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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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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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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Minkowsky 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 ...
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Number of solution pairs $(i,j)$ such that $i+jk \leq l$

I have show that the number of solutions $\left(\, i,j\,\right)$ of non-negative integers to $i + jk \leq l$ is $$ \left(\,\left\lfloor\, l \over k\,\right\rfloor +1\,\right) {2l + 2 - k\left\lfloor\, ...
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Line and distance to lattice points?

Consider the lattice $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$ and a line (not necessarily through the origin). What conditions can be placed on the slope of the line that is necessary and sufficient so ...
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Lattices in $\mathbb C$ as modules of the ring of integers in an imaginary quadratic field

Let $K$ be an imaginary quadratic number field and let $O_K\subset K$ be the ring of algebraic integers in $K$. Let us call a lattice $\Lambda\subset\mathbb C$ normalized if the tori $\mathbb ...
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Number of self-avoiding rook walks in a rectangular grid

I was wondering how many self-avoiding rook walks there are on an $m×n$ grid. A self-avoiding rook walk is a path from the bottom left corner to the top right corner of the grid, composed only of ...
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Leech Lattice and Golay Code

Consider the following Miracle Octat Generator or MOG. Choose the sign $\pm 3$ and fill in the blanks $\pm 1$ to create a point $x$ in the Leech lattice $\Lambda_{24}$ with $||x||^2=8$. $ ...
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Is there a theorem or axiom which shows that permutations of step sequences through a lattice graph result in the same destination?

I have been searching for a theorem, lemma, or even an axiom which shows that the permutations of a step sequence in Taxicab Geometry result in the same destination in such a lattice graph. To ...
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Dual Lattice definition

A lattice is a free abelian group $L$ together with a symmetric non-degenerate bilinear form $(,): L\times L\rightarrow \Bbb{Z}$. Let $H=\mathrm{Hom}(L,\Bbb{Z})$. I often see the dual of $L$ being ...
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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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$\Delta$ vs. $\delta$ in Lattice Theory

I'm learning about lattices and I'd like to confirm the difference between the density $\Delta$ and another density $\delta$. I would greatly appreciate if someone could confirm, correct, or even ...
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Orthogonal Vectors in a 2D Lattice with minimum area

I came across an interesting problem in my research (not a mathematician). Here it goes: Suppose, there is a 2D lattice $\Lambda$ in the X-Y plane with basis vectors $\vec{a}$ and $\vec{b}$, which ...
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238 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
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Generic structure in $\mathbb{Z}^n$

As a quick definition, a subset of $A\subseteq\mathbb{Z}^n$ is generic if for every pair of elements $\alpha_1,\alpha_2\in A$ that have an equal coordinate, there is a third element $\gamma\in A$ that ...
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What does the notation for a group ${A_n}^*$ mean?

I was on a website that catalouges lattices. I was looking through the alternating group lattices and the dihedral lattices and there were two kinds for each. For example, there was $A_2$ and ...
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Number of lattice paths

In my notes I have this: A lattice path is path consisting of step points $(x_0,y_0),(x_1,y_1),\ldots,(x_m,y_m).$ where either $x_{i+1}=x_{i}$ and $y_{i+1}=y_i+1$ or $x_{i+1}=x_{i}+1$ and ...
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Estimate for the co-volume of discs centered at lattice points in the plane?

Suppose I have a unimodular lattice $\Lambda = A \mathbb{Z^2}$ ($A\in SL(2,\mathbb{R})$) in the plane. I place a disc of fixed radius, $r$, around each point of $\Lambda$, so that I have a union of ...
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estimates for the largest disc not intersecting a unimodular lattice?

Are there any nice estimates for the size of the largest disc (centered anywhere) not intersecting a unimodular (i.e. covolume = 1) lattice in the plane? Maybe estimates in terms of the shortest ...
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Why changing a vector in a basis for a lattice can lead to a new lattice if $gcd(a_{m+1}, \ldots, a_n) = 1$?

On the book "Lattice basis reduction" by M.R. Bremner published by Taylor & Francis Corollary 1.26. Let $L$ be an $n$-dimensional lattice in $\mathbb{R}^n$ with basis $x_1, x_2, \ldots , x_n$. ...
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Help in Understanding the Formula for The Lattice Point Counting in Triangles with Rational Coordinates

Yesterday I have found this paper while searching Google. However, since the author of this paper gave no examples of implementing the following formula, I don't understand how to implement it in ...
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{0,1}-solutions for integer equations via lattice base reduction?

I would like to find $\{0,1\}$-solutions of a system of equations of the form $$\left\{\begin{array}{c}\sum_{i\in I_1}x_i=1\\\sum_{i\in I_2}x_i=1\\\vdots\\\sum_{i\in I_k}x_i=1\end{array}\right.$$ ...
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Points of a lattice inside square of side $N$

Let $\Lambda\subseteq \mathbb Z^m$ be a full-rank lattice of index $h$. I would like to know an upper bound for the quantity $H_N=|\Lambda\cap [-N,N[^m|$ where $[-N,N[^m=\{(a_1,\dots,a_m)\in \mathbb ...
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Number of integer lattice points within a circle

I am trying to solve a problem on codeforces, to be precised, this problem. I was able to figure out that the solution is $N(n) - N(n-1)$ where $N(n)$ is the number of lattice points withing a circle ...
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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 ...
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Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?