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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Verify if linear combination of vectors is in lattice

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be vectors in $\Bbb{R}^3$. How do I verify if there is a linear combination of them that belongs in the lattice $\mathcal{L}(B)$ where $B = \{(1,1,1)\}$?
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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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What happens when the basis vectors of an integer lattice are not linearly independent?

The definition of a lattice requires basis vectors that are not linearly independent. Why? Specifically the following three vectors are linearly independent and form the basis of a lattice: ...
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Existence of Unimodular Congruence Transformation for Symmetric, Integer matrices

Two symmetric, integer valued matrices, $K_1$ and $K_2$, are congruent if there exists a unimodular integer matrix, $X$, such that $$X^T K_1 X = K_2$$ What are the conditions on the existence of such ...
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How can I find a linearly independent subset of a set of vectors in $\mathbb{Z}^n$?

I have a set of $M$ vectors in the module $\mathbb{Z}^n$ ($M>n$) over $\mathbb{Z}$. Question 1: How can I find a linearly independent subset of these vectors? (so that others can be written as a ...
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A problem about successive minima of a lattice in $\mathbb{R}^n$

The problem: Given $\Lambda$ be a full-rank lattice in $\mathbb{R}^n$, which has $\lambda_1 < \lambda_2 < \; ... < \lambda_n$ as successive minima. There exist $\textbf{x}_1, \textbf{x}_2, ...
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128 views

Lattice in $\Bbb R^2$

Let $(a,b)$ be a lattice basis of a lattice $L$ in $\mathbb{R}^2$. Prove that every other lattice basis has the form $(a',b')=(a,b)P$, where $P$ is a $2\times 2$ integer matrix with determinant $1$ or ...
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Gram-Schmidt process - Division by zero (ERROR)

I'm working with full-rank lattice basis, and I need to compute the Gram-Schmidt norms and coefficients to measure its quality. But during the process I have a division by 0. The division by zero is ...
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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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How can I see that we have $\Lambda_a=s\Lambda_a\mathbin{\dot\cup} \big(s\Lambda_a+(1+s)\big)\mathbin{\dot\cup}s\Lambda_b$ (Silver mean substitution)

Consider the (Silver mean) substitution $$\varrho:\begin{aligned}&a\mapsto aba\\&b\mapsto a\end{aligned}.$$ If we take $w^{(1)}=a|a$ and $w^{i+1}=\varrho(w^{(i)})$, then we get: ...
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Estimate for the discrete Green's function associated with the discrete Laplace operator $\Delta$

Consider the discrete lattice $\mathbb{Z}^{d}$ with $d\geq 2$. Let $x\in\mathbb{Z}^{d}$. We define the discrete Laplace operator $\Delta$ as follows: $$-\Delta u(x) := \sum_{\vert ...
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The lattice points in the real cone of some semigroups are just the integer cone of that semigroup.

I'm trying to solve an exercise in Fulton's book on toric varieties, and have reduced it to the following: Let $M$ be a lattice of rank $n$ with $M \otimes \mathbb{R} = V$, and $S$ be a finitely ...
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0answers
29 views

Intersection of two Lattices

In a script by Daniele Micciancio there is the following statement on page 8: It is easy to show that if $\mathcal{L}(D)$ and $\mathcal{L}(D')$ are the dual lattices of $\mathcal{L}(B)$ and ...
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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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1answer
69 views

Largest possible value of a sum

This question has been confusing me and I would love some help. If $M$ is $n$ by $n$, symmetric, positive definite and integer valued and $n$ is a fixed positive integer, what is the large possible ...
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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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How to maximise the minimum distance of a lattice using orthogonal matrices?

Given an $n$-dimensional real lattice $\Lambda$ with generator matrix ${\bf L}_{n\times n}$ (basis vectors are columns of ${\bf L}$). What is the solution to the following optimisation problem? ...
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1answer
50 views

Integer lattice: find all solutions

In general, how do you solve the following kind of problems borrowing techniques from Group Theory? Describe all points (if any) in the affine integral lattice $$ \mathcal{L} = \{(x, y, z, t) : ...
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Vertex ideal in graphs?

Vertex ideal originates from lattices here. Is there some relationship to relate it to graphs such as series-parallel graphs?
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72 views

A parallelogram between two points on a hexagonal lattice containing all the shortest paths

For any two points on a hexagonal grid with integer coordinates there is a unique parallelogram which contains all of the shortest paths (in terms of taxicab norm) between these points. See the ...
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2answers
86 views

Define all points in the affine integral lattice.

Define all points in the affine integral lattice $\mathcal{L}=\{(x,y,z,t) : x+y+z+t=5$ and $x-z \equiv 0$ (mod $12$)$\} \subset \mathbb{Z}^4$. This is a question from a practice exam I have with no ...
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1answer
28 views

Show that $SO_3(\mathbb{Z}) \simeq SO_3(\mathbb{Z}/3\mathbb{Z}) $

I have read the surprising fact that $SO_3(\mathbb{Z}) \simeq SO_3(\mathbb{Z}/3\mathbb{Z}) $. At first I could only come up with diagonal elements of $SO_3$ such as: $$\left[ \begin{array}{rrr} -1 ...
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Size of closed loop on a (bipartite) hexagonal lattice with equal number of enclosed A and B sublattice sites.

If I draw closed loops on a hexagonal lattice such that it always encloses equal number of A and B sublattice sites, I seem to get loops of sizes 4n+2. Is there a way this can be proved in general? ...
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89 views

Understanding the Cholesky decomposition

I'm attempting to understand the Cholesky decomposition via the following site: http://en.wikipedia.org/wiki/Cholesky_decomposition If I have a matrix, say $$A = \begin{bmatrix} 2 & -1 & ...
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29 views

Do all geometric lattices admit an order-theoretic lattice structure?

Wikipedia defines the geometric notion of a lattice as a discrete subgroup of $\mathbb{R}^n$ (i.e. a subgroup isomorphic to $\mathbb{Z}^n$. This can be viewed as the span of a basis for $\mathbb{R}^n$ ...
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28 views

Combinatoric problem in a lattice

You are given a 2-dimensional lattice (you can as well consider it to be a square grid graph) with dimensions $L_x$ and $L_y$. The lattice is filled with two types of elements, A and B. For an cell ...
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Unable to find referenced theorem

I've am reading the article "Finitely summable Fredholm modules over higher rank groups and lattices" http://arxiv.org/abs/0806.2759 . Theorem 4.3 here refers to the article Property (T) and rigidity ...
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lower bound for the length of the longest path contained in a subset of $\mathbb{Z}^d$

Let $A \subset \mathbb{Z}^d$ be a finte connected set. Let $\gamma$ be the longest self-avoiding path of nearest neighbors which is entirely contained in $A$. Can you provide a lower bound for the ...
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Minimal diameter of a connected subset of $\mathbb{Z}^d$

Let $A \subset \mathbb{Z}^d$ be a connected set of cardinality $|A|$. Let the diameter of the $A$ be defined as the length of the longest path of distinct, nearest neighbor sites which is entirely ...
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37 views

Jumping on the Coordinate lattice grid

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show ...
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Gaussian Circle Problem. What does the area of the error term corresponds to?

In a paper of proving an upper bound of the error term E(R), it was stated that $$|E(R)|=|N(R)-\pi R^2|\le A(R)$$ where A(R) denote the are of the unit squares intersecting the boundary of the circle ...
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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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Unique sub-lattices of a given volume

I have an interesting problem from my research that I have been struggling to solve. I am not a mathematician so please bear with me. The question is as follows: The way I think about a lattice is as ...
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1answer
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Must a polynomial function of $x$ pass through infinitely many integer lattice points?

I made a mistake in my formulation of this question when I last asked it and got downvoted because the answer was actually trivial. However, I think the intended question is actually an interesting ...
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Does every line through the origin in the plane intersect the integer lattice an infinite number of times? [closed]

Question is in the title. What about every algebraic curve through the origin? Does every line through the origin in the complex numbers pass through an infinite number of Gaussian primes? EDIT: Just ...
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Definition of q-ary lattices

Lattices defined over the vector space over $\mathbb{R}^n$, whereas q-ary lattices consists of only integers i.e., Let A be a $\mathbb{Z}_q^{n\times m}$ then q-ary lattice is defined as $$\Lambda(A) = ...
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Symmetric, commuting matrices in $\mathrm{SL}(3,\mathbb{Z})$

Can someone evaluate whether the following is true: Given two symmetric, commuting matrices $M_1,M_2 \in \mathrm{SL}(3,\mathbb{Z})$ (integer entries with determinant 1), where $M_2$ is not ...
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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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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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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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Approximate root of $\alpha x - \beta y$ over $\mathbb Z$ except origin

Consider the polynomial $ f(x,y) = \alpha x^2 - \beta y^2 $ Prove or disprove: For any choice of $\alpha, \beta \in \mathbb R_0^+$, the polynomial $f$ gets arbitrarily close to $0$ over $\mathbb Z ...
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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 ...