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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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 ...
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Construction of lattice in a given genus

Recently I needed to find a lattice in a given genus (which I attained as an orthogonal sum of lower dimensional lattices), which led me to the question how one can costruct such a lattice in general. ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 $\sigma_1,..\sigma_n$ ...
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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 ...
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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 ...
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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 ...
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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} & ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: $$ ...
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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 ...
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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 ...
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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 ...
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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 - ...
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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?
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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, ...
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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 : ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $L\subset\mathbb R^n$ ...
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Simple question about lattices in $\mathbb{C}$

Milne's Modular Functions and Modular Forms states, at the bottom of page 10: We can normalize our lattices so they are of the form $$\Lambda(\tau) := \mathbb{Z} \cdot 1 + \mathbb{Z} \cdot ...
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How to eliminate some edges of a lattice to get exactly k paths?

We have an $n$ by $n$ lattice. We want to find a way to eliminate some edges, so that there are exactly $k$ paths from $(1,1)$ to $(n,n)$ of length $2n-2$. (this means our paths should be NE). I don't ...
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Lattice in $\Bbb R^2$

Let (a,b) be a lattice basis of a lattice L in R2. Prove that every other lattice basis has the form (a',b')=(a,b)P, where P is a 2x2 integer matrix with determinant 1 or - 1.
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Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
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Number of Dyck paths from $(0,0)$ to $(2n,k_1)$ if allowed to go below the $x$ axis

What is the number of (general?) Dyck paths from $(0,0)$ to $(2n,k_1)$, where $k_1\geq0$, allowing the path to go below the $x$ axis and touch the negative horizontal line at $k_2\leq0$ an arbitrary ...
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Lattice generated by vectors orthogonal to an integer vector

Given a non-zero vector $\boldsymbol{v}$ composed of integers, imagine the set of all non-zero integer vectors $\boldsymbol{u}$, such that $\boldsymbol{u} \cdot \boldsymbol{v} = 0$, i.e., the integer ...
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Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...
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Minkowski Theorem

Theorem: Let $L$ be an $n$-dimensional lattice in $\mathbb R^n$ with fundamental domain $T$, and let $X$ be a bounded symmetric convex subset of $\mathbb R^n$. If $Vol(X)>2^nVol(T)$ then $X$ ...
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Faber-Krahn inequality for domain in Z^d with nearest-neighbor connections

In $\mathbb{R}^d$ there is a theorem that if you are looking for the first Dirichlet eigenvalue $\lambda_1$ of a domain $D \subset \mathbb{R}^d$ with a given volume $V$, then $\lambda_1$ will be ...