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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The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
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Equivalent definitions of a lattice in a real vector space of finite dimension

I'm currently trying to work my way through chapter seven of Serre's book "A Course in Arithmetic" with a view to learning about modular forms. During the course of this chapter the book begins to ...
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Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
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Scalar multiple of one lattice contained in another

I believe my question boils down to the following question: Given lattices $L$ and $L'$ in $k^{n}$, does there exist $\lambda \in k^{\times}$ so that $\lambda L' \subseteq L$ and $\lambda L' ...
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Number of paths from $(0,0)$ to $(n,k)$ where all four directions are allowed, using a specific number of steps

It is well known that the number of paths from $(0,0)$ to $(n,k)$ in $\mathbb{N^2}$ with the set of steps $\{(1,0),(0,1)\}$ is ${n+k \choose k}$. This is the minimum number of steps needed to get to ...
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“All vertices of a convex pentagon are lattice points, and its sides have integral length. Show that its perimeter is even.”

"All vertices of a convex pentagon are lattice points, and its sides have integral length. Show that its perimeter is even." - Problem Solving Strategies, Arthur Engel, pg. 27. I have proven the ...
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Minimum difference of angles between points on square lattice

I have integer grid of size $N \times N$. If I calculate angles between all point triples - is it possible analytically find minimal non-zero difference between those angles?
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Are there 2D analogues for integer division and modular arithmetic?

Let's say you have a "parallelogram" of points $P = \{(0, 0), (0, 1), (1, 1), (0, 2), (1, 2)\}$. This parallelogram lies between $u = (2, 1)$ and $v = (-1, 2)$. Then for any point $n \in ...
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58 views

show that L^+ is non-empty

I want to show that $L^+$ is non-empty where $L$ is a full-rank integer lattice and $L^+$ denotes the set of elements of $L$ having positive coordinates I have an indication that I did not understand ...
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Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
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moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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Lattices and Elliptic curves and number fields

Let $K$ be a number field with ring of integers $O_K$. If $K$ is totally real, then $O_K$ is a lattice in $\mathbb R$. If $K$ is imaginary quadratic, then $O_K$ is a lattice in $\mathbb C$. If $K$ ...
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smallest integer contained in sublattice $\Rightarrow$ $L'=[q,r\tau+s]$

Let $L'$ be a sublattice of the lattice $[1,\tau]$ in a imaginary quadratic field. Reminder: a lattice $L$ consists of the $\mathbb{Z}$-linear combinations of $1$ and $\tau$, with $\{1,\tau \}$ linear ...
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Helping solve mod problems

I am having trouble solving the below problems. My teacher taught us to write out the solutions by hand.. but I really think there is an easier way to do the higher numbers. Thanks!
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Counting sum of lattice points

Assume a set $S$ with $|S|$ entries. Indeed, $S$ is the set of lattice points inside a $k$-sphere. Assume $V=S\oplus S$ where $\oplus$ is the Minkowski sum of two sets. Do you know any lower bound on ...
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Is this the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$?

I have been attempting to create the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$. I have, so far, this: http://www.scribd.com/doc/223680804/Subgroup-Lattice-of-Z-4-x-Z-8 While I have calculated ...
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Correctness of the size of an planar integer lattice unknot

A planar integer lattice unknot is a polygon drawn over a two dimensional integer lattice. Here is an example: Given a number $N$, a planar unknot is not always possible. For example, a planar ...
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Integer polynomial with two aligned roots

Anyone have an example of a monic polynomial $f(x) \in \mathbb{Z}[x]$ with $f(0)=1$ such that for $\alpha \in \mathbb{C}\backslash\mathbb{R}$ and $t \in R, t>0, t\neq 1$ both $\alpha$ and $t\alpha$ ...
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254 views

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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459 views

Lattice Definition

I see two Lattice definitions in Mathematics. Partial order set with each pair of elements have a least least upper bound and greatest lower bound. Integer linear combinations of vectors. Is ...
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Integer Points on Circles

Let $r(n)$ denote the number of integral solutions to $a^2+b^2 = n$ where $a,b,n$ are integers. Furthermore, we count the pairs with regard to order and signs. (So if $(a,b)$ is a solution, so are ...
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Generators of Special Linear Groups

Linear algebra and special-linear group experts please help: I learn that in principle one can generate this $M$ matrix form the $B_1$ and $B_2$ matrix below. Here $$ M=\begin{pmatrix} 0& 1& ...
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Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
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Which of the following sets are sub lattices of $\mathbb{Z}^2$?

Here are the first three sets: $\{(x, y) \in \mathbb Z^2 : x + y = 1\}$. $\{(x, y) \in Z^2 : x + y = 0\} = S^2$. $\{(x,y) \in Z^2 :2\mid x\} = S^3$ I found that the first one is not a subgroup ...
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Obtaining a new basis for a lattice with one of the new basis vectors fixed

Suppose that a lattice $L$ in $\mathbb{R}^3$ is given with a basis $B = \mathbf{ \{ v_1, v_2, v_3 \} }$. Is there an algorithm that would help me obtain a new basis $B' = \mathbf{ \{ v_1', v_2', v_3' ...
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Orbits of action of $SL_2(\mathbb{Z})$ on lattice

I'm interested in the action of $SL_2(\mathbb{Z})$ on $\mathbb{Z}^2$: if $A\in SL_2(\mathbb{Z})$ and $v\in\mathbb{Z}^2$, then $Av\in\mathbb{Z}^2$. Specifically, what are the orbits of this action?
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Which lattices are ideals of a number field?

Let $K$ be a number field, then its ring of integers $\mathcal{O}_K$ in the Minkowski space of $K$ is a lattice $\Lambda$. Is there some geometric descrpition/intuition that describes sublattices of ...
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Number of points in the diagonal of an $X \times Y$ square lattice box.

So assuming we have a $X \times Y$ lattice, Say for example a $3 \times 5$ like so * * * * * * * * * * * * * * * I need to find the number of points that each ...
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Non zero Unimodular matrices

I have a question about the lattice reduction algorithms like LLL algorithm. Lattice reduction algorithms like LLL generate a unimodular matrix which makes more orthogonal basises for a given matrix. ...
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Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
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Factoring an integer matrix into unimodular and upper triangular matrices

I'm stuck at the following problem. I've read this somewhere, but the author did not provide a proof, probably assuming that this is `clear'. Let $A \in \mathbb{Z}^{n \times n}$. Then, there exist a ...
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Is the Voronoi region of a lattice symmetric around origion?

Assume an n-dimensional lattice. Is the Voronoi region of the lattice symmetric around origion? In other word, is the following statement true? "if $x\in \mathcal{V}$ then $-x\in \mathcal{V}$" where ...
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Lattice definition as finite density infinite set of vectors closed under addition?

This question came up in the definition of lattices (in the crystallography/group theory sense, not the ordered set sense) in our condensed matter lectures, but I believe it's more appropriate here ...
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Weyl group and weight lattice chambers.

Consider two simple Lie groups $G_1$ and $G_2$. Let $G_1$ have $W_1$ as a Weyl group and $G_2$ have $W_2$ as a Weyl group. Is it true that the Weyl group of $G_1 \times G_2$ is $W_1 \times W_2$? ...
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Closest vector problem

Given is a vector $v=\begin{pmatrix}2,&-1,&0,&1\end{pmatrix}$ as the shortest vector of the lattice $\Lambda (B)$, where $B$ is determined as $B=\begin{pmatrix}4 &-3 & 2 & 0\\ ...
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Continuity of the mapping of a lattice in $\mathbb{R}^{d}$ onto the length of the shortest vector

I have the following problem: Let $X_{d}:=\{\mathbb{Z}^{d}g;g\in\operatorname{SL}_{d}(\mathbb{R})\}$ and define $\lambda:X_{d}\to(0,\infty)$ by: $$ \lambda(\Lambda):=\min\left\{r>0;\Lambda\cap ...
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isotropic sublattice

In Scattone " Compactification of Moduli Spaces of Algebraic K3 Surfaces" the author cites a correspondance between primitive isotropic sublattices of some lattice L and the rational boundary ...
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Equivalence of Lattices

Let $\Gamma=\{mw_1+nw_2:m,n\in\mathbb{Z}\}$ and $\Gamma'=\{mw_1'+nw_2':m,n\in\mathbb{Z}\}$. Show that $\Gamma=\Gamma'$ if and only if there exists a matrix $A\in SL(2,\mathbb{Z})$ such that $\left( ...
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Which matrices generate the same lattices?

Given are the following matrices: $A_{1}=\begin{bmatrix} 4 & 7 &9 \\ 3 & 5 &3 \\ 1 &2 &0 \end{bmatrix}$ , $A_{2}=\begin{bmatrix} 10 & 9 &6 \\ 11 & 7 &2 \\ ...
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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 $x$ meet $y = \operatorname{gcd}(x,y)$ and $x$ joint ...
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How to count lattice points on a line.

How can we count the number of lattice point on a line, given that the endpoints of the lines are themselves lattice points? I really can't think of how counting lattice points would work, so please ...
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Shortest Non-Zero Vector in Integer Lattices with Given Points

There are two questions related to the shortest non zero vector problem that have left me scratching my head. Please bear with me as I describe the problem. Disclaimer: this is homework. For the ...
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Show $\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$ by block-walking interpretation of Pascal's triangle

A combinatorial proof for the identity $$\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$$ is the following "committee" argument, which seems the most common. There are $\binom{n}{k}$ ...
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integer programming with bounded dimension

We know that integer programming with bounded dimension or fixed number of variables can be solved in polynomial time by Lenstra's result(from results of the LLL algorithm). After heavy foraging i ...
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Nonplanar equilateral lattice “pentagons”

It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html) The same is true for lattices in $\mathbb{R}^n$, ...
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Lattice points in spheres

Let $\mathbb{R}^n$ have the standard Euclidean metric and call a point $P = (x_1, \ldots,x_n)\in\mathbb{R}^n$ a lattice point if for all $i$, $x_i\in\mathbb{Z}$. Allowing small number theoretic ...
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What is a good introduction to quantities such as the norm of a lattice and of short vectors in the context of lattice reduction?

I am trying to make sense of different notations used in measuring lattices, in particular before and after a basis reduction. In particular, I am trying to get bounds and estimates for the size of ...
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Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
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Inclusion-minimality of a lattice basis

An integer lattice is a subgroup of $\mathbb{Z}^n$. Since $\mathbb{Z}$ is PID, each lattice has a well-defined rank and a generating set of rank many elements is a basis. I wonder if there is a way ...
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Question regarding polygons

Can you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?