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.

learn more… | top users | synonyms

0
votes
0answers
28 views

Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
0
votes
0answers
8 views

Are lattice approximations of lines always tile-able patterns of lattice points?

I'm interested in sets of the form: $L = \left\{(\left\lfloor x \right\rfloor, \left\lfloor y\right\rfloor) \mid ax + bx + c = 0; x, y \in \mathbf{R}\right\}$. I want to know if we can always write ...
0
votes
1answer
67 views

Finding the quotient ring $\mathbb{Z}[i]/(4+i)$

Find the quotient ring $\mathbb{Z}[i]/(4+i)$ by identifying elements with the lattice points in the square generated by $4+i$. I know that $N(4+i) = 17$. Therefore, $4+i$ is irreducible. Now ...
0
votes
1answer
20 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 ...
3
votes
1answer
28 views

How Many Rational Slopes?

Given an $N$ by $M$ grid with integer coordinates (e.g. like pixels in an image), how many slopes are defined by the set of lines passing through the each grid point pair? Note that because the ...
0
votes
0answers
63 views

Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...
2
votes
1answer
33 views

Is $\mathbb{Z}[\sqrt{2}]$ a lattice?

Is $\mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ a discrete subgroup of $\mathbb{R}$? How to prove that?
2
votes
1answer
34 views

Solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$

How to solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$? The origin of this problem is the following question: Show that rank-2 free $\mathbb Z$ module $\mathbb Z^2$ has $p+1$ ...
1
vote
0answers
25 views

Center of mass of voronoi cells of 3d lattice

Let $v_1,v_2,v_3$ be linearly independent vectors in $\mathbb{R}^3$, and let $A$ be a matrix whose columns are $v_1,v_2,v_3$. i.e. $A = [v_1,v_2,v_3]$ Then, define a lattice $\Lambda$ as $\Lambda = ...
0
votes
0answers
5 views

Dual lattice and extreme value

Suppose I have a periodic function which has minimums only at lattice sites. Say the lattice is a honeycomb lattice, do I have my maximums at sites of a triangular lattice (which is the dual lattice ...
0
votes
1answer
15 views

Sacle the distance of lattice points

I know that for a hexagonal lattice generated by (0,1) and ($\sqrt{3}/2$,1/2) (i.e., when the distance between lattice points is 1), the number of lattice points in a circle of given radius $r$ can be ...
0
votes
0answers
21 views

Finite differences on a hexagonal/triangular lattice with Cartesian coordinates

So, I've been thinking recently about how to approximate the Laplacian operator using finite differences on a non-square lattice. For example, on a typical square lattice, in a Cartesian coordinate ...
4
votes
1answer
65 views

Determining if $\mathbb{Z}[a]$ is a discrete subring of $\mathbb{C}$.

Let $a \in \mathbb{C}$ and consider the ring $\mathbb{Z}[a]$. Is there some nice criterion which will tell me whether $\mathbb{Z}[a]$ is discrete in the sense that there is some $\delta >0$ such ...
0
votes
0answers
34 views

What conditions are necessary for this property?

Let L be an ideal lattice. choose four elements a,b,c and d in L Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ 2 by 2 matrix. I think that $A=BCB^{-1}$ for some invetible matrix B ...
0
votes
0answers
6 views

Given a random unit cell's defining vectors in a random basis, what kind of lattice am I looking at?

So, as stated in the title, I have a lattice basis $B \in \mathbb{R}^{3\times 3}$ (given as a non-singular Matrix containing its unit cell vectors) which is arbitrarily rotated, scaled and reflected. ...
0
votes
0answers
30 views

Possible areas within an integer grid

Given a 1x1 grid with 4 lattice points $[(0,0),(0,1),(1,0),(1,1)]$ (equivalent to a $2 \times 2$ grid of vertices), there are 2 shapes and areas that can be formed: a triangle and a square. There are ...
2
votes
0answers
27 views

lattice walks with primes and composites

In the regular square lattice create a walk moving according the value of a counter. Consider two types of walks: In the first walk advance forward one unit if the counter is a composite number and ...
0
votes
0answers
15 views

Volume of the lattice generated by an ideal

Let $F$ be a totally real number field, $\mathfrak a \subset F$ a fractional ideal. Consider a lattice in $\mathbb R^n$ consisting of vectors $(\sigma_1(v),..\sigma_n(v))$, where $\sigma_1,..\sigma_n$ ...
6
votes
2answers
45 views

Which planar angles on an integer lattice are possible?

As shown in this question, you can construct an angle $A$ on 3 integer points on a plane only if $\tan A$ is rational. A natural generalization is to ask which values can planar angles based on 3 ...
2
votes
1answer
43 views

Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
4
votes
1answer
28 views

What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
0
votes
0answers
12 views

Fast method or direct generation of random upper triangular matrix using integer-restricted gauss elimination

I'd like to generate non-singular random upper triangular matrices of the following form: $$ \left[\begin{matrix} 1 & r_1 \cos{\alpha_1} & r_2 \cos{\alpha_2} & r_3 \cos{\alpha_3} & ...
0
votes
0answers
10 views

The size of the vertex hull of a lattice

Let $L$ be a lattice defined by $d$ vectors in $\mathbb{R}^d$. The Vertex Hull of a node $a$ at radius $r$ (denoted $VH(a, r)$) is defined to be the set of vertices that define the convex hull of $L ...
1
vote
0answers
27 views

Prove that $a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n $ are all satisfied by a nonzero $n-tuple$ of integers.

My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$ I want to prove that the n-linear ...
1
vote
1answer
17 views

Is it true that for a lattice $L$, $\mathbb{R}L = \mathbb{R}^{n}$?

I have as a definition A lattice $L \subseteq \mathbb{R}^{n}$ is a subgroup that is free of rank $n$ such that $\mathbb{R}L = \mathbb{R}^{n}$. I don't know if I am misinterpreting the statement, ...
-1
votes
1answer
133 views

Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is an "easy estimator" if any point on $f$, $(x_0,y_0)$, is near a lattice point $([x_0],[y_0])$ then $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$. In ...
0
votes
1answer
35 views

When does a real inverrtible matrix send a lattice in $\mathbb{R}^2$ to itself?

There is a question in Artin asking when does an inverrtible 2x2 real matrix send a lattice to itself. The issue I have is with this question not being specific as to what kind of answer is ...
0
votes
0answers
43 views

When do three complex numbers define a lattice?

Given three complex numbers, when is the set of all integer linear combinations of them a lattice? Since lattices can be defined by only two basis elements, I suppose the answer must be that one ...
0
votes
0answers
27 views

Intersection of two convex lattices polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice. Let P and Q two convex lattice polygons with n ,(resp. m) vertices. Let R be the convex lattice polygon ...
0
votes
0answers
28 views

Is this a known lattice?

I am wondering if the lattice generated by $$ \mathbf{B}=\mathbf{I}_N-\frac{\mathbf{n}\, \mathbf{n}^{\rm T}}{\mathbf{n}^{\rm T} \,\mathbf{n}} $$ where $\mathbf{n}=\left[n_1,n_2,\ldots,n_N\right]^{\rm ...
0
votes
0answers
15 views

Lattices, compact orbits, and admissible boxes

A box in $\mathbb{R}^n$ is a set of the form $[-b_1,b_1]\times \cdots \times [-b_n,b_n]$ with $b_i>0$ for all $i$. For any unimodular lattice $\Lambda$ define ...
0
votes
1answer
26 views

Finding an integral basis for a lattice defined in terms of an equation modulo p

Let p be a prime number, u relatively prime to p, and $\Lambda := \lbrace (a, b) \in \mathbb{Z}^2 : b \equiv au$ (mod p)$\rbrace$. How then can I find an integral basis $v_1, v_2$ for $\Lambda$? I ...
1
vote
1answer
29 views

How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$?

According to what I found on Wikipedia[1,2], you can represent any quasi-crystal structure in $\mathbb{R}^n$ by cutting a space $\mathbb{R}^N, N>n$ at an angle with the $\mathbb{R}^n$ space and ...
2
votes
2answers
52 views

Expression from generators of Special Linear Groups II

I wonder whether one can generate this $t$ matrix form the $A_1$ and $A_2$ matrix below. Here $$ t=\begin{pmatrix} 1& 1& 0\\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix} $$ from: $$ ...
9
votes
1answer
137 views

Modular parametrization of elliptic curve

Let $f$ be a cusp form of weight $2$ on $\Gamma_0(N)$ and assume that $f$ is a Hecke form and a newform. Then, we easily see that $$C(\gamma)=2i\pi \int_{\tau}^{\gamma \tau}{f(\tau')d\tau'} \quad ...
2
votes
2answers
177 views

Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains

For a non-square integer $d$ such that $d \equiv 1 \mod 4,$ we define the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}.$$ Prove that ...
0
votes
1answer
34 views

Maximal even sub-lattice in $\mathbb{Z}^n$ (reference request)

Lemma: The maximal even sub-lattice in $\mathbb{Z^n}$ is $$ \Big\{ (x_1,\cdots,x_n)\in \mathbb{Z}^n ~~\big|~~ \sum_{1 \le i \le n} x_i \in \mathbb{2Z} \Big\} $$ I found the above lemma in an ...
0
votes
0answers
33 views

Minimize multivariate (multivariable) polynomial over the integers

I'd like to minimize the following polynomial in 6 variables $h_0,h_1,g_0,g_1,g_2,g_3$: $$ g_3^2\cdot h_0^3\cdot h_1^3 - g_2\cdot g_3\cdot h_0^2\cdot h_1^4 + g_1\cdot g_3\cdot h_0\cdot h_1^5 - ...
0
votes
1answer
43 views

How do we balance the chemical equation and give an integer solution? [closed]

$$\require{mhchem}\ce{NaOH + H2SO4 -> Na2SO4 + H2O}$$ I'm trying all my best to come up with one solution but cant. in giving the integer solution to this chemical equation is it compulsory?
1
vote
1answer
33 views

Why ins't $\mathfrak{h}$ enough to parametrize complex elliptic curves?

this a pretty idiot question and of course there is a mistake in my way of thinking. Let $E$ be a elliptic curve, $E (\mathbb{C}) \cong \mathbb{C} / \Lambda$, where $\Lambda = \langle \omega_1, ...
1
vote
0answers
20 views

Coexistance of certain vectors in a lattice

I'm currently working on the SVP (Shortest Lattice Vector Problem) as a part of a paper that I'm writing. I've been trying to prove ( or disprove) the following without too much success : Question : ...
0
votes
1answer
18 views

Is it easy to find a lattice vector whose length is in a specific interval?

Say $L$ is a lattice of ${\bf R}^n$ with rank $m$. Let $\alpha, \beta \ (\alpha<\beta)$ be positive real numbers. Set $A_{L}=\{{\bf x}\in L: \alpha\leq \|{\bf x}\|\leq \beta\}.$ It may be difficult ...
3
votes
1answer
40 views

Finding a basis for the integer lattice points in a subspace

When writing this answer, one subgoal involved finding all integer solutions to the equations $\sum_{i=1}^5x_i=\sum_{i=1}^5ix_i=0$ in $\Bbb Z^5$. Since this is a linear system, the solutions form a ...
1
vote
0answers
28 views

Lower bounds on possible integer relations from the PSLQ algorithm

For the equation: $$ \sum_{i=1}^na_ix_i=0 $$ where all $x_i$ are real numbers and all $a_i$ are integers, the PSLQ algorithm can either find an integer relation or give lower bounds on the norm of ...
0
votes
0answers
31 views

Isoperimetric inequality for Lattice points

Here is the problem I was wondering if it was solved: Given a simple polygon with edges being on lattice points such as the one below. Does the circle with the same area intersect strictly less ...
3
votes
1answer
34 views

Measure theoretic proof of $|\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)|$

Let $A \in \Bbb{Z}^{d\times d}$ be an invertible matrix with entries in $\Bbb{Z}$. It is well-known (and can be proved using algebraic properties of matrices) that the index of the group $A \Bbb{Z}^d ...
2
votes
2answers
56 views

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$ The questions: Give an example of a nonempty subset of $\mathbb{R^2}$ (noted $M$) which is closed under addition and for all $m\in M$ we have $-m\in ...
4
votes
1answer
43 views

Integer solutions to a two variable equation.

For $m, n \in \mathbb{Z}$, show the only integer solutions to $f(m,n) = \displaystyle \frac{3^m(2^n+1)-2^{m+n}}{2^{m+n}-3^{m+1}}$ are $f(1, 2) = -7$, $f(0, 1) = -1$, and $f(0, 2) = 1$. More ...
2
votes
1answer
80 views

Why is Minkowski's Theorem so powerful?

Minkowksi's Convex Body Theorem is evidently pretty powerful, as it yields swift proofs of Fermat's Two Square and Lagrange's Four Square Theorems. Also, Minkowski's bound on class number and the ...
1
vote
0answers
53 views

Finding a basis for a particular integer lattice

The following problem arose in the context of string theory. I hope someone here might provide some guidance or a solution... Our starting point is: i) an integer-lattice $L\subset\mathbb R^n$ ...