Tagged Questions

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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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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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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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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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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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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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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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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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 ...