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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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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The number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
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number of lattice points in an n-ball

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems. Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that ...
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440 views

Counting lattice points interior to a polygon

If I define an integer lattice $\Lambda \subseteq \mathbb{Z}^2$ with a basis given by $$\omega_{1} = a \hat{i} + b\hat{j}, \;\;\; \omega_{2} = -b \hat{i} + a\hat{j}$$ How can I count how many lattice ...
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150 views

Mathematical Results from Counting Points in Lattices

I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the ...
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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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137 views

Show that $(\Bbb{N}, |)$ is a distributive lattice.

Show that the set of Natural numbers with divisibility form a distributive Lattice where for any $x, y\in\mathbb{N}$ we have $x\wedge y = \operatorname{gcd}(x,y)$ and $x\vee y=\operatorname{lcm}(x,y)...
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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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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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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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29 views

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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16 views

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: \begin{...
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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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36 views

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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130 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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37 views

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 $\vec x$...
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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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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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32 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 $L_1\...
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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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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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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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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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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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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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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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91 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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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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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 ...