lattice as a manifold of negative dimension

Various fractal dimensions coincide on self-similar fractals as the logarithm of self-copies the fractal includes divided by the logarithm of the factor by which the copies are smaller than the ...
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31 views

Finding total number of lattice points on a circle and one of the coordinate values

I was wondering if it was possible to quickly find the total amount of lattice points on a circle, given its equation (origin and radius), and I was also a bit confused on how to find a particular ...
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24 views

Maximize the product of pairwise distances between particles on a lattice

I am trying to solve the following problem : Consider an $N\times N$ square lattice ($N$ even integer, we can assume that N is large) on the complex plane, with lattice sites at position $j+ik$, $j$ ...
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Number of Lattice points in the intersection between an N-Sphere and an N-Cube

I have a hyper-Cube lattice with coordinates between 0 and 10, therefore 11^N lattice points. I take one of this point, C, and then another point P at distance D from C. I need to know a good ...
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Compute a reduced basis for the lattice with basis matrix

For an exercise I need to compute a reduced basis for the lattice with basis matrix: \begin{pmatrix} 1 & 0 \\ 1414 & 1000 \\ \end{pmatrix} I found the algorithm for ...
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Ideals of $\mathbb{Z}[i]$ geometrically

It is pretty easy to visualize the ideals of $\mathbb{Z}$ in the "integer line". Let's go up to $\mathbb{Z}[i]$ and consider the ideal $3\cdot\mathbb{Z}[i]$. We can visualize it as a "sub-lattice" ...
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Finding the number of lattice paths

Find the number of lattice path of length $2n$ that starts on $(0, 0)$ such that for all the points $(x, y)$ in the path, $x < y$. So pretty much all the points besides the origin are strictly ...
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Representation of integers by sum of squares with linear constraints of a special form

I would like to know what integers $d$ can be written as a sum $d=\sum_{i=1}^N\sum_{j=1}^M a_{ij}^2 $ with $a_{ij} \in \mathbb{Z}$ and where the row and column sums of $a_{ij}$ are fixed $\sum_{i=1}^N ...
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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 ...
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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 ...
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Set of $K > N$ integer-valued vectors so that any N-subset is a base for $\mathbb{R}^N$

Be $N$, $K$ natural numbers so that $K > N > 1$. Be $V$ a set of $K$ vectors in $\mathbb{R}^N$: $V = \{v_1, \ldots, v_K \in \mathbb{R}^N\}$. First of all, I need to find a $V$ so that, for any ...
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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 $n$ ...
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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).
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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 ...
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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 ...
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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 ...
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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. ...
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Hypercube and Hyperspheres

Let $n,k\in\mathbb{N}$. In this problem, the geometry of $\mathbb{R}^n$ is the usual Euclidean geometry. The lattice hypercube $ Q(n,k)$ is defined to be the set $ \{1,2,...,k\}^n ...
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n-dimensional lattice as a collection of lower dimensional spaces.

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 ...
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Covering a family of sets of $\mathbb{Z}^d$ with boxes of a given diameter

Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite ...
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Why are there only so many Bravais Lattices?

I am in doubt as to why there are exactly five 2d Bravais lattices? For example, I could take the square lattice and place a lattice point at the midpoint on every side of each square. Shouldn't ...
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Determine the isomorphism class of $\mathbb Z^3 / M$ for the subgroup $M$ of $\mathbb Z^3$generated by $(13,9,2),(29,21,5),(2,2,2)$

The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations". Please see the ...
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Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
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Newton-Raphson For Integer Factorization

Per my earlier question on Naive Grouping for factorization here, below is the modified Newton-Raphson method (integers only) for the polynomial $N -x^2 - yx - x = 0$. ...