Questions tagged [euclidean-lattices]
The euclidean-lattices tag has no usage guidance.
96
questions
2
votes
0
answers
58
views
Configurations of minimal vectors for a 4-dimensional symplectic lattice
The possible configurations of minimal vectors for a 4-dimensional lattice are known for ages, but what about symplectic lattices ? If a 4-dimensional symplectic lattice $\Lambda$ has two minimal ...
1
vote
0
answers
103
views
On dimension of Segre embedding of lattice translations
Consider three lattices $L_1$, $L_2$ in $\Bbb Z^{n+1}$ and $L$ in $\Bbb Z^{2n+1}$.
Let $L_1+v_1$, $L_2+v_2$ and $L+v$ be their respective translationsfor some $v_1,v_2\in\Bbb Z^{n+1}\backslash\{(0,\...
0
votes
2
answers
304
views
Lattice question
Consider a lattice $\mathcal{L} = \mathbb{Z}v_1 \oplus \ldots \oplus \mathbb{Z}v_l$ in $\mathbb{R}^n$ and let $S_0$ be the set of edges of the fundamental unit of $\mathcal{L}$. We call a region $X$ ...
5
votes
0
answers
128
views
Lattice paths in polytopes
Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
5
votes
1
answer
183
views
lattice with Voronoi cell inside a circle
This considers real-valued lattices in two dimensions.
I need to find the densest lattice $\Lambda$, i.e., the one with the smallest determinant of its generator matrix, such that the Voronoi cell of ...
2
votes
2
answers
606
views
What is the spinor genus of the Leech lattice?
The Leech lattice and the 23 Niemeier lattices make a single genus. How does it break up into spinor genera?
6
votes
0
answers
266
views
Bound on the determinant of a quadratic form restricted to a subspace
Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose:
The minimum of $Q(x,y)$ as $y$ varies ...
2
votes
1
answer
237
views
Smallest angle among two lines in an n × n grid
Does anybody have a reference answering the following (at least for me surprisingly non trivial) question?
Given an $n \times n$ integer grid, what is the minimum angle between any two distinct lines,...
22
votes
1
answer
703
views
Does $E_8$ know $Spin(7)$?
One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...
1
vote
0
answers
379
views
How to count the number of shortest paths in a 2x2 grid? [closed]
Say that we have a 2x2 regular grid or network. We label the nodes 0 to 3 row-wise. Then, for each node, we want to compute the number of shortest paths that pass through them.
I have a Python code ...
16
votes
2
answers
452
views
what is the equivalent of the Euler constant for higher dimensional lattices
Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$. Then there are constants such that
$$\sum_{\substack{\gamma\in \Lambda\\0<|\gamma|<R\\}} \frac{1}{|\gamma|^d} = c_1 \log R + c_2 + o(1).$$...
1
vote
0
answers
52
views
Lattice points with next-largest norm [closed]
In a 2D integer grid, the points in increasing distance from the origin are:
$(0,0)$
$(\pm1,0)$ and $(0,\pm1)$
$(\pm1,\pm1)$
etc
By symmetry we need only consider one-eighth of the lattice, $x\ge0$ ...
10
votes
0
answers
183
views
Boomerangs in Polya's orchard
Polya's orchard problem asks for what radius $r$ of trees
at each lattice point within a distance $R$
of the origin block all lines of sight to the exterior of the orchard.
The answer is known; $r$ ...
0
votes
0
answers
228
views
Writing integers in ring of integers of number fields
Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...
2
votes
0
answers
147
views
Listing all Lattice Points in a Box
Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
3
votes
0
answers
131
views
Enumerating 1-Lipschitz functions on an integer grid
Let $G$ denote an integer grid consisting of $\{0,\dots,m\}\times\{0,\dots,n\}$. An integer-valued function $f:G\to\mathbb{Z}$ is said to be 1-Lipschitz if it satisfies $|f(x) - f(y)| \leq \| x-y \|$...
7
votes
1
answer
563
views
Upper bound for the number of integral points in a convex set
Let $K \subset \mathbb{R}^3$ be a bounded convex set such that the points with integer
coordinates in $K$ are not all coplanar. Is it true that $|K \cap \mathbb{Z}^3| \leq 6{\rm Vol}(K) + 3$?
0
votes
0
answers
882
views
Number of lattice points in a given triangle
Given a triangle with real coordinates, does anybody know how to find the number of lattice points contained within it? What if the points are only rational? I know Pick's formula can be used for the ...
6
votes
3
answers
921
views
Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
The probability that a random walk on $\mathbb{Z}^3$ returns to the origin is about 34%.
This is (part of)
Pólya's theorem.
I have been looking for an analogous (numerical) result for the probability
...
8
votes
1
answer
1k
views
Given a positive-definite integral unimodular Gram matrix, how to find a basis of the associated lattice (over $\mathbf Q$)?
Let $G$ be a $n\times n-$symmetric matrix with integral coefficients and determinant $1$ (i.e. unimodular) such that the associated quadratic form is positive-definite.
I am interested in having an ...
3
votes
1
answer
333
views
Covering points with a shortest lattice spiral
Let $S$ be a finite set of lattice points in $\mathbb{Z}^2$.
My question is, roughly:
Q. How can a shortest lattice spiral that passes through
every point of $S$ be found?
A lattice spiral (my ...
10
votes
1
answer
616
views
A random variation on Polya's orchard problem
Polya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, e.g.,
...
6
votes
2
answers
302
views
Covolume of the row span of a matrix and of the kernel of a matrix
Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The covolume
$\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a
fundamental domain for $L$, i.e., the volume of the parallelopiped
...
41
votes
2
answers
2k
views
Can we find lattice polyhedra with faces of area 1,2,3,...?
I asked this question two months ago on MSE, where it earned the rare
Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with ...
6
votes
2
answers
415
views
Bound on Minimal Length of Vectors in Lattice and its Dual Lattice
Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
2
votes
1
answer
259
views
The number of different lattice triangles
Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number ...
5
votes
2
answers
492
views
Even unimodular lattices with root system $32 A_1$
I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question.
For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...
13
votes
0
answers
562
views
Multiplicity of ball covering
Background. My questions are motivated by the following:
A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete
and Computational Geometry, 8 (1992) 109-130) considered the ...
3
votes
0
answers
135
views
Lattices achieving best density
Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...
49
votes
4
answers
4k
views
What fraction of the integer lattice can be seen from the origin?
Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$.
Say that a point $(x,y)$ of $Q$ is visible from the origin if the
segment from $(0,0)$ to $(x,y) \in Q$ passes ...
3
votes
2
answers
183
views
Lattice-point-free buffers around circles
Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
&...
7
votes
0
answers
203
views
Lattice radial-step (ratchet) spirals
(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...
7
votes
2
answers
901
views
Is there a 3d equivalent of this picture?
This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin:
Fitting a mesh to a density function
The picture below is the image of a regular ...
9
votes
4
answers
930
views
Applications of n-dimensional crystallographic groups
I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups)
1) in mathematics
2) outside of mathematics,
besides the applications to $...
3
votes
2
answers
591
views
Primitive orthogonal vectors/Unimodular matrices
Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice.
A set of integer vectors $v_1,\ldots,...
3
votes
1
answer
475
views
The right conformal map to make a certain picture
This is a follow-up to a question I asked a year ago, which was helpfully answered by Anton Petrunin: Fitting a mesh to a density function.
I am trying to come up with a way to make a picture of an ...
3
votes
0
answers
163
views
A non-commutative ring from SU(2)
$SU(2)$, which will be regarded here as the group of unit quaternions under multiplication, has 3 conjugacy classes of finite subgroups which don't have cyclic subgroups of index 1 or 2. They are:
...
0
votes
0
answers
245
views
upper bound on the size of sumset of lattice points
Let $\Lambda$ be a lattice (discrete additive subgroup) in $\mathbb R^n$ ($n\geq 2$). In my problem, $\Lambda$ lies in a $k$ dimensional ($1< k\leq n$) subspace of $\mathbb R^n$. Let $A\subset \...
4
votes
2
answers
492
views
Empty lattice simplex or White's theorem
White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem:
If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of $...
10
votes
1
answer
358
views
Identifying lattices
I wrote a program that numerically searches for lattices in $\mathbb{R}^d$ with high sphere packing densities. As I have been running the program, it has been able to find, in addition to well-known ...
5
votes
2
answers
276
views
Doubly covering an even lattice
I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be the Leech lattice, ...
10
votes
2
answers
790
views
Fitting a mesh to a density function
Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
0
votes
0
answers
176
views
Quadratic forms and 0-1 points.
I have a quadratic form $Q(u) = \langle Du , u \rangle = 0$, where $D$ is circulant-symmetric from $\mathbb{R}^{n \times n}$ $D$ has all entries $0$ or $1$ except the diagonal which a negative real ...
5
votes
1
answer
546
views
A 'generalized Four Squares Theorem'?
The $4$-dimensional lattice $\mathbb{Z}^{4}$ has vectors of length $\sqrt{n}$ for any positive integer $n$ by the Four Squares Theorem, but this need not be true for higher-dimensional integral, ...
1
vote
0
answers
135
views
A bounded function of the packing and covering density of lattices
Given a (finite-dimensional) lattice $L$ of an Euclidean vector-space, the function
$$L\longmapsto -\log(\hbox{packing density of }L)/
\log(\hbox{covering density of }L)$$
is bounded and bounded away ...
3
votes
2
answers
281
views
Extremal lattices
Denote by $\mu_n$ the largest value such that there exists a lattice of determinant $1$
in $\mathbb R^n$ for which the distances between different lattice points are greater or equal to $\mu_n$.
...