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
12 views

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 $X$ ...
0
votes
0answers
5 views

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 ...
1
vote
1answer
25 views

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\, ...
1
vote
0answers
17 views

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 ...
1
vote
0answers
70 views

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 ...
3
votes
1answer
106 views

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 ...
2
votes
0answers
20 views

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 $||x||^2=8$. $ ...
0
votes
3answers
28 views

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 ...
2
votes
0answers
20 views

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 ...
0
votes
0answers
30 views

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 & ...
1
vote
0answers
16 views

$\Delta$ 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 ...
3
votes
3answers
76 views

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 ...
1
vote
2answers
200 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 ...
0
votes
0answers
19 views

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 ...
1
vote
1answer
29 views

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 ...
2
votes
2answers
139 views

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 ...
1
vote
0answers
29 views

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 ...
1
vote
1answer
18 views

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 ...
0
votes
0answers
15 views

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 $x_1, x_2, \ldots , x_n$. ...
2
votes
1answer
85 views

{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 $$\left\{\begin{array}{c}\sum_{i\in I_1}x_i=1\\\sum_{i\in I_2}x_i=1\\\vdots\\\sum_{i\in I_k}x_i=1\end{array}\right.$$ ...
0
votes
0answers
7 views

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 ...
0
votes
1answer
37 views

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 ...
1
vote
0answers
31 views

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 ...
1
vote
0answers
24 views

Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
0
votes
1answer
39 views

$(a \vee b)\wedge c=b\wedge c$ implies $(c\wedge b)\wedge a= b \vee c$

Show that for any elements a,b,c in a modular lattice $(a \vee b)\wedge c=b\wedge c$ implies $(c\wedge b)\wedge a= b \vee c$ ? $\wedge$ is meet and $\vee$ is join operations respectively .
-1
votes
1answer
14 views

What's the relation between the number of sites and the number of links for a 2d square lattice? [closed]

How about the case of 3d and 4d... Is there some general relation of the number of links and number of sites?
2
votes
0answers
27 views

lattice symmetries portrayed in animation

Below is an animated gif by Dave Whyte: What is the orbifold/fibrifold notation for the symmetries of the lattice depicted below? (a la /The Symmetries of Things/? Are there related theta function ...
3
votes
2answers
57 views

Placing a circle in a square lattice

Two part question. Consider the square lattice $\mathbb{Z}^2$: Imagine you are going to place a circle of radius $r$ somewhere in $\mathbb{R}^2$. Question 1: What is the radius of the largest ...
3
votes
2answers
34 views

Smallest linear combination of a set of vectors

I'm searching for an algorithm to accomplish a (hopefully) simple task. If I have a set of vetors, (e.g. $\left( ...
2
votes
2answers
75 views

$x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)

I've looked for help with this question but I have not found anything, I hope this is not a duplicate. Define the set $A=\{\mid x+y\sqrt{2}\mid \ : x,y\in \mathbb{Z}\ \mbox{and} \mid ...
5
votes
1answer
153 views

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 ...
1
vote
1answer
72 views

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 ...
3
votes
2answers
130 views

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}= ...
3
votes
1answer
27 views

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' ...
2
votes
1answer
61 views

Lattice Path Spaces.

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 ...
1
vote
1answer
40 views

“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 ...
0
votes
0answers
26 views

Finding integer vectors in the column space of a matrix

Consider a given set $S \subset Z$. $S$ is a finite set. Matrix $A \in S^{N \times M}$ is also given. Does there exist an algorithm to find all the vectors belonging to the space Col$(A)\cap S^N$ ...
0
votes
2answers
51 views

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?
3
votes
2answers
172 views

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 ...
0
votes
1answer
56 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 ...
3
votes
1answer
190 views

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 ...
1
vote
0answers
19 views

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 ...
1
vote
1answer
41 views

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 $K$ ...
0
votes
0answers
19 views

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 ...
0
votes
1answer
50 views

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!
1
vote
1answer
52 views

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 ...
2
votes
1answer
50 views

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 ...
0
votes
1answer
13 views

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 ...
0
votes
1answer
33 views

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 $t\alpha$ ...
0
votes
0answers
34 views

Automorphism group of torus fixing origin

I've got a short question: Suppose that you have some lattice $\Lambda$, say $\Lambda=\mathbb{Z}+\mathbb{Z}i$, and let $T$ be the torus $\mathbb{C}/\Lambda$, coming with the quotient map ...