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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positive definite integral lattice and existence of $\mathbb{Z}$-basis with a certain kind of “shortest vector” property [on hold]

Let $L$ be a positive definite integral lattice ($\mathbb{Z}$-module with positive definite symmetric binlinear form $\langle\cdot,\cdot\rangle$). Define the set $\Phi=\{\alpha\in L\,|\,\langle\alpha,...
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Quantizing a matrix of reals while preserving row and column sums

Assume $E = [\epsilon_{i,j}]$, $i=1,2,\dotsc,m$, $j=1,2,\dotsc,n$, is an $m\times n$ matrix of reals. We know that $\forall i,j$, $\epsilon_{i,j}\in[-1/2,1/2]$. Moreover we know that both row-sums ...
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Why is $ \theta(m) \propto \zeta(2) $ if it is counting lattice points in a hyperbola?

I found this lattice point identity in a derivation of $\zeta(2)$: $$ \theta(x) = \sum_{mr \leq x} m = \sum_{r \leq x}\sum_{m=1}^{[x/r]} m = \sum_{r \leq x} \left( [x/r]^2 + [x/r] \right) = \sum_{...
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Perfect pairing co-weight lattice and root lattice

Let $\Phi$ be a root system and let $\Lambda_R$ and $\Lambda_W$ denote root lattice and weight lattice. I know that there is a perfect pairing $\Lambda_W \times \Lambda_R^\vee \to \mathbb{Z}$, where $\...
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An interesting connection between the Möbius function and the parity of the number of sublattices of index $n$ in generic $3$-dimensional lattice

I recently discovered an interesting connection between the following two On-Line Encyclopedia of Integer Sequences (OEIS) sequences: A001001 and A209635. More specifically, there seems to be an ...
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Intersection of two sets of rationals

I'm looking to see if anyone has any solutions or references for this problem. I'm not even sure of a proper category. It seems like it should be trivial, perhaps I'm missing something obvious. ...
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Isomorphism of invariant factor decomposition

By the structure theorem, for every finite abelian group $A$, we have an isomorphism $A \cong \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_n}$ for unique $d_i$, s.t. $d_i | d_{i+1}$. My question ...
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Conjecture about circles in plane lattices

A plane lattice $\Lambda$ is a set $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, where $A,B$ are linearly independent vectors in $\mathbb R^2$. The set of all circles in $\Lambda$ is $$\mathcal K(\...
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How interpret the dual lattice $\Gamma^*$?

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$ - $29$, they talk about the lattice $\Gamma$ and it is defined as $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \...
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Norms of lattices in $\mathbb{R}^n$

Consider a lattice $\Gamma \subset \mathbb{R}^{n}$. Let its norm $\operatorname{Nm}(\Gamma)$ be defined by: $$\operatorname{Nm}(\Gamma) = \inf_{x \in \Gamma \setminus \{ 0 \}} \prod_{i=1}^{n} \vert ...
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image of composition of upper triangular integral matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $im(AB)$ in terms of $im(A)$, $im(B)$, unions, intersections, determinants, etc?
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How to draw a lattice for the divisors of big numbers?

An exercise ask to find atoms and join-irreducible elements for the set of divisors of 360. I know how to find them by drawing the lattice but it seems difficult in this case. Is there another way to ...
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Decomposition of quotient group of lattices

By the Chinese remainder theorem, we know that $\mathbb{Z}_m \cong \prod_{i=1}^l \mathbb{Z}_{p_i^{k_i}}$, where $m=p_1^{k_1} ... p_l^{k_l}$. Now, let $\Lambda = A(\mathbb{Z}^n) \subseteq \mathbb{Z}^...
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Character group and lattices

Let $\Lambda$ be the complex n-th dimensional lattice over Eisenstein integers ($\mathbb{Z}[\omega]$)). The map $R: \mathbb{C} \mapsto \mathbb{R}$ is defined as following: $R(z)=R(z_{a}+\omega z_b)=...
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Basis of weight lattice in terms of root lattice

Let $\Lambda_R = \bigoplus_{i \in I} \mathbb{Z} \cdot \alpha_i$ be the root lattice of a root system $\Phi$ with simple roots $\alpha_i$ and let $\Lambda_W$ denote the corresponding weight lattice. ...
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Counting balls in face centred cubic close packing

Possibly too easy for stack exchange, but... Consider a cubic close packing, or face centred cubic, arrangement of balls or radius $1$ in dimension $3$. Suppose that the origin is the centre of one ...
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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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53 views

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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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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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: $$a|a\...
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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 x-y\vert=1}\left(u(...
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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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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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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? $$\arg\...
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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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92 views

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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55 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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81 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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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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30 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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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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39 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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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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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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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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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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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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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 \...