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Questions tagged [integer-lattices]

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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Is Pick's formula true for a general (non-integer-vertices) lattice?

Does Pick's formula hold for non-integer lattices (nodes with non-integer coordinates)? I heard that it holds for any lattice (given lattice is: we take two groups of parallel lines and intersect ...
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How to calculate the mobius function of a Poset using Hall's theorem

Hall's Theorem states that: $u(x,y) = C_0-C_1+C_2-C_3+...$ where $C_k$ is number of chains of length $k$ If $x\neq y$ then $C_0=0$ and $C_1=1$ But my question is why does $C_1$ have to equal ...
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The automorphism group of the lattice $E_6$

Let $E_6$ be the root lattice, and $G$ be its automorphism group as a lattice (i.e. as $\mathbb Z$-module together with the inner product). Let $W(E_6)$ be the Weyl group. Apparently $W(E_6)\subset G$....
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lattices in the complex plane from 3 elements

I am asked to find when, for some nonzero complex numbers $\alpha,\beta,\gamma$, the set defined by $\{l\alpha+m\beta+n\gamma|l,m,n\in \mathbb{Z}\}$ is a lattice. Assuming it is a lattice, I ...
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Does E8 sphere packing proof use a computer

Does the proof by Maryna S. Viazovska that the E8 sphere packing is optimal use a computer?
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Number of ways to choose a closed path of given length on a square lattice

Also known as self-avoiding polygons, this is an unsolved problem. However, to leading order in the asymptotic limit, the number of polygons of a given perimeter scales exponentially with perimeter ...
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Unimodular lattice

We are considering $\mathbb{R}^n$ with the inner product (the usual one), and $L$ an unimodular lattice (so $covolume(L)=1$ and $<u,v> \in \mathbb{Z}$ for all u,v in L). I have to show that, ...
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On the number of cosets of a sublattice in a lattice

In several questions, e.g. 1, 2, it has been asked why the index of a sublattice $M\mathbb{Z}^n$ in $\mathbb{Z}^n$ is equal to $\det(M)$, that is $|\mathbb{Z}^n/M\mathbb{Z}^n|=\det(M)$. The answer to ...
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Are the only Riemann surfaces which are quotients of $\mathbb{C}$the cylinder and the toruses? Why?

Consider the Riemann surfaces $\mathbb{C}^\times=\mathbb{C}\setminus\{0\}$ and $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice in the complex plane (i.e. a discrete additive subgroup of $\mathbb{C}$...
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Prove that there exists a $m×m$ lattice square in the $x-y$ plane such that none of its coordinates are visible [duplicate]

Call a lattice point 'visible' if the $gcd$ of its coordinates is 1. Then there exists a $m×m$ square in the $x-y$ plane such that none of its coordinates are visible. You can actually define such ...
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Lattice points below a curve

Assume a curve represented by a function $f(b)=0.5(\sqrt{N-b^2}-b+1)$ with $1\leq b \leq \sqrt{\frac{N}{2}}$. I want to count the lattice points below this curve, more specifically, I would like a ...
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Integer points in parallelepiped [closed]

There is a parallelepiped in $n$ dimensional vector space over $\mathbb{R}$. All its vertex are integer. Its volume $V>1$. How to prove that there is an integer point which belongs to the ...
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Lattice associated to 4th Jacobi theta function?

For a lattice (specifically the dual lattice of a torus) there is associated a theta function $ \theta_{\Gamma}(w)=\sum_{\gamma\in\Gamma}w^{||\gamma||^2},\text{ where $w=e^{-4\pi^2t}$ and $t\in(0,\...
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List integer points in fundamental parallelepiped

Assume that we have a fundamental parallelepiped in $\mathbb{R}^3$ $\Pi$ := {$\lambda_1$w$_1$ + $\lambda_2$w$_2$ + $\lambda_3$w$_3$: 0 $\leq$ $\lambda_1,\lambda_2, \lambda_3$ < 1}, w$_1$,w$_2$, w$...
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Example of two-dimensional diagonal integer-lattice

In some article describes the method of generating a family of sequences of integers (chaotic mixing) using Toral Automorphisms theory and integer-lattices. The main mathematical structure in this ...
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Number of integer lattice points between two rational points

I'm trying to find a way to compute the number of integer lattice points $\left ( (x, y) \in \mathbb{Z}^2 \right )$ between two rational points $\left ( (p, q) \in \mathbb{Q}^2 \right )$. Let $(p_1, ...
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Can a regular $n$-simplex have vertices in $\mathbb Z^n$ for $n > 1$?

Trivially, a regular $0$-simplex (point) and $1$-simplex (line segment) can have integer vertices in $0$ and $1$ dimensional Euclidean space respectively. On the other hand, a regular $2$-simplex (...
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Relation between norm of a vector and sublattice to which it belongs.

Given four rational numbers $m, n, p$, and $q$, define the lattices $L_1 = \{ (m\cdot x, n\cdot x) : x \in \mathbb{Z} \}$ and $L_2 = \{ (p\cdot y, q\cdot y) : y \in \mathbb{Z} \}$. That is, the ...
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Horocylic flow and the velocity at which $c_t$ fills space

The horocyclic flow is the dynamics generated by the matrices $h^s_+,$ which are stable manifolds. Here I am considering the flow on the space of lattices with area $1.$ $$ h^s_+= \begin{pmatrix} 1&...
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Any vector shorter than $\lambda_1^*$ must belong to a sublattice.

Let $R = \mathbb{Z}[x] / \langle x^n -1 \rangle$ and $f, g, F,$ and $G$ be polynomials in $R$. Let $\Lambda_h = \{ (f, g)u + (F, G)v : u, v \in R \}$ be the $R$-module (lattice) generated by linear ...
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Symmetric integer matrix and odd entry in the diagonal.

K is a 2N$\times$2N symmetric integer matrix with at least one odd element in the diagonal. Suppose $\mathcal{M}$ be a set of integer vectors satisfying the following two properties: 1) $m^{T}K^{-1}m'...
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Divisibility of kissing numbers

Denote by $ K(d) $ the kissing number in dimension $ d $. I have two questions : 1) does $ d\mid K(d) $ for all $ d $? 2) does $ d\mid D $ imply $K(d)\mid K(D) $?
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Counting Lattice Points

Lattice points are of great importance. I encountered a problem as follows:- given a circle of radius 'r' as x²+y²=r² the number of lattice points can take values . The options were (0,72,69,140). I ...
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Sublattice of a lattice

Show that for any two lattice bases $B \in \mathbb{R}^{d \times k}$ and $C \in \mathbb{R}^{d \times n}$, the first generates a sublattice of the second $(L(B) \subseteq L(C))$ if and only if there is ...
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Range of one-dimensional lattice paths of a given length

How many lattice paths in $\mathbb{Z}$ of length $n$ with steps in $\{-1,0,+1\}$ visit $m$ distinct points? Notice that this is just the number of lattice paths $P$ such that $\max P - \min P + 1 = m$...
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show that $f(x)=a^2 + b^2 + c^2 + d^2$ is smooth on $\text{SL}_2(\mathbb{Z})\backslash \text{SL}_2(\mathbb{R})$

The space of lattices in the Euclidean plane $X=\text{SL}_2(\mathbb{Z})\backslash \text{SL}_2(\mathbb{R})$ can it have smooth functions? I'm trying to find examples of $C^\infty$ functions. Consider ...
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Proof clarification: Catalan numbers and lattice paths

I'm reading a proof of the fact that the number of monotonic lattice paths from (0,0) to (n,n) not crossing over the diagonal y=x is given by the Catalan numbers. The proof uses the reflection ...
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How to estimate the numbers of the integer points of $\frac{(m-1)(m-2)}{2}+k\leq N$, $a\leq\frac{k}{m}\leq b$

How to estimate the numbers of the integer lattices of $$\frac{(m-1)(m-2)}{2}+k\leq N$$ $$a\leq\frac{k}{m}\leq b$$ Here , m,k are the variables , b,a are selected, N can tend to the infinity, well, ...
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Counting lattice paths which use fixed steps but can end in different places

Consider a lattice path where one starts at $(0,0)$ and can move only right or up in integer steps. The total number of steps made is $4$, but the maximum steps in one direction $3$. How many paths ...
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Enumerating the image of an integer matrix applied to a lattice

Let $A\in\mathbb{Z}^{n \times m}$ and $x\in \mathbb{Z}^m$. Also suppose we are given $l,u\in\mathbb{Z}^m$. I would like to efficiently enumerate $$\{Ax \,|\, l_i \le x_i \le u_i \text{ for all } i\}$$ ...
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King of the Centre - Is this an existing game?

Consider an $n$-player infinitely repeated game. First stage nature chooses for each player, $i$, a radius $r_{i}$. For each later stage $t$ each player $i$: The payer chooses a "target" $p_{i, t}$...
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second theorem of Minkowski proof

I am wondering if anyone have a proof of second theorem of Minkowski. He says that if $vol(K) = 2^{n} det(L)$ and $K$ is compact and symmetric and convex then $K$ contains a non zero lattice point. I ...
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dimension of dual lattice over a lattice formula

A lattice $L \subset \mathbb{R}^{n}$ is the set of integer linear combinations of $k$-linearly independent vectors. Formally $$ L = \{ \sum_{j=1}^{k} x_{i} e_{i} \text{ such that } x_i \in \mathbb{Z} \...
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Odd version of 24 squares formula?

The Leech lattice is related to the formula: $$1^2+2^2+3^2+....+24^2=70^2$$ This is related in turn to 26D bosonic string theory. Is there another known formula that involves the integers 1..8, ...
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Help finding the result of a sum over a lattice with two variables

I would need some help for finding a closed form for $$f(a,b):=\sum_{(k,l)\in\mathbb{Z}^2,\ l\ne0}\frac{1}{(2 i k a \pi + l b) (2 i (k - l) a \pi + l b)}$$ In general, I do not know many ...
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“Intersection” of quotient groups, modulo $\mathbb{Z}^n$

Let's say that $M,N$ are two invertible $n\times n$ integer matrices, such that all their eigenvalues are greater than $1$ in magnitudes. Then we define $$K_M:=(M^{-1}\mathbb{Z}^d)\cap[0,1)^n,\quad ...
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Is there a higher dimension equivalent of lattices with lines instead of points?

Looking at some interesting lattices like $E_8$ and the Leech lattice, I was thinking these are all made of 0D objects. Would there be an equivalent object made of 1D or even nD objects. One might ...
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Integral points on a line

In this post What is the number of integer coordinates on a line segment? , we have the formula for the number of integral points strictly between $(x_1,y_1)$ and $(x_2,y_2)$ (both are integral points)...
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Quasi-crystal/aperiodic pattern?

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. A ...
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Calculate the number of nonnegative integer solutions of $ax+by\leq c$.

If $a$, $b$, and $c$ are known, and $x$ and $y$ are integers greater than or equal to zero, how many possible values of ($x$, $y$) exist that satisfy the equation $$ax + by \le c\,?$$ I have ...
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Upper bound on lattice coefficients for elements within a sphere

Here is the situation: I have a set of $n$ independent vectors ${\bf b}_i \in \mathbb{R}^n$, and I am interested in the lattice vectors that lie in a ball of radius $L$ centered around ${\bf 0}$, i.e. ...
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A holomorphic map between complex tori (proposition 1.3.2 in Diamond–Shurman)

The following is from Diamond and Shurman's A First Course in Modular Forms book: I had studied Munkres Topology a few years ago but for lifting I had to review the materials again, but I still don't ...
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What are the constructible angles and regular/uniform/semiregular/etc polytopes in n-dimensional integer lattices?

In a 2-dimensional regular square lattice, using only lattice points, lines connecting lattice points, and points where such lines intersect, I am aware it is impossible to make a 30-degree angle ...
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Primitive embedding of lattices

I am wondering is there a way to know if one lattice (finite free $Z$-module with inner product) can be embeded into another? For example, I have a lattice $L:=E_6 \oplus D_4(-1)\oplus D_4(-1)\oplus ...
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How to prove that two $p$-adic lattices are isomorphic?

Let $\mathbb{Z}_{p}$ be the p-adic number. a pair $(L,<>)$ is called lattice if $L$ be a free $\mathbb{Z}_{p}$ module of finite rank and $<>:L×L \to \mathbb{Z}_{p}$be a nondegenerate ...
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free ultrafilters on $\mathbb{N}$

Let $\mathbb{N}$ be the set of natural numbers, I know the fact that the number of free ultrafilters on $\mathbb{N}$ is uncountable. I have two questions: 1.How to construct these uncountable ...
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A contradictory relation between probability and number of paths

Consider an urn containing $c$ balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We perform $n$ trials with replacement of one ...
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How to generalize Neumann's minmax theorem to the case where one of the input's domain is discrete?

I know the Nuemann's minimax theorem requires that both of the input's domain to be convex set,but I encounter this problem of the following form Here A={1,2,...n},V takes continious values,My ...
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Rational probabilities and discrete paths on a lattice

Consider an urn containing $c$ elements, $b$ of which are black. If we perform $n$ trials with replacement of one element at a time from the urn, the probability to get $n$ times a black ball is $\...
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How do you notate a set of neighbouring locations in an integer lattice?

So I'm trying to write a paper and in it I'm describing the movement of an agent in a multi-agent simulation. Space is treated as a 2 or 3 dimensional lattice. I need some notation for the set of ...